Some topics in algebra

Stephen Semmes Rice University

Preface

These informal notes deal with some topics in algebra that seem to be related to various areas. The reader is expected to have some familiarity with abstract algebra, although a fair amount of details are often given in the arguments being considered. At the same time, much of the discussion may be somewhat more algebraic than in some related matters involving interactions between algebra and analysis.

Although the term “functor” is not normally used here, there are of course a number of functors being considered. Often these involve spaces of homomorphisms between modules over rings or algebras, or tensor products of modules. Many of their basic properties illustrate arguments that work more broadly under analogous conditions. Sometimes various cases are discussed in some detail, to indicate the roles of the properties or conditions being used.

In particular, the term “derived functor” is not normally used here, although some related notions and examples are discussed. More precisely, derived functors may be defined using projective and injective resolutions, depending on the circumstances, and these are discussed in Chapter 10. This is related to the notion of satellites, as in Chapter 32.

Of course, vector spaces over (commutative) fields are basic types of modules. Left and right vector spaces over division rings that may not be commutative are fundamental as well. Many basic notions related to vector spaces can also be used for modules, such as linear mappings or homomorphisms between modules. However, there are many important properties of vector spaces and linear mappings between them that do not necessarily work for arbitrary modules and their homomorphisms. Thus one may consider additional conditions on modules, such as projectivity and injectivity, as in Chapter 2.

Even for vector spaces over the real or complex numbers, the situation is more complicated when one also considers norms, topologies, or other types of additional structure. This includes additional conditions on linear mappings, such as continuity. If V is a topological vector space over the real or complex numbers, for instance, such as a vector space with a norm, and if W is a closed linear subspace of V, then one might be interested in the possibility of finding a closed linear subspace Z of V such that V corresponds to the direct sum of W and Z. This should include compatibility with the topology, so that the corresponding projections of V onto W and Z should be continuous. If V is a Hilbert space, then one can simply take Z to be the orthogonal complement of W in V.

Modules with differentiation are discussed in Chapter 5. These may be considered as modules over suitable rings or associative algebras of dual numbers, as in Section 5.4. This is a basic way in which modules can be more complicated than vector spaces. In particular, the homology of a module with differentiation may be defined as a module over the initial ring or associative algebra, as in Section 5.1.

Graded modules are defined in Section 5.9. A complex is a graded module with differentiation, with a compatibility property of differentiation with respect to the grading, as in Section 5.10. It is helpful to consider double gradings and double complexes as well, as in Sections 5.13 and 5.14. These are used on tensor products of graded modules, and spaces of homomorphisms between graded modules, as in Chapter 6. Left and right complexes are discussed in Chapter 10, and are used to define projective and injective resolutions.

Chapter 9 is rather classical, while fitting with many of the broad themes considered here. This includes the notion of left and right hereditary rings, as in Section 9.3. Similarly, Chapter 11 deals with aspects of integral domains that are related to a number of topics discussed here. In particular, a Dedekind ring may be defined as an integral domain that is a hereditary ring, as on p134 of [50]. This is also known as a Dedekind domain, as in [8, 263], although there are some slight variations about whether fields are permitted in the definition. Some background material has been included after that, to try to clarify the relationship between these perspectives. There are a lot of connections between these matters and the rest too.

Note that some of the sections and chapters here are somewhat independent of some of the others. Often earlier sections will be mentioned when notions or results from those sections are being used, to make it easier to look at later chapters without necessarily reading all of the previous ones.

Contents

I Algebras, modules, and tensor products 1

1 Modules and tensor products 2 1.1 Sums and products of modules . . . . . . . . . . . . . . . . . . . 2 1.2 Module-valued functions . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Tensor products over commutative rings . . . . . . . . . . . . . . 6 1.4 Some properties of tensor products . . . . . . . . . . . . . . . . . 8 1.5 Modules over associative algebras . . . . . . . . . . . . . . . . . . 10 1.6 Sums, products, and compositions . . . . . . . . . . . . . . . . . 11 1.7 Spaces of homomorphisms . . . . . . . . . . . . . . . . . . . . . . 13 1.8 Additional structure on Hom(V, W) . . . . . . . . . . . . . . . . 15 1.9 Tensor products over algebras . . . . . . . . . . . . . . . . . . . . 16 1.10 Additional properties and structure . . . . . . . . . . . . . . . . . 18 1.11 Opposite algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.12 Some double tensor products . . . . . . . . . . . . . . . . . . . . 22 1.13 Homomorphisms and tensor products . . . . . . . . . . . . . . . . 23 1.14 Bilinearity over A, B . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.15 Direct families of homomorphisms . . . . . . . . . . . . . . . . . 27

2 Modules and tensor products, 2 29 2.1 Compositions and Hom(·, W) . . . . . . . . . . . . . . . . . . . . 29 2.2 Some converse statements . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Compositions and Hom(V, ·) . . . . . . . . . . . . . . . . . . . . . 32 2.4 Some more converse statements . . . . . . . . . . . . . . . . . . . 34 2.5 Compositions and tensor products . . . . . . . . . . . . . . . . . 35 2.6 Another approach . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.7 Projective modules . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.8 Injective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.9 Covariant ϕ-extensions . . . . . . . . . . . . . . . . . . . . . . . . 44 2.10 More on covariant ϕ-extensions . . . . . . . . . . . . . . . . . . . 45 2.11 Contravariant ϕ-extensions . . . . . . . . . . . . . . . . . . . . . 46 2.12 ϕ-Projectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.13 ϕ-Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.14 A ϕ-projectivity property . . . . . . . . . . . . . . . . . . . . . . 52 2.15 A ϕ-injectivity property . . . . . . . . . . . . . . . . . . . . . . . 54 3 Modules and tensor products, 3 57

3.1 Semisimple modules . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Direct systems of modules . . . . . . . . . . . . . . . . . . . . . . 59 3.3 Direct systems of submodules . . . . . . . . . . . . . . . . . . . . 61 3.4 Homomorphisms between direct systems . . . . . . . . . . . . . . 62 3.5 Limits of bilinear mappings . . . . . . . . . . . . . . . . . . . . . 64 3.6 Direct limits and tensor products . . . . . . . . . . . . . . . . . . 66 3.7 Double limits and bilinear mappings . . . . . . . . . . . . . . . . 68 3.8 Double limits and tensor products . . . . . . . . . . . . . . . . . 69 3.9 Iterated direct limits . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.10 Another property of compositions . . . . . . . . . . . . . . . . . . 75 3.11 Compositions and Hom(·, ·) . . . . . . . . . . . . . . . . . . . . . 78 3.12 Finitely-generated submodules . . . . . . . . . . . . . . . . . . . 81 3.13 An interesting homomorphism . . . . . . . . . . . . . . . . . . . . 82 3.14 An abstract construction . . . . . . . . . . . . . . . . . . . . . . . 85 3.15 Another abstract construction . . . . . . . . . . . . . . . . . . . . 86

4 Some associative algebras 89 4.1 Tensor products of algebras . . . . . . . . . . . . . . . . . . . . . 89 4.2 Modules over tensor products . . . . . . . . . . . . . . . . . . . . 91 4.3 Formal power series . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4 Polynomial functions . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Invertibility in A[[T_{1}, …, T_{n}]] . . . . . . . . . . . . . . . . . . . . 96 4.6 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.7 Algebras of module homomorphisms . . . . . . . . . . . . . . . . 99 4.8 Modules over semigroups . . . . . . . . . . . . . . . . . . . . . . 102 4.9 Semigroup algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.10 Free semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.11 Tensor algebras using bimodules . . . . . . . . . . . . . . . . . . 105 4.12 Tensor algebras and homomorphisms . . . . . . . . . . . . . . . . 107 4.13 Local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.14 Local rings and power series . . . . . . . . . . . . . . . . . . . . . 109 4.15 Exterior rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

II Differentiation and complexes 113

5 Differentiation and dual numbers 114 5.1 Modules with differentiation . . . . . . . . . . . . . . . . . . . . . 114 5.2 Mappings and differentiation . . . . . . . . . . . . . . . . . . . . 116 5.3 Submodules and quotients . . . . . . . . . . . . . . . . . . . . . . 118 5.4 Dual numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.5 Easy differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.6 Some isomorphisms with W^{x} . . . . . . . . . . . . . . . . . . . . 125 5.7 Some related properties of (V, d_{V}) . . . . . . . . . . . . . . . . . 126 5.8 Some more properties of (V, d_{V}) . . . . . . . . . . . . . . . . . . 128 5.9 Graded modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.10 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.11 Maps between complexes . . . . . . . . . . . . . . . . . . . . . . 133 5.12 Some related homomorphisms . . . . . . . . . . . . . . . . . . . . 135 5.13 Double gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.14 Double complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.15 Maps between double complexes . . . . . . . . . . . . . . . . . . 141

6 More on differentiation 143 6.1 Tensor products and gradings . . . . . . . . . . . . . . . . . . . . 143 6.2 Tensor products of complexes . . . . . . . . . . . . . . . . . . . . 144 6.3 Gradings and homomorphisms . . . . . . . . . . . . . . . . . . . 147 6.4 Induced homomorphisms on Hom_{A}^{gr}(V, W) . . . . . . . . . . . . . 149 6.5 Complexes and homomorphisms . . . . . . . . . . . . . . . . . . 150 6.6 Maps of any degree . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.7 Double complexes and any bidegree . . . . . . . . . . . . . . . . . 154 6.8 Tensor products and any degrees . . . . . . . . . . . . . . . . . . 157 6.9 Homomorphisms and any degrees . . . . . . . . . . . . . . . . . . 159 6.10 The complex M(V, W) . . . . . . . . . . . . . . . . . . . . . . . . 161 6.11 Tensor products and homology . . . . . . . . . . . . . . . . . . . 162 6.12 Some homomorphisms related to α . . . . . . . . . . . . . . . . . 165 6.13 Homology and Hom_{A}^{gr}(V, W) . . . . . . . . . . . . . . . . . . . . 168 6.14 Some properties of Hom_{A}^{gr}(Z′(V), Z(W)) . . . . . . . . . . . . . . 170 6.15 Some properties of Hom_{A}^{gr}(Z(V), Z′(W)) . . . . . . . . . . . . . . 171

7 More on differentiation, 2 174 7.1 Some splitting and α . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.2 More splitting and α . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.3 Some splitting and α′ . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4 More splitting and α′ . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.5 Transforming modules and complexes . . . . . . . . . . . . . . . 184 7.6 Induced mappings and homotopies . . . . . . . . . . . . . . . . . 185 7.7 An analogue of α . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.8 Analogues of τ, ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.9 Splitting Z′(V) and α . . . . . . . . . . . . . . . . . . . . . . . . 190 7.10 An injectivity condition and α . . . . . . . . . . . . . . . . . . . 192 7.11 Injectivity of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7.12 Differentiation and Hom(V, ·) . . . . . . . . . . . . . . . . . . . . 196 7.13 Induced mappings and Hom(V, ·) . . . . . . . . . . . . . . . . . . 199 7.14 An analogue of α′ . . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.15 Complexes and α′ . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8 More on differentiation, 3 207

8.1 Splitting Z(W) and α′ . . . . . . . . . . . . . . . . . . . . . . . . 207 8.2 Surjectivity conditions and α′ . . . . . . . . . . . . . . . . . . . . 211 8.3 Surjectivity conditions and complexes . . . . . . . . . . . . . . . 212 8.4 Differentiation and Hom(·, W) . . . . . . . . . . . . . . . . . . . . 214 8.5 Induced mappings and Hom(·, W) . . . . . . . . . . . . . . . . . 216 8.6 Another analogue of α′ . . . . . . . . . . . . . . . . . . . . . . . . 218 8.7 Another α′ and complexes . . . . . . . . . . . . . . . . . . . . . . 220 8.8 Splitting Z′(V) and α′ . . . . . . . . . . . . . . . . . . . . . . . . 224 8.9 Additional conditions and α′ . . . . . . . . . . . . . . . . . . . . 228 8.10 Additional conditions and complexes . . . . . . . . . . . . . . . . 230 8.11 Easy complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8.12 Sums, products, and differentiation . . . . . . . . . . . . . . . . . 234 8.13 Sums, products, and complexes . . . . . . . . . . . . . . . . . . . 236 8.14 Direct limits and differentiation . . . . . . . . . . . . . . . . . . . 238 8.15 Direct sums and limits . . . . . . . . . . . . . . . . . . . . . . . . 245

III More rings, modules, and complexes 249

9 Some rings and modules 250 9.1 A reformulation of projectivity . . . . . . . . . . . . . . . . . . . 250 9.2 A reformulation of injectivity . . . . . . . . . . . . . . . . . . . . 251 9.3 Hereditary rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 9.4 Kaplansky’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 254 9.5 Semi-hereditary rings . . . . . . . . . . . . . . . . . . . . . . . . . 255 9.6 Hereditary rings and differentiation . . . . . . . . . . . . . . . . . 256 9.7 Noetherian conditions . . . . . . . . . . . . . . . . . . . . . . . . 257 9.8 A criterion for finite generation . . . . . . . . . . . . . . . . . . . 259 9.9 Direct sums and injective modules . . . . . . . . . . . . . . . . . 260 9.10 Some finitely-generated modules . . . . . . . . . . . . . . . . . . 262 9.11 Some remarks about division rings . . . . . . . . . . . . . . . . . 263 9.12 Polynomials and division rings . . . . . . . . . . . . . . . . . . . 264 9.13 An interesting family of rings . . . . . . . . . . . . . . . . . . . . 265 9.14 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 9.15 Graded modules over graded rings . . . . . . . . . . . . . . . . . 269

10 Left and right complexes 272 10.1 Left complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.2 Projective resolutions . . . . . . . . . . . . . . . . . . . . . . . . 274 10.3 Maps over homomorphisms . . . . . . . . . . . . . . . . . . . . . 275 10.4 Some related homotopies . . . . . . . . . . . . . . . . . . . . . . . 277 10.5 Some remarks about compositions . . . . . . . . . . . . . . . . . 278 10.6 Right complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.7 Injective resolutions . . . . . . . . . . . . . . . . . . . . . . . . . 280 10.8 Maps and right complexes . . . . . . . . . . . . . . . . . . . . . . 282 10.9 Homotopies and right complexes . . . . . . . . . . . . . . . . . . 284 10.10Compositions and right complexes . . . . . . . . . . . . . . . . . 285 10.11Tensor products and left complexes . . . . . . . . . . . . . . . . . 286 10.12Left complexes in one factor . . . . . . . . . . . . . . . . . . . . . 287 10.13Homomorphisms and resolutions . . . . . . . . . . . . . . . . . . 290 10.14Left complexes and Hom(·, W) . . . . . . . . . . . . . . . . . . . 292 10.15Right complexes and Hom(V, ·) . . . . . . . . . . . . . . . . . . . 293

11 Integral domains 296 11.1 Torsion and divisible elements . . . . . . . . . . . . . . . . . . . . 296 11.2 More on torsion and divisibility . . . . . . . . . . . . . . . . . . . 297 11.3 Some easy homomorphisms . . . . . . . . . . . . . . . . . . . . . 300 11.4 Fields of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11.5 Some injective mappings . . . . . . . . . . . . . . . . . . . . . . . 303 11.6 Another injective mapping . . . . . . . . . . . . . . . . . . . . . . 305 11.7 Fractional and invertible ideals . . . . . . . . . . . . . . . . . . . 307 11.8 Inversible ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 11.9 Prüfer rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 11.10Dedekind rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 11.11Some injective modules . . . . . . . . . . . . . . . . . . . . . . . 314 11.12Some divisible modules . . . . . . . . . . . . . . . . . . . . . . . . 314

12 Commutative rings and fractions 316 12.1 Rings of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 12.2 Modules of fractions . . . . . . . . . . . . . . . . . . . . . . . . . 318 12.3 Fractions and tensor products . . . . . . . . . . . . . . . . . . . . 320 12.4 Fractions and prime ideals . . . . . . . . . . . . . . . . . . . . . . 322 12.5 The nilradical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 12.6 Using powers to get fractions . . . . . . . . . . . . . . . . . . . . 324 12.7 Extensions and contractions of ideals . . . . . . . . . . . . . . . . 325 12.8 Products and ideal quotients . . . . . . . . . . . . . . . . . . . . 326 12.9 More on extensions and contractions . . . . . . . . . . . . . . . . 327 12.10Radicals of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 12.11Ring homomorphisms and fractions . . . . . . . . . . . . . . . . . 329 12.12More on rings of fractions . . . . . . . . . . . . . . . . . . . . . . 330 12.13Extensions, contractions, and fractions . . . . . . . . . . . . . . . 331 12.14More on modules of fractions . . . . . . . . . . . . . . . . . . . . 334 12.15Contractions of prime ideals . . . . . . . . . . . . . . . . . . . . . 335

IV More on rings and modules 337

13 Nakayama’s lemma and other matters 338 13.1 The Jacobson radical . . . . . . . . . . . . . . . . . . . . . . . . . 338 13.2 Ideals, modules, and homomorphisms . . . . . . . . . . . . . . . . 339 13.3 Nakayama’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 341 13.4 A noncommutative version . . . . . . . . . . . . . . . . . . . . . . 342 13.5 Ideals and fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 343 13.6 Modules over quotient rings . . . . . . . . . . . . . . . . . . . . . 344 13.7 Another noncommutative version . . . . . . . . . . . . . . . . . . 345 13.8 Some quotients by ideals . . . . . . . . . . . . . . . . . . . . . . . 346 13.9 Some freely-generated modules . . . . . . . . . . . . . . . . . . . 348 13.10Complementary ideals . . . . . . . . . . . . . . . . . . . . . . . . 349 13.11A class of modules . . . . . . . . . . . . . . . . . . . . . . . . . . 351 13.12Quotients and gradings . . . . . . . . . . . . . . . . . . . . . . . 352 13.13Some graded free modules . . . . . . . . . . . . . . . . . . . . . . 353 13.14Some more freely-generated modules . . . . . . . . . . . . . . . . 354 13.15Some quotients as tensor products . . . . . . . . . . . . . . . . . 356

14 Fractions, dimension, and valuations 358 14.1 Pairs of multiplicatively closed sets . . . . . . . . . . . . . . . . . 358 14.2 Fractions and integral domains . . . . . . . . . . . . . . . . . . . 360 14.3 Fractions and invertible ideals . . . . . . . . . . . . . . . . . . . . 361 14.4 Fractions and (k : M)_{Q_{k}} . . . . . . . . . . . . . . . . . . . . . . . 362 14.5 Primary ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 14.6 More on S^{−1} k . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 14.7 Powers of radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 14.8 Dimension via prime ideals . . . . . . . . . . . . . . . . . . . . . 368 14.9 Noetherian local rings . . . . . . . . . . . . . . . . . . . . . . . . 369 14.10Local rings of dimension 0 . . . . . . . . . . . . . . . . . . . . . . 370 14.11One-dimensional integral domains . . . . . . . . . . . . . . . . . 372 14.12One-dimensional local domains . . . . . . . . . . . . . . . . . . . 373 14.13Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 14.14Some related subgroups and ideals . . . . . . . . . . . . . . . . . 376 14.15Valuations on integral domains . . . . . . . . . . . . . . . . . . . 377

15 Discrete valuations and Dedekind domains 379 15.1 Discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . 379 15.2 Getting discrete valuation rings . . . . . . . . . . . . . . . . . . . 380 15.3 Some remarks about integral domains . . . . . . . . . . . . . . . 381 15.4 More on principal ideal domains . . . . . . . . . . . . . . . . . . 383 15.5 Local rings and valuations . . . . . . . . . . . . . . . . . . . . . . 384 15.6 More on the additional condition . . . . . . . . . . . . . . . . . . 385 15.7 Discrete valuation rings and invertibility . . . . . . . . . . . . . . 386 15.8 Local domains and fractional ideals . . . . . . . . . . . . . . . . . 387 15.9 Local domains and invertible ideals . . . . . . . . . . . . . . . . . 388 15.10A criterion for invertibility . . . . . . . . . . . . . . . . . . . . . . 389 15.11Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . 391 15.12Dedekind domains and invertibility . . . . . . . . . . . . . . . . . 392 15.13Dedekind domains from invertibility . . . . . . . . . . . . . . . . 393 15.14Primary ideals in Dedekind domains . . . . . . . . . . . . . . . . 393 15.15One-dimensional Noetherian domains . . . . . . . . . . . . . . . . 394 16 Valuation rings and integral elements 397

16.1 Valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 16.2 Noetherian valuation rings . . . . . . . . . . . . . . . . . . . . . . 398 16.3 Integral elements of larger rings . . . . . . . . . . . . . . . . . . . 399 16.4 Some properties of integral elements . . . . . . . . . . . . . . . . 400 16.5 Some corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 16.6 Getting a discrete valuation . . . . . . . . . . . . . . . . . . . . . 403 16.7 Another argument using fractional ideals . . . . . . . . . . . . . . 404 16.8 Powers in the denominator . . . . . . . . . . . . . . . . . . . . . 405 16.9 Using powers in the denominator . . . . . . . . . . . . . . . . . . 407 16.10Fractions and integral elements . . . . . . . . . . . . . . . . . . . 407 16.11More on fractions, integral elements . . . . . . . . . . . . . . . . 409 16.12Dedekind domains and integral elements . . . . . . . . . . . . . . 410 16.13Fields, quotients, and integral elements . . . . . . . . . . . . . . . 411 16.14Two corollaries, and another argument . . . . . . . . . . . . . . . 412 16.15Getting prime ideals in k_{1} . . . . . . . . . . . . . . . . . . . . . . 414

V Rings, modules, and field extensions 416

17 Some properties of ideals 417 17.1 Sums, products, and intersections . . . . . . . . . . . . . . . . . . 417 17.2 Finitely many ideals . . . . . . . . . . . . . . . . . . . . . . . . . 418 17.3 Finite unions and intersections . . . . . . . . . . . . . . . . . . . 420 17.4 Irreducible ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 17.5 More on primary ideals . . . . . . . . . . . . . . . . . . . . . . . 423 17.6 Primary decompositions . . . . . . . . . . . . . . . . . . . . . . . 424 17.7 More on primary decompositions . . . . . . . . . . . . . . . . . . 426 17.8 Surjective ring homomorphisms . . . . . . . . . . . . . . . . . . . 427 17.9 Some remarks about zero-divisors . . . . . . . . . . . . . . . . . . 430 17.10Primary decompositions of {0} . . . . . . . . . . . . . . . . . . . 431 17.11Zero-divisors and prime ideals . . . . . . . . . . . . . . . . . . . . 432 17.12Saturated multiplicatively closed sets . . . . . . . . . . . . . . . . 433 17.13Saturations of ideals . . . . . . . . . . . . . . . . . . . . . . . . . 434 17.14Primary decompositions and fractions . . . . . . . . . . . . . . . 436 17.15The second uniqueness theorem . . . . . . . . . . . . . . . . . . . 437

18 Some properties of ideals, 2 440 18.1 Intersections of prime ideals . . . . . . . . . . . . . . . . . . . . . 440 18.2 Primary decompositions and (I : x) . . . . . . . . . . . . . . . . 442 18.3 Primary decompositions and products . . . . . . . . . . . . . . . 443 18.4 Products and Dedekind domains . . . . . . . . . . . . . . . . . . 445 18.5 Artinian modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 18.6 Artinian and Noetherian rings . . . . . . . . . . . . . . . . . . . . 447 18.7 Commutative Artin rings . . . . . . . . . . . . . . . . . . . . . . 448 18.8 Nilpotency of the nilradical . . . . . . . . . . . . . . . . . . . . . 449 18.9 More on commutative Artin Rings . . . . . . . . . . . . . . . . . 450 18.10Finite collections of prime ideals . . . . . . . . . . . . . . . . . . 451 18.11Some arguments for Dedekind domains . . . . . . . . . . . . . . . 452 18.12Valuations on Dedekind domains . . . . . . . . . . . . . . . . . . 454 18.13Some remarks about ideals . . . . . . . . . . . . . . . . . . . . . 455 18.14Ideals in Dedekind domains . . . . . . . . . . . . . . . . . . . . . 456 18.15Another ideal in k . . . . . . . . . . . . . . . . . . . . . . . . . . 458

19 Ideals and field extensions 460 19.1 Products of ideals in k . . . . . . . . . . . . . . . . . . . . . . . . 460 19.2 More on ideals in k . . . . . . . . . . . . . . . . . . . . . . . . . . 461 19.3 Fractional ideals of k . . . . . . . . . . . . . . . . . . . . . . . . . 462 19.4 More on v_{I}(M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 19.5 More on fractional ideals . . . . . . . . . . . . . . . . . . . . . . . 466 19.6 Using some ideals in k . . . . . . . . . . . . . . . . . . . . . . . . 468 19.7 Reducing to the previous case . . . . . . . . . . . . . . . . . . . . 469 19.8 A result about linear independence . . . . . . . . . . . . . . . . . 471 19.9 Algebraic elements and extensions . . . . . . . . . . . . . . . . . 471 19.10Distinguished classes and the compositum . . . . . . . . . . . . . 473 19.11A couple of distinguished classes . . . . . . . . . . . . . . . . . . 474 19.12Some remarks about embeddings . . . . . . . . . . . . . . . . . . 476 19.13Algebraically closed fields . . . . . . . . . . . . . . . . . . . . . . 477 19.14Splitting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 19.15Galois groups and normal extensions . . . . . . . . . . . . . . . . 480

20 More on field extensions 482 20.1 Some properties of normal extensions . . . . . . . . . . . . . . . . 482 20.2 More on normal extensions . . . . . . . . . . . . . . . . . . . . . 484 20.3 Separable polynomials . . . . . . . . . . . . . . . . . . . . . . . . 485 20.4 Separable elements and extensions . . . . . . . . . . . . . . . . . 486 20.5 Separable extensions and embeddings . . . . . . . . . . . . . . . 488 20.6 Separable extensions are distinguished . . . . . . . . . . . . . . . 489 20.7 Minimal normal extensions . . . . . . . . . . . . . . . . . . . . . 490 20.8 Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 20.9 Separable extensions and the trace . . . . . . . . . . . . . . . . . 493 20.10Traces and integral elements . . . . . . . . . . . . . . . . . . . . . 495 20.11Traces and integral closures . . . . . . . . . . . . . . . . . . . . . 495 20.12Dedekind domains and integral closures . . . . . . . . . . . . . . 497 20.13Some remarks about algebras . . . . . . . . . . . . . . . . . . . . 498 20.14A result of Emmy Noether . . . . . . . . . . . . . . . . . . . . . . 500 20.15Some remarks about multiple roots . . . . . . . . . . . . . . . . . 500

VI Rings, fields, modules, and derivations 503

21 More on field extensions, 2 504 21.1 More on separable elements . . . . . . . . . . . . . . . . . . . . . 504 21.2 More on algebraic extensions . . . . . . . . . . . . . . . . . . . . 506 21.3 Maximal separable subextensions . . . . . . . . . . . . . . . . . . 509 21.4 Some additional corollaries . . . . . . . . . . . . . . . . . . . . . 511 21.5 Some remarks about Gal(K/k) . . . . . . . . . . . . . . . . . . . 513 21.6 Separability of K over K_{1} . . . . . . . . . . . . . . . . . . . . . . 514 21.7 Normality and separability . . . . . . . . . . . . . . . . . . . . . 515 21.8 Primitive elements of finite extensions . . . . . . . . . . . . . . . 516 21.9 Existence of primitive elements . . . . . . . . . . . . . . . . . . . 517 21.10Separability and primitive elements . . . . . . . . . . . . . . . . . 518 21.11More on Galois extensions . . . . . . . . . . . . . . . . . . . . . . 519 21.12Some fields of rational functions . . . . . . . . . . . . . . . . . . . 520 21.13Derivations on algebras . . . . . . . . . . . . . . . . . . . . . . . 523 21.14Derivations on rings and fields . . . . . . . . . . . . . . . . . . . 524 21.15Some more derivations . . . . . . . . . . . . . . . . . . . . . . . . 526

22 Some more rings and fields 528 22.1 Semigroups with cancellation . . . . . . . . . . . . . . . . . . . . 528 22.2 Gaussian semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 530 22.3 Greatest common divisors . . . . . . . . . . . . . . . . . . . . . . 531 22.4 More on unique factorization domains . . . . . . . . . . . . . . . 531 22.5 Some remarks about polynomials . . . . . . . . . . . . . . . . . . 533 22.6 The content of a polynomial . . . . . . . . . . . . . . . . . . . . . 535 22.7 Some remarks about prime elements . . . . . . . . . . . . . . . . 536 22.8 Prime elements and valuations . . . . . . . . . . . . . . . . . . . 538 22.9 Unique factorization domains and v_{p} . . . . . . . . . . . . . . . . 539 22.10Another type of content . . . . . . . . . . . . . . . . . . . . . . . 541 22.11Another version of Gauss’ lemma . . . . . . . . . . . . . . . . . . 543 22.12Unique factorization in k[X] . . . . . . . . . . . . . . . . . . . . . 544 22.13Gauss’ lemma and Dedekind domains . . . . . . . . . . . . . . . 545 22.14Some remarks about fractions . . . . . . . . . . . . . . . . . . . . 547 22.15Some more remarks about fractions . . . . . . . . . . . . . . . . . 548

23 More on derivations 551 23.1 Some more properties of derivations . . . . . . . . . . . . . . . . 551 23.2 Another type of derivation . . . . . . . . . . . . . . . . . . . . . . 552 23.3 A related notion of derivation . . . . . . . . . . . . . . . . . . . . 554 23.4 More on V -valued derivations . . . . . . . . . . . . . . . . . . . . 555 23.5 Some algebras from modules . . . . . . . . . . . . . . . . . . . . . 556 23.6 Some more derivations on polynomials . . . . . . . . . . . . . . . 557 23.7 Some extensions of derivations . . . . . . . . . . . . . . . . . . . 559 23.8 Existence of the extension . . . . . . . . . . . . . . . . . . . . . . 561 23.9 The n = 1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 23.10Some derivations on power series . . . . . . . . . . . . . . . . . . 563 23.11A property of some derivations . . . . . . . . . . . . . . . . . . . 564 23.12More on derivations on A[[X_{1}, …, X_{n}]] . . . . . . . . . . . . . . . 565 23.13Some remarks about centralizers . . . . . . . . . . . . . . . . . . 567 23.14More on centralizers . . . . . . . . . . . . . . . . . . . . . . . . . 569 23.15Derivations and partial derivatives . . . . . . . . . . . . . . . . . 570

24 More on field extensions, 3 573 24.1 More on formal polynomials . . . . . . . . . . . . . . . . . . . . . 573 24.2 More on polynomial functions . . . . . . . . . . . . . . . . . . . . 575 24.3 Some examples with I = Z_{+} . . . . . . . . . . . . . . . . . . . . . 577 24.4 Algebraic independence . . . . . . . . . . . . . . . . . . . . . . . 578 24.5 More on algebraic independence . . . . . . . . . . . . . . . . . . . 579 24.6 Transcendence bases . . . . . . . . . . . . . . . . . . . . . . . . . 580 24.7 A lemma about polynomials . . . . . . . . . . . . . . . . . . . . . 581 24.8 Noether’s normalization theorem . . . . . . . . . . . . . . . . . . 583 24.9 Some remarks about some extensions . . . . . . . . . . . . . . . . 584 24.10A result mentioned earlier . . . . . . . . . . . . . . . . . . . . . . 586 24.11Reducing to normal extensions . . . . . . . . . . . . . . . . . . . 587 24.12The purely inseparable case . . . . . . . . . . . . . . . . . . . . . 589 24.13Some remarks about nonzero polynomials . . . . . . . . . . . . . 590 24.14Another lemma about polynomials . . . . . . . . . . . . . . . . . 591 24.15Extending some homomorphisms . . . . . . . . . . . . . . . . . . 592

VII Some rings, modules, and polynomials 594

25 Composition series and chain conditions 595 25.1 Composition series of modules . . . . . . . . . . . . . . . . . . . . 595 25.2 More on composition series . . . . . . . . . . . . . . . . . . . . . 596 25.3 A result about lengths . . . . . . . . . . . . . . . . . . . . . . . . 598 25.4 Some initial steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 25.5 Some induced homomorphisms . . . . . . . . . . . . . . . . . . . 601 25.6 Some properties of M and N . . . . . . . . . . . . . . . . . . . . 602 25.7 Not necessarily finitely generated modules . . . . . . . . . . . . . 604 25.8 The Krull–Akizuki theorem . . . . . . . . . . . . . . . . . . . . . 605

26 Some examples of Nagata 608 26.1 Some commutative Noetherian rings . . . . . . . . . . . . . . . . 608 26.2 Some sets of indeterminates . . . . . . . . . . . . . . . . . . . . . 610 26.3 Pairs of complementary subsets . . . . . . . . . . . . . . . . . . . 610 26.4 Some ideals and homomorphisms . . . . . . . . . . . . . . . . . . 611 26.5 Some more ideals and homomorphisms . . . . . . . . . . . . . . . 612 26.6 Ideals, homomorphisms, and fractions . . . . . . . . . . . . . . . 614 26.7 These examples of Nagata . . . . . . . . . . . . . . . . . . . . . . 615 26.8 A helpful fact about ideals in A . . . . . . . . . . . . . . . . . . . 616 26.9 Ideals in S^{−1} A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 26.10Some Noetherian conditions . . . . . . . . . . . . . . . . . . . . . 618 26.11Some more properties of S^{−1} A . . . . . . . . . . . . . . . . . . . 620 26.12Some properties of Dedekind domains . . . . . . . . . . . . . . . 622 26.13Nontrivial ideals in k . . . . . . . . . . . . . . . . . . . . . . . . . 623

27 Some polynomials and formal sums 625 27.1 Polynomials in one variable . . . . . . . . . . . . . . . . . . . . . 625 27.2 The factor theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 626 27.3 More on one-variable polynomials . . . . . . . . . . . . . . . . . . 628 27.4 More on several-variable polynomials . . . . . . . . . . . . . . . . 629 27.5 Families of indeterminates . . . . . . . . . . . . . . . . . . . . . . 630 27.6 Some modules of formal sums . . . . . . . . . . . . . . . . . . . . 632 27.7 Some algebras of formal sums . . . . . . . . . . . . . . . . . . . . 634 27.8 Formal sums and algebraic independence . . . . . . . . . . . . . . 635 27.9 Another algebraic dependence property . . . . . . . . . . . . . . 637 27.10Some subrings of A[X^{E}] . . . . . . . . . . . . . . . . . . . . . . . 638 27.11Quotients of formal sums . . . . . . . . . . . . . . . . . . . . . . 639

28 A few related results 642 28.1 Some remarks about graded rings . . . . . . . . . . . . . . . . . . 642 28.2 Finite groups of field automorphisms . . . . . . . . . . . . . . . . 643 28.3 A lemma about separable extensions . . . . . . . . . . . . . . . . 645 28.4 Another approach to Artin’s result . . . . . . . . . . . . . . . . . 646 28.5 General and special linear groups . . . . . . . . . . . . . . . . . . 647 28.6 Projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 28.7 Projective lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 28.8 Some formal compositions . . . . . . . . . . . . . . . . . . . . . . 651 28.9 Some irreducible formal polynomials . . . . . . . . . . . . . . . . 653 28.10Some formal rational functions . . . . . . . . . . . . . . . . . . . 654 28.11Some more formal compositions . . . . . . . . . . . . . . . . . . . 655 28.12More on these formal compositions . . . . . . . . . . . . . . . . . 657

29 Formal series and other matters 659 29.1 Some formal series . . . . . . . . . . . . . . . . . . . . . . . . . . 659 29.2 More on these formal series . . . . . . . . . . . . . . . . . . . . . 660 29.3 Some derivations on formal series . . . . . . . . . . . . . . . . . . 661 29.4 More on derivations on A((T)) . . . . . . . . . . . . . . . . . . . 663 29.5 Some derivations on A[[T]] . . . . . . . . . . . . . . . . . . . . . . 664 29.6 Some remarks about modules . . . . . . . . . . . . . . . . . . . . 665 29.7 Hilbert’s basis theorem . . . . . . . . . . . . . . . . . . . . . . . . 667 29.8 Noetherian conditions and power series . . . . . . . . . . . . . . . 669 29.9 Some remarks about Noetherian rings . . . . . . . . . . . . . . . 670 29.10Sums of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 29.11Some more spaces of matrices . . . . . . . . . . . . . . . . . . . . 673 29.12More on M_{n}(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 29.13More on M_{m,n}(A) . . . . . . . . . . . . . . . . . . . . . . . . . . 676 29.14Some tensor products using matrices . . . . . . . . . . . . . . . . 678 29.15Some spaces of homomorphisms . . . . . . . . . . . . . . . . . . . 680

VIII Some semisimplicity and other topics 683

30 Some vector spaces and topologies 684 30.1 Some basic topological spaces . . . . . . . . . . . . . . . . . . . . 684

IX Rings, modules, satellites, and augmentations 686

31 Some more rings and modules 687 31.1 Some local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 31.2 Some remarks about a(T) B_{1} . . . . . . . . . . . . . . . . . . . . 688 31.3 Some right modules over B_{1} . . . . . . . . . . . . . . . . . . . . . 689 31.4 Some related Notherian conditions . . . . . . . . . . . . . . . . . 691 31.5 Some more local rings . . . . . . . . . . . . . . . . . . . . . . . . 691 31.6 More on W_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 31.7 Some more Noetherian conditions . . . . . . . . . . . . . . . . . . 694 31.8 Another family of local rings . . . . . . . . . . . . . . . . . . . . 695 31.9 Some right modules over B_{3} . . . . . . . . . . . . . . . . . . . . . 697 31.10Noetherian conditions related to B_{3} . . . . . . . . . . . . . . . . 698 31.11Direct systems of sets . . . . . . . . . . . . . . . . . . . . . . . . 699 31.12Some limits of free modules . . . . . . . . . . . . . . . . . . . . . 701 31.13More on systems of modules . . . . . . . . . . . . . . . . . . . . . 702 31.14Projective resolutions of direct limits . . . . . . . . . . . . . . . . 704

32 Some satellites 705 32.1 Some basic comparisons . . . . . . . . . . . . . . . . . . . . . . . 705 32.2 Comparisons and tensor products . . . . . . . . . . . . . . . . . . 706 32.3 Left satellites of tensor products . . . . . . . . . . . . . . . . . . 709 32.4 Comparisons and Hom_{A}(·, W) . . . . . . . . . . . . . . . . . . . . 710 32.5 The right satellite of Hom_{A}(·, W) . . . . . . . . . . . . . . . . . . 713 32.6 Necessity of injectivity . . . . . . . . . . . . . . . . . . . . . . . . 714 32.7 Some more comparisons . . . . . . . . . . . . . . . . . . . . . . . 715 32.8 Comparisons and Hom_{A}(V, ·) . . . . . . . . . . . . . . . . . . . . 717 32.9 The right satellite of Hom_{A}(V, ·) . . . . . . . . . . . . . . . . . . 719 32.10Necessity of projectivity . . . . . . . . . . . . . . . . . . . . . . . 720 32.11A type of double satellite . . . . . . . . . . . . . . . . . . . . . . 721 32.12Another type of double satellite . . . . . . . . . . . . . . . . . . . 722 32.13Left and right ideals . . . . . . . . . . . . . . . . . . . . . . . . . 723 32.14More on left, right ideals . . . . . . . . . . . . . . . . . . . . . . . 725 33 Augmented rings 726

33.1 Left augmentations . . . . . . . . . . . . . . . . . . . . . . . . . . 726 33.2 Right augmentations, and related matters . . . . . . . . . . . . . 728 33.3 Augmentations and homology . . . . . . . . . . . . . . . . . . . . 729 33.4 Two-sided ideals and homology . . . . . . . . . . . . . . . . . . . 731 33.5 Augmentations and cohomology . . . . . . . . . . . . . . . . . . . 732 33.6 Dual numbers and augmentations . . . . . . . . . . . . . . . . . . 734 33.7 Dual numbers and homology . . . . . . . . . . . . . . . . . . . . 735 33.8 Dual numbers and cohomology . . . . . . . . . . . . . . . . . . . 736 33.9 Augmentations and homomorphisms . . . . . . . . . . . . . . . . 737 33.10The Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . 741 33.11Some interesting conditions . . . . . . . . . . . . . . . . . . . . . 742 33.12A related left complex . . . . . . . . . . . . . . . . . . . . . . . . 744 33.13Some more left complexes . . . . . . . . . . . . . . . . . . . . . . 745 33.14Some nice projective resolutions . . . . . . . . . . . . . . . . . . . 749 33.15Using these projective resolutions . . . . . . . . . . . . . . . . . . 750

34 Some helpful facts 753 34.1 Projectivity and tensor products . . . . . . . . . . . . . . . . . . 753 34.2 Injectivity of Hom_{A}(V, Z) . . . . . . . . . . . . . . . . . . . . . . 755 34.3 Another injectivity property . . . . . . . . . . . . . . . . . . . . . 756 34.4 An associativity isomorphism . . . . . . . . . . . . . . . . . . . . 758 34.5 Some more isomorphisms . . . . . . . . . . . . . . . . . . . . . . 759 34.6 Another projectivity property . . . . . . . . . . . . . . . . . . . . 760 34.7 A couple of corollaries . . . . . . . . . . . . . . . . . . . . . . . . 762 34.8 Some alternate approaches . . . . . . . . . . . . . . . . . . . . . . 763 34.9 Another result about injectivity . . . . . . . . . . . . . . . . . . . 765 34.10Two corollaries about injectivity . . . . . . . . . . . . . . . . . . 766 34.11Augmentations and tensor products . . . . . . . . . . . . . . . . 767 34.12Homomorphisms induced by ϕ . . . . . . . . . . . . . . . . . . . 769 34.13Semigroups and tensor products . . . . . . . . . . . . . . . . . . 772

X Associative and supplemented algebras 774

35 A family of augmentations 775 35.1 The enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . 775 35.2 The augmentation ideal J . . . . . . . . . . . . . . . . . . . . . . 776 35.3 Crossed homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 778 35.4 Principal crossed homomorphisms . . . . . . . . . . . . . . . . . 780 35.5 Homology and cohomology of A . . . . . . . . . . . . . . . . . . . 781 35.6 Some basic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 35.7 A right augmentation on A^{en} . . . . . . . . . . . . . . . . . . . . 784 35.8 Some modules S_{n}(A) . . . . . . . . . . . . . . . . . . . . . . . . . 785 35.9 Some homomorphisms d_{n} . . . . . . . . . . . . . . . . . . . . . . 787 35.10The standard complex S(A) . . . . . . . . . . . . . . . . . . . . . 788 35.11Using the standard complex . . . . . . . . . . . . . . . . . . . . . 790 35.12Some modules N_{n}(A) . . . . . . . . . . . . . . . . . . . . . . . . 792 35.13Analogues of s_{n}, d_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 794 35.14The normalized standard complex N(A) . . . . . . . . . . . . . . 795 35.15Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

36 A family of augmentations, 2 799 36.1 Homomorphisms between algebras . . . . . . . . . . . . . . . . . 799 36.2 The case where B = k_{1} ⨂_{k} A . . . . . . . . . . . . . . . . . . . . 800 36.3 Projectivity of A over A^{en} . . . . . . . . . . . . . . . . . . . . . . 802 36.4 Some examples with matrices . . . . . . . . . . . . . . . . . . . . 804 36.5 Some remarks about polynomial algebras . . . . . . . . . . . . . 805 36.6 More on polynomial algebras . . . . . . . . . . . . . . . . . . . . 806 36.7 Some remarks about free algebras . . . . . . . . . . . . . . . . . . 807 36.8 More on semigroup algebras . . . . . . . . . . . . . . . . . . . . . 808 36.9 Some remarks about tensor products . . . . . . . . . . . . . . . . 810 36.10More on the analogue of s_{n} . . . . . . . . . . . . . . . . . . . . . 811 36.11More on the analogue of d_{n} . . . . . . . . . . . . . . . . . . . . . 814 36.12Adjoining an identity element . . . . . . . . . . . . . . . . . . . . 814

37 Supplemented algebras 817 37.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 37.2 Some related modules . . . . . . . . . . . . . . . . . . . . . . . . 818 37.3 Crossed homomorphisms into W_{ε} . . . . . . . . . . . . . . . . . . 819 37.4 Modules with trivial operators . . . . . . . . . . . . . . . . . . . 822 37.5 Homomorphisms and supplemented algebras . . . . . . . . . . . . 823 37.6 Homomorphisms from tensor products . . . . . . . . . . . . . . . 824 37.7 Connection with A^{en} . . . . . . . . . . . . . . . . . . . . . . . . . 826 37.8 A helpful isomorphism . . . . . . . . . . . . . . . . . . . . . . . . 828 37.9 Some verifications . . . . . . . . . . . . . . . . . . . . . . . . . . 830 37.10Some related projective resolutions . . . . . . . . . . . . . . . . . 832 37.11Standard complexes and resolutions . . . . . . . . . . . . . . . . 833 37.12Differentiation on these resolutions . . . . . . . . . . . . . . . . . 836 37.13Some related complexes . . . . . . . . . . . . . . . . . . . . . . . 837 37.14Standard complexes and \overline{X} . . . . . . . . . . . . . . . . . . . . . 839

38 Semigroups and augmentations 842 38.1 Semigroup rings and augmentations . . . . . . . . . . . . . . . . 842 38.2 Semigroup algebras and augmentations . . . . . . . . . . . . . . . 843 38.3 Semigroups and homomorphisms . . . . . . . . . . . . . . . . . . 845 38.4 Some related surjectivity conditions . . . . . . . . . . . . . . . . 848 38.5 Some augmentations and homomorphisms . . . . . . . . . . . . . 849 38.6 More on the unit augmentation . . . . . . . . . . . . . . . . . . . 850 38.7 Free semigroups and augmentations . . . . . . . . . . . . . . . . 851

Bibliography 852 xviii CONTENTS Index 871

Part I

Chapter 1

Algebras, modules, and tensor products Modules and tensor products

1.1 Sums and products of modules

Let k be a commutative ring with a multiplicative identity element, and let I be a nonempty set. Suppose that for each j ∈ I, V_{j} is a module over k. It is easy to see that the Cartesian product

∏_{j∈I} V_{j} (1.1.1)

is a module over k too, with respect to coordinatewise addition and scalar multiplication. This is the direct product of the V_{j}’s, j ∈ I.

If v ∈ ∏_{j∈I} V_{j} and l ∈ I, then let v_{l} be the lth coordinate of v in V_{l}. Put

(1.1.2) ⨁_{j∈I} V_{j} = {v ∈ ∏_{j∈I} V_{j} : v_{l} = 0 for all but finitely many l ∈ I}.

This is the direct sum of the V_{j}’s, j ∈ I. Of course, this is a submodule of ∏_{j∈I} V_{j}, as a module over k. If I has only finitely many elements, then (1.1.2) is the same as ∏_{j∈I} V_{j}.

Let Z be another module over k, and suppose that ϕ_{j} is a homomorphism from V_{j} in Z, as modules over k, for each j ∈ I. If v ∈ ⨁_{j∈I} V_{j}, then

ϕ(v) = ∑_{j∈I} ϕ_{j}(v_{j}) (1.1.3)

defines an element of Z, because ϕ_{j}(v_{j}) = 0 for all but finitely many j ∈ I. It is easy to see that

(1.1.4) ϕ is a homomorphism from ⨁_{j∈I} V_{j} into Z, as modules over k. Conversely, if ϕ is any homomorphism from ⨁_{j∈I} V_{j} into Z, as modules over k, then ϕ corresponds to a unique family of module homomorphisms ϕ_{j} from V_{j} into Z, j ∈ I, in this way.

If V is any module over k, then let

Hom(V, Z) = Hom_{k}(V, Z) (1.1.5)

be the space of homomorphisms from V into Z, as modules over k. This is a module over k too, with respect to pointwise addition and scalar multiplication of mappings from V into Z. The remarks in the previous paragraph show that there is a natural isomorphism

(1.1.6) from Hom_{k}( ⨁_{j∈I} V_{j}, Z) onto ∏_{j∈I} Hom_{k}(V_{j}, Z),

as modules over k.

Similarly, let V be a module over k, and suppose that Z_{j} is a module over k for each j ∈ I. Under these conditions, a module homomorphism from V into ∏_{j∈I} Z_{j} corresponds exactly to a family of module homomorphisms from V into Z_{j}, j ∈ I. This leads to a natural isomorphism

(1.1.7) from Hom_{k}(V, ∏_{j∈I} Z_{j}) onto ∏_{j∈I} Hom_{k}(V, Z_{j}),

as modules over k.

Suppose that ϕ_{j} is a homomorphism from V into Z_{j}, as modules over k, for each j ∈ I. Let ϕ be the element of ∏_{j∈I} Hom_{k}(V, Z_{j}) whose jth component is ϕ_{j} for each j ∈ I. This corresponds to the homomorphism

(1.1.8) from V into ∏_{j∈I} Z_{j},

as modules over k, whose jth coordinate is equal to ϕ_{j}, as in the preceding paragraph.

Suppose for the moment that

ϕ ∈ ⨁_{j∈I} Hom_{k}(V, Z_{j}), (1.1.9)

so that (1.1.10) ϕ_{j} = 0 for all but finitely many j ∈ I.

In this case, we get a homomorphism

(1.1.11) from V into ⨁_{j∈I} Z_{j},

as modules over k.

Suppose now that ϕ corresponds to a homomorphism as in (1.1.11), as modules over k. This means that for each v ∈ V,

(1.1.12) ϕ_{j}(v) = 0 for all but finitely many j ∈ I.

If V is finitely generated as a module over k, then one can verify that (1.1.10) holds, so that (1.1.9) holds.

Of course, k may be considered as a module over itself. If z ∈ Z, then

t ↦ t · z (1.1.13)

is a homomorphism from k into Z, as modules over k. It is easy to see that every homomorphism from k into Z, as modules over k, is of this form for a unique z ∈ Z. This defines an isomorphism

(1.1.14) between Hom_{k}(k, Z) and Z,

as modules over k.

A module W over k is said to be free if there is a family {w_{j}}_{j∈I} of elements of W such that every element of W can be expressed in a unique way as a linear combination of the w_{j}’s, j ∈ I, with coefficients in k, and where all but finitely many of the coefficients are equal to 0. In this case, {w_{j}}_{j∈I} may be called a basis for W as a module over k, and one may say that W is freely generated by the w_{j}’s, j ∈ I, as a module over k. Equivalently, a free module over k can be expressed as a direct sum of copies of k, as a module over itself.

1.2 Module-valued functions

Let k be a commutative ring with a mutliplicative identity element, and let Z be a module over k. Also let X be a nonempty set, and let

c(X, Z) (1.2.1)

be the space of all Z-valued functions on X. This is a module over k, with respect to pointwise addition and scalar multiplication of functions. This is the same as the direct product of copies of Z indexed by X.

If f ∈ c(X, Z), then the support of f is defined to be the set of x ∈ X such that f(x) ≠ 0. One may consider X to be equipped with the discrete topology, to be consistent with the analogous definition for functions on topological spaces. Let c_{00}(X, Z) (1.2.2)

be the set of such functions whose support has only finitely many elements. This is a submodule of c(X, Z), as a module over k, which corresponds to the direct sum of copies of Z indexed by X. If X has only finitely many elements, then this is the same as c(X, Z).

In particular, c_{00}(X, k) is a free module over k. If x ∈ X, then let δ_{x} be the k-valued function on X equal to 1 at x and to 0 at every other element of X. Thus δ_{x} ∈ c_{00}(X, k), and c_{00}(X, k) is freely generated by δ_{x}, x ∈ X.

If f ∈ c_{00}(X, Z), then

∑_{x∈X} f(x) (1.2.3)

reduces to a finite sum, and defines an element of Z. This defines a homomorphism from c_{00}(X, Z) into Z, as modules over k.

Let ζ be a Z-valued function on X. If f ∈ c_{00}(X, k), then f · ζ ∈ c_{00}(X, Z), so that

∑_{x∈X} f(x) · ζ(x) (1.2.4)

defines an element of Z, as in the preceding paragraph. This defines a homomorphism from c_{00}(X, k) into Z, as modules over k. One can check that every homomorphism ϕ from c_{00}(X, k) into Z, as modules over k, corresponds to a unique ζ ∈ c(X, Z) in this way, with

ζ(x) = ϕ(δ_{x}) (1.2.5)

for every x ∈ X. This defines an isomorphism between Hom_{k}(c_{00}(X, k), Z) and c(X, Z), as modules over k.

Let W be another module over k, and suppose that for each x ∈ X, ϕ_{x} is a homomorphism from Z into W, as modules over k. Thus

x ↦ ϕ_{x} (1.2.6)

is an element of c(X, Hom_{k}(Z, W)). If z ∈ Z, then

x ↦ ϕ_{x}(z) (1.2.7)

is an element of c(X, W). The mapping from z ∈ Z to (1.2.7) defines a homomorphism from Z into c(X, W), as modules over k. Every homomorphism from Z into c(X, W), as modules over k, corresponds to a unique element of c(X, Hom_{k}(Z, W)) in this way, as in the previous section, which leads to a natural isomorphism

(1.2.8) from Hom_{k}(Z, c(X, W)) onto c(X, Hom_{k}(Z, W)),

as modules over k, as before.

Let (1.2.6) be an element of c(X, Hom_{k}(Z, W)) again. If f ∈ c_{00}(X, k) and z ∈ Z, then f(x) · ϕ_{x}(z) (1.2.9)

defines an element of c_{00}(X, W), so that

∑_{x∈X} f(x) · ϕ_{x}(z) (1.2.10)

defines an element of W, as before. It is easy to see that this defines a mapping from c_{00}(X, k)×Z into W that is bilinear over k. Conversely, if μ is any mapping from c_{00}(X, k) × Z into W that is bilinear over k, then

ϕ_{x} = μ(δ_{x}, ·) (1.2.11)

is a homomorphism from Z into W, as modules over k, for each x ∈ X. One can check that μ is the same as the mapping from c_{00}(X, k) × Z into W defined by (1.2.10) under these conditions.

Similarly, if (1.2.6) is an element of c(X, Hom_{k}(Z, W)) and F ∈ c_{00}(X, Z), then ϕ_{x}(F (x)) (1.2.12)

defines an element of c_{00}(X, W), so that

∑_{x∈X} ϕ_{x}(F (x)) (1.2.13)

defines an element of W. In particular, if f ∈ c_{00}(X, k) and z ∈ Z, then

F (x) = f(x) · z (1.2.14)

defines an element of c_{00}(X, Z), in which case (1.2.12) and (1.2.13) are the same as (1.2.9) and (1.2.10), respectively. The mapping from F to (1.2.13) defines a homomorphism from c_{00}(X, Z) into W, as modules over k. One can verify that every homomorphism Φ from c_{00}(X, Z) into W, as modules over k, corresponds to a unique element (1.2.6) of c(X, Hom_{k}(Z, W)) in this way. More precisely, if x ∈ X and z ∈ Z, then δ_{x} z defines an element of c_{00}(X, Z), and one can take

ϕ_{x}(z) = Φ(δ_{x} z). (1.2.15)

1.3 Tensor products over commutative rings

Let k be a commutative ring with a multiplicative identity element, and let V_{1}, V_{2} be modules over k. A tensor product of V_{1} and V_{2} is a module

V_{1} ⨂_{k}V_{2} = V_{1} ⨂ V_{2} (1.3.1)

over k with the following two properties. First, V_{1} ⨂ V_{2} should be equipped with a mapping from V_{1} × V_{2} into V_{1} ⨂ V_{2} that is bilinear over k. The image of (v_{1}, v_{2}) ∈ V_{1} × V_{2} under this mapping may be expressed as v_{1} ⊗ v_{2}.

Second, let W be a module over k, and let μ be a mapping from V_{1} × V_{2} into W that is bilinear over k. Under these conditions, μ can be expressed in a unique way as the composition of the mapping from V_{1} × V_{2} into V_{1} ⨂ V_{2} mentioned in the preceding paragraph with a homomorphism from V_{1} ⨂ V_{2} into W, as modules over k. This means that there is a unique module homomorphism μ̃ from V_{1} ⨂ V_{2} into W such that

μ̃(v_{1} ⊗ v_{2}) = μ(v_{1}, v_{2}) (1.3.2) for every v_{1} ∈ V_{1} and v_{2} ∈ V_{2}.

It is well known and not too difficult to show that such a tensor product is unique up to a suitable isomorphic equivalence. We also have that V_{1} ⨂ V_{2} is generated by the associated image of V_{1} × V_{2}, as a module over k, or simply as a commutative group with respect to addition. This is easily seen using the standard construction of tensor products, or the uniqueness of tensor product up to isomorphism.

Let X be a nonempty set, and let Z be a module over k. If f ∈ c_{00)}(X, k) and z ∈ Z, then f z ∈ c_{00}(X, Z). This defines a mapping

(1.3.3) from c_{00}(X, k) × Z into c_{00}(X, Z)

that is bilinear over k. One can check that c_{00}(X, Z) satisfies the requirements of a tensor product of c_{00}(X, k) and Z, as modules over k, with respect to this mapping. This uses the description of bilinear mappings from c_{00}(X, k)×Z into other modules over k mentioned in the previous section.

Let W_{1}, W_{2} be another pair of modules over k, and let ϕ_{j} be a homomorphism from V_{j} into W_{j} for j = 1, 2, as modules over k. This leads to a mapping

(v_{1}, v_{2}) ↦ ϕ_{1}(v_{1}) ⊗ ϕ_{2}(v_{2}) (1.3.4)

from V_{1} × V_{2} into W_{1} ⨂ W_{2} that is bilinear over k. Thus we get a unique homomorphism ϕ (1.3.5) from V_{1} ⨂ V_{2} into W_{1} ⨂ W_{2},

as modules over k, such that

ϕ(v_{1} ⊗ v_{2}) = ϕ_{1}(v_{1}) ⊗ ϕ_{2}(v_{2}) (1.3.6)

for every v_{1} ∈ V_{1}, v_{2} ∈ V_{2}.

If ϕ_{j}(V_{j}) = W_{j} for j = 1, 2, then

ϕ(V_{1} ⨂ V_{2}) = W_{1} ⨂ W_{2}. (1.3.7)

This follows from the fact that W_{1} ⨂ W_{2} is generated by the associated image of W × W_{2}, as a module over k.

Suppose for the moment that V_{j} = W_{j} for j = 1, 2, and that ϕ_{j} is the identity mapping on V_{j}, j = 1, 2. In this case, the identity mapping on V_{1} ⨂ V_{2} satisfies (1.3.6), so that ϕ is the identity mapping, by uniqueness.

Let Z_{1}, Z_{2} be a pair of modules over k, and let ψ_{j} be a homomorphism from W_{j} into Z_{j}, j = 1, 2, as modules over k. Also let ψ be the corresponding homomorphism (1.3.8) from W_{1} ⨂ W_{2} into Z_{1} ⨂ Z_{2},

as modules over k, as in (1.3.6). Of course, ψ_{j} ∘ ϕ_{j} is a module homomorphism from V_{j} into Z_{j}, j = 1, 2. One can verify that ψ ∘ ϕ is the same as the module homomorphism (1.3.9) from V_{1} ⨂ V_{2} into Z_{1} ⨂ Z_{2} associated to ψ_{j} ∘ ϕ_{j}, j = 1, 2, as before.

Suppose now that V_{j} = Z_{j} for j = 1, 2, and that ψ_{j} ∘ ϕ_{j} is the identity mapping on V_{j}, j = 1, 2. Under these conditions, we get that

(1.3.10) ψ ∘ ϕ is the identity mapping on V_{1} ⨂ V_{2},

as before.

Similarly, if ϕ_{j} ∘ ψ_{j} is the identity mapping on W_{j} for j = 1, 2, then ϕ ∘ ψ is the identity mapping on W_{1} ⨂ W_{2}. If ϕ_{j} is a module isomorphism from V_{j} onto W_{j} for j = 1, 2, then it follows that

(1.3.11) ϕ is a module isomorphism from V_{1} ⨂ V_{2} onto W_{1} ⨂ W_{2}.

1.4 Some properties of tensor products

Let k be a commutative ring with a multiplicative identity element, and let V_{1}, V_{2} be modules over k. Observe that

(v_{1}, v_{2}) ↦ v_{2} ⊗ v_{2} (1.4.1)

defines a mapping from V_{1} × V_{2} into V_{2} ⨂ V_{1} that is bilinear over k, which leads to a homomorphism (1.4.2) from V_{1} ⨂ V_{2} into V_{2} ⨂ V_{1},

as modules over k. Similarly,

(v_{2}, v_{1}) ↦ v_{1} ⊗ v_{2} (1.4.3)

defines a mapping from V_{2} × V_{1} into V_{1} ⊗ V_{2} that is bilinear over k, which leads to a homomorphism (1.4.4) from V_{2} ⨂ V_{1} into V_{1} ⨂ V_{2},

as modules over k. One can check that the compositions of these two homomorphisms in either order is equal to the identity mapping on V_{1} ⨂ V_{2} or V_{2} ⨂ V_{1}, as appropriate. Thus these homomorphisms are isomorphisms.

Note that scalar multiplication on V_{1} defines a mapping from V_{1} × k into V_{1} that is bilinear over k, where k is considered as a module over itself. One can verify that (1.4.5) V_{1} satisfies the requirements of V_{1} ⨂ k,

using this mapping. Similarly,

(1.4.6) V_{2} satisfies the requirements of k ⨂ V_{2},

with respect to the bilinear mapping from k ×V_{2} into V_{2} corresponding to scalar multiplication on V_{2}.

If V_{3} is another module over k, then it is well known that there is a unique isomorphism

(1.4.7) from (V_{1} ⨂ V_{2}) ⨂ V_{3} onto V_{1} ⨂(V_{2} ⨂ V_{3}),

as modules over k, with

(v_{1} ⊗ v_{2}) ⊗ v_{3} ↦ v_{1} ⊗ (v_{2} ⊗ v_{3}) (1.4.8)

for every v_{1} ∈ V_{1}, v_{2} ∈ V_{2}, and v_{3} ∈ V_{3}. Alternatively, one can define tensor products of finitely many modules over k directly, and get unique isomorphisms with V_{1} ⨂ V_{2} ⨂ V_{3} such that (v_{1} ⊗ v_{2}) ⊗ v_{3} and v_{1} ⊗ (v_{2} ⊗ v_{3}) correspond to v_{1} ⊗ v_{2} ⊗ v_{3} for every v_{1} ∈ V_{1}, v_{2} ∈ V_{2}, and v_{3} ∈ V_{3}.

Let I be a nonempty set, let V_{j} be a module over k for every j ∈ I, and let W, Z be modules over k too. It is easy to see that a bilinear mapping

(1.4.9) from ( ⨁_{j∈I} V_{j}) × W into Z

corresponds exactly to a family of bilinear mappings from V_{j} × W into Z, j ∈ I. In particular, if Z_{j} is a module over k for each j ∈ I, then a family of bilinear mappings from V_{j} × W into Z_{j}, j ∈ I, leads to a bilinear mapping

(1.4.10) from ( ⨁_{j∈I} V_{j}) × W into ⨁_{j∈I} Z_{j}.

It is well known that there is a natural isomorphism

(1.4.11) from ( ⨁_{j∈I} V_{j}) ⨂ W onto ⨁_{j∈I}(V_{j} ⨂ W),

as modules over k. To see this, one can define natural mappings in both directions, as follows.

If j ∈ I, then V_{j} ⨂ W comes with a bilinear mapping from V_{j} × W into V_{j} ⨂ W, as in the previous section. This leads to a bilinear mapping

(1.4.12) from ( ⨁_{j∈I} V_{j}) × W into ⨁_{j∈I}(V_{j} ⨂ W),

as before. Using this, we get a homomorphism

(1.4.13) from ( ⨁_{j∈I} V_{j}) ⨂ W into ⨁_{j∈I}(V_{j} ⨂ W),

as modules over k, as in the previous section. If v ∈ ⨁_{j∈I} V_{j} and w ∈ W, then this mapping sends v⊗w to the element of ⨁_{j∈I}(V_{j} ⨂ W) whose jth coordinate is equal to v_{j} ⊗ w for every j ∈ I. Note that v_{j} = 0 for all but finitely many j, so that v_{j} ⊗ w = 0 for all but finitely many j.

If l ∈ I, then there is an obvious inclusion mapping from V_{l} into ⨁_{j∈I} V_{j}, which sends v_{l} ∈ V_{l} to the element of ⨁_{j∈I} V_{j} whose lth coordinate is equal to v_{l}, and whose other coordinates are equal to 0. This leads to a homomorphism

(1.4.14) from V_{l} ⨂ W into ( ⨁_{j∈I} V_{j}) ⨂ W,

as modules over k, using the identity mapping on W. We may combine these mappings to get a homomorphism

(1.4.15) from ⨁_{j∈I}(V_{j} ⨂ W) into ( ⨁_{j∈I} V_{j}) ⨂ W,

as modules over k, as in Section 1.1. More precisely, if l ∈ I, v_{l} ∈ V_{l}, and w ∈ W, then v_{l} ⊗ w ∈ V_{l} ⨂ W, and we get an element of ⨁_{j∈I}(V_{j} ⨂ W) by taking the jth coordinate in V_{j} ⨂ W to be 0 when j ≠ l. The mapping just described sends this element of ⨁_{j∈I}(V_{j} ⨂ W) to an element of ( ⨁_{j∈I} V_{j}) ⨂ W of the form v ⊗ w, where v ∈ ⨁_{j∈I} V_{j} has lth coordinate equal to v_{l}, and jth coordinate equal to 0 when j ≠ l.

One can verify that the mappings described in the previous two paragraphs are inverses of each other, so that we get a module isomorphism, as before. In particular, if V is a free module over k, then V ⨂ W corresponds to a direct sum of copies of W. If W is also free as a module over k, then V ⨂ W is free as well.

1.5 Modules over associative algebras

Let k be a commutative ring with a multiplicative identity element, let A_{1} be an associative algebra over k with a multiplicative identity element e_{1} = e_{A_{1}}, and let V_{1} be a module over k. (Some basic notions related to algebras over k are mentioned in Section 20.13.) Suppose that for every a_{1} ∈ A_{1} and v_{1} ∈ V_{1}, a_{1} · v_{1} is defined as an element of V_{1}, and that the corresponding mapping from A_{1} × V_{1} into V_{1} is bilinear over k. If we also have that

a_{1} · (b_{1} · v_{1}) = (a_{1} b_{1}) · v_{1} (1.5.1)

and e_{1} · v_{1} = v_{1} (1.5.2)

for every a_{1}, b_{1} ∈ A_{1} and v_{1} ∈ V_{1}, then V_{1} is said to be a left module over A_{1}. Let W_{1} be a submodule of V_{1}, as a module over k. If

a_{1} · w_{1} ∈ W_{1} (1.5.3)

for every a_{1} ∈ A_{1} and w_{1} ∈ W_{1}, then W_{1} is said to be a submodule of V_{1}, as a left module over A_{1}. Of course, this means that W_{1} is a left module over A_{1} too.

Let Z_{1} be another module over k that is also a left module over A_{1}. A mapping ϕ_{1} from V_{1} into Z_{1} is said to be a homomorphism from V_{1} into Z_{1} as left modules over A_{1} if ϕ is linear over k and

ϕ_{1}(a_{1} · v_{1}) = a_{1} · ϕ(v_{1}) (1.5.4)

for every a_{1} ∈ A_{1} and v_{1} ∈ V_{1}. We may also say that ϕ_{1} is linear over A_{1} on the left in this case. Note that the kernel of ϕ_{1} is a submodule of V_{1}, as a left module over A_{1}, under these conditions. If W_{1} is a submodule of V_{1}, as a left module over A_{1}, then the obvious inclusion mapping from W_{1} into V_{2} may be considered as a homomorphism from W_{1} into V_{1}, as left modules over A_{1}.

Let W_{1} be a submodule of V_{1} again, as a left module over A_{1}, and consider the quotient V_{1}/W_{1}, defined initially as a module over k. If a_{1} ∈ A_{1}, then we would like to define the action of a_{1} on V_{1}/W_{1} on the left by putting

a_{1} · q_{1}(v_{1}) = q_{1}(a_{1} · v_{1}) (1.5.5)

for every v_{1} ∈ V_{1}, where q_{1} is the natural quotient mapping from V_{1} onto V_{1}/W_{1}. One can check that the right side depends only on a_{1} and q_{1}(v_{1}), and that V_{1}/W_{1} becomes a left module over A_{1} in this way. By construction, q_{1} is a homomorphism from V_{1} onto V_{1}/W_{1}, as left modules over A_{1}.

Similarly, let A_{2} be an associative algebra over k with a multiplicative identity element e_{2} = e_{A_{2}}, and let V_{2} be a module over k. Suppose that for every a_{2} ∈ A_{2} and v_{2} ∈ V_{2}, v_{2} · a_{2} is defined as an element of V_{2}, and that the corresponding mapping from V_{2} × A_{2} into V_{2} is bilinear over k. If

(v_{2} · a_{2}) · b_{2} = v_{2} · (a_{2} b_{2}) (1.5.6)

and v_{2} · e_{2} = v_{2} (1.5.7)

for every a_{2}, b_{2} ∈ A_{2} and v_{2} ∈ V_{2}, then V_{2} is a right module over A_{2}. Of course, left and right modules over commutative algebras are basically the same. One can define submodules of right modules, homomorphisms between right modules, and quotients of right modules in essentially the same ways as for left modules.

1.6 Sums, products, and compositions

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e = e_{A}, and let V, W, and Z be modules over k. Suppose that ϕ is a homomormophism from V into W, and that ψ is a homomorphism from W into Z, as modules over k, so that the composition ψ ∘ ϕ of ϕ and ψ is a homomorphism from V into Z, as modules over k. If V, W, and Z are left modules over A, ϕ is a homomorphism from V into W as left modules over A, and ψ is a homomorphism from W into Z as left modules over A, then ψ ∘ ϕ is a homomorphism from V into Z, as left modules over A. Similarly, if V, W, and Z are right modules over A, ϕ is a homomorphism from V into W as right modules over A, and ψ is a homomorphism from W into Z as right modules over A, then ψ ∘ ϕ is a homomorphism from V into Z, as right modules over A.

Suppose that ϕ is a one-to-one homomorphism from V onto W, as modules over k, so that the inverse ϕ^{−1} of ϕ is a homomorphism from W into V, as modules over k. Equivalently, this means that ϕ is an isomorphism from V onto W, as modules over k. If V, W are left modules over A, and ϕ is a homomorphism from V onto W, as left modules over A, then ϕ^{−1} is a homomorphism from W onto V, as left modules over A. Similarly, if V and W are right modules over A, and ϕ is a homomorphism from V onto W, as right modules over A, then ϕ^{−1} is a homomorphism from W onto V, as right modules over A. In each case, ϕ is said to be an isomorphism from V onto W, as left or right modules over A, as appropriate.

Note that A may be considered as both a left and right module over itself, in the obvious way. Let w ∈ W be given, and suppose for the moment that W is a left module over A. In this case,

a ↦ a · w (1.6.1)

defines a homomorphism from A into W, as left modules over A. It is easy to see that every homomorphism ϕ from A into W as left modules over A corresponds to a unique w ∈ W in this way, with w = ϕ(e).

Similarly, if W is a right module over A, then

a ↦ w · a (1.6.2)

defines a homomorphism from A into W, as right modules over A. As before, every homomorphism ϕ from A into W as right modules over A corresponds to a unique w ∈ W in this way, with w = ϕ(e).

Let I be a nonempty set, and let V_{j} be a module over k for each j ∈ I, so that the direct sum and product of the V_{j}’s may be defined as modules over k as in Section 1.1. If V_{j} is a left module over A for every j ∈ I, then ∏_{j∈I} V_{j} is a left module over A too, where the action of A on the left is defined coordinatewise. Under these conditions, ⨁_{j∈I} V_{j} is a submodule of ∏_{j∈I} V_{j}, as a left module over A. Similarly, if V_{j} is a right module over A for each j ∈ I, then ∏_{j∈I} V_{j} is a right module over A as well, where the action of A on the right is defined coordinatewise, and ⨁_{j∈I} V_{j} is a submodule of ∏_{j∈I} V_{j}, as a right module over A.

A left module over A is said to be free as a left module over A if it is isomorphic to the direct sum of a family of copies of A, as a left module over itself. Similarly, a right module over A is said to be free as a right module over A if it is isomorphic to the direct sum of a family of copies of A, as a right module over itself.

If X is a nonempty set, and V is a left or right module over A, then the space c(X, V) of V -valued functions on X is a left or right module over A, as appropriate, where the action of A is defined pointwise. This corresponds to the direct product of copies of V indexed by X. Similarly, the space c_{00}(X, V) of V -valued functions on X with finite support is a submodule of c(X, V), as a left or right module over A, as appropriate. This corresponds to the direct sum of copies of V indexed by X.

1.7 Spaces of homomorphisms

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e = e_{A}, and let V, W be modules over k. If V and W are both left moduules over A, then the space of homomorphisms from V into W as left modules over A may be denoted

Hom(V, W) = Hom_{A}(V, W). (1.7.1)

Similarly, if V and W are both right modules over A, then the space of homomorphisms from V into W as right modules over A may be denoted in the same way. In both cases, Hom_{A}(V, W) is a module over k, with respect to pointwise addition and scalar multiplication of mappings from V into W.

If W is a left module over A, and A is considered as a left module over itself, then the characterization of homomorphisms from A into W, as left modules over A, mentioned in the previous section defines an isomorphism

(1.7.2) between Hom_{A}(A, W) and W,

as modules over k. Similarly, if W is a right module over A, and A is considered as a right module over itself, then the characterization of homomorphisms from A into W, as right modules over A, discussed in the previous section defines an isomorphism as in (1.7.2) again, as modules over k.

Let I be a nonempty set, and let V_{j} be a module over k for each j ∈ I. If V_{j} is a left module over A for every j ∈ I, and W is a left module over A, then there is a natural isomorphism

(1.7.3) from Hom_{A}( ⨁_{j∈I} V_{j}, W ) onto ∏_{j∈I} Hom_{A}(V_{j}, W),

as modules over k, as in Section 1.1. Similarly, if V_{j} is a right module over A for every j ∈ I, and W is a right module over A, then there is a natural isomorphism as in (1.7.3), as modules over k.

Let V be a module over k again, and let W_{j} be a module over k for every j ∈ I. If V is a left module over A, and W_{j} is a left module over A for every j ∈ I, then there is a natural isomorphism

(1.7.4) from Hom_{A}(V, ∏_{j∈I} W_{j}) onto ∏_{j∈I} Hom_{A}(V, W_{j}),

as modules over k, as in Section 1.1. Similarly, if V is a right module over A, and W_{j} is a right module over A for every j ∈ I, then there is a natural isomorphism as in (1.7.4), as modules over k.

Let V, W, Z be modules over k, and let ϕ_{0} be a homomorphism from V into W, as modules over k. Observe that

ψ ↦ ψ ∘ ϕ_{0} (1.7.5)

defines a homomorphism

(1.7.6) from Hom_{k}(W, Z) into Hom_{k}(V, Z),

as modules over k. If ϕ_{0}(V) = W, (1.7.7)

then this homomorphism is injective. If ϕ_{0} is an isomorphism from V onto W, as modules over k, then (1.7.5) defines an isomorphism as in (1.7.6), as modules over k.

If ψ_{0} is a homomorphism from W into Z, as modules over k, then

ϕ ↦ ψ_{0} ∘ ϕ (1.7.8)

defines a homomorphism

(1.7.9) from Hom_{k}(V, W) into Hom_{k}(V, Z),

as modules over k. If (1.7.10) ψ_{0} is injective,

then this homomorphism is injective. If ψ_{0} is an isomorphism from W onto Z, as modules over k, then (1.7.5) defines an isomorphism as in (1.7.9), as modules over k.

Suppose now that V, W, and Z are left modules over A. If ϕ_{0} is a homomorphism from V into W, as left modules over A, then (1.7.5) defines a homomorphism (1.7.11) from Hom_{A}(W, Z) into Hom_{A}(V, Z),

as modules over k. This homomorphism is injective when ϕ_{0} is surjective, as before. If ϕ_{0} is an isomorphism from V onto W, as left modules over A, then (1.7.5) defines an isomorphism as in (1.7.11), as modules over k.

Similarly, if ψ_{0} is a homomorphism from W into Z as left modules over A, then (1.7.8) defines a homomorphism

(1.7.12) from Hom_{A}(V, W) into Hom_{A}(V, Z),

as modules over k. This homomorphism is injective when ψ_{0} is injective. If ψ_{0} is an isomorphism from W onto Z, as modules over A, then (1.7.8) defines an isomorphism as in (1.7.12), as modules over k. Of course, there are analogous statements when V, W, and Z are right modules over A, ϕ_{0} is a homomorphism from V into W as right modules over A, and ψ_{0} is a homomorphism from W into Z as right modules over A.

1.8. ADDITIONAL STRUCTURE ON HOM(V, W) 15

1.8 Additional structure on Hom(V, W)

Let k be a commutative ring with a multiplicative identity element, let A, B be associative algebras over k with multiplicative identity elements e_{A}, e_{B}, respectively, and let V, W be modules over k. If W is a left or right module over B, then Hom_{k}(V, W) is a left or right module over B as well, respectively, where the action of B is defined pointwise. Similarly, if V is a left or right module over B, then Hom_{k}(V, W) is a right or left module over B, respectively, in a natural way. More precisely, if ϕ ∈ Hom_{k}(V, W), then the action of an element b of B on ϕ is defined by composing ϕ with the action of b on V.

Suppose now that V and W are left modules over A, and that W is a left or right module over B, where the actions of A and B on W commute with each other. In this case, one may say that W is a bimodule over A and B. Equivalently, this means that the action of a ∈ A on W is a homomorphism from W into itself, as a module over B. This is the same as saying that the action of b ∈ B on W is a homomorphim from W into itself, as a module over A. Under these conditions, the space Hom_{A}(V, W) of homomorphisms from V into W, as left modules over A, is a left or right module over B, where the action of B is defined pointwise, as usual. Of course, there is an analogous statement when V and W are right modules over A. This corresponds to some remarks on p22 of [50].

Suppose that V and W are left modules over A again, and now that V is a left or right module over B, and that the actions of A and B on V commute with each other. Under these conditions, the space Hom_{A}(V, W) of homomorphisms from V into W, as left modules over A, is a right or left module over B, respectively. As before, the action of b ∈ B on ϕ ∈ Hom_{A}(V, W) is defined by composing ϕ with the action of b on V. There is an analogous statement when V and W are right modules over A. This corresponds to some more remarks on p22 of [50].

The left and right actions of A on itself commute, by associativity, so that A is a bimodule as a left and right module over itself. If W is a left module over A, then the space Hom_{A}(A, W) of homomorphisms from A into W, as left modules over A, is a left module over A too, with respect to the action determined by the action of A on itself on the right. Remember that there is a natural isomorphism between Hom_{A}(A, W) and W, as modules over k, as in the previous section. One can check that this is an isomorphism

(1.8.1) between Hom_{A}(A, W) and W, as left modules over A,

as mentioned on p21 of [50]. Similarly, if W is a right module over A, then the space Hom_{A}(A, W) of homomorphisms from A into W, as right modules over A, is a right module over A, with respect to the action determined by the action of A on itself on the left. One can verify that the isomorphism between Hom_{A}(A, W) and W, as modules over k, mentioned in the previous section is an isomorphism

(1.8.2) between Hom_{A}(A, W) and W, as right modules over A.

Let Z be another module over k, and let ϕ_{0} be a homomorphism from V into W, as modules over k. Suppose for the moment that V, W, and Z are left modules over A, and that ϕ_{0} is a homomorphism from V into W, as left modules over A. Suppose also that V and W are both left or both right modules over B, where the actions of A and B on each of V and W commute, and that ϕ_{0} is a homomorphism from V into W as left or right modules over B too. Under these conditions, one can check that composition with ϕ_{0} defines a homomorphism

(1.8.3) from Hom_{A}(W, Z) into Hom_{A}(V, Z), as right or left modules over B,

as appropriate, as in (1.7.5).

Suppose now that W and Z are both left or both right modules over B, where the actions of A and B on each of W and Z commute. Let ψ_{0} be a homomorphism from W into Z, as modules over k, left modules over A, and left or right modules over B, as appropriate. In this case, composition with ψ_{0} defines a homomorphism

(1.8.4) from Hom_{A}(V, W) into Hom_{A}(V, Z), as left or right modules over B,

as appropriate, as in (1.7.8).

As usual, there are analogous statements when V, W, and Z are right modules over A, and ϕ_{0}, ψ_{0} are homomorphisms from V into W and W into Z, respectively, as right modules over A.

1.9 Tensor products over algebras

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e = e_{A}. Also let V_{1}, V_{2} be modules over k, with V_{1} a right module over A, and V_{2} a left module over A. A tensor product of V_{1} and V_{2} over A is a module V_{1} ⨂_{A} V_{2} over k with the following two properties.

First, V_{1} ⨂_{A} V_{2} should be equipped with a mapping from V_{1} × V_{2} into V_{1} ⨂_{A} V_{2} that is bilinear over k. If v_{1} ∈ V_{1} and v_{2} ∈ V_{2}, then the image of (v_{1}, v_{2}) under this mapping may be denoted v_{1} ⊗ v_{2}, or v_{1} ⊗_{A} v_{2} to indicate the role of A. This mapping should also satisfy

(v_{1} · a) ⊗ v_{2} = v_{1} ⊗ (a · v_{2}) (1.9.1)

for every a ∈ A, v_{1} ∈ V_{1}, and v_{2} ∈ V_{2}.

Second, let W be a module over k, and let b be a mapping from V_{1} × V_{2} into W that is bilinear over k. Suppose that

b(v_{1} · a, v_{2}) = b(v_{1}, a · v_{2}) (1.9.2)

for every a ∈ A, v_{1} ∈ V, and v_{2} ∈ V_{2}. Under these conditions, there should be a unique homomorphism c from V_{1} ⨂_{A} V_{2} into W, as modules over k, such that

b(v_{1}, v_{2}) = c(v_{1} ⊗ v_{2}) (1.9.3) for every v_{1} ∈ V_{1} and v_{2} ∈ V_{2}.

Of course, this reduces to the definition in Section 1.3 when A = k. As before, V_{1} ⨂_{A} V_{2} is unique up to a suitable isomorphic equivalence, and V_{1} ⨂_{A} V_{2} is generated by the associated image of V_{1} ×V_{2}, as a module over k, or even as a commutative group with respect to addition. The latter follows from the standard construction of tensor products, or their uniqueness up to isomorphism, as in the previous case. Note that the bilinear mapping from V_{1} ×V_{2} into V_{1} ⨂_{A} V_{2} mentioned earlier leads to a homomorphism

(1.9.4) from V_{1} ⨂_{k}V_{2} into V_{1} ⨂_{A}V_{2},

as modules over k. This homomorphism is a surjection, by the preceding remark.

Let W_{1}, W_{2} be another pair of modules over k, with W_{1} a right module over A, and W_{2} and left module over k. Suppose that ϕ_{1} is a homomorphism from V_{1} into V_{2}, as right modules over A, and that ϕ_{2} is a homomorphism from V_{2} into W_{2}, as left modules over A. Note that

(v_{1}, v_{2}) ↦ ϕ_{1}(v_{1}) ⊗ ϕ_{2}(v_{2}) (1.9.5)

is bilinear over k as a mapping from V_{1} × V_{2} into W_{1} ⨂_{A} W_{2}. If a ∈ A, v_{1} ∈ V_{1}, and v_{2} ∈ V_{2}, then

ϕ_{1}(v_{1} · a) ⊗ ϕ_{2}(v_{2}) = (ϕ_{1}(v_{1}) · a) ⊗ ϕ_{2}(v_{2})
(1.9.6)  = ϕ_{1}(v_{1}) ⊗ (a · ϕ_{2}(v_{2})) = ϕ_{1}(v_{1}) ⊗ ϕ_{2}(a · v_{2}).

Using this, we get a unique homomorphism ϕ

(1.9.7) from V_{1} ⨂_{A}V_{2} into W_{1} ⨂_{A}W_{2},

as modules over k, such that

ϕ(v_{1} ⊗ v_{2}) = ϕ_{1}(v_{1}) ⊗ ϕ_{2}(v_{2}) (1.9.8)

for every v_{1} ∈ V_{1} and v_{2} ∈ V_{2}.

If ϕ_{j}(V_{j}) = W_{j} for j = 1, 2, then

ϕ(V_{1} ⨂_{A}V_{2}) = W_{1} ⨂_{A}W_{2}, (1.9.9)

because W_{1} ⨂_{A} W_{2} is generated by the associated image of W_{1} × W_{2}, as a module over k.

Let Z_{1}, Z_{2} be a pair of modules over k, with Z_{1} a right module over A, and Z_{2} a left module over A. Also let ψ_{1} be a homomorphism from W_{1} into Z_{1}, as right modules over A, and let ψ_{2} be a homomorphism from W_{2} into Z_{2}, as left modules over A. This leads to a homomorphism ψ

(1.9.10) from W_{1} ⨂_{A}W_{2} into Z_{1} ⨂_{A}Z_{2}, as modules over k, as before. Note that ψ_{1} ∘ ϕ_{1} is a homomorphism from V_{1} into Z_{1}, as right modules over A, and that ψ_{2} ∘ ϕ_{2} is a homomorphism from V_{2} into Z_{2}, as left modules over A. One can check that ψ ∘ ϕ is the same as the homomorphism (1.9.11) from V_{1} ⨂_{A}V_{2} into Z_{1} ⨂_{A}Z_{2},

as modules over k, associated to ψ_{1} ∘ ϕ_{1}, ψ_{2} ∘ ϕ_{2}, as before.

Suppose that ϕ_{1} is an isomorphism from V_{1} onto W_{1}, as right modules over A, and that ϕ_{2} is an isomorphism from V_{2} onto W_{2}, as left modules over A. Under these conditions,

(1.9.12) ϕ is an isomorphism from V_{1} ⨂_{A}V_{2} onto W_{1} ⨂_{A}W_{2},

as modules over k. More precisely, ϕ^{−1} can be obtained from ϕ_{1}^{−1}, ϕ_{2}^{−1} as a mapping from W_{1} ⨂_{A} W_{2} into V_{1} ⨂_{A} V_{2} in the same way as before.

1.10 Additional properties and structure

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e = e_{A}. Remember that A may be considered as both a left module and a right module over itself. If V is a module over k that is a right module over A, then one can check that (1.10.1) V satisfies the requirements of V ⨂_{A}A,

as a module over k, using the mapping (v, a) ↦ v·a from V ×A into V. Similarly, if W is a module over k that is a left module over A, then

(1.10.2) W satisfies the requirements of A ⨂_{A}W,

as a module over k, using the mapping (a, w) ↦ a · w from A × W into W.

Let I be a nonempty set, and let V_{j} be a module over k for each j ∈ I. Suppose that V_{j} is a right module over A for each j ∈ I, and that W is a module over k that is a left module over A. Under these conditions, one can get a natural isomorphism

(1.10.3) from ( ⨁_{j∈I} V_{j}) ⨂_{A}W onto ⨁_{j∈I}(V_{j} ⨂_{A}W),

as modules over k, as in Section 1.4. Similarly, if V_{j} is a left module over A for every j ∈ I, and W is a right module over A, then there is a natural isomorphism

(1.10.4) from W ⨂_{A}( ⨁_{j∈I} V_{j}) onto ⨁_{j∈I}(W ⨂_{A}V_{j}),

as modules over k.

If V is a free right module over A, and W is any left module over A, then it follows that V ⨂_{A} W corresponds to a direct sum of copies of W, as a module over k. Similarly, if V is any right module over A, and W is a free left module over A, then V ⨂_{A} W corresponds to a direct sum of copies of V, as a module over k.

Let V be a right module over A, let W be a left module over A, and let B be another associative algebra over k with a multiplicative identity element e_{B}. If V or W is also a left or right module over B, where the actions of A and B commute, then

(1.10.5) V ⨂_{A}W is a left or right module over B,

as appropriate, as on p23 of [50].

More precisely, suppose that V is a left or right module over B, where the actions of A and B on V commute. If b ∈ B, then the action of b on V is a homomorphism from V into itself, as a right module over A. Of course, the identity mapping on W is a homomorphism from W into itself, as a left module over A. One can use these two module homomorphisms to get a homomorphism from V ⨂_{A} W into itself, as a module over k, as in the previous section. This is used to define the action of b on V ⨂_{A} W. If v ∈ V and w ∈ W, then the action of b on v ⊗_{A} w is obtained by combining the action of b on v with w using ⊗_{A}, and the action of b on V ⨂_{A} W is uniquely determined by this property. One can check that this makes V ⨂_{A} W a left or right module over B, as appropriate. If W is a left or right module over B, where the actions of A and B on W commute, then the action of B on V ⨂_{A} W can be defined analogously.

In particular, V ⨂_{A} A is a right module over A in a natural way, because A is a left and right module over itself, and the actions of A on itself on the left and right commute. One can check that

(1.10.6) V ⨂_{A}A corresponds to V as a right module over A,

as in (1.10.1). Similarly,

(1.10.7) A ⨂_{A}W corresponds to W as a left module over A,

as in (1.10.2).

Let V_{1}, V_{2}, W_{1}, W_{2} be modules over k, with V_{1}, W_{1} right modules over A, and V_{2}, W_{2} left modules over A. Also let ϕ_{1} be a homomorphism from V_{1} into W_{1}, as right modules over A, and let ϕ_{2} be a homomorphism from V_{2} into W_{2}, as left modules over A. This leads to a unique homomorphism ϕ from V_{1} ⨂_{A} V_{2} into W_{1} ⨂_{A} W_{2}, as modules over k, that satisfies (1.9.8). Suppose that V_{1}, W_{1} are both left modules over B, or both right modules over B, where the actions of A and B on V_{1} and W_{1} commute. Thus V_{1} ⨂_{A} V_{2} and W_{1} ⨂_{A} W_{2} may be considered as left or right modules over B, as appropriate. If

(1.10.8) ϕ_{1} is a homomorphism from V_{1} into W_{1}, as modules over B, then one can check that

(1.10.9) ϕ is a homomorphism from V_{1} ⨂_{A}V_{2} into W_{1} ⨂_{A}W_{2},

as modules over B.

Similarly, suppose instead that V_{2}, W_{2} are both left or both right modules over B, where the actions of A and B on V_{2}, W_{2} commute, so that V_{1} ⨂_{A} V_{2} and W_{1} ⨂_{A} W_{2} may be considered as left or right modules over B, as appropriate. If (1.10.10) ϕ_{2} is a homomorphism from V_{2} into W_{2}, as modules over B,

then one can verify that (1.10.9) holds again.

1.11 Opposite algebras

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k. The corresponding opposite algebra A^{op} is defined to be the same as A as a module over k. The product of x, y ∈ A^{op} in A^{op} is defined to be the same as the product y x of y and x in A. It is easy to see that A^{op} is an associative algebra over k too. If A has a multiplicative identity element e_{A}, then e_{A} is the multiplicative identity element in A^{op} as well.

Of course, if A is a commutative algebra, then A^{op} is the same as A. Note that the opposite algebra (A^{op})^{op} of A^{op} is always the same as A.

If x ∈ A, then it may be helpful to use x^{op} to indicate that a is being considered as an element of A^{op}. This is similar to the definition on p109 of [50], with slightly different notation. Using this, multiplication in A^{op} is given by x^{op} y^{op} = (y x)^{op}. (1.11.1)

Let B be another associative algebra over k, and let ϕ be a homomorphism from A into B, as modules over k. Note that ϕ is a homomorphism from A into B as algebras over k if and only if ϕ is a homomorphism from A^{op} into B^{op}, as algebras over k. Let us say that ϕ is an opposite algebra homomorphism from A into B if ϕ(x y) = ϕ(y) ϕ(x) (1.11.2)

for every x, y ∈ A. This is equivalent to asking that ϕ be a homomorphism from A into B^{op}, or from A^{op} into B, as algebras over k. If A, B have multiplicative identity elements e_{A}, e_{B}, respectively, then one may wish to ask that

ϕ(e_{A}) = e_{B}. (1.11.3)

If ϕ is a one-to-one opposite algebra homomorphism from A onto B, then the inverse mapping ϕ^{−1} is an opposite algebra homomorphism from B onto A. In this case, we may say that ϕ is an opposite algebra isomorphism from A onto B. If A has a multiplicative identity element e_{A}, then it follows that ϕ(e_{A}) is the multiplicative identity element in B. An opposite algebra isomorphism from A onto itself may also be called an opposite algebra automorphism of A.

An algebra involution on A is an opposite algebra homomorphism from A into itself whose composition with itself is the identity mapping on A. If k is the field of complex numbers, then one may also be interested in the analogous notion for conjugate-linear mappings.

Let n be a positive integer, so that the space M_{n}(A) of n × n matrices with entries in A is an associative algebra over k with respect to matrix multiplication. Of course, M_{n}(A^{op}) is an associative algebra over k too. One can check that the usual mapping from a matrix to its transpose defines an opposite algebra isomorphism from M_{n}(A) onto M_{n}(A^{op}).

Suppose that A has a multiplicative identity element e_{A}, and let V be a module over k. If V is a left module over A, then V may be considered as a right module over A^{op}, with v · a^{op} = a · v (1.11.4) for every a ∈ A and v ∈ V, as on p109 of [50]. Similarly, if V is a right module over A, then V may be considered as a left module over A^{op}.

Suppose that V is a right module over A, and that W is a module over k that is a left module over A. Thus V may be considered as a left module over A^{op}, and W may be considered as a right module over A^{op}, as in the preceding paragraph. Let V ⨂_{A} W be a tensor product of V and W over A, and let W ⨂_{A^{op}} V be a tensor product of W and V over A^{op}. If v ∈ V and w ∈ W, then let v ⊗_{A} w and w ⊗_{A^{op}} v be the corresponding elements of V ⨂_{A} W and W ⨂_{A^{op}} V, respectively.

Of course, (v, w) ↦ w ⊗_{A^{op}} v (1.11.5) defines a mapping from V × W into W ⨂_{A^{op}} V that is bilinear over k. If a ∈ A, v ∈ V, and w ∈ W, then this mapping sends

(v · a, w) = (a^{op} · v, w) (1.11.6)

and (v, a · w) = (v, w · a^{op}) (1.11.7) to (w · a^{op}) ⊗_{A^{op}} v = w ⊗_{A^{op}} (a^{op} · v). (1.11.8) It follows that there is a unique homomorphism from V ⨂_{A} W into W ⨂_{A^{op}} V, as modules over k, with v ⊗_{A} w ↦ w ⊗_{A^{op}} v (1.11.9) for every v ∈ V and w ∈ W.

Similarly, there is a unique homomorphism from W ⨂_{A^{op}} V into V ⨂_{A} W, as modules over k, with w ⊗_{A^{op}} v ↦ v ⊗_{A} w (1.11.10) for every v ∈ V and w ∈ W. It is easy to see that this is the inverse of the mapping mentioned in the preceding paragraph. Thus we get an isomorphism

(1.11.11) from V ⨂_{A}W onto W ⨂_{A^{op}}V,

as on p109 of [50].

1.12 Some double tensor products

Let k be a commutative ring with a multiplicative identity element, and let A, B be associative algebras over k, with multiplicative identity elements e_{A}, e_{B}, respectively. Also let V, W, and Z be modules over k, and suppose that

(1.12.1) V is a right module over A, W is a left module over A, W is a right module over B, and Z is a left module over B.

More precisely, the actions of A and B on W should commute, so that W is a bimodule.

Let V ⨂_{A} W be a tensor product of V and W over A, and let W ⨂_{B} Z be a tensor product of W and Z over B. Thus

(1.12.2) V ⨂_{A}W may be considered as a right module over B,

and W ⨂_{B}Z may be considered as a left module over A,

as in Section 1.10. Let (V ⨂_{A}W ) ⨂_{B}Z (1.12.3)

be a tensor product of V ⨂_{A} W and Z over B, and let

V ⨂_{A}(W ⨂_{B}Z) (1.12.4)

be a tensor product of V and W ⨂_{B} Z over A.

Under these conditions, it is well known that there is a unique homomorphism from (1.12.3) into (1.12.4), as modules over k, with

(v ⊗_{A} w) ⊗_{B} z ↦ v ⊗_{A} (w ⊗_{B} z) (1.12.5)

for every v ∈ V, w ∈ W, and z ∈ Z. To see this, let z ∈ Z be given, and consider the mapping from V × W into (1.12.4) with

(v, w) ↦ v ⊗_{A} (w ⊗_{B} z) (1.12.6)

for every v ∈ V and w ∈ W. This mapping is bilinear over k, and we have that

(v · a) ⊗_{A} (w ⊗_{B} z) = v ⊗_{A} (a · (w ⊗_{B} z)) = v ⊗_{A} ((a · w) ⊗_{B} z) (1.12.7)

for every a ∈ A, v ∈ V, and w ∈ W. This leads to a unique homomorphism from V ⨂_{A} W into (1.12.4), as modules over k, with

v ⊗_{A} w ↦ v ⊗_{A} (w ⊗_{B} z) (1.12.8)

for every v ∈ V and w ∈ W. It is easy to see that this homomorphism is linear in z over k, by uniqueness.

If b ∈ B, then (v ⊗_{A} w) · b = v ⊗_{A} (w · b) (1.12.9) is sent to v ⊗_{A} ((w · b) ⊗_{B} z) = v ⊗_{A} (w ⊗_{B} (b · z)) (1.12.10)

for every v ∈ V and w ∈ W. Thus we get a mapping from (V ⨂_{A} W ) × Z into (1.12.4) that is bilinear over k, and for which the action of b ∈ B on (V ⨂_{A} W ) on the right corresponds to the action of b on Z on the left. This leads to a unique homomorphism from (1.12.3) into (1.12.4), as modules over k, associated to the bilinear mapping on (V ⨂_{A} W ) × Z just mentioned. In particular, this homomorphism satisfies (1.12.5), and one can check that it is uniquely determined by this property.

Similarly, there is a unique homomorphism from (1.12.4) into (1.12.3), as modules over k, with

v ⊗_{A} (w ⊗_{B} z) ↦ (v ⊗_{A} w) ⊗_{B} z (1.12.11)

for every v ∈ V, w ∈ W, and z ∈ Z. One can verify that these homomorphisms are inverses of each other, and thus isomorphisms. This corresponds to Proposition 5.1 on p27 of [50]. This was also mentioned in Section 1.4 when A = B = k.

1.13 Homomorphisms and tensor products

Let k be a commutative ring with a multiplicative identity element, and let A, B be associative algebras over k with multiplicative identity elements e_{A}, e_{B}, respectively, again. Also let V, W, and Z be modules over k, and suppose now that

(1.13.1) V is a left module over A, W is a right module over A, W is a left module over B, and Z is a left module over B.

The actions of A and B on W should commute, so that W is a bimodule, as usual.

Let W ⨂_{A} V be a tensor product of W and V over A, which may be considered as a left module over B, as in Section 1.10. Similarly, the space Hom_{B}(W, Z) of homomorphisms from W into Z, as left modules over B, may be considered as a left module over A, as in Section 1.8. Thus the space

Hom_{A}(V, Hom_{B}(W, Z)) (1.13.2)

of homomorphisms from V into Hom_{B}(W, Z), as left modules over A, can be defined as a module over k in the usual way. The space

Hom_{B}((W ⨂_{A}V ), Z) (1.13.3)

of homomorphisms from W ⨂_{A} V into Z, as left modules over B, is defined as a module over k as well.

An element of (1.13.2) can be evaluated at v ∈ V to get an element of Hom_{B}(W, Z), which can be evaluated at w ∈ W to get an element β(w, v) of Z. This defines a mapping β from W × V into Z, which is bilinear over k. One can check that β(w · a, v) = β(w, a · v) (1.13.4)

for every a ∈ A, v ∈ V, and w ∈ W. Similarly,

β(b · w, v) = b · β(w, v) (1.13.5)

for every b ∈ B, v ∈ V, and w ∈ W. Conversely, if β is a mapping from W × V into Z that is bilinear over k and satsfies these two conditions, then β determines an element of (1.13.2) in this way.

If β is a mapping from W × V into Z that is bilinear over k and satisfies (1.13.4), then there is a unique homomorphism from W ⨂_{A} V into Z, as modules over k, with w ⊗_{A} v ↦ β(w, v) (1.13.6)

for every v ∈ V and w ∈ W, as usual. If β satisfies (1.13.5) too and b ∈ B, then we get that

b · (w ⊗_{A} v) = (b · w) ⊗_{A} v ↦ β(b · w, v) = b · β(w, v) (1.13.7)

for every v ∈ V and w ∈ W. This implies that β corresponds to a homomorphism from W ⨂_{A} V into Z, as modules over B. Using this and the remarks in the preceding paragraph, we get a homomorphism from (1.13.2) into (1.13.3), as modules over k.

Conversely, if v ∈ V and w ∈ W, then an element of (1.13.3) can be evaluated at w ⊗_{A} v to get an element β(w, v) of Z. This defines a mapping β from W ×V into Z that is bilinear over k, and one can check that it satisfies (1.13.4) and (1.13.5). Thus β determines an element of (1.13.2), as before. This defines a homomorphism from (1.13.3) into (1.13.2), as modules over k.

It is easy to see that the homomorphisms between (1.13.2) and (1.13.3) described in the previous two paragraphs are inverses of each other. This corresponds to Proposition 5.2 on p28 of [50].

Similarly, suppose instead that

(1.13.8) V is a right module over A, W is a left module over A, W is a right module over B, and Z is a right module over B,

where the actions of A and B on W commute. Let V ⨂_{A} W be a tensor product of V and W over A, which may be considered as a right module over B. The space Hom_{B}(W, Z) of homomorphisms from W into Z, as right modules over B, may be considered as a right module over A. This means that the space (1.13.2) of homomorphisms from V into Hom_{B}(W, Z), as right modules over A, is defined as a module over k. The space

Hom_{B}((V ⨂_{A}W ), Z) (1.13.9)

1.14. BILINEARITY OVER A, B 25

of homomorphisms from V ⨂_{A} W into Z, as right modules over B, is defined as a module over k too.

As before, an element of (1.13.2) can be evaluated at v ∈ V and w ∈ W to get an element β(v, w) of Z. This defines β as a mapping from V × W into Z that is bilinear over k. One can verify that

β(v · a, w) = β(v, a · w) (1.13.10)

for every a ∈ A, v ∈ V, and w ∈ W, and that

β(v, w · b) = β(v, w) · b (1.13.11)

for every b ∈ B, v ∈ V, and w ∈ W. Conversely, a mapping β from V × W into Z that is bilinear over k and satisfies (1.13.10) and (1.13.11) determines an element of (1.13.2) in this way.

Of course, if β is a mapping from V × W into Z that is bilinear over k and satisfies (1.13.10), then there is a unique homomorphism from V ⨂_{A} W into Z, as modules over k, such that

v ⊗_{A} w ↦ β(v, w) (1.13.12)

for every v ∈ V and w ∈ W. If β satisfies (1.13.11) as well and b ∈ B, then

(v ⊗_{A} w) · b = v ⊗_{A} (w · b) ↦ β(v, w · b) = β(v, w) · b (1.13.13)

for every v ∈ V and w ∈ W. This implies that β corresponds to a homomorphism from V ⨂_{A} W into Z as right modules over B, as before. This leads to a homomorphism from (1.13.2) into (1.13.9), as modules over k.

Conversely, an element of (1.13.9) corresponds to a mapping β from V × W into Z that is bilinear over k, and one can verify that β satisfies (1.13.12) and (1.13.13). This determines an element of (1.13.2), as before. Using this, we get a homomorphism from (1.13.9) into (1.13.2), as modules over k. One can check that this is the inverse of the homomorphism mentioned in the preceding paragraph. This corresponds to Proposition 5.2’ on p28 of [50].

1.14 Bilinearity over A, B

Let k, A, and B be as in the previous section, and let V, W, and Z be modules over k again. In this section, we suppose that

(1.14.1) V is a left module over A, W is a right module over B, Z is a left module over A, and Z is a right module over B,

where the actions of A and B on Z commute, as usual.

Let V ⨂_{k} W be a tensor product of V and W, as modules over k. The actions of A on V and B on W lead to actions of A and B on V ⨂_{k} W, so that V ⨂_{k} W is a left module over A and a right module over B, as before. One can check that these actions of A and B on V ⨂_{k} W commute.

Consider the space Hom_{A,B}((V ⨂_{k}W ), Z) (1.14.2)

of homomorphisms from V ⨂_{k} W into Z, as both left modules over A, and right modules over B. This is a module over k, with respect to pointwise addition and scalar multiplication of mappings, as usual.

Let β be a mapping from V × W into Z that is bilinear over k. This leads to a unique homomorphism β̃ from V ⨂_{k} W into Z, as modules over k, such that

β̃(v ⊗ w) = β(v, w) (1.14.3)

for every v ∈ V and w ∈ W. Suppose that

(1.14.4) for each w ∈ W, β(·, w) is a homomorphism from V into Z, as left modules over A.

In this case, one can check that β̃ is a homomorphism from V ⨂_{k} W into Z, as left modules over A. Similarly, if

(1.14.5) for each v ∈ V, β(v, ·) is a homomorphism from W into Z, as right modules over B,

then β̃ is a homomorphism from V ⨂_{k} W into Z, as right modules over B.

Thus, if (1.14.4) and (1.14.5) both hold, then β̃ is an element of (1.14.2). Conversely, every element of (1.14.2) corresponds to a mapping from V × W into Z that is bilinear over k and satisfies (1.14.4) and (1.14.5) in this way. This can be seen by composing with the natural mapping from V × W into V ⨂_{k} W.

Note that the space Hom_{B}(W, Z) of homomorphisms from W into Z, as right modules over B, may be considered as a left module over A, as in Section 1.8. This means that the space

Hom_{A}(V, Hom_{B}(W, Z)) (1.14.6)

of homomorphisms from V into Hom_{B}(W, Z), as left modules over A, can be defined in the usual way. Similarly, the space Hom_{A}(V, Z) of homomorphisms from V into Z, as left modules over A, may be considered as a right module over B. This permits us to define the space

Hom_{B}(W, Hom_{A}(V, Z)) (1.14.7)

of homomorphisms from W into Hom_{A}(V, Z), as right modules over B, in the usual way.

It is easy to see that the elements of both (1.14.6) and (1.14.7) correspond exactly to mappings from V × W into Z that are bilinear over k and satisfy (1.14.4) and (1.14.5). This defines a natural isomorphism between (1.14.6) and (1.14.7), as modules over k, as in Exercise 4 on p32 of [50]. More precisely, (1.14.6) and (1.14.7) are both isomorphic to (1.14.2), as modules over k, in a natural way.

1.15 Direct families of homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose for the moment that W and Z are modules over k, that are either both left modules over A, or both right modules over A. Suppose also that ϕ is a homomorphism from W into Z, and that ψ is a homomorphism from Z into W, as modules over A, such that

(1.15.1) ψ ∘ ϕ is the identity mapping on W.

This implies that ϕ is injective on W, ψ(Z) = W, and

(1.15.2) ϕ ∘ ψ is the identity mapping on ϕ(W).

Under these conditions, one can check that

(1.15.3) Z corresponds to the direct sum of ϕ(W) and ker ψ,

as a module over A.

Now let V be a module over k, let I be a nonempty set, and let V_{j} be a module over k for each j ∈ I. Suppose that these are either all left modules over A, or all right modules over A. Suppose also that for each j ∈ I, i_{j} is a homomorphism from V_{j} into V, and that π_{j} is a homomorphism from V into V_{j}, as modules over A, with

(1.15.4) π_{j} ∘ i_{j} equal to the identity mapping on V_{j}.

If j, l ∈ I and j ≠ l, then we suppose in addition that

π_{l} ∘ i_{j} = 0. (1.15.5)

Under these conditions, the family of i_{j}, π_{j}, j ∈ I, is said to be a direct family of homomorphisms, as on p4 of [50].

Suppose for the moment that

⨁_{j∈I} V_{j} ⊆ V ⊆ ∏_{j∈I} V_{j}, (1.15.6)

with V a submodule of ∏_{j∈I} V_{j}, as a module over A. If l ∈ I, then let i_{l} be the mapping from V_{l} into V that sends v_{l} ∈ V_{l} to the element of ⨁_{j∈I} V_{j} with lth coordinate equal to v_{l}, and all other coordinates equal to 0. If π_{l} is the restriction of the lth coordinate projection on ∏_{j∈I} V_{j} to V, then it is easy to see that we get a direct family of homomorphisms.

Let V be any left or right module over A again, and suppose that we have a direct family of homomorphisms i_{j}, π_{j}, j ∈ I, as before. Using the i_{j}’s, we get a homomorphism (1.15.7) from ⨁_{j∈I} V_{j} into V, as modules over A. We can also use the π_{j}’s to get a homomorphism

(1.15.8) from V into ∏_{j∈I} V_{j},

as modules over A. It is easy to see that

(1.15.9) the composition of these two homomorphisms

is the identity mapping on ⨁_{j∈I} V_{j},

by (1.15.4) and (1.15.5). In particular, this means that the homomorphism mentioned in (1.15.7) is an injection.

Suppose that I has only finitely many elements, so that ⨁_{j∈I} V_{j} and ∏_{j∈I} V_{j} are the same, and the homomorphism mentioned in (1.15.8) is a surjection. In this case, it is easy to see that the homomorphism mentioned in (1.15.7) is a surjection if and only if the homomorphism mentioned in (1.15.8) is an injection. This happens if and only if

(1.15.10)  ∑_{j∈I} i_{j} ∘ π_{j} is the identity mapping on V,

as on p4 of [50].

Modules and tensor products, 2

2.1 Compositions and Hom(·, W)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that V_{1}, V_{2}, V_{3}, and W are modules over k that are either all left modules over A, or all right modules over A. Let ϕ_{1} be a homomorphism from V_{1} into V_{2}, and let ϕ_{2} be a homomorphism from V_{2} into V_{3}, as modules over A.

If ψ_{2} is a homomorphism from V_{2} into W, as modules over A, then

Φ_{1}(ψ_{2}) = ψ_{2} ∘ ϕ_{1} (2.1.1)

is a homomorphism from V_{1} into W, as modules over A. Similarly, if ψ_{3} is a homomorphism from V_{3} into W, as modules over A, then

Φ_{2}(ψ_{3}) = ψ_{3} ∘ ϕ_{2} (2.1.2)

is a homomorphism from V_{2} into W, as modules over A. Note that Φ_{1} defines a homomorphism from Hom_{A}(V_{2}, W) into Hom_{A}(V_{1}, W), and that Φ_{2} defines a homomorphism from Hom_{A}(V_{3}, W) into Hom_{A}(V_{2}, W), as modules over k.

Thus Φ_{1} ∘ Φ_{2} is a homomorphism from Hom_{A}(V_{3}, W) into Hom_{A}(V_{1}, W), as modules over k. If ψ_{3} ∈ Hom_{A}(V_{3}, W), then

(Φ_{1} ∘ Φ_{2})(ψ_{3}) = Φ_{1}(Φ_{2}(ψ_{3})) = (Φ_{2}(ψ_{3})) ∘ ϕ_{1} = ψ_{3} ∘ (ϕ_{2} ∘ ϕ_{1}). (2.1.3)

Of course, ϕ_{2} ∘ ϕ_{1} is a homomorphism from V_{1} into V_{3}, as modules over A. If

ϕ_{2} ∘ ϕ_{1} = 0, (2.1.4)

then we get that Φ_{1} ∘ Φ_{2} = 0. (2.1.5)

30 CHAPTER 2. MODULES AND TENSOR PRODUCTS, 2

Note that (2.1.4) is the same as saying that ϕ_{1}(V_{1}) ⊆ ker ϕ_{2}. (2.1.6) Similarly, (2.1.5) is the same as saying that (2.1.7) Φ_{2}(Hom_{A}(V_{3}, W)) ⊆ ker Φ_{1}. Alternatively, (2.1.8) ker Φ_{1} = {ψ_{2} ∈ Hom_{A}(V_{2}, W) : ϕ_{1}(V_{1}) ⊆ ker ψ_{2}}, by construction. If ψ_{3} ∈ Hom_{A}(V_{3}, W), then (2.1.9) ker Φ_{2}(ψ_{3}) = ker ψ_{3} ∘ ϕ_{2} ⊇ ker ϕ_{2}. Thus (2.1.6) implies that ϕ_{1}(V_{1}) ⊆ ker Φ_{2}(ψ_{3}), (2.1.10) so that Φ_{2}(ψ_{3}) ∈ ker Φ_{1}.

Suppose now that

V_{1} ⟶^{ϕ_{1}} V_{2} ⟶^{ϕ_{2}} V_{3} ⟶ 0 (2.1.11)

is an exact sequence, so that ϕ_{1}(V_{1}) = ker ϕ_{2}, ϕ_{2}(V_{2}) = V_{3}. (2.1.12) It is well known that (2.1.13) 0 ⟶ Hom_{A}(V_{3}, W) ⟶^{Φ_{2}} Hom_{A}(V_{2}, W) ⟶^{Φ_{1}} Hom_{A}(V_{1}, W) is exact under these conditions, so that Φ_{2} is injective and (2.1.14) Φ_{2}(Hom_{A}(V_{3}, W)) = ker Φ_{1}. This is known as left exactness of Hom_{A}(·, W), and is part of Proposition 4.4 on p26 of [50].

The injectivity of Φ_{2} follows from the surjectivity of ϕ_{2}, as in Section 1.7. We also have that (2.1.7) holds when (2.1.6) holds, as before.

Suppose that ψ_{2} ∈ Hom_{A}(V_{2}, W) is an element of the kernel of Φ_{1}. We would like to find ψ_{3} ∈ Hom_{A}(V_{3}, W) such that ψ_{2} = Φ_{2}(ψ_{3}) = ψ_{3} ∘ ϕ_{2} (2.1.15) when (2.1.12) holds. Equivalently, this means that ψ_{2}(v_{2}) = ψ_{3}(ϕ_{2}(v_{2})) (2.1.16) for every v_{2} ∈ V_{2}.

If ker ϕ_{2} ⊆ ϕ_{1}(V), then ker ϕ_{2} ⊆ ker ψ_{2}, (2.1.17)

by (2.1.8). If v_{2}, v′_{2} ∈ V_{2} and ϕ_{2}(v_{2}) = ϕ_{2}(v′_{2}), then v_{2} − v′_{2} ∈ ker ϕ_{2}, so that v_{2} − v′_{2} ∈ ker ψ_{2}, and thus ψ_{2}(v_{2}) = ψ_{2}(v′_{2}). This implies that ψ_{3} is uniquely determined on ϕ_{2}(V_{2}) by (2.1.16). It follows that ψ_{3} is uniquely determined on V_{3} by (2.1.16), because ϕ_{2}(V_{2}) = V_{3}, by hypothesis. It is easy to see that ψ_{3} is a homomorphism from V_{3} into W, as modules over A, because of the analogous properties of ϕ_{2} and ψ_{2}, as desired.

2.2 Some converse statements

Let us continue with the same notation and hypotheses as in the previous section. The remarks in the previous section correspond to the “only if” part (i) of Proposition 2.9 on p22 of [8]. The “if” part of the proposition deals with the necessity of the various conditions on ϕ_{1}, ϕ_{2}, in order that the corresponding conditions on Φ_{1}, Φ_{2} hold for all W.

If ϕ_{2} is not surjective, for instance, then ϕ_{2}(V_{2}) is a proper submodule of V_{3}, as a module over A. This means that the quotient

W = V_{3}/ϕ_{2}(V_{2}) (2.2.1)

is a nontrivial module over A. Let ψ_{3} be the natural quotient mapping from V_{3} onto W, which is a nonzero homomorphism from V_{3} onto W, as modules over A. In this case, Φ_{2}(ψ_{3}) = ψ_{3} ∘ ϕ_{2} = 0, (2.2.2)

so that Φ_{2} is not injective. This shows that the surjectivity of ϕ_{2} is necessary to get the injectivity of Φ_{2} for all W.

Similarly, if (2.1.5) holds for every W, then (2.1.4) holds. To see this, one can take W = V_{3}, and ψ_{3} to be the identity mapping on V_{3}, considered as a homomorphism from V_{3} into W, as modules over A. Thus

(Φ_{1} ∘ Φ_{2})(ψ_{3}) = ψ_{3} ∘ (ϕ_{2} ∘ ϕ_{1}) = ϕ_{2} ∘ ϕ_{1}, (2.2.3)

as in (2.1.3). If (2.1.5) holds, then we get that (2.1.4) holds, as desired. Equivalently, this shows that if (2.1.7) holds for every W, then (2.1.6) holds.

Suppose now that

(2.2.4) ker Φ_{1} ⊆ Φ_{2}(Hom_{A}(V_{3}, W))

for every W, and let us check that

ker ϕ_{2} ⊆ ϕ_{1}(V_{1}). (2.2.5)

Of course, ϕ_{1}(V_{1}) is a submodule of V_{2}, as a module over A, so that

W = V_{2}/ϕ_{1}(V_{1}) (2.2.6) is a module over A as well. Let ψ_{2} be the natural quotient mapping from V_{2} onto W, which is a homomorphism from V_{2} onto W, as modules over A. Observe that Φ_{1}(ψ_{2}) = ψ_{2} ∘ ϕ_{1} = 0, (2.2.7)

by construction. This means that ψ_{2} ∈ ker Φ_{1}.

Using (2.2.4), we get that there is a ψ_{3} ∈ Hom_{A}(V_{3}, W) such that ψ_{2} = Φ_{2}(ψ_{3}) = ψ_{3} ∘ ϕ_{2}. In particular, it follows that the kernel of ϕ_{2} is contained in the kernel of ψ_{2}. This implies (2.2.5), because

ker ψ_{2} = ϕ_{1}(V_{1}), (2.2.8)

by construction.

2.3 Compositions and Hom(V, ·)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A} again. In this section, we suppose that V, W_{1}, W_{2}, and W_{3} are modules over k that are either all left modules over A, or all right modules over A. Let ψ_{1} be a homomorphism from W_{1} into W_{2}, and let ψ_{2} be a homomorphism from W_{2} into W_{3}, as modules over A.

If ϕ_{1} is a homomorphism from V into W_{1}, as modules over A, then

Ψ_{1}(ϕ_{1}) = ψ_{1} ∘ ϕ_{1} (2.3.1)

is a homomorphism from V into W_{2}, as modules over A. Similarly, if ϕ_{2} is a homomorphism from V into W_{2}, as modules over A, then

Ψ_{2}(ϕ_{2}) = ψ_{2} ∘ ϕ_{2} (2.3.2)

is a homomorphism from V into W_{3}, as modules over A. As usual, Ψ_{1} defines a homomorphism from Hom_{A}(V, W_{1}) into Hom_{A}(V, W_{2}), and Ψ_{2} defines a homomorphism from Hom_{A}(V, W_{2}) into Hom_{A}(V, W_{3}), as modules over k.

Thus Ψ_{2} ∘ Ψ_{1} is a homomorphism from Hom_{A}(V, W_{1}) into Hom_{A}(V, W_{3}), as modules over k. If ϕ_{1} ∈ Hom_{A}(V, W_{1}), then

(Ψ_{2} ∘ Ψ_{1})(ϕ_{1}) = Ψ_{2}(Ψ_{1}(ϕ_{1})) = ψ_{2} ∘ (Ψ_{1}(ϕ_{1})) = (ψ_{2} ∘ ψ_{1}) ∘ ϕ_{1}. (2.3.3)

Note that ψ_{2} ∘ ψ_{1} is a homomorphism from W_{1} into W_{3}, as modules over A. If

ψ_{2} ∘ ψ_{1} = 0, (2.3.4)

then it follows that Ψ_{2} ∘ Ψ_{1} = 0. (2.3.5)

Of course, (2.3.4) is the same as saying that

ψ_{1}(W_{1}) ⊆ ker ψ_{2}, (2.3.6)

2.3. COMPOSITIONS AND HOM(V, ·) 33 and (2.3.5) is the same as saying that (2.3.7) Ψ_{1}(Hom_{A}(V, W_{1})) ⊆ ker Ψ_{2}. Observe that (2.3.8) ker Ψ_{2} = {ϕ_{2} ∈ Hom_{A}(V, W_{2}) : ϕ_{2}(V) ⊆ ker ψ_{2}}. If ϕ_{1} ∈ Hom_{A}(V, W_{1}), then (Ψ_{1}(ϕ_{1}))(V) = (ψ_{1}(ϕ_{1}(V))) ⊆ ψ_{1}(W_{1}). (2.3.9) This implies that (Ψ_{1}(ϕ_{1}))(V) ⊆ ker ψ_{2} (2.3.10) when (2.3.6) holds. This is another way to get that Ψ_{1}(ϕ_{1}) ∈ ker Ψ_{2}, using (2.3.8).

Suppose that

0 ⟶ W_{1} ⟶^{ψ_{1}} W_{2} ⟶^{ψ_{2}} W_{3} (2.3.11)

is an exact sequence, so that ψ_{1} is injective, and ψ_{1}(W_{1}) = ker ψ_{2}. (2.3.12) It is well known that (2.3.13) 0 ⟶ Hom_{A}(V, W_{1}) ⟶^{Ψ_{1}} Hom_{A}(V, W_{2}) ⟶^{Ψ_{2}} Hom_{A}(V, W_{3}) is exact in this case, so that Ψ_{1} is injective, and (2.3.14) Ψ_{1}(Hom_{A}(V, W_{1})) = ker Ψ_{2}. This means that Hom_{A}(V, ·) is left exact, which is another part of Proposition 4.4 on p26 of [50].

The injectivity of Ψ_{1} follows immediately from the injectivity of ψ_{1}, as in Section 1.7. We have also seen that (2.3.7) holds when (2.3.6) holds.

Let ϕ_{2} ∈ Hom_{A}(V, W_{2}) be an element of the kernel of Ψ_{2}. We would like to find ϕ_{1} ∈ Hom_{A}(V, W_{1}) such that ϕ_{2} = Ψ_{1}(ϕ_{1}) = ψ_{1} ∘ ϕ_{1} (2.3.15) when ψ_{1} is injective and (2.3.12) holds. More precisely, ϕ_{2}(V) ⊆ ψ_{1}(W_{1}) (2.3.16) when ker ψ_{2} ⊆ ψ_{1}(W_{1}), by (2.3.8). Using this, we can get ϕ_{1} ∈ Hom_{A}(V, W_{1}) as in (2.3.15), because ψ_{1} is injective.

2.4 Some more converse statements

Let us continue with the same notation and hypotheses as in the previous section. The remarks in the previous section correspond to the “only if” part of part (ii) of Proposition 2.9 on p23 of [8]. As before, the “if” part of the proposition concerns the necessity of the conditions on ψ_{1}, ψ_{2}, for the corresponding conditions on Ψ_{1}, Ψ_{2} to hold for all V.

If ψ_{1} is not injective, then the kernel of ψ_{1} is a nontrivial submodule of W_{1}, as a module over A. Let us take

V = ker ψ_{1}, (2.4.1)

and let ϕ_{1} be the natural inclusion mapping from V into W_{1}. Thus ϕ_{1} is a nonzero homomorphism from V into W_{1}, as modules over A. Clearly

Ψ_{1}(ϕ_{1}) = ψ_{1} ∘ ϕ_{1} = 0, (2.4.2)

by construction, so that Ψ_{1} is not injective. It follows that the injectivity of ψ_{1} is needed for Ψ_{1} to be injective for any V.

Suppose that (2.3.5) holds for every V, and let us check that (2.3.4) holds. To do this, we take V = W_{1}, and ϕ_{1} to be the identity mapping on W_{1}, considered as a homomorphism from V into W_{1}, as a module over A. In this case,

(Ψ_{2} ∘ Ψ_{1})(ϕ_{1}) = (ψ_{2} ∘ ψ_{1}) ∘ ϕ_{1} = ψ_{2} ∘ ψ_{1}, (2.4.3)

as in (2.3.3). Thus (2.3.5) implies (2.3.4), as desired. Equivalently, if (2.3.7) holds for every V, then (2.3.6) holds.

If (2.4.4) ker Ψ_{2} ⊆ Ψ_{1}(Hom_{A}(V, W_{1}))

for every V, then we would like to verify that

ker ψ_{2} ⊆ ψ_{1}(W_{1}). (2.4.5)

Let us take V = ker ψ_{2}, (2.4.6)

considered as a module over A. More precisely, V is a submodule of W_{2}, and we take ϕ_{2} to be the natural inclusion mapping from V into W_{2}. This is a homomorphism from V into W_{2}, as modules over A, with

Ψ_{2}(ϕ_{2}) = ψ_{2} ∘ ϕ_{2} = 0. (2.4.7)

Thus ϕ_{2} ∈ ker Ψ_{2}.

If (2.4.4) holds, then there is a ϕ_{1} ∈ Hom_{A}(V, W_{1}) such that ϕ_{2} = Ψ_{1}(ϕ_{1}) = ψ_{1} ∘ ϕ_{1}. This implies that

ker ψ_{2} = V = ϕ_{2}(V) ⊆ ψ_{1}(W_{1}), (2.4.8)

as desired.

2.5 Compositions and tensor products

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that V_{1}, V_{2}, and V_{3} are right modules over A, and that W is a left module over A. Let V_{1} ⨂_{A} W, V_{2} ⨂_{A} W, and V_{3} ⨂_{A} W be tensor products of V_{1}, V_{2}, and V_{3}, respectively, with W over A.

Suppose that ϕ_{1} is a homomorphism from V_{1} into V_{2}, and that ϕ_{2} is a homomorphism from V_{2} into V_{3}, as right modules over A. Using ϕ_{1}, we get a unique homomorphism Φ_{1} from V_{1} ⨂_{A} W into V_{2} ⨂_{A} W, as modules over k, such that Φ_{1}(v_{1} ⊗ w) = ϕ_{1}(v_{1}) ⊗ w (2.5.1)

for every v_{1} ∈ V_{1} and w ∈ W. More precisely, Φ_{1} is obtained from ϕ_{1} and the identity mapping on W as in Section 1.9. Similarly, there is a unique homomorphism Φ_{2} from V_{2} ⨂_{A} W into V_{3} ⨂_{A} W, as modules over k, such that

Φ_{2}(v_{2} ⊗ w) = ϕ_{2}(v_{2}) ⊗ w (2.5.2)

for every v_{2} ∈ V_{2} and w ∈ W.

Of course, ϕ_{2} ∘ ϕ_{1} is a homomorphism from V_{1} into V_{3}, as right modules over A. This leads to a unique homomorphism from V_{1} ⨂_{A} W into V_{3} ⨂_{A} W, as modules over k, with

v_{1} ⊗ w ↦ ϕ_{2}(ϕ_{1}(v_{1})) ⊗ w (2.5.3)

for every v_{1} ∈ V_{1} and w ∈ W, as before. This homomorphism is the same as

Φ_{2} ∘ Φ_{1}, (2.5.4)

as in Section 1.9. In particular, if

ϕ_{2} ∘ ϕ_{1} = 0, (2.5.5)

then Φ_{2} ∘ Φ_{1} = 0. (2.5.6)

Equivalently, if ϕ_{1}(V_{1}) ⊆ ker ϕ_{2}, (2.5.7)

then Φ_{1}(V_{1} ⨂_{A}W ) ⊆ ker Φ_{2}. (2.5.8)

Suppose that

V_{1} ⟶^{ϕ_{1}} V_{2} ⟶^{ϕ_{2}} V_{3} ⟶ 0 (2.5.9)

is an exact sequence, so that ϕ_{2} is surjective, and

ϕ_{1}(V_{1}) = ker ϕ_{2}. (2.5.10)

Under these conditions, it is well known that

V_{1} ⨂_{A}W ⟶^{Φ_{1}} V_{2} ⨂_{A}W ⟶^{Φ_{2}} V_{3} ⨂_{A}W ⟶ 0 (2.5.11)

is an exact sequence, so that Φ_{2} is surjective and

Φ_{1}(V_{1} ⨂_{A}W ) = ker Φ_{2}. (2.5.12)

This is known as right exactness of V ↦ V ⨂_{A} W, which is part of Proposition 4.5 on p26 of [50]. More precisely, the surjectivity of Φ_{2} follows from the surjectivity of ϕ_{2}, as in Section 1.9.

To show that (2.5.12) holds, note first that (2.5.8) holds, as before. Of course, Φ_{1}(V_{1} ⨂_{A} W ) is a submodule of V_{2} ⨂_{A} W, as a module over k. Thus the quotient Y = (V_{2} ⨂_{A}W )/Φ_{1}(V_{1} ⨂_{A}W ) (2.5.13)

is defined as a module over k. Let q_{Y} be the natural quotient mapping from V_{2} ⨂_{A} W onto Y.

Using (2.5.8), we get that there is a unique homomorphism Φ̃_{2} from Y into V_{3} ⨂_{A} W, as modules over k, such that

Φ̃_{2} ∘ q_{Y} = Φ_{2}. (2.5.14)

We would like to show that Φ̃_{2} is an isomorphism.

If v_{2} ∈ V_{2} and w ∈ W, then

q_{Y} (v_{2} ⊗ w) (2.5.15)

is an element of Y. Observe that

q_{Y} (ϕ_{1}(v_{1}) ⊗ w) = 0 (2.5.16)

for every v_{1} ∈ V_{1} and w ∈ W, by (2.5.1) and the definition of Y. This implies that (2.5.15) only depends on ϕ_{2}(v_{2}) and w, because of (2.5.10). This leads to a mapping from V_{3} × W into Y, with

(ϕ_{2}(v_{2}), w) ↦ q_{Y} (v_{2} ⊗ w) (2.5.17)

for every v_{2} ∈ V_{2} and w ∈ W, because ϕ_{2}(V_{2}) = V_{3}, by hypothesis. It is easy to see that this mapping is bilinear over k.

If a ∈ A, v_{2} ∈ V_{2}, and w ∈ W, then ϕ_{2}(v_{2} · a) = ϕ_{2}(v_{2}) · a and (v_{2} · a) ⊗ w = v_{2} ⊗ (a · w) in V_{2} ⨂_{A} W. This implies that

(ϕ_{2}(v_{2}) · a, w) and (ϕ_{2}(v_{2}), a · w) (2.5.18)

are mapped to the same element of Y by the mapping described in the preceding paragraph. This leads to a unique homomorphism from V_{3} ⨂_{A} W into Y, as modules over k, with ϕ_{2}(v_{2}) ⊗ w ↦ q_{Y} (v_{2} ⊗ w) (2.5.19) for every v_{2} ∈ V_{2} and w ∈ W.

One can check that the composition of the mapping from V_{3} ⨂_{A} W into Y defined in the previous paragraph with Φ̃_{2} is the identity mapping on V_{3} ⨂_{A} W. Similarly, one can verify that the composition of Φ̃_{2} with the mapping from V_{3} ⨂_{A} W into Y defined in the previous paragraph is the identity mapping on Y. Thus Φ̃_{2} is an isomorphism from Y onto V_{3} ⨂_{A} W, as modules over k.

In particular, the kernel of Φ̃_{2} is trivial. This implies (2.5.12), as desired. Of course, there is an analogous right exactness property for W ↦ V ⨂_{A} W,

which is also part of Proposition 4.5 on p26 of [50].

2.6 Another approach

Let us continue with the same notation and hypotheses as in the previous section. The fact that the exactness of (2.5.9) implies the exactness of (2.5.11) corresponds to Proposition 2.18 on p28 of [8]. The argument in [8] uses the characterization of exactness of sequences like these in Sections 2.1 and 2.2, as follows.

Let Z be an arbitrary module over k. Thus, for each j = 1, 2, 3, the space

Hom_{k}(V_{j} ⨂_{A}W, Z) (2.6.1)

of homomorphisms from V_{j} ⨂_{A} W into Z, as modules over k, is a module over k with respect to pointwise addition and scalar multiplication of mappings into Z, as usual. Let Φ̂_{1} be the homomorphism

(2.6.2) from Hom_{k}(V_{2} ⨂_{A}W, Z) into Hom_{k}(V_{1} ⨂_{A}W, Z),

as modules over k, that sends an element of Hom_{k} (V_{2} ⨂_{A} W, Z) to its composition with Φ_{1}. Similarly, let Φ̂_{2} be the homomorphism

(2.6.3) from Hom_{k}(V_{3} ⨂_{A}W, Z) into Hom_{k}(V_{2} ⨂_{A}W, Z),

as modules over k, that sends an element of Hom_{k} (V_{3} ⨂_{A} W, Z) to its composition with Φ_{2}.

The exactness of (2.5.11) implies that

0 ⟶ Hom_{k}(V_{3} ⨂_{A}W, Z) \underrightarrow{_{Φ_{2}}} Hom_{k}(V_{2} ⨂_{A}W, Z) (2.6.4)

\underrightarrow{_{Φ_{1}}} Hom_{k}(V_{1} ⨂_{A}W, Z)

is exact, as in Section 2.1. Conversely, in order to show that (2.5.11) is exact, it sufficies to verify that (2.6.4) is exact for every Z, as in Section 2.2.

Remember that the space Hom_{k}(W, Z) of homomorphisms from W into Z, as modules over k, may be considered as a right module over A in a natural way, because W is a left module over A, as in Section 1.8. We have also seen that (2.6.1) is isomorphic to

Hom_{A}(V_{j}, Hom_{k}(W, Z)), (2.6.5)

as modules over k, in a natural way for j = 1, 2, 3, as in Section 1.13. Let Φ′_{1} be the homomorphism

(2.6.6) from Hom_{A}(V_{2}, Hom_{k}(W, Z)) into Hom_{A}(V_{1}, Hom_{k}(W, Z)),

as modules over k, that sends an element of Hom_{A} (V_{2}, Hom_{k}(W, Z)) to its composition with ϕ_{1}. Similarly, let Φ′_{2} be the homomorphism

(2.6.7) from Hom_{A}(V_{3}, Hom_{k}(W, Z)) into Hom_{A}(V_{2}, Hom_{k}(W, Z)),

as modules over k, that sends an element of Hom_{A} (V_{3}, Hom_{k}(W, Z)) to its composition with ϕ_{2}.

One can check that Φ′_{1}, Φ′_{2} correspond exactly to Φ̂_{1}, Φ̂_{2}, respectively, with respect to the isomorphisms between (2.6.1) and (2.6.5), j = 1, 2, 3, mentioned in the preceding paragraph. This uses the way that Φ_{1}, Φ_{2} are obtained from ϕ_{1}, ϕ_{2}, respectively, as in the previous section.

It follows that the exactness of (2.6.4) is equivalent to the exactness of

(2.6.8) 0 ⟶ Hom_{A}(V_{3}, Hom_{k}(W, Z)) ⟶^{Φ′_{2}} Hom_{A}(V_{2}, Hom_{k}(W, Z))

⟶^{Φ′_{1}} Hom_{A}(V_{1}, Hom_{k}(W, Z)).

The exactness of (2.5.9) implies the exactness of (2.6.8), as in Section 2.1. Thus the exactness of (2.5.9) implies the exactness of (2.6.4). More precisely, this works for every module Z over k. This shows that the exactness of (2.5.9) implies the exactness of (2.5.11), as before.

2.7 Projective modules

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. A left module V over A is said to be projective if it has the following property. Let W and Z be left modules over A, and let ψ be a homomorphism from W onto Z, as left modules over A. If ϕ_{Z} is a homomorphism from V into Z, as left modules over A, then there should be a homomorphism ϕ_{W} from V into W, as left modules over A, such that

ϕ_{Z} = ψ ∘ ϕ_{W} . (2.7.1)

Of course, projectivity of right modules is defined analogously.

It is easy to see that A is projective as a left module over itself. Indeed, suppose that W, Z, ψ, and ϕ_{Z} are as in the preceding paragraph, with V = A. Note that ϕ_{Z} is uniquely determined by ϕ_{Z}(e_{A}), and similarly that any homomorphism ϕ_{W} from A into W, as left modules over A, is determined by ϕ_{W} (e_{A}). Thus it suffices to choose ϕ_{W} (e_{A}) ∈ W such that

ϕ_{Z}(e_{A}) = ψ(ϕ_{W} (e_{A})), (2.7.2)

and to take ϕ_{W} to be the corresponding module homomorphism from A into W.

Let I be a nonempty set, and let V_{j} be a left module over A for each j ∈ I. It is well known that

(2.7.3) V_{j} is projective for every j ∈ I

if and only if

(2.7.4) V = ⨁_{j∈I} V_{j} is projective as a left module over A,

as in Proposition 2.1 on p6 of [50]. To show that (2.7.3) implies (2.7.4), let W, Z, ψ, and ϕ_{Z} be given as before. If l ∈ I, then one can ge a homomorphism ϕ_{l,Z} from V_{l} into Z, as left modules over A, by composing ϕ_{Z} with the natural inclusion mapping from V_{l} into V. Using the projectivity of V_{l}, we get a homomorphism ϕ_{l,W} from V_{l} into W, as left modules over A, such that

ϕ_{l,Z} = ψ ∘ ϕ_{l,W} . (2.7.5)

One can combine the homomorphisms ϕ_{l,W}, l ∈ I, to get a homomorphism ϕ_{W} from V into W, as left modules over A, that satisfies (2.7.1). In particular, if V is free as a left module over A, then it follows that V is projective.

Conversely, suppose that (2.7.4) holds, and let us check that (2.7.3) holds. Let l ∈ I be given, let W, Z, and ψ be as before, and let ϕ_{l,Z} be a homomorphism from V_{l} into Z, as left modules over A. Using ϕ_{l,Z}, we can get a homomorphism ϕ_{Z} from V into Z, as left modules over A, whose composition with the natural inclusion mapping from V_{l} into V is equal to ϕ_{l,Z}, and whose composition with the natural inclusion mapping from V_{j} into V is equal to 0 when j ≠ l. If V is projective, then there is a homomorphism ϕ_{W} from V into W, as left modules over A, that satisfies (2.7.1). Let ϕ_{l,W} be the composition of ϕ_{W} with the natural inclusion mapping from V_{l} into V, which is a homomorphism from V_{l} into W, as left modules over A. Under these conditions, (2.7.5) follows by composing both sides of (2.7.1) with the natural inclusion mapping from V_{l} into V. This implies that V_{l} is projective, as desired.

If V is any left module over A, then it is well known that

(2.7.6) there is a homomorphism from a free left module over A onto V,

as left modules over A. Indeed, if X is a free left module over A, then any mapping from the generators of X, as a free left module over A, into V can be extended to a homomorphism from X into V, as left modules over A. This homomorphism maps X onto the submodule of V, as a left module over A, generated by the images of the generators of X. If X is freely generated by sufficiently many elements, then one can use this to get a homomorphism from X onto V.

Let Y be a left module over A, and let η be a homomorphism from Y onto V, as left modules over A. If V is projective, then there is a homomorphism ϕ from V into Y, as modules over A, such that

(2.7.7) η ∘ ϕ is the identity mapping on V.

This corresponds to taking W = Y, Z = V, ψ = η, and ϕ_{Z} equal to the identity mapping on V, in the definition of projectivity. In particular, (2.7.7) implies that ϕ is injective, and that

(2.7.8) ϕ ∘ η is the identity mapping on ϕ(V).

Note that ker η = ker(ϕ ∘ η) (2.7.9)

is a submodule of Y. Under these conditions,

(2.7.10) Y corresponds to the direct sum of ϕ(V) and ker η,

as a left module over A, by standard arguments. This corresponds to part of Proposition 2.4 on p7 of [50].

It is well known that a left module V over A is projective if and only if

(2.7.11) there is a free left module over A that is isomorphic to the direct sum of V and another left module over A,

as left modules over A, as in Theorem 2.2 on p6 of [50]. Of course, the “if” part follows from the fact that (2.7.4) implies (2.7.3). The “only if” part follows from (2.7.6) and (2.7.10). Using the “if” part and (2.7.6), we get that projectivity of V is necessary for the property mentioned in the preceding paragraph, as in Proposition 2.4 on p7 of [50].

Let V be a left module over A, let U_{1}, U_{2} be right modules over A, and let θ be a homomorphism from U_{1} into U_{2}, as right modules over A. Also let U_{1} ⨂_{A} V, U_{2} ⨂_{A} V be tensor products of U_{1}, U_{2} with V over A, respectively. Thus we get a homomorphism

(2.7.12) Θ from U_{1} ⨂_{A}V into U_{2} ⨂_{A}V,

as modules over k, using θ and the identity mapping on V, as in Section 1.9. If V is projective and θ is injective, then it is well known that

(2.7.13) Θ is injective.

More precisely, let U_{3} be another right module over A, let θ′ be a homomorphism from U_{2} into U_{3}, as right modules over A, and let U_{3} ⨂_{A} V be a tensor product of U_{3} and V over A. As before, we get a homomorphism

(2.7.14) Θ′ from U_{2} ⨂_{A}V into U_{3} ⨂_{A}V,

as modules over k, using θ′ and the identity mapping on V. Suppose that

U_{1} ⟶^{θ} U_{2} ⟶^{θ′} U_{3} (2.7.15)

is exact, so that θ(U_{1}) = ker θ′. If V is projective, then

U_{1} ⨂_{A}V ⟶^{Θ} U_{2} ⨂_{A}V ⟶^{Θ′} U_{3} ⨂_{A}V (2.7.16)

is exact, so that Θ(U_{1} ⨂_{A} V ) = ker Θ′.

This corresponds to Proposition 1.1a on p106 of [50], where “left balanced” is defined on p97 of [50]. If V is a free module over A, then this can be verified directly. Otherwise, one can use (2.7.11) to reduce to the case of free modules. This is related to Exercise 4 on p31 of [8]. There are analogous statements with the roles of right and left modules interchanged, as usual.

2.8 Injective modules

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. A left module W over A is said to be injective if it has the following property. Let V be a left module over A, let V_{0} be a submodule of V, and let ϕ_{0} be a homomorphism from V_{0} into W, as left modules over A. Under these conditions, there should be a homomorphism ϕ from V into W, as left modules over A, such that ϕ = ϕ_{0} on V_{0}. (2.8.1)

Injectivity of a right module over A is defined in the same way.

Let I be a nonempty set, and let W_{j} be a left module over A for each j ∈ I. It is well known that

(2.8.2) W_{j} is injective for every j ∈ I

if and only if

(2.8.3)  ∏_{j∈I} W_{j} is injective,

as in Proposition 3.1 on p8 of [50]. To see this, let V be a left module over A, let V_{0} be a submodule of V, and suppose that (2.8.2) holds. If ϕ_{0} is a homomorphism from V_{0} into ∏_{j∈I} W_{j}, as left modules over A, then for each l ∈ I we get a homomorphism ϕ_{0,l} from V_{0} into W_{l}, as left modules over A, in the usual way. If (2.8.2) holds, then for every l ∈ I, ϕ_{0,l} can be extended to a homomorphism ϕ_{l} from V into W_{l}, as left modules over A. This leads to a homomorphism ϕ from V into ∏_{j∈I} W_{j}, as left modules over A. The restriction of ϕ to V_{0} is equal to ϕ_{0}, by construction.

Conversely, suppose that (2.8.3) holds, and let us check that (2.8.2) holds. Let l ∈ I be given, let V be a left module over A again, let V_{0} be a submodule of V, and let ϕ_{0,l} be a homomorphism from V_{0} into W_{l}, as left modules over A. We can get a homomorphism ϕ_{0} from V_{0} into ∏_{j∈I} W_{j}, as left modules over A, using the zero homomorphism from V_{0} into W_{j} when j ≠ l. If (2.8.3) holds, then there is an extension ϕ of ϕ_{0} to a homomorphism from V into ∏_{j∈I} W_{j}, as left modules over A. The lth component ϕ_{l} of ϕ is an extension of ϕ_{0,l} to a homomorphism from V into W_{l}, as left modules over A, as desired.

Let W be a left module over A again, and consider the following condition:

(2.8.4) if I is a left ideal in A, and ϕ_{I} is a homomorphism from I into W, then there is a w_{0} ∈ W such that ϕ_{I}(a) = a · w_{0} for every a ∈ I.

More precisely, I may be considered as a left module over A, and ϕ_{I} is supposed to be a homomorphism from I into W, as left modules over A. It is well known that W is injective as a left module over A if and only if this condition holds, as in Theorem 3.2 on p8 of [50].

To get the necessity of this condition, let I and ϕ_{I} be given as before. Note that A may be considered as a left module over itself, so that I is a submodule of A. If W is injective, then there is an extension ϕ of ϕ_{I} to a homomorphism from A into W, as left modules over A. In this case, the condition holds with

w_{0} = ϕ(e_{A}). (2.8.5)

To show the sufficiency of the condition, let V be a left module over A, let V_{0} be a submodule of V, and let ϕ_{0} be a homomorphism from V_{0} into W, as left modules over A. Also let v_{1} ∈ V be given, and observe that

V_{1} = {a · v_{1} + v_{0} : a ∈ A, v_{0} ∈ V_{0}} (2.8.6)

is a submodule of V that contains V_{0}. Put

I_{1} = {a ∈ A : a · v_{1} ∈ V_{0}}, (2.8.7)

which is a left ideal in A. If a ∈ I_{1}, then put

ϕ_{I_{1}}(a) = ϕ_{0}(a · v_{1}). (2.8.8)

It is easy to see that this defines a homomorphism from I_{1} into W, as left modules over A.

By hypothesis, there is a w_{1} ∈ W such that

ϕ_{0}(a · v_{1}) = ϕ_{I_{1}}(a) = a · w_{1} (2.8.9) for every a ∈ I_{1}. If a ∈ A and v_{0} ∈ V_{0}, then we would like to put

ϕ_{1}(a · v_{1} + v_{0}) = a · w_{1} + ϕ_{0}(v_{0}). (2.8.10)

One can check that ϕ_{1} is well-defined as a mapping from V_{1} into W, using (2.8.9). More precisely, ϕ_{1} is a homomorphism from V_{1} into W, as left modules over A. Of course, ϕ_{1} = ϕ_{0} on V_{0}, (2.8.11)

by construction.

If V = V_{1}, then there is nothing more to do. If V can be generated by V_{0} and finitely many other elements as a left module over A, then one can extend ϕ_{0} to V by repeating this argument finitely many times. Similarly, if V can be generated by V_{0} and an infinite sequence of other elements, as a left module over A, then one can repeat this argument to extend ϕ_{0} to an increasing sequence of submodules of V, whose union is V.

Otherwise, let E be the collection of all ordered pairs (U, ϕ_{U}), where U is a submodule of V that contains V_{0}, and ϕ_{U} is a homomorphism from U into W, as left modules over A, that extends ϕ_{0}. If (U_{1}, ϕ_{U_{1}}), (U_{2}, ϕ_{U_{2}}) ∈ E, then put

(U_{1}, ϕ_{U_{1}}) ⪯ (U_{2}, ϕ_{U_{2}}) (2.8.12)

when U_{1} ⊆ U_{2} and ϕ_{U_{2}} = ϕ_{U_{1}} on U_{1}. This defines a partial ordering on E, and one can use Zorn’s lemma or Hausdorff’s maximality principle to get that E has a maximal element with respect to this partial order. If a maximal element of E did not correspond to a homomorphism from V into W, then the earlier construction could be used to get an extension to a larger submodule of V.

It is well known that every left module over A is isomorphic to a submodule of an injective module, as in Theorem 3.3 on p9 of [50]. We shall return to this in Sections 3.14 and 3.15.

Let W and Z be left modules over A, and let ζ be an injective homomorphism from W into Z, as left modules over A. If W is injective as a left module over A, then there is a homomorphism ϕ from Z into W, as left modules over A, such that (2.8.13) ϕ ∘ ζ is the identity mapping on W.

More precisely, one can use the injectivity of W to extend the inverse of ζ on ζ(W) to a homomorphism ϕ from Z into W. Note that ϕ(Z) = W, and that

(2.8.14) ζ ∘ ϕ is the identity mapping on ζ(W).

Of course, ker ϕ = ker(ζ ∘ ϕ) (2.8.15)

is a submodule of Z. One can check that

(2.8.16) Z corresponds to the direct sum of ζ(W) and ker ϕ

as a left module over A, using standard arguments. This corresponds to part of Proposition 3.4 on p10 of [50].

In fact, the injectivity of W is necessary for this property to hold, as in Proposition 3.4 on p10 of [50]. Indeed, if W is any left module over A, then there is an injective homomorphism from W into an injective module Z, as mentioned earlier. In this case, (2.8.16) implies that ζ(W) is injective, as before, so that W is injective.

Of course, there are analogous statements for right modules over A.

2.9 Covariant ϕ-extensions

Let k be a commutative ring with a multiplicative identity element, and let A, B be associative algebras over k with multiplicative identity elements e_{A}, e_{B}, respectively. Suppose that ϕ is an algebra homomorphism from A into B, with ϕ(e_{A}) = e_{B}.

Let V be a module over k. If V is a left module over B, then V may be considered as a left module over A, where the action of a ∈ A on V is defined to be the given action of ϕ(a) ∈ B on V. Similarly, if V is a right module over B, then V may be considered as a right module over A. This corresponds to some remarks on p28f of [50]. If A and B are commutative, then this is called restriction of scalars, as on p27 of [8].

Of course, B may be considered as a left and right module over itself. Thus B may be considered as a left and right module over A, as in the preceding paragraph. Note that the left action on B by A or B commutes with the right action by A or B. We also have that

(2.9.1) ϕ may be considered as a homomorphism from A into B, as left and right modules over A.

If V is a right module over A, then let

V_{(ϕ)} = V ⨂_{A}B (2.9.2)

be a tensor product of V and B over A, where B is considered as a left module over A, as in the preceding paragraph. More precisely, B may be considered as a right module over itself too, and the left action of A on B commutes with the right action of B on itself. This means that (2.9.2) may be considered as a right module over B, as in Section 1.10.

Similarly, if V is a left module over A, then let

(ϕ)V = B ⨂_{A}V (2.9.3)

be a tensor product of B and V over A, where B is considered as a right module over A. This may be considered as a left module over B, as before. In each of these two cases, the resulting module over B is called the covariant ϕ-extension of V, as on p29 of [50]. This corresponds to extension of scalars when A and B are commutative, as on p28 of [8].

2.10. MORE ON COVARIANT ϕ-EXTENSIONS 45

Let V be a right module over A again, and remember that V ⨂_{A} A is isomorphic to V, as a right module over A, in a natural way, as in Section 1.10. If we consider ϕ as a homomorphism from A into B, as left modules over A, then we get a unique homomorphism from V ⨂_{A} A into (2.9.2), as modules over k, with v ⊗_{A} a ↦ v ⊗_{A} ϕ(a) (2.9.4)

for every a ∈ A and v ∈ V, as in Section 1.9. More precisely, one can check that this is a homomorphism from V ⨂_{A} A into (2.9.2), as right modules over A. This may be identified with a homomorphism

(2.9.5) from V into V_{(ϕ)},

as right modules over A, as on p29 of [50].

If V is a left module over A, then A ⨂_{A} V is isomorphic to V as a left module over A in a natural way, as before. If we consider ϕ as a homomorphism from A into B, as right modules over A, then we get a unique homomorphism from A ⨂_{A} V into (2.9.3) with

a ⊗_{A} v ↦ ϕ(a) ⊗_{A} v (2.9.6)

for every a ∈ A and v ∈ V. One can verify that this is a homomorphism from A ⨂_{A} V into (2.9.3), as left modules over A. This may be identified with a homomorphism (2.9.7) from V into _{(ϕ)}V,

as left modules over A, as before.

2.10 More on covariant ϕ-extensions

Let us continue with the same notation and hypotheses as at the beginning of the previous section. Suppose that V is a right module over A, and let Z be a left module over B. Thus B ⨂_{B} Z corresponds to Z, as a left module over B, in a natural way, which may be considered as a left module over A, as before. This means that V ⨂_{A}(B ⨂_{B}Z) (2.10.1)

corresponds to V ⨂_{A}Z (2.10.2)

in a natural way, as modules over k. We also have that (2.10.1) is isomorphic to V_{(ϕ)} ⨂_{B}Z = (V ⨂_{A}B) ⨂_{B}Z, (2.10.3)

as modules over k, as in Section 1.12. Here B is considered as a left module over A, and a right module over itself. Thus (2.10.2) is isomorphic to the left side of (2.10.3), as modules over k, as in (1) on p29 of [50].

Similarly, suppose that V is a right module over B, and that Z is a left module over A. One can check that there is a natural isomorphism between (2.10.2) and V ⨂_{B}_{(ϕ)}Z, (2.10.4)

as modules over k, using the remarks in Section 1.12, as before. This corresponds to (2) on p29 of [50].

Suppose now that V is a left module over A, and that Z is a left module over B. There is a natural isomorphism between

Hom_{B}(_{(ϕ)}V, Z) = Hom_{B}(B ⨂_{A}V, Z) (2.10.5)

and Hom_{A}(V, Hom_{B}(B, Z)), (2.10.6)

as modules over k, as in Section 1.13. We also have that Hom_{B}(B, Z) corresponds to Z in a natural way, as a left module over B, as in Section 1.8. This leads to a natural isomorphism between the left side of (2.10.5) and

Hom_{A}(V, Z), (2.10.7)

as modules over k. This corresponds to (3) on p29 of [50].

Similarly, suppose that V is a right module over A, and that Z is a right module over B. One can get a natural isomorphism between

Hom_{B}(V_{(ϕ)}, Z) (2.10.8)

and (2.10.7), as modules over k, using the remarks in Section 1.13, as in the preceding paragraph. This corresponds to (3’) on p30 of [50].

If V is a right module over A that is projective as a module over A, then Proposition 6.1 on p30 of [50] states that V_{(ϕ)} is projective as a right module over B. More precisely, one can use the isomorphism between (2.10.8) and (2.10.7) to reduce the projectivity condition for V_{(ϕ)} to the projectivity condition for V. Similarly, if V is a left module over A that is projective as a module over A, then _{(ϕ)}V is projective as a left module over B.

2.11 Contravariant ϕ-extensions

Let us continue with the same notation and hypotheses as at the beginning of Section 2.9 again.

If V is a right module over A again, then let

V^{(ϕ)} = Hom_{A}(B, V) (2.11.1)

be the space of homomorphisms from B into V, as right modules over A. This may be considered as a right module over B, as in Section 1.8, because B is a 2.11. CONTRAVARIANT ϕ-EXTENSIONS 47

left module over itself, and the left action commutes with the right action by A. Similarly, if V is a left module over A, then let

(ϕ)V = Hom_{A}(B, V) (2.11.2)

be the space of homomorphisms from B into V, as left modules over A. This may be considered as a left module over B, because B is a right module over itself, and the right action commutes with the left action by A, as before. In both cases, the resulting module over B is called the contravariant ϕ-extension of V, as on p29 of [50].

Suppose that V is a right module over A again, and remember that the space Hom_{A}(A, V) of homomorphisms from A into V, as right modules over A, is isomorphic to V as a right module over A in a natural way, as in Section 1.8. There is a natural homomorphism

(2.11.3) from Hom_{A}(B, V) into Hom_{A}(A, V),

as modules over k, defined by

ψ ↦ ψ ∘ ϕ, (2.11.4)

as in Section 1.7. In fact, this is a homomorphism as in (2.11.3), as right modules over A, because ϕ is a homomorphism from A into B, as both left and right modules over A. This leads to a homomorphism

(2.11.5) from V^{(ϕ)} into V,

as right modules over A, using the isomorphism between Hom_{A}(A, V) and V mentioned earlier. This corresponds to a remark on p29 of [50].

If V is a left module over A, then the space Hom_{A}(A, V) of homomorphisms from A into V, as left modules over A, is isomorphic to V as a left module over A, as in Section 1.8 again. As before, there is a natural homomorphism as in (2.11.3), as modules over k, defined by (2.11.4). More precisely, this is a homomorphism as in (2.11.3), as left modules over A, because ϕ is a homomorphism from A into B, as both left and right modules over A. This leads to a homomorphism (2.11.6) from ^{(ϕ)}V into V,

as left modules over A, using the isomorphism between Hom_{A}(A, V) and V.

Suppose now that V is a left module over B, and that Z is a left module over A. There is a natural isomorphism between

(2.11.7) Hom_{B}(V,^{(ϕ)}Z) = Hom_{B}(V, Hom_{A}(B, Z))

and Hom_{A}((B ⨂_{B}V ), Z), (2.11.8)

as modules over k, as in Section 1.13. Remember that B ⨂_{B} V corresponds to V in a natural way, as a left module over B, as in Section 1.10. Thus (2.11.8) is isomorphic to Hom_{A}(V, Z) (2.11.9) in a natural way, as modules over k. This means that the left side of (2.11.7) is isomorphic to (2.11.9) in a natural way, as modules over k, as in (4) on p29 of [50].

Similarly, suppose that V is a right module over B, and that Z is a right module over A. One can use the remarks in Section 1.13 to get a natural isomorphism between Hom_{B}(V, Z^{(ϕ)}) (2.11.10)

and (2.11.9), as modules over k, as before. This corresponds to (4’) on p30 of [50].

If Z is a right module over A that is injective as a module over A, then Proposition 6.1a on p30 of [50] states that Z^{(ϕ)} is injective as a right module over B. This uses the isomorphism between (2.11.10) and (2.11.9) to reduce the injectivity of Z^{(ϕ)} to the injectivity of Z. Similarly, if Z is a left module over A that is injective as a module over A, then ^{(ϕ)}Z is injective as a left module over B.

2.12 ϕ-Projectivity

We continue with the same notation and hypotheses as at the beginning of Section 2.9.

Let V be a right module over B. Thus V may be considered as a right module over A, and we use V (A) to refer to V as a module over A. As a right module over A, we may define its covariant ϕ-extension

V (A)_{(ϕ)} = V (A) ⨂_{A}B, (2.12.1)

as before.

Consider the mapping from V (A) × B into V defined by

(v, b) ↦ v · b (2.12.2)

for every v ∈ V (A) and b ∈ B, using the action of B on V on the right. If a ∈ A, then a acts on V (A) on the right by the action of ϕ(a) ∈ B on V on the right, as a module over B. Similarly, a acts on B on the left, as a left module over A, by multiplication on the left by ϕ(a). If we let a act on the right on v ∈ V (A) in this way, then we get the same result in (2.12.2) as when we let a act on the left on b ∈ B. More precisely, the result is

(v · ϕ(a)) · b = v · (ϕ(a) b), (2.12.3)

in terms of multiplication on B and the action of B on V on the right, as a module over B.

It follows that there is a unique homomorphism g from (2.12.1) into V, as modules over k, such that g(v ⊗ b) = v · b (2.12.4)

2.12. ϕ-PROJECTIVITY 49

for every v ∈ V (A) and b ∈ B. More precisely, g is a homomorphism from V (A)_{(ϕ)} into V, as right modules over B, using the action of B on V (A)_{(ϕ)} as in Section 2.9. This corresponds to the homomorphism g defined on p30 of [50].

We may consider V (A)_{(ϕ)} as a right module over A, as usual. Remember that there is a natural homomorphism from V (A) into V (A)_{(ϕ)}, as right modules over A, as in Section 2.9. This mapping was obtained from the homomorphism from V (A) ⨂_{A} A into V (A) ⨂_{A} B, as right modules over A, that corresponds to the identity mapping on V (A) and ϕ, as a homomorphism from A into B, as left modules over A. One can check that

(2.12.5) the composition of the natural mapping from V (A) into V (A)_{(ϕ)} with g is equal to the identity mapping on V (A),

as on p30 of [50]. In particular, this implies that

g(V (A)_{(ϕ)}) = V. (2.12.6)

Of course, the kernel of g is a submodule of V (A)_{(ϕ)}, as a right module over B. Thus ker g may be considered as a submodule of V (A)_{(ϕ)}, as a right module over A. In fact, V (A)_{(ϕ)} corresponds to the direct sum of ker g and the image of the natural mapping from V (A) into V (A)_{(ϕ)}, as a right module over A, by the remarks in the preceding paragraph.

Note that the natural mapping from V (A) in V (A)_{(ϕ)} is injective, by (2.12.5). This and the statement in the previous paragraph are mentioned in Exercise 13 on p32 of [8], for modules over commutative rings. More precisely, the natural mapping from V (A) into V (A)_{(ϕ)} is called g in [8], and the mapping g here is called p in [8].

If V is projective as a right module over B, then

(2.12.7) V (A)_{(ϕ)} corresponds to the direct sum of ker g and another submodule of V (A)_{(ϕ)}, as a right module over B,

as in Section 2.7. If (2.12.7) holds, then V is said to be ϕ-projective as a right module over B, as on p30 of [50]. In this case, the restriction of g to the other submodule of V (A)_{(ϕ)} is an isomorphism from that submodule onto V, as right modules over B.

If V (A) is projective as a right module over A, then V (A)_{(ϕ)} is projective as a right module over B, as in Section 2.10. If V is also ϕ-projective, as a right module over B, then it follows that

(2.12.8) V is projective as a right module over B,

as in Section 2.7. This corresponds to the first part of Proposition 6.2 on p30 of [50].

If B is projective as a right module over A, and V is projective as a right module over B, then the second part of Proposition 6.2 on p30 of [50] states that (2.12.9) V (A) is projective as a right module over A.

To see this, let W and Z be right modules over A, and let ψ be a homomorphism from W onto Z, as right modules over A. Consider the contravariant ϕ-extensions W^{(ϕ)} and Z^{(ϕ)} of W and Z, respectively, as in the previous section. Using ψ, we get a homomorphism Ψ from W^{(ϕ)} into Z^{(ϕ)}, as right modules over B. More precisely, Ψ sends a homomorphism from B into W, as right modules over A, to its composition with ψ, to get a homomorphism from B into Z, as right modules over A.

If B is projective as a right module over A, then

Ψ(W^{(ϕ)}) = Z^{(ϕ)}. (2.12.10)

Using Ψ, we get a homomorphism Ψ̂

(2.12.11) from Hom_{B}(V, W^{(ϕ)}) into Hom_{B}(V, Z^{(ϕ)}),

as modules over k. As before, Ψ̂ sends a homomorphism from V into W^{(ϕ)}, as right modules over B, to its composition with Ψ, to get a homomorphism from V into Z^{(ϕ)}, as right modules over B. If V is projective, as a right module over B, then Ψ̂(Hom_{B}(V, W^{(ϕ)})) = Hom_{B}(V, Z^{(ϕ)}). (2.12.12)

Similarly, we can use ψ to get a homomorphism Ψ̃

(2.12.13) from Hom_{A}(V (A), W) into Hom_{A}(V (A), Z),

as modules over k. As usual, Ψ̃ sends a homomorphism from V (A) into W, as right modules over A, to its composition with ψ, to get a homomorphism from V (A) into Z, as right modules over A. To show that V is projective, as a right module over A, we would like to check that

Ψ̃(Hom_{A}(V (A), W)) = Hom_{A}(V (A), Z). (2.12.14)

Remember that there are natural isomorphisms between Hom_{A}(V (A), W), Hom_{A}(V (A), Z) and Hom_{B}(V, W^{(ϕ)}), Hom_{B}(V, Z^{(ϕ)}), respectively, as modules over k, as in the previous section. Using this, we get that (2.12.14) follows from (2.12.12), as desired.

Of course, there are analogous statements for left modules.

2.13 ϕ-Injectivity

We continue with the same notation and hypotheses as at the beginning of Section 2.9 again.

Let Z be a right module over B, which may be considered as a right module over A. To be precise, we use Z(A) to refer to Z as a module over A, whose contravariant ϕ-extension is defined by

Z(A)^{(ϕ)} = Hom_{A}(B, Z(A)), (2.13.1)

2.13. ϕ-INJECTIVITY 51

as before.

If z ∈ Z, then b ↦ z · b (2.13.2)

defines a homomorphism from B into Z, as right modules over B. This may also be considered as a homomorphism from B into Z(A), as right modules over A, which is to say an element of (2.13.1). This defines a mapping h from Z into (2.13.1), with h(z) = (2.13.2) (2.13.3)

for every z ∈ Z, as on p30 of [50]. One can check that h is a homomorphism from Z into Z(A)^{(ϕ)}, as right modules over B, where the action of B on Z(A)^{(ϕ)} on the right is defined using the action of B on itself on the left, as in Section 2.11.

Remember that there is a natural homomorphism from Z(A)^{(ϕ)} into Z(A), as right modules over A, as in Section 2.11. This mapping was obtained from the homomorphism from Hom_{A}(B, Z(A)) into Hom_{A}(A, Z(A)) defined by composition with ϕ, by identifying Hom_{A}(A, Z(A)) with Z(A) in the usual way. This is the same as evaluating an element of Z(A)^{(ϕ)} at e_{B}, as a homomorphism from B into Z(A).

Observe that

(2.13.4) the composition of h with the natural mapping from Z(A)^{(ϕ)} into Z(A) is equal to the identity mapping on Z(A),

as on p30 of [50]. In particular, this implies that

(2.13.5) h is injective on Z.

Note that h(Z) is a submodule of Z(A)^{(ϕ)}, as a right module over B, and thus as a right module over A. We also have that Z(A)^{(ϕ)} corresponds to the direct sum of h(Z) and the kernel of the natural mapping from Z(A)^{(ϕ)} into Z(A), as a right module over A, because of (2.13.4).

If Z is injective as a right module over B, then

(2.13.6) Z(A)^{(ϕ)} corresponds to the direct sum of h(Z) and another submodule of Z(A)^{(ϕ)}, as a right module over B,

as in Section 2.8. If (2.13.6) holds, then Z is said to be ϕ-injective as a right module over Z, as on p31 of [50].

If Z(A) is injective as a right module over A, then Z(A)^{(ϕ)} is injective as a right module over B, as in Section 2.11. If Z is also ϕ-injective, as a right module over B, then it follows that

(2.13.7) Z is injective as a right module over B,

as in Section 2.8. This corresponds to the first part of Proposition 6.2a on p31 of [50].

Suppose that B is projective as a left module over A, and that Z is injective as a right module over B. Under these conditions, the second part of Proposition 6.2a on p31 of [50] states that

(2.13.8) Z(A) is injective as a right module over A.

Let V be a right module over A, and let V_{0} be a submodule of V. Of course, there is a natural mapping

(2.13.9) from Hom_{A}(V, Z(A)) into Hom_{A}(V_{0}, Z(A)),

which sends a homomorphism from V into Z(A), as right modules over A, to its restriction to V_{0}. This defines a homomorphism as in (2.13.9), as a mapping between modules over k, and we would like to show that this homomorphism is a surjection.

Let V_{(ϕ)}, (V_{0})_{(ϕ)} be the covariant ϕ-extensions of V, V_{0}, respectively, as in Section 2.9. The natural inclusion mapping from V_{0} into V leads to a homomorphism (2.13.10) from (V_{0})_{(ϕ)} into V_{(ϕ)},

as right modules over B, using the identity mapping on B. This homomorphism is injective, because B is projective as a left module over A, as in Section 2.7.

There are natural isomorphisms between Hom_{A}(V, Z(A)), Hom_{A}(V_{0}, Z(A)) and Hom_{B}(V_{(ϕ)}, Z), Hom_{B}((V_{0})_{(ϕ)}, Z), respectively, as modules over k, as in Section 2.10. There is also a natural homomorphism

(2.13.11) from Hom_{B}(V_{(ϕ)}, Z) into Hom_{B}((V_{0})_{(ϕ)}, Z),

as modules over k, which sends a homomorphism from V_{(ϕ)} into Z, as right modules over B, to its composition with the homomorphism as in (2.13.10). This homomorphism maps Hom_{B}(V_{(ϕ)}, Z) onto Hom_{B}((V_{0})_{(ϕ)}, Z), because Z is injective as a right module over B, and the homomorphism as in (2.13.10) is injective, as before.

It follows that the natural homomorphism as in (2.13.9) is surjective, as desired.

2.14 A ϕ-projectivity property

We continue with the notation and hypotheses mentioned at the beginning of Section 2.9.

Let V be a right module over A, so that V_{(ϕ)} is a right module over B, as in Section 2.9. The first part of Proposition 6.3 on p31 of [50] states that

(2.14.1) V_{(ϕ)} is ϕ-projective,

in the sense described in Section 2.12.

2.14. A ϕ-PROJECTIVITY PROPERTY 53

Let V_{(ϕ)}(A) be V_{(ϕ)}, considered as a right module over A, as in Section 2.12. The covariant ϕ-extension of V_{(ϕ)}(A) is defined by

(V_{(ϕ)}(A))_{(ϕ)} = V_{(ϕ)}(A) ⨂_{A}B, (2.14.2)

as before. Let g_{V_{(ϕ)}} be the homomorphism from (2.14.2) into V_{(ϕ)}, as right modules over B, defined in Section 2.12. In fact, g_{V_{(ϕ)}} maps (2.14.2) onto V_{(ϕ)}, as before. We would like to show that (2.14.2) corresponds to the direct sum of the kernel of g_{V_{(ϕ)}} and another submodule of (2.14.2), as a right module over B.

Let B ⨂_{A} B be a tensor product of B with itself over A, where the first B is considered as a right module over A, and the second B is considered as a left module over A. Note that B ⨂_{A} B may be considered as a left and right module over B, by considering the first B as a left module over itself, and the second B as a right module over itself. It is easy to see that these two actions of B on B ⨂_{A} B on the left and right commute with each other. It follows that B ⨂_{A} B may be considered as a left and right module over A as well.

Let α be the mapping from B into B ⨂_{A} B defined by

α(b) = e_{B} ⊗ b (2.14.3)

for every b ∈ B. One can check that α is a homomorphism from B into B ⨂_{A} B, as left modules over A, and right modules over B, as on p31 of [50].

It is easy to see that there is a unique homomorphism β from B ⨂_{A} B into B, as modules over k, such that

β(b_{1} ⊗ b_{2}) = b_{1} b_{2} (2.14.4)

for every b_{1}, b_{2} ∈ B. More precisely, this is a homomorphism from B ⨂_{A} B into B as left and right modules over B. In particular, β may be considered as a homomorphism from B ⨂_{A} B into B, as left modules over A, as on p31 of [50]. Note that (2.14.5) β ∘ α is the identity mapping on B,

by construction.

Using the identity mapping on V and α on B, we get a homomorphism α′

(2.14.6) from V_{(ϕ)} = V ⨂_{A}B into V ⨂_{A}(B ⨂_{A}B),

as modules over k. More precisely, this is a homomorphism as in (2.14.6), as right modules over B, as on p31 of [50]. Similarly, we can use the identity mapping on V and β on B ⨂_{A} B to get a homomorphism β′

(2.14.7) from V ⨂_{A}(B ⨂_{A}B) into V ⨂_{A}B = V_{(ϕ)},

as right modules over B. It is easy to see that

(2.14.8) β′ ∘ α′ is the identity mapping on V_{(ϕ)}, because of (2.14.5).

As in Section 1.12, there is a natural isomorphism

(2.14.9) between V ⨂_{A}(B ⨂_{A}B) and (V ⨂_{A}B) ⨂_{A}B = (V_{(ϕ)}(A))_{(ϕ)},

as modules over k. More precisely, this is an isomorphism as in (2.14.9), as right modules over B. Thus β′ corresponds to a homomorphism

(2.14.10) from (V_{(ϕ)}(A))_{(ϕ)} into V_{(ϕ)},

as right modules over B. In fact, β′ corresponds to the homomorphism g_{V_{(ϕ)}} mentioned earlier, as on p31 of [50].

Similarly, α′ corresponds to a homomorphism

(2.14.11) from V_{(ϕ)} into (V_{(ϕ)}(A))_{(ϕ)},

as right modules over B. Using (2.14.8), we get that

(2.14.12) the composition of this homomorphism with g_{V_{(ϕ)}} is the identity mapping on V_{(ϕ)}.

This implies that

(2.14.13) (V_{(ϕ)}(A))_{(ϕ)} corresponds to the direct sum of ker g_{V_{(ϕ)}} and the image of V_{(ϕ)} under the homomorphism corresponding to α′ as in (2.14.11),

as a right module over B, as desired.

Alternatively, there is a natural homomorphism from V into V_{(ϕ)}, as right modules over A, as in Section 2.9. This homomorphism may be described equivalently by v ↦ v ⊗ ϕ(e_{a}) = v ⊗ e_{B} (2.14.14)

for each v ∈ V. This leads to a homomorphism from V_{(ϕ)} = V ⨂_{A} B into (2.14.2), as right modules on B, using the identity mapping on B. This is the same as the homomorphism corresponding to α′ as in the preceding paragraph. The composition of this homomorphism with g_{V_{(ϕ)}} is the identity mapping on V_{(ϕ)}, as before.

There are analogous statements for left modules over A, as in Proposition 6.3 on p31 of [50].

2.15 A ϕ-injectivity property

We continue with the notation and hypotheses mentioned at the beginning of Section 2.9 again.

Let V be a right module over A. The second part of Proposition 6.3 on p31 of [50] states that (2.15.1) V^{(ϕ)} is ϕ-injective 2.15. A ϕ-INJECTIVITY PROPERTY 55

as a right module over B, in the sense described in Section 2.13. Remember that V^{(ϕ)} is defined as in Section 2.11, and let V^{(ϕ)}(A) be V^{(ϕ)}, considered as a right module over A, as before. Thus the contravariant ϕ-extension of V^{(ϕ)}(A) is defined by (V^{(ϕ)}(A))^{(ϕ)} = Hom_{A}(B, V^{(ϕ)}(A)). (2.15.2)

Let h_{V^{(ϕ)}} be the homomorphism from V^{(ϕ)} into (2.15.2), as right modules over B, defined in Section 2.13. Remember that this homomorphism is injective, as before. We would like to show that (2.15.2) corresponds to the direct sum of the image of h_{V^{(ϕ)}} and another submodule of (2.15.2), as a right module over B.

Observe that

(V^{(ϕ)}(A))^{(ϕ)} = Hom_{A}(B, Hom_{A}(B, V)) (2.15.3)

is isomorphic to Hom_{A}(B ⨂_{A}B, V ), (2.15.4)

as modules over k, in a natural way, as in Section 1.13. More precisely, this is an isomorphism between right modules over B. Here B ⨂_{A} B is considered as a right module over A, and as a left module over B, so that the space (2.15.4) of homomorphisms from B ⨂_{A} B into V, as right modules over A, may be considered as a right module over B.

Let α̃ be the mapping from B into B ⨂_{A} B defined by

α̃(b) = b ⊗ e_{B} (2.15.5)

for every b ∈ B. This is a homomorphism from B into B ⨂_{A} B, as left modules over B, and right modules over A.

Let α′ be the mapping

(2.15.6) from (2.15.4) into V^{(ϕ)} = Hom_{A}(B, V)

defined by composing an element of (2.15.4) with α̃, to get a homomorphism from B into V, as right modules over A. This defines a homomorphism from (2.15.4) into V^{(ϕ)}, as right modules over B. Using the isomorphism between (2.15.3) and (2.15.4) mentioned earlier, we get that α′ corresponds to a homomorphism (2.15.7) from (V^{(ϕ)}(A))^{(ϕ)} into V^{(ϕ)},

as right modules over B.

Remember that β is the homomorphism from B ⨂_{A} B into B determined by (2.14.4). Let β̃′ be the mapping

(2.15.8) from V^{(ϕ)} = Hom_{A}(B, V) into (2.15.4)

defined by composing an element of Hom_{A}(B, V) with β. This defines a homomorphism as in (2.15.8), as right modules over B. One can verify that β̃′ corresponds to h_{V^{(ϕ)}}, using the isomorphism between (2.15.3) and (2.15.4) mentioned earlier. This uses the way that the action of B on V^{(ϕ)} on the right is defined.

Of course, (2.15.9) β ∘ α̃ is the identity mapping on B,

by construction. This implies that

(2.15.10) α′ ∘ β̃′ is the identity mapping on V^{(ϕ)}.

Equivalently, this means that the composition of h_{V^{(ϕ)}} with the homomorphism from (V^{(ϕ)}(A))^{(ϕ)} into V^{(ϕ)} corresponding to α′ is the identity mapping on V^{(ϕ)}. It follows that (V^{(ϕ)}(A))^{(ϕ)} corresponds to the direct sum of the image of h_{V^{(ϕ)}} and the kernel of the homomorphism corresponding to α′, as a right module over B, as desired.

Alternatively, there is a natural homomorphism from V^{(ϕ)} into V, as right modules over A, as in Section 2.11. Equivalently, this homomorphism sends an element of V^{(ϕ)} = Hom_{A}(B, V) to its value at e_{B}. This leads to a natural mapping from (2.15.2) into V^{(ϕ)}, by composing an element of Hom_{A}(B, V^{(ϕ)}(A)) with the homomorphism just mentioned to get an element of Hom_{A}(B, V). This defines a homomorphism from (V^{(ϕ)}(A))^{(ϕ)} into V^{(ϕ)}, as right modules over B. This is the same as the homomorphism corresponding to α′ mentioned earlier.

One can check that that composition of h_{V^{(ϕ)}} with the homomorphism just mentioned is the same as the identity mapping on V^{(ϕ)}, as before. This basically reduces to evaluating an element of V^{(ϕ)} = Hom_{A}(B, V) at b e_{B} = b for each b ∈ B, because of the way that the action of b ∈ B on V^{(ϕ)} on the right is defined.

There are analogous statements for left modules over A, as usual.

Modules and tensor products, 3

3.1 Semisimple modules

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a module over k that is a left or right module over A. We say that V is simple as a module over A if A ≠ {0}, and V does not have any proper nonzero submodules, as a module over A. We say that V is semisimple as a module over A if it is isomorphic to the direct sum of a family of simple modules, as a module over A.

Let V_{0} be a submodule of V, as a module over A. It is well known that

(3.1.1) V corresponds to the direct sum of V_{0} and another submodule, as a module over A

if and only if

(3.1.2) there is a homomorphism ϕ_{0} from V onto V_{0}, as modules over A, that is equal to the identity mapping on V_{0}.

More precisely, if (3.1.2) holds, then V corresponds to the direct sum of V_{0} and the kernel of ϕ_{0}, as a module over A.

It is well known that V is semisimple as a module over A if and only if (3.1.1) holds for every submodule V_{0} of V, as a module over A, as in Proposition 4.1 on p11 of [50]. To show the “only if” part, suppose that V corresponds to the direct sum of a family {V_{j}}_{j∈I} of simple submodules, as a module over A. If I_{0} is a subset of I, then let V (I_{0}) be the submodule of V generated by V_{j}, j ∈ I_{0}.

Let W be a submodule of V, as a module over A. Let us say that I_{0} ⊆ I is admissible with respect to W if

V (I_{0}) ∩ W = {0}. (3.1.3)

We would like to use a subset I_{1} of I that is admissible with respect to W and maximal with respect to inclusion. If I has only finitely or countably many elements, then one can get a maximal admissible subset of I using fairly straightforward arguments. Otherwise, one can use Zorn’s lemma or Hausdorff’s maximality principle to get a maximal admissible subset of I.

If j ∈ I \ I_{1}, then I_{1} ∪ {j} is not admissible with respect to W, so that

V (I_{1} ∪ {j}) ∩ W ≠ {0}. (3.1.4)

This means that there is a nonzero element of W that can be expressed as the sum of elements of V (I_{1}) and V_{j}. This element of V_{j} has to be nonzero, because I_{1} is admissible with respect to W. It follows that there is a nonzero element of V_{j} that can be expressed as the sum of elements of V (I_{1}) and W. Thus

V_{j} ∩ (V (I_{1}) + W) ≠ {0}, (3.1.5)

which implies that V_{j} ⊆ V (I_{1}) + W, (3.1.6)

because V_{j} ∩ (V (I_{1}) + W) is a submodule of V_{j}, as a module over A, and V_{j} is simple, by hypothesis. This shows that

V = V (I_{1}) + W, (3.1.7)

because (3.1.6) holds for every j ∈ I \ I_{1}. This means that V corresponds to the direct sum of V (I_{1}) and W, as a module over A, because I_{1} is admissible with respect to W.

Conversely, suppose that (3.1.1) holds for every submodule V_{0} of V, as a module over A, so that (3.1.2) holds for every submodule V_{0} of V. Let W be a submodule of V, and let W_{0} be a submodule of W, as modules over A. In particular, W_{0} is a submodule of V, so that there is a homomorphism from V onto W_{0}, as modules over A, that is equal to the identity mapping on W_{0}. The restriction of this homomorphism to W is a homomorphism from W onto W_{0}, as modules over A, that is equal to the identity mapping on W_{0}. It follows that

(3.1.8) W corresponds to the direct sum of W_{0} and another submodule, as a module over A,

as before.

If k is a field, and V has finite dimension as a vector space over k, then one can use this repeatedly to express V as the direct sum of finitely many simple submodules, as a module over A. Otherwise, one can use the following argument.

Let Z be a nonzero submodule of V, as a module over A. We would like to show that Z contains a nonzero simple submodule, as a module over A. Let z be a nonzero element of Z. One can use Zorn’s lemma or Hausdorff’s maximality principle to show that there is a submodule Z_{0} of Z, as a module over A, such that z ∉ Z_{0}, (3.1.9) and Z_{0} is maximal with respect to inclusion. By hypothesis, there is a submodule Z_{1} of Z such that Z corresponds to the direct sum of Z_{0} and Z_{1}, as a module over A, as in (3.1.8). Note that Z_{1} ≠ {0}, because of (3.1.9). We would like to check that Z_{1} is simple, as a module over A.

If Z_{1} is not simple, then there is a nonzero proper submodule Z_{2} of Z_{1}. This means that Z_{1} corresponds to the direct sum of Z_{2} and another nonzero proper submodule Z_{3}, as a module over A, by hypothesis. Thus Z corresponds to the direct sum of Z_{0}, Z_{2}, and Z_{3}, as a module over A. It is easy to see that z cannot be an element of both Z_{0} + Z_{2} and Z_{0} + Z_{3}, because of (3.1.9). This contradicts the maximality of Z_{0}, as desired.

Let us say that a collection C of submodules of V is admissble if every element of C is simple as a module over A, and the submodule of V generated by C corresponds to the direct sum of the elements of C, as a module over A. One can use Zorn’s lemma or Hausdorff’s maximality principle to show that there is an admissble collection C_{0} that is maximal with respect to inclusion. Let W be the submodule of V generated by the elements of C_{0}. If W ≠ V, then V corresponds to the direct sum of W and a nonzero submodule Z, as a module over A, by hypothesis. Let Z_{0} be a submodule of Z that is simple, as a module over A, as in the previous paragraphs. Under these conditions, C_{0} ∪ {Z_{0}} is admissible, contradicting the maximality of C_{0}. It follows that V = W, so that V is semisimple as a module over A.

3.2 Direct systems of modules

Let I be a set. A binary relation ⪯ on I is said to be a pre-order on I if it is reflexive and transitive on I. In this case, the binary relation on I defined by

j ≃ l when j ⪯ l and l ⪯ j (3.2.1)

defines an equivalence relation on I. If (3.2.1) holds if and only if j = l, then ⪯ is said to be a partial ordering on I.

Let us say that (I, ⪯) is a pre-directed set or a pre-directed system if ⪯ is a pre-ordering on I, and for every j, l ∈ I there is an r ∈ I such that j, l ⪯ r. Thus a directed set or system is the same as a pre-directed set or system, where the pre-ordering is a partial ordering. Of course, slightly different conventions are sometimes used.

One can define direct limits using pre-directed sets instead of directed sets, and we shall do this now for modules. Let k be a commutative ring with a multiplicative identity element, and let (I, ⪯) be a nonempty pre-directed set. Also let V_{j} be a module over k for every j ∈ I. Suppose that for every j, l ∈ I with j ⪯ l we have a homomorphism ν_{j,l} from V_{j} into V_{l}, as modules over k, with the following two properties. First, ν_{j,j} is the identity mapping on V_{j} for every j ∈ I. Second, if j, l, r ∈ I and j ⪯ l ⪯ r, then

ν_{l,r} ∘ ν_{j,l} = ν_{j,r}. (3.2.2)

Under these conditions, the family of modules V_{j} and homomorphisms ν_{j,l} is said to form a direct or inductive system over (I, ⪯), as before. If j, l ∈ I satisfy (3.2.1), then it follows that ν_{j,l} and ν_{l,j} are inverses of each other.

If l ∈ I, then let ι_{l} be the natural injection from V_{l} into the direct sum ⨁_{j∈I} V_{j}, so that for each v_{l} ∈ V_{l}, the jth coordinate of ι_{l}(v_{l}) is equal to v_{l} when j = l, and to 0 when j ≠ l. Let U be the subset of ⨁_{j∈I} V_{j} consisting of finite sums of the form ι_{l}(v_{l}) − ι_{r}(ν_{l,r}(v_{l})), (3.2.3)

where l, r ∈ I, l ⪯ r, and v_{l} ∈ V_{l}. It is easy to see that U is a submodule of ⨁_{j∈I} V_{j}, as a module over k. Put

lim_{⟶} V_{j} = ( ⨁_{j∈I} V_{j})/U, (3.2.4)

where the quotient on the right defines a module over k. This is the direct or inductive limit of the direct system of V_{j}’s, j ∈ I, as a module over k.

Let q be the natural quotient mapping from ⨁_{j∈I} V_{j} into (3.2.4), and for each l ∈ I, put ν_{l} = q ∘ ι_{l}. (3.2.5)

This is a homomorphism from V_{l} into (3.2.4), as modules over k. Note that

ν_{l} = ν_{r} ∘ ν_{l,r} (3.2.6)

for every l, r ∈ I with l ⪯ r, by construction. The direct limit consists more precisely of the module (3.2.4) together with the homomorphisms ν_{l}, as in Exercise 14 on p32f of [8].

Every element of (3.2.4) can be expressed as ν_{l}(v_{l}) for some l ∈ I and v_{l} ∈ V_{l}, as before. This uses the hypothesis that (I, ⪯) be a pre-directed set. If

ν_{l}(v_{l}) = 0, (3.2.7)

then there is an r ∈ I such that l ⪯ r and

ν_{l,r}(v_{l}) = 0. (3.2.8)

This corresponds to Exercise 15 on p33 of [8].

Let Z be another module over k, and let ζ_{l} be a homomorphism from V_{l} into Z, as modules over k, for each l ∈ I. Suppose that for every l, r ∈ I with l ⪯ r, we have that ζ_{l} = ζ_{r} ∘ ν_{l,r}. (3.2.9)

Under these conditions, there is a unique homomorphism ζ from (3.2.4) into Z, as modules over k, such that ζ ∘ ν_{l} = ζ_{l} (3.2.10)

for every l ∈ I. The direct limit is uniquely determined up to isomorphism by this property, as in Exercise 16 on p33 of [8].

Let A be an associative algebra over k, with a multiplicative identity element e_{A}. Suppose either that V_{j} is a left module over A for every j ∈ I, or a right module over A for every j ∈ I. Suppose also that for every j, l ∈ I, ν_{j,l} is a homomorphism from V_{j} into V_{l}, as modules over A. Of course, ⨁_{j∈I} V_{j} is a left or right module over A too, as appropriate. It is easy to see that the set U defined earlier is a submodule of ⨁_{j∈I} V_{j}, as a module over A, in this case. Thus (3.2.4) may be considered as a left or right module over A, as appropriate. Similarly, (3.2.5) defines a homomorphism from V_{l} into (3.2.4), as modules over A, for each l ∈ I. There are analogues of the remarks in the previous paragraph for modules over A as well.

3.3 Direct systems of submodules

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a left or right module over A, and let (I, ⪯) be a nonempty pre-directed set. Suppose that V_{j} is a submodule of V for each j ∈ I, with

V_{l} ⊆ V_{r} (3.3.1)

when l, r ∈ I and l ⪯ r. Note that

⋃_{j∈I} V_{j} (3.3.2)

is a submodule of V, because (I, ⪯) is a pre-directed set. We can get a direct system by taking ν_{j,l} to be the natural inclusion mapping from V_{j} into V_{l} when j, l ∈ I and j ⪯ l. The corresponding direct limit is isomorphic to (3.3.2), as in Exercise 17 on p33 of [8]. Of course, any collection of submodules of V is partially ordered by inclusion, and one can simpy ask that such a collection be a directed set with respect to inclusion.

Now let I be a nonempty set, and let V_{j} be a left module over A for each j ∈ I. Thus the direct sum V = ⨁_{j∈I} V_{j} is a left module over A as well. Let I be the collection of nonempty finite subsets of I. It is easy to see that I is a directed set with respect to the partial ordering defined by inclusion.

If I_{1} ∈ I, then let V (I_{1}) be the submodule of V consisting of elements whose jth coordinate is equal to 0 when j ∈ I \ I_{1}. If I_{2} ∈ I and I_{1} ⊆ I_{2}, then

V (I_{1}) ⊆ V (I_{2}). (3.3.3)

Clearly

V = ⋃_{I_{1}∈I} V (I_{1}). (3.3.4)

As before, we may consider the family of V (I_{1})’s, I_{1} ∈ I, as a direct system of modules, using the natural inclusion mapping from V (I_{1}) into V (I_{2}) when I_{1}, I_{2} ∈ I and I_{1} ⊆ I_{2}. The corresponding direct limit is isomorphic to V, as a left module over A, as in the preceding paragraph.

Alternatively, if I_{1} ∈ I, then put

V_{I_{1}} = ∏_{j∈I_{1}} V_{j}, (3.3.5)

which is a left module over A. If I_{2} ∈ I and I_{1} ⊆ I_{2}, then there is a natural injective homomorphism from V_{I_{1}} into V_{I_{2}}, as left modules over A. This homomorphism sends an element of V_{I_{1}} to the element of V_{I_{2}} with the same jth coordinates when j ∈ I_{1}, and with all other coordinates equal to 0. Using these homomorphisms, the family of V_{I_{1}}’s, I_{1} ∈ I, becomes a direct system of modules. Of course, V_{I_{1}} is isomorphic to V (I_{1}), as left modules over A, in a natural way for each I_{1} ∈ I, and the homomorphism from V_{I_{1}} into V_{I_{2}} just mentioned corresponds exactly to the inclusion of V (I_{1}) into V (I_{2}) when I_{1} ⊆ I_{2}. This leads to an isomorphism between the direct limit of the V_{I_{1}}’s and the direct limit of the V (I_{1})’s, as left modules over A. There are analogous statements for right modules over A.

3.4 Homomorphisms between direct systems

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k, with a multiplicative identity element e_{A}. Also let (I, ⪯) be a nonempty directed set, and let V_{j}, W_{j} be left modules over A for every j ∈ I. Suppose that for every j, l ∈ I with j ⪯ l, we have homomorphisms ν_{j,l}, μ_{j,l} from V_{j}, W_{j} into V_{l}, W_{l}, respectively, as left modules over A, and that these homomorphisms satisfy the two conditions mentioned in Section 3.2. Thus the direct limits of the V_{j}’s and W_{j}’s can be defined as left modules over A, as before. If l ∈ I, then we let ν_{l}, μ_{l} be the corresponding homomorphisms from V_{l}, W_{l} into lim_{⟶} V_{j}, lim_{⟶} W_{j}, respectively.

Let ϕ_{j} be a homomorphism from V_{j} into W_{j}, as left modules over A, for each j ∈ I. If j, l ∈ I and j ⪯ l, then we ask that

ϕ_{l} ∘ ν_{j,l} = μ_{j,l} ∘ ϕ_{j}. (3.4.1)

Under these conditions, the family of ϕ_{j}’s, j ∈ I, is considered to define a homomorphism between the direct systems of V_{j}’s and W_{j}’s, j ∈ I, as in Exercise 18 on p33 of [8].

In this case, there is a unique homomorphism ϕ = lim_{⟶} ϕ_{j} from lim_{⟶} V_{j} into lim_{⟶} W_{j}, as left modules over A, such that

ϕ ∘ ν_{l} = μ_{l} ∘ ϕ_{l} (3.4.2)

for every l ∈ I, as in [8]. Indeed, μ_{l} ∘ ϕ_{l} defines a homomorphism from V_{l} into lim_{⟶} W_{j}, as left modules over A, for each l ∈ I. If l, r ∈ I and l ⪯ r, then

(μ_{r} ∘ ϕ_{r}) ∘ ν_{l,r} = μ_{r} ∘ μ_{l,r} ∘ ϕ_{l} = μ_{l} ∘ ϕ_{l}. (3.4.3)

Using this, we can get the homomorphism ϕ from the homomorphisms μ_{l} ∘ ϕ_{l}, l ∈ I, as in Section 3.2.

Let Z_{j} be a left module over A for every j ∈ I, and suppose that for every j, l ∈ I with j ⪯ l, we have a homomorphism ζ_{j,l} from Z_{j} into Z_{l}, as left modules over A, that satisfy the two conditions mentioned in Section 3.2. This means that the direct limit of the Z_{j}’s can be defined as a left module over A as before, and we let ζ_{l} be the corresponding homomorphism from Z_{l} into lim_{⟶} Z_{j} for each l ∈ I.

Let ψ_{j} be a homomorphism from W_{j} into Z_{j}, as left modules over A, for each j ∈ I, and suppose that

ψ_{l} ∘ μ_{j,l} = ζ_{j,l} ∘ ψ_{j} (3.4.4)

for every j, l ∈ with j ⪯ l. This leads to a unique homomorphism ψ = lim_{⟶} ψ_{j} from lim_{⟶} W_{j} into lim_{⟶} Z_{j}, as left modules over A, such that

ψ ∘ μ_{l} = ζ_{l} ∘ ψ_{l} (3.4.5)

for every l ∈ I, as before.

Of course, ψ_{j} ∘ ϕ_{j} is a homomorphism from V_{j} into Z_{j}, as left modules over A, for every j ∈ I. If j, l ∈ I and j ⪯ l, then

(ψ_{l} ∘ ϕ_{l}) ∘ ν_{j,l} = ψ_{l} ∘ μ_{j,l} ∘ ϕ_{j} = ζ_{j,l} ∘ (ψ_{j} ∘ ϕ_{j}). (3.4.6)

We also have that

(ψ ∘ ϕ) ∘ ν_{l} = ψ ∘ μ_{l} ∘ ϕ_{l} = ζ_{l} ∘ (ψ_{l} ∘ ϕ_{l}) (3.4.7)

for every l ∈ I. This means that

lim_{⟶}(ψ_{j} ∘ ϕ_{j}) = ψ ∘ ϕ, (3.4.8)

as homomorphisms from lim_{⟶} V_{j} into lim_{⟶} Z_{j}, as left modules over A.

Suppose that ψ_{j} ∘ ϕ_{j} = 0 (3.4.9)

for every j ∈ I, and observe that

ψ ∘ ϕ = 0. (3.4.10)

In fact, suppose that

V_{j} ⟶^{ϕ_{j}} W_{j} ⟶^{ψ_{j}} Z_{j} (3.4.11)

is exact for each j ∈ I, so that we have an exact sequence of homomorphisms between direct systems, as in Exercise 19 on p33 of [8]. We would like to check that

(3.4.12)  lim_{⟶} V_{j} ⟶^{ϕ} lim_{⟶} W_{j} ⟶^{ψ} lim_{⟶} Z_{j}

64 CHAPTER 3. MODULES AND TENSOR PRODUCTS, 3 is exact, as in [8].

Remember that an arbitary element of lim_{⟶} W_{j} can be expressed as μ_{l}(w_{l}) for some l ∈ I and w_{l} ∈ W_{l}, as in Section 3.2. If ψ(μ_{l}(w_{l})) = 0, (3.4.13) then ζ_{l}(ψ_{l}(w_{l})) = 0, (3.4.14) by (3.4.5). This implies that there is an r ∈ I such that l ⪯ r and ζ_{l,r}(ψ_{l}(w_{l})) = 0, (3.4.15) as in Section 3.2. It follows that ψ_{r}(μ_{l,r}(w_{l})) = 0, (3.4.16) by (3.4.4).

Using our exactness hypothesis, we get that there is a v_{r} ∈ V_{r} such that μ_{l,r}(w_{l}) = ϕ_{r}(v_{r}). (3.4.17) Remember that μ_{l}(w_{l}) = μ_{r}(μ_{l,r}(w_{l})), (3.4.18) as in (3.2.6). Thus μ_{l}(w_{l}) = μ_{r}(ϕ_{r}(v_{r})) = ϕ(ν_{r}(v_{r})), (3.4.19) using (3.4.2) in the second step. This shows that μ_{l}(w_{l}) is in the image of lim_{⟶} V_{j} under ϕ, as desired.

Of course, there are analogous statements for right modules.

3.5 Limits of bilinear mappings

Let k be a commutative ring with a multiplicative identity element, and let (I, ⪯) be a nonempty directed set. Also let V_{j} be a module over k for each j ∈ I, and suppose that for every j, l ∈ I with j ⪯ l we have a homomorphism ν_{j,l} from V_{j} into V_{l}, as modules over k, that satisfies the two conditions mentioned in Section 3.2. This means that the direct limit lim_{⟶} V_{j} can be defined as a module over k as in Section 3.2, and we let ν_{l} be the corresponding homomorphism from V_{l} into the direct limit, as modules over k, for each l ∈ I, as before.

Let W and Z be modules over k, and suppose that for each j ∈ I, b_{j} is a mapping from V_{j} × W into Z that is bilinear over k. If w ∈ W, then b_{j,w}(v) = b_{j}(v, w) (3.5.1)

3.5. LIMITS OF BILINEAR MAPPINGS 65 defines a homomorphism from V_{j} into Z, as modules over k. Suppose that for every l, r ∈ I with l ⪯ r and w ∈ W we have that b_{l,w} = b_{r,w} ∘ ν_{l,r}. (3.5.2) This implies that for each w ∈ W there is a unique homomorphism b_{w} from lim_{⟶} V_{j} into Z such that b_{w} ∘ ν_{l} = b_{l,w} (3.5.3) for every l ∈ I, as in Section 3.2. Of course, (3.5.2) is the same as saying that b_{l}(v_{l}, w) = b_{r}(ν_{l,r}(v_{l}), w) (3.5.4) for every l, r ∈ I with l ⪯ r, v_{l} ∈ V_{l}, and w ∈ W.

Consider the mapping b (3.5.5) from ( lim_{⟶} V_{j}) × W into Z defined by b(v, w) = b_{w}(v) (3.5.6) for every v ∈ lim_{⟶} V_{j} and w ∈ W. Of course, b(v, w) is linear over k in v, as in the preceding paragraph. One can check that b(v, w) is linear over k in w too, because b_{w} is uniquely determined by (3.5.3). Note that b is uniquely determined by the condition that b(ν_{l}(v_{l}), w) = b_{l}(v_{l}, w) (3.5.7) for every l ∈ I, v_{l} ∈ V_{l}, and w ∈ W.

Let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose for the moment that V_{j} is a right module over A for every j ∈ I, and that ν_{j,l} is a homomorphism from V_{j} into V_{l}, as right modules over A, for every j, l ∈ I with j ⪯ l. This implies that lim_{⟶} V_{j} is a right module over A, and that ν_{l} is a homomorphism from V_{l} into lim_{⟶} V_{j}, as right modules over A, for every l ∈ I. Suppose also that W is a left module over A, and that b_{j}(v_{j} · a, w) = b_{j}(v_{j}, a · w) (3.5.8) for every j ∈ I, a ∈ A, v_{j} ∈ V_{j}, and w ∈ W. If l ∈ I, a ∈ A, v_{l} ∈ V_{l}, and w ∈ W, then it follows that

b(ν_{l}(v_{l}) · a, w) = b(ν_{l}(v_{l} · a), w) = b_{l}(v_{l} · a, w)
(3.5.9)  = b_{l}(v_{l}, a · w) = b(ν_{l}(v_{l}), a · w).

This means that b(v · a, w) = b(v, a · w) (3.5.10) for every a ∈ A, v ∈ lim_{⟶} V_{j}, and w ∈ W, because every element of lim_{⟶} V_{j} can be expressed as ν_{l}(v_{l}) for some l ∈ I and v_{l} ∈ V_{l}. This leads to a homomorphism β

(3.5.11) from ( lim_{⟶} V_{j}) ⨂_{A}W into Z,

as modules over k, in the usual way. More precisely,

β(v ⊗ w) = b(v, w) (3.5.12)

for every v ∈ lim_{⟶} V_{j} and w ∈ W, and β is uniquely determined by this property. Equivalently, β(ν_{l}(v_{l}) ⊗ w) = b(ν_{l}(v_{l}), w) = b_{l}(v_{l}, w) (3.5.13)

for every l ∈ I, v_{l} ∈ V_{l}, and w ∈ W. Of course, there are analogous statements when V_{j} is a left module over A for every j ∈ I, W is a right module over A, and so on.

3.6 Direct limits and tensor products

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (I, ⪯) be a nonempty pre-directed set, and let V_{j} be a right module over A for each j ∈ I. Suppose that for each j, l ∈ I with j ⪯ l, we have a homomorphism ν_{j,l} from V_{j} into V_{l}, as right modules over A, that satisfies the two conditions mentioned in Section 3.2. Thus the direct limit of the V_{j}’s can be defined as a right module over A as before.

Let W be a left module over A, and for each j ∈ I, let V_{j} ⨂_{A} W be a tensor product of V_{j} and W over A. If j, l ∈ I and j ⪯ l, then we get a homomorphism θ_{j,l} (3.6.1) from V_{j} ⨂_{A}W into V_{l} ⨂_{A}W,

as modules over k, using ν_{j,l} and the identity mapping on W. Note that θ_{j,j} is the identity mapping on V_{j} ⨂_{A} W for every j ∈ I, because ν_{j,j} is the identity mapping on V_{j}. If j, l, r ∈ I and j ⪯ l ⪯ r, then

θ_{l,r} ∘ θ_{j,l} = θ_{j,r}, (3.6.2)

because of the analogous property of ν_{j,r}. Thus the family of V_{j} ⨂_{A} W, j ∈ I, is a direct system of modules over k with respect to these homomorphisms.

This means that the direct limit

lim_{⟶} (V_{j} ⨂_{A}W ) (3.6.3)

may be defined in the usual way, as a module over k. If l ∈ I, then we let θ_{l} be the corresponding homomorphism from V_{l} ⨂_{A} W into (3.6.3), as modules over k. If l, r ∈ I and l ⪯ r, then θ_{l} = θ_{r} ∘ θ_{l,r}, (3.6.4)

3.6. DIRECT LIMITS AND TENSOR PRODUCTS 67

as before.

If l ∈ I, then let ν_{l} be the natural homomorphism from V_{l} into lim_{⟶} V_{j}, as right modules over A. This leads to a homomorphism ρ_{l} from V_{l} ⨂_{A} W into

( lim_{⟶} V_{j}) ⨂_{A}W, (3.6.5)

as modules over k, using the identity mapping on W. If r ∈ I satisfies l ⪯ r, then ρ_{r} ∘ θ_{l,r} = ρ_{l}, (3.6.6) because of the analogous property for ν_{l,r}. It follows that there is a unique homomorphism ρ from (3.6.3) into (3.6.5), as modules over k, such that ρ ∘ θ_{l} = ρ_{l} (3.6.7) for every l ∈ I. We would like to show that (3.6.8) ρ is an isomorphism from (3.6.3) onto (3.6.5), as modules over k, as in Exercise 20 on p33f of [8].

If l ∈ I, v_{l} ∈ V_{l}, and w ∈ W, then put b_{l}(v_{l}, w) = θ_{l}(v_{l} ⊗ w) (3.6.9) This defines a mapping from V_{l} × W into (3.6.3) that is bilinear over k. This is the same as the composition of the natural bilinear mapping from V_{l} × W into V_{l} ⨂_{A} W with θ_{l}. In particular, b_{l}(v_{l} · a, w) = b_{l}(v_{l}, a · w) (3.6.10) for every a ∈ A. If l, r ∈ I, l ⪯ r, v_{l} ∈ V_{l}, and w ∈ W, then

(3.6.11) b_{r}(ν_{l,r}(v_{l}), w) = θ_{r}(ν_{l,r}(v_{l}) ⊗ w) = θ_{r}(θ_{l,r}(v_{l} ⊗ w))
= θ_{l}(v_{l} ⊗ w) = b_{l}(v_{l}, w).

This leads to a mapping b from

( lim_{⟶} V_{j}) × W (3.6.12)

into (3.6.3) that is bilinear over k, as in the previous section. More precisely, b satisfies (3.5.10), which leads to a homomorphism β from (3.6.5) into (3.6.3), as modules over k, as before. If l ∈ I, v_{l} ∈ V_{l}, and w ∈ W, then β(ν_{l}(v_{l}) ⊗ w) = b_{l}(v_{l}, w) = θ_{l}(v_{l} ⊗ w). (3.6.13) In this case, we also have that ρ(θ_{l}(v_{l} ⊗ w)) = ρ_{l}(v_{l} ⊗ w) = ν_{l}(v_{l}) ⊗ w. (3.6.14) One can use this to check that β and ρ are inverses of each other.

There are analogous statements when the limit is taken in the second factor in the tensor product.

3.7 Double limits and bilinear mappings

Let (I_{1}, ⪯_{1}), (I_{2}, ⪯_{2}) be nonempty pre-directed sets, and put I = I_{1} × I_{2}. Consider the binary relation ⪯ defined on I by

(j_{1}, j_{2}) ⪯ (l_{1}, l_{2}) when j_{1} ⪯_{1} l_{1} and j_{2} ⪯_{2} l_{2}, (3.7.1)

where j_{1}, l_{1} ∈ I_{1} and j_{2}, l_{2} ∈ I_{2}. It is easy to see that (I, ⪯) is a pre-directed set.

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V_{j_{1}} be a right module over A for every j_{1} ∈ I_{1}, and let W_{j_{2}} be a left module over A for every j_{2} ∈ I_{2}. Suppose that for every j_{1}, l_{1} ∈ I_{1} with j_{1} ⪯_{1} l_{1} we have a homomorphism ν_{j_{1},l_{1}} from V_{j_{1}} into V_{j_{2}}, as right modules over A, and that these homomorphisms satisfy the two conditions mentioned in Section 3.2. Similarly, suppose that for every j_{2}, l_{2} ∈ I_{2} with j_{2} ⪯_{2} l_{2} we have a homomorphism μ_{j_{2},l_{2}} from W_{j_{2}} into W_{l_{2}}, as left modules over A, that satisfy the same two conditions.

The direct limits of the V_{j_{1}}’s and W_{j_{2}}’s may be defined as right and left modules over A, respectively, as before. If l_{1} ∈ I_{1} and l_{2} ∈ I_{2}, then we let ν_{l_{1}}, μ_{l_{2}} be the corresponding homomorphisms from V_{l_{1}}, W_{l_{2}} into lim_{⟶} V_{j_{1}}, lim_{⟶} W_{j_{2}}, respectively.

Let Z be a module over k, and for each j_{1} ∈ I_{1} and j_{2} ∈ I_{2}, let b_{j_{1},j_{2}} be a mapping from V_{j_{1}} × W_{j_{2}} into Z that is bilinear over k. If l_{1}, r_{1} ∈ I_{1}, l_{2}, r_{2} ∈ I_{2}, l_{1} ⪯_{1} r_{1}, l_{2} ⪯_{2} r_{2}, v_{l_{1}} ∈ V_{l_{1}}, and w_{l_{2}} ∈ W_{l_{2}}, then we ask that

b_{l_{1},l_{2}}(v_{l_{1}}, w_{l_{2}}) = b_{r_{1},r_{2}}(ν_{l_{1},r_{1}}(v_{l_{1}}), μ_{l_{2},r_{2}}(w_{l_{2}})). (3.7.2)

We also ask that

b_{j_{1},j_{2}}(v_{j_{1}} · a, w_{j_{2}}) = b_{j_{1},j_{2}}(v_{j_{1}}, a · w_{j_{2}}) (3.7.3)

for every j_{1} ∈ I_{1}, j_{2} ∈ I_{2}, a ∈ A, v_{j_{1}} ∈ V_{j_{1}}, and w_{j_{2}} ∈ W_{j_{2}}.

Let j_{2} ∈ I_{2} be given. As in Section 3.5, there is a unique mapping b_{j_{2}}

(3.7.4) from ( lim_{⟶} V_{j_{1}}) × W_{j_{2}} into Z

such that b_{j_{2}}(ν_{l_{1}}(v_{l_{1}}), w_{j_{2}}) = b_{l_{1},j_{2}}(v_{l_{1}}, w_{j_{2}}) (3.7.5)

for every l_{1} ∈ I_{1}, v_{l_{1}} ∈ V_{l_{1}}, and w_{j_{2}} ∈ W_{j_{2}}. This mapping is bilinear over k, and satisfies b_{j_{2}}(v · a, w_{j_{2}}) = b_{j_{2}}(v, a · w_{j_{2}}) (3.7.6)

for every a ∈ A, v ∈ lim_{⟶} V_{j_{1}}, and w_{j_{2}} ∈ W_{j_{2}}.

Similarly, there is a unique mapping b

(3.7.7) from ( lim_{⟶} V_{j_{1}}) × ( lim_{⟶} W_{j_{2}}) into Z such that b(v, μ_{l_{2}}(w_{l_{2}})) = b_{l_{2}}(v, w_{l_{2}}) (3.7.8)

for every v ∈ lim_{⟶} V_{j_{1}}, l_{2} ∈ I_{2}, and w_{l_{2}} ∈ W_{l_{2}}. This mapping is bilinear over k, and satisfies b(v · a, w) = b(v, a · w) (3.7.9)

for every a ∈ A, v ∈ lim_{⟶} V_{j_{1}}, and w ∈ lim_{⟶} W_{j_{2}}. Note that (3.7.8) is equivalent to saying that b(ν_{l_{1}}(v_{l_{1}}), μ_{l_{2}}(w_{l_{2}})) = b_{l_{1},l_{2}}(v_{l_{1}}, w_{l_{2}}) (3.7.10)

for every l_{1} ∈ I_{1}, l_{2} ∈ I_{2}, v_{l_{1}} ∈ V_{l_{1}}, and w_{l_{2}} ∈ W_{l_{2}}.

Using b, we get a unique homomorphism β

(3.7.11) from ( lim_{⟶} V_{j_{1}}) ⨂_{A}( lim_{⟶} W_{j_{2}}) into Z,

as modules over k, such that

β(v ⊗ w) = b(v, w) (3.7.12)

for every v ∈ lim_{⟶} V_{j_{1}} and w ∈ lim_{⟶} W_{j_{2}}. Equivalently, this means that

β(ν_{l_{1}}(v_{l_{1}}) ⊗ μ_{l_{2}}(w_{l_{2}})) = b(ν_{l_{1}}(v_{l_{1}}), μ_{l_{2}}(w_{l_{2}})) = b_{l_{1},l_{2}}(v_{l_{1}}, w_{l_{2}}) (3.7.13)

for every l_{1} ∈ I_{1}, l_{2} ∈ I_{2}, v_{l_{1}} ∈ V_{l_{1}}, and w_{l_{2}} ∈ W_{l_{2}}.

3.8 Double limits and tensor products

Let us continue with the same notation and hypotheses as in the first three paragraphs of the previous section.

If j_{1} ∈ I_{1} and j_{2} ∈ I_{2}, then let V_{j_{1}} ⨂_{A} W_{j_{2}} be a tensor product of V_{j_{1}} and W_{j_{2}} over A. If j = (j_{1}, j_{2}), l = (l_{1}, l_{2}) ∈ I and j ⪯ l, then we get a homomorphism θ_{j,l}

(3.8.1) from V_{j_{1}} ⨂_{A}W_{j_{2}} into V_{l_{1}} ⨂_{A}W_{l_{2}},

as modules over k, using ν_{j_{1},l_{1}} and μ_{j_{2},l_{2}}. In particular, θ_{j,j} is the identity mapping on V_{j_{1}} ⨂_{A} W_{j_{2}}, because ν_{j_{1},j_{1}} is the identity mapping on V_{j_{1}}, and μ_{j_{2},j_{2}} is the identity mapping on W_{j_{2}}. If j, l, r ∈ I and j ⪯ l ⪯ r, then

θ_{l,r} ∘ θ_{j,l} = θ_{j,r}, (3.8.2)

beause of the analogous properties of ν_{j_{1},r_{1}} and μ_{j_{2},r_{2}}. This shows that the family of V_{j_{1}} ⨂_{A} W_{j_{2}}, j = (j_{1}, j_{2}) ∈ I, is a direct system of modules over k with respect to these homomorphisms.

Thus the direct limit

lim_{⟶} (V_{j_{1}} ⨂_{A}W_{j_{2}}) (3.8.3)

may be defined as a module over k in the usual way. If l = (l_{1}, l_{2}) ∈ I, then let θ_{l} be the corresponding homomorphism from V_{l_{1}} ⨂_{A} W_{l_{2}} into (3.8.3), as modules over k. If l, r ∈ I and l ⪯ r, then

θ_{l} = θ_{r} ∘ θ_{l,r}, (3.8.4)

as usual.

If l = (l_{1}, l_{2}) ∈ I, then let ρ_{l} be the homomorphism from V_{l_{1}} ⨂_{A} W_{l_{2}} into

( lim_{⟶} V_{j_{1}}) ⨂_{A}( lim_{⟶} W_{j_{2}}), (3.8.5)

as modules over k, corresponding to ν_{l_{1}} and μ_{l_{2}}. If r = (r_{1}, r_{2}) ∈ I and l ⪯ r, then ρ_{r} ∘ θ_{l,r} = ρ_{l}, (3.8.6) because of the analogous properties for ν_{l_{1},r_{1}} and μ_{l_{2},r_{2}}. This implies that there is a unique homomorphism ρ from (3.8.3) into (3.8.5), as modules over k, such that ρ ∘ θ_{l} = ρ_{l} (3.8.7) for every l ∈ I. We would like to show that

(3.8.8) ρ is an isomorphism from (3.8.3) onto (3.8.5),

as modules over k, as in Proposition 9.2* on p99 of [50].

If l = (l_{1}, l_{2}) ∈ I, v_{l_{1}} ∈ V_{l_{1}}, and w_{l_{2}} ∈ W_{l_{2}}, then put

b_{l_{1},l_{2}}(v_{l_{1}}, w_{l_{2}}) = θ_{l}(v_{l_{1}} ⊗ w_{l_{2}}), (3.8.9)

which defines a mapping from V_{l_{1}} × W_{l_{2}} into (3.8.3) that is bilinear over k. We also have that b_{l_{1},l_{2}}(v_{l_{1}} · a, w_{l_{2}}) = b_{l_{1},l_{2}}(v_{l_{1}}, a · w_{l_{2}}) (3.8.10) for every a ∈ A, because b_{l_{1},l_{2}} is the same as the composition of the natural bilinear mapping from V_{l_{1}} × W_{l_{2}} into V_{l_{1}} ⨂_{A} W_{l_{2}} with θ_{l}. If r = (r_{1}, r_{2}) ∈ I and l ⪯ r, then

b_{r_{1},r_{2}}(ν_{l_{1},r_{1}}(v_{l_{1}}), μ_{l_{2},r_{2}}(w_{l_{2}})) = θ_{r}(ν_{l_{1},r_{1}}(v_{l_{1}}) ⊗ μ_{l_{2},r_{2}}(w_{l_{2}}))
(3.8.11)  = θ_{r}(θ_{l,r}(v_{l_{1}} × w_{l_{2}}))
= θ_{l}(v_{l_{1}} ⊗ w_{l_{2}}) = b_{l_{1},l_{2}}(v_{l_{1}}, w_{l_{2}}).

Using this, we get a mapping b from

( lim_{⟶} V_{j_{1}}) × ( lim_{⟶} W_{j_{2}}) (3.8.12)

into (3.8.3) that is bilinear over k, as in the previous section. Remember that b satisfies (3.7.9), which leads to a homomorphism β from (3.8.5) into (3.8.3), as before. If l = (l_{1}, l_{2}) ∈ I, v_{l_{1}} ∈ V_{l_{1}}, and w_{l_{2}} ∈ W_{l_{2}}, then

β(ν_{l_{1}}(v_{l_{1}}) ⊗ μ_{l_{2}}(w_{l_{2}})) = b_{l_{1},l_{2}}(v_{l_{1}}, w_{l_{2}}) = θ_{l}(v_{l_{1}} ⊗ w_{l_{2}}). (3.8.13)

Observe that

ρ(θ_{l}(v_{l_{1}} ⊗ w_{l_{2}})) = ρ_{l}(v_{l_{1}} ⊗ w_{l_{2}}) = ν_{l_{1}}(v_{l_{1}}) ⊗ μ_{l_{2}}(w_{l_{2}}). (3.8.14)

One can use this to verify that β and ρ are inverses of each other.

3.9. ITERATED DIRECT LIMITS 71

3.9 Iterated direct limits

Let (I_{1}, ⪯_{1}), (I_{2}, ⪯_{2}) be nonempty pre-directed sets again, and let I = I_{1} × I_{2} be equipped with the corresponding pre-order ⪯, as in Section 3.7. Also let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}.

Suppose that for each j = (j_{1}, j_{2}) ∈ I, V_{j} = V_{(j_{1},j_{2})} (3.9.1) is a left module over A. If j, l ∈ I, then we ask that ν_{j,l} be a homomorphism from V_{j} into V_{l}, as left modules over A, and that these homomorphisms satisfy the two conditions mentioned in Section 3.2. Thus the direct limit

lim_{⟶} V_{j} = lim_{⟶}^{I}V_{j} (3.9.2)

of the V_{j}’s, j ∈ I, can be defined in the usual way, as a left module over A. If l = (l_{1}, l_{2}) ∈ I, then we get a natural homomorphism (3.9.3) ν_{l} = ν_{(l_{1},l_{2})} from V_{l} = V_{(l_{1},l_{2})} into (3.9.2), as left modules over A, as before. If r = (r_{1}, r_{2}) ∈ I and l ⪯ r, then ν_{(l_{1},l_{2})} = ν_{l} = ν_{r} ∘ ν_{l,r} = ν_{(r_{1},r_{2})} ∘ ν_{l,r}, (3.9.4) by construction.

Let j_{2} ∈ I_{2} be given. If j_{1}, l_{1} ∈ I_{1} and j_{1} ⪯_{1} l_{1}, then put j = (j_{1}, j_{2}) and l = (l_{1}, j_{2}) (3.9.5) for the moment, and observe that j ⪯ l. In this case, ν_{j_{1},l_{1}}^{j_{2}} = ν_{j,l} (3.9.6) is a homomorphism from V_{j} = V_{(j_{1},j_{2})} into V_{l} = V_{(l_{1},j_{2})}, as left modules over A. It is easy to see that these homomorphisms satisfy the usual two conditions in Section 3.2, so that the V_{(j_{1},j_{2})}’s, j_{1} ∈ I_{1}, form a direct system of modules. Let

lim_{⟶}^{I_{1}}V_{(j_{1},j_{2})} (3.9.7)

be the corresponding direct limit, which is a left module over A.

If l_{1} ∈ I_{1}, then we get a natural homomorphism (3.9.8) ν_{l_{1}}^{j_{2}} from V_{(l_{1},j_{2})} into (3.9.7), as left modules over A, as in Section 3.2. If r_{1} ∈ I_{1} and l_{1} ⪯_{1} r_{1}, then ν_{l_{1}}^{j_{2}} = ν_{r_{1}}^{j_{2}} ∘ ν_{l_{1},r_{1}}^{j_{2}}, (3.9.9)

72 CHAPTER 3. MODULES AND TENSOR PRODUCTS, 3

as before.

Let l_{2} ∈ I_{2} be given, with j_{2} ⪯_{2} l_{2}. If j_{1} ∈ I_{1}, then put j = (j_{1}, j_{2}) and l = (j_{1}, l_{2}) (3.9.10) for the moment, and observe that j ⪯ l. Under these conditions, ϕ_{j_{1}}^{j_{2},l_{2}} = ν_{j,l} (3.9.11) is a homomorphism from V_{j} = V_{(j_{1},j_{2})} into V_{l} = V_{(j_{1},l_{2})}, as left modules over A. If l_{1} ∈ I_{1} and j_{1} ⪯_{1} l_{1}, then one can check that ϕ_{l_{1}}^{j_{2},l_{2}} ∘ ν_{j_{1},l_{1}}^{j_{2}} = ν_{j_{1},l_{1}}^{l_{2}} ∘ ϕ_{j_{1}}^{j_{2},l_{2}}. (3.9.12) More precisely, both sides are the same as the homomorphism ν_{(j_{1},j_{2}),(l_{1},l_{2})} (3.9.13) from V_{(j_{1},j_{2})} into V_{(l_{1},l_{2})}.

Of course, V_{(j_{1},l_{2})}, j_{1} ∈ I_{1}, forms a direct system of left modules over A, as before. As in Section 3.4, there is a unique homomorphism

(3.9.14) ϕ^{j_{2},l_{2}} = lim_{⟶}^{I_{1}}ϕ_{j_{1}}^{j_{2},l_{2}} from lim_{⟶}^{I_{1}}V_{(j_{1},j_{2})} into lim_{⟶}^{I_{1}}V_{(j_{1},l_{2})}

such that ϕ^{j_{2},l_{2}} ∘ ν_{l_{1}}^{j_{2}} = ν_{l_{1}}^{l_{2}} ∘ ϕ_{l_{1}}^{j_{2},l_{2}} (3.9.15) for every l_{1} ∈ I_{1}.

One can use the uniqueness of ϕ^{j_{2},l_{2}} to get that these homomorphisms satisfy the two conditions mentioned in Section 3.2. This means that the family of modules of the form (3.9.7), with j_{2} ∈ I_{2}, is a direct system. Thus the corresponding direct limit

lim_{⟶}^{I_{2}}(lim_{⟶}^{I_{1}}V_{(j_{1},j_{2})}) (3.9.16)

can be defined as a left module over A in the usual way.

If l_{2} ∈ I_{2}, then there is a natural homomorphism (3.9.17) ϕ^{l_{2}} from lim_{⟶}^{I_{1}}V_{(j_{1},l_{2})} into (3.9.16), as before. If r_{2} ∈ I_{2} and l_{2} ⪯ r_{2}, then ϕ^{l_{2}} = ϕ^{r_{2}} ∘ ϕ^{l_{2},r_{2}}, (3.9.18) as usual.

Let l = (l_{1}, l_{2}) ∈ I be given, and remember that ν_{l_{1}}^{l_{2}} is a homomorphism from V_{(l_{1},l_{2})} into lim_{⟶}^{I_{1}}V_{(j_{1},l_{2})}, as left modules over A, as in (3.9.8). Thus ξ_{l} = ϕ^{l_{2}} ∘ ν_{l_{1}}^{l_{2}} (3.9.19) defines a homomorphism from V_{l} = V_{(l_{1},L_{2})} into (3.9.16), as left modules over A. Suppose that r = (r_{1}, r_{2}) ∈ I satisfies l ⪯ r, so that

ν_{l,r} = ν_{l_{1},r_{2}}^{r_{2}} ∘ ϕ_{l_{1}}^{l_{2},r_{2}}, (3.9.20)

as in the equality between (3.9.12) and (3.9.13). It follows that

ξ_{r} ∘ ν_{l,r} = ϕ^{r_{2}} ∘ ν_{r_{1}}^{r_{2}} ∘ ν_{l_{1},r_{2}}^{r_{2}} ∘ ϕ_{l_{1}}^{l_{2},r_{2}} = ϕ^{r_{1}} ∘ ν_{l_{1}}^{r_{2}} ∘ ϕ_{l_{1}}^{l_{2},r_{2}}, (3.9.21)

where the second step is as in (3.9.9). This implies that

ξ_{r} ∘ ν_{l,r} = ϕ^{r_{2}} ∘ ϕ^{l_{2},r_{2}} ∘ ν_{l_{1}}^{l_{2}} = ϕ^{l_{2}} ∘ ν_{l_{1}}^{l_{2}} = ξ_{l}, (3.9.22)

using (3.9.15) in the first step, and (3.9.18) in the second step.

As in Section 3.2, there is a unique homomorphism

(3.9.23) ξ from (3.9.2) into (3.9.16),

as left modules over A, such that

ξ ∘ ν_{l} = ξ_{l} (3.9.24)

for every l ∈ I. Here ν_{l} is the natural homomorphism from V_{l} into (3.9.2), as before.

Let l_{2} ∈ I_{2} be given. If l_{1} ∈ I_{1}, so that l = (l_{1}, l_{2}) ∈ I, then ν_{l} = ν_{(l_{1},l_{2})} maps V_{l} = V_{(l_{1}.l_{2})} into (3.9.2), as before. If r_{1} ∈ I_{1} and l_{1} ⪯ r_{1}, then ν_{l_{1},r_{1}}^{l_{2}} maps V_{(l_{1},l_{2})} into V_{(r_{1},l_{2})}, as in (3.9.6). Note that

ν_{(l_{1},l_{2})} = ν_{(r_{1},l_{2})} ∘ ν_{l_{1},r_{1}}^{l_{2}}, (3.9.25)

as in (3.9.4).

Using ν_{(l_{1},l_{2})}, l_{1} ∈ I_{1}, we get a homomorphism

(3.9.26) η^{l_{2}} from lim_{⟶}^{I_{1}}V_{(j_{1},l_{2})} into (3.9.2),

as left modules over A, as in Section 3.2. This homomorphism is characterized by the property that η^{l_{2}} ∘ ν_{l_{1}}^{l_{2}} = ν_{(l_{1},l_{2})} (3.9.27)

for every l_{1} ∈ I_{1}. Here ν_{l_{1}}^{l_{2}} is the natural homomorphism from V_{(l_{1},l_{2})} into

lim_{⟶}^{I_{1}}V_{(j_{1},l_{2})}, as in (3.9.8).

Suppose that r_{2} ∈ I_{2} satisfies l_{2} ⪯_{2} r_{2}, and let ϕ^{l_{2},r_{2}} be as in (3.9.14). We would like to verify that η^{r_{2}} ∘ ϕ^{l_{2},r_{2}} = η^{l_{2}}. (3.9.28)

To do this, observe that

η^{r_{2}} ∘ ϕ^{l_{2},r_{2}} ∘ ν_{l_{1}}^{l_{2}} = η^{r_{2}} ∘ ν_{l_{1}}^{r_{2}} ∘ ϕ_{l_{1}}^{l_{2},r_{2}}, (3.9.29) by (3.9.15). This implies that

η^{r_{2}} ∘ ϕ^{l_{2},r_{2}} ∘ ν_{l_{1}}^{l_{2}} = ν_{(l_{1},r_{2})} ∘ ϕ_{l_{1}}^{l_{2},r_{2}}, (3.9.30)

by (3.9.27). Using the definition (3.9.11) of ϕ_{l_{1}}^{l_{2},r_{2}}, we get that

η^{r_{2}} ∘ ϕ^{l_{2},r_{2}} ∘ ν_{l_{1}}^{l_{2}} = ν_{(l_{1},r_{2})} ∘ ν_{(l_{1},l_{2}),(l_{1},r_{2})} = ν_{(l_{1},l_{2})}. (3.9.31)

Here the second step is as in (3.9.4). It follows that (3.9.28) holds, because η^{l_{2}} is uniquely determined by (3.9.27).

Thus we can use the η^{l_{2}}’s, l_{2} ∈ I_{2}, to get a homomorphism

(3.9.32) η from (3.9.16) into (3.9.2),

as left modules over A, as in Section 3.2. This homomorphism is uniquely determined by the property that

η ∘ ϕ^{l_{2}} = η^{l_{2}} (3.9.33)

for each l_{2} ∈ I_{2}, where ϕ^{l_{2}} is as in (3.9.17). We would like to check that the homomorphism ξ from (3.9.23) and η are inverses of each other.

To show that η ∘ ξ is the identity mapping on (3.9.2), it suffices to verify that η ∘ ξ ∘ ν_{l} = ν_{l} (3.9.34)

on V_{l} for every l = (l_{1}, l_{2}) ∈ I. Note that

η ∘ ξ ∘ ν_{l} = η ∘ ξ_{l} = η ∘ ϕ^{l_{2}} ∘ ν_{l_{1}}^{l_{2}}, (3.9.35)

using (3.9.24) in the first step, and (3.9.19) in the second step. Thus

η ∘ ξ ∘ ν_{l} = η^{l_{2}} ∘ ν_{l_{1}}^{l_{2}} = ν_{(l_{1},l_{2})} = ν_{l}, (3.9.36)

as desired, using (3.9.33) in the first step, and (3.9.27) in the second step.

To show that ξ ∘ η is the identity mapping on (3.9.16), it suffices to check that ξ ∘ η ∘ ϕ^{l_{2}} = ϕ^{l_{2}} (3.9.37)

on lim_{⟶}^{I_{1}}V_{(j_{1},l_{2})} for every l_{2} ∈ I_{2}, where ϕ^{l_{2}} is as in (3.9.17). Equivalently, this means that ξ ∘ η^{l_{2}} = ϕ^{l_{2}} (3.9.38)

on lim_{⟶}^{I_{1}}V_{(j_{1},l_{2})}, because of (3.9.33). In order to show this, it suffices to verify that ξ ∘ η^{l_{2}} ∘ ν_{l_{1}}^{l_{2}} = ϕ^{l_{2}} ∘ ν_{l_{1}}^{l_{2}} (3.9.39)

on V_{(l_{1},l_{2})} for every l_{1} ∈ I_{1}, where ν_{l_{1}}^{l_{2}} is as in (3.9.8). Observe that

ξ ∘ η^{l_{2}} ∘ ν_{l_{1}}^{l_{2}} = ξ ∘ ν_{l} = ξ_{l}, (3.9.40)

using (3.9.27) in the first step, and (3.9.24) in the second step. Thus (3.9.39) follows from the definition (3.9.19) of ξ_{l}, as desired.

Of course, there are analogous statements for right modules over A.

3.10 Another property of compositions

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V_{1}, V_{2}, and V_{3} be right modules over A, and let W_{1}, W_{2}, and W_{3} be left modules over A. Suppose that V_{j} ⨂_{A} W_{l} is a tensor product of V_{j} and W_{l} over A for each j, l = 1, 2, 3.

Let ϕ_{1}, ψ_{1} be homomorphisms from V_{1}, W_{1} into V_{2}, W_{2}, respectively, and let ϕ_{2}, ψ_{2} be homomorphisms from V_{2}, W_{2} into V_{3}, W_{3}, respectively, as modules over A. If j = 1 or 2 and l = 1, 2, or 3, then there is a unique homomorphism

(3.10.1) Φ_{j,l} from V_{j} ⨂_{A}W_{l} into V_{j+1} ⨂_{A}W_{l},

as modules over k, such that

Φ_{j,l}(v_{j} ⊗ w_{l}) = ϕ_{j}(v_{j}) ⊗ w_{l} (3.10.2)

for every v_{j} ∈ V_{j} and w_{l} ∈ W_{l}. Equivalently, Φ_{j,l} is obtained from ϕ_{j} and the identity mapping on W_{l} as in Section 1.9. Similarly, if j = 1, 2, or 3 and l = 1 or 2, then there is a unique homomorphism

(3.10.3) Ψ_{j,l} from V_{j} ⨂_{A}W_{l} into V_{j} ⨂_{A}W_{l+1},

as modules over k, such that

Ψ_{j,l}(v_{j} ⊗ w_{l}) = v_{j} ⊗ ψ_{l}(w_{l}) (3.10.4)

for every v_{j} ∈ V_{j} and w_{l} ∈ W_{l}. This is the same as the homomorphism obtained from ψ_{l} and the identity mapping on V_{j}, as before.

In the same way, we can use ϕ_{2} and ψ_{2} to get a unique homomorphism

(3.10.5) Θ_{2} from V_{2} ⨂_{A}W_{2} into V_{3} ⨂_{A}W_{3},

as modules over k, such that

Θ_{2}(v_{2} ⊗ w_{2}) = ϕ_{2}(v_{2}) ⊗ ψ_{2}(w_{2}) (3.10.6)

for every v_{2} ∈ V_{2} and w_{2} ∈ W_{2}. Note that

Θ_{2} = Ψ_{3,2} ∘ Φ_{2,2} = Φ_{2,3} ∘ Ψ_{2,2}. (3.10.7)

If ϕ_{2} ∘ ϕ_{1} = 0, (3.10.8)

then Φ_{2,2} ∘ Φ_{1,2} = 0. (3.10.9)

This is because Φ_{2,2} ∘ Φ_{1,2} is the same as the homomorphism from V_{1} ⨂_{A} W_{2} into V_{3} ⨂_{A} W_{2} obtained from ϕ_{2} ∘ ϕ_{1} and the identity mapping on W_{2}. This implies that Θ_{2,2} ∘ Φ_{1,2} = 0, (3.10.10) by (3.10.7). It follows that

Φ_{1,2}(V_{1} ⨂_{A}W_{2}) ⊆ ker Θ_{2,2} (3.10.11)

in this case.

Similarly, if ψ_{2} ∘ ψ_{1} = 0, (3.10.12)

then Ψ_{2,2} ∘ Ψ_{2,1} = 0. (3.10.13)

This implies that Θ_{2,2} ∘ Ψ_{2,1} = 0, (3.10.14)

so that Ψ_{1,2}(V_{2} ⨂_{A}W_{1}) ⊆ ker Θ_{2,2}, (3.10.15)

as before.

Suppose now that

V_{1} ⟶^{ϕ_{1}} V_{2} ⟶^{ϕ_{2}} V_{3} ⟶ 0 (3.10.16)

and

W_{1} ⟶^{ψ_{1}} W_{2} ⟶^{ψ_{2}} W_{3} ⟶ 0 (3.10.17)

are exact sequences, so that ϕ_{2}, ψ_{2} are surjective, and

ϕ_{1}(V_{1}) = ker ϕ_{2}, ψ_{1}(W_{1}) = ker ψ_{2}. (3.10.18)

In this case, it is well known that we get an exact sequence

(V_{1} ⨂_{A}W_{2}) ⨁ (V_{2} ⨂_{A}W_{1}) ⟶ V_{2} ⨂_{A}W_{2} (3.10.19)

⟶^{Θ_{2,2}} V_{3} ⨂_{A}W_{3} ⟶ 0,

using the mapping obtained from Φ_{1,2} and Ψ_{2,1} in the first step. This corresponds to part of Proposition 4.3 on p24f of [50], which is stated more abstractly and under slightly different conditions, as mentioned at the top of p26 of [50].

The exactness of (3.10.19) means that Θ_{2,2} is surjective, and that

Φ_{1,2}(V_{1} ⨂_{A}W_{2}) + Ψ_{2,1}(V_{2} ⨂_{A}W_{1}) = ker Θ_{2,2}. (3.10.20)

Of course, Φ_{1,2}(V_{1} ⨂_{A}W_{2}) + Ψ_{2,1}(V_{2} ⨂_{A}W_{1}) ⊆ ker Θ_{2,2}, (3.10.21)

by (3.10.11) and (3.10.15).

To get the exactness of (3.10.19), the statement and argument in [50] uses the right exactness of the tensor product in each factor, as in Section 2.5. The surjectivity of Θ_{2,2} can be obtained from the surjectivity of Φ_{2,2} and Ψ_{3,2}, or from the surjectivity of Φ_{2,3} and Ψ_{2,2}, by (3.10.7). The surjectivity of these mappings follows from the surjectivity of ϕ_{2} and ψ_{2}, as before. If an element u_{2,2} of V_{2} ⨂_{A} W_{2} is in the kernel of Θ_{2,2}, then

Ψ_{2,2}(u_{2,2}) ∈ ker Φ_{2,3}, (3.10.22)

by (3.10.7). This implies that there is an element u_{1,3} of V_{1} ⨂_{A} W_{3} such that

Ψ_{2,2}(u_{2,2}) = Φ_{1,3}(u_{1,3}), (3.10.23)

as in Section 2.5.

Note that Ψ_{1,2} is surjective, because ψ_{2} is surjective, as in Section 1.9. This means that there is an element u_{1,2} of V_{1} ⨂_{A} W_{2} such that

Ψ_{1,2}(u_{1,2}) = u_{1,3}. (3.10.24)

It is easy to see that

Φ_{1,3} ∘ Ψ_{1,2} = Ψ_{2,2} ∘ Φ_{1,2}, (3.10.25)

and more precisely that both sides correspond to the mapping from V_{1} ⨂_{A} W_{2} into V_{2} ⨂_{A} W_{3} obtained from ϕ_{1} and ψ_{2} in the usual way. Thus

Φ_{1,3}(u_{1,3}) = Φ_{1,3}(Ψ_{1,2}(u_{1,2})) = Ψ_{2,2}(Φ_{1,2}(u_{1,2})). (3.10.26)

This means that Ψ_{2,2}(u_{2,2}) = Ψ_{2,2}(Φ_{1,2}(u_{1,2})), (3.10.27)

by (3.10.23).

Equivalently, u_{2,2} − Φ_{1,2}(u_{1,2}) ∈ ker Ψ_{2,2}. (3.10.28)

It follows that there is an element u_{2,1} of V_{2} ⨂_{A} W_{1} such that

u_{2,2} − Φ_{1,2}(u_{1,2}) = Ψ_{2,1}(u_{2,1}), (3.10.29)

as in Section 2.5. This shows that u_{2,2} is contained in the left side of (3.10.20), as desired.

Alternatively, we can use an argument like the one in Section 2.5. The surjectivity of Θ_{2,2} follows from the surjectivity of ϕ_{2} and ψ_{2}, as in Section 1.9. The left side of (3.10.20) is a submodule of V_{2} ⨂_{A} W_{2}, as a module over k, so that the quotient

Y = (V_{2} ⨂_{A}W_{2})/(Φ_{1,2}(V_{1} ⨂_{A}W_{2}) + Ψ_{2,1}(V_{2} ⨂_{A}W_{1})) (3.10.30)

is defined as a module over k. Let q_{Y} be the natural quotient mapping from V_{2} ⨂_{A} W_{2} onto Y.

Because of (3.10.21), there is a unique homomorphism Θ̃_{2,2} from Y into V_{3} ⨂_{A} W_{3}, as modules over k, such that

Θ̃_{2,2} ∘ q_{Y} = Θ_{2,2}. (3.10.31)

It suffices to show that Θ̃_{2,2} is an isomorphism.

If v_{2} ∈ V_{2} and w_{2} ∈ W_{2}, then

q_{Y} (v_{2} ⊗ w_{2}) (3.10.32)

is an element of Y. One can check that (3.10.32) only depends on ϕ_{2}(v_{2}) and ψ_{2}(w_{2}), because of (3.10.18) and (3.10.30). This leads to a mapping from V_{3}×W_{3} into Y, with (ϕ_{2}(v_{2}), ψ_{2}(w_{2})) ↦ q_{Y} (v_{2} ⊗ w_{2}) (3.10.33)

for every v_{2} ∈ V_{2} and w_{2} ∈ W_{2}, because ϕ_{2} and ψ_{2} are surjective, by hypothesis. One can also verify that this mapping is bilinear over k.

If a ∈ A, v_{2} ∈ V_{2}, and w_{2} ∈ W_{2}, then it is easy to see that

(ϕ_{2}(v_{2}) · a, ψ_{2}(w_{2})) and (ϕ_{2}(v_{2}), a · ψ_{2}(w_{2})) (3.10.34)

are mapped to the same element of Y by the mapping described in the preceding paragraph. This implies that there is a unique homomorphism from V_{3} ⨂_{A} W_{3} into Y, as modules over k, with

ϕ_{2}(v_{2}) ⊗ ψ_{2}(w_{2}) ↦ q_{Y} (v_{2} ⊗ w_{2}) (3.10.35)

for every v_{2} ∈ V_{2} and w_{2} ∈ W_{2}.

One can check that this homomorphism is the inverse of Θ̃_{2,2}. In particular, Θ̃_{2,2} is an isomorphism, as desired.

3.11 Compositions and Hom(·, ·)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V_{j} and W_{l} be all left or all right modules over A, for j, l = 1, 2, 3. Suppose that ϕ_{j}, ψ_{j} are homomorphisms from V_{j}, W_{j} into V_{j+1}, W_{j+1}, respectively, as modules over A, for j = 1, 2.

If j = 1 or 2 and l = 1, 2, or 3, then

Φ_{j,l}(α_{j+1,l}) = α_{j+1,l} ∘ ϕ_{j} (3.11.1)

defines a homomorphism

(3.11.2) from Hom_{A}(V_{j+1}, W_{l}) into Hom_{A}(V_{j}, W_{l}),

as modules over k. Similarly, if j = 1, 2, or 3 and l = 1 or 2, then

Ψ_{j,l}(α_{j,l}) = ψ_{l} ∘ α_{j,l} (3.11.3)

defines a homomorphism

(3.11.4) from Hom_{A}(V_{j}, W_{l}) into Hom_{A}(V_{j}, W_{l+1}), 3.11. COMPOSITIONS AND HOM(·, ·) 79 as modules over k.

If α_{3,1} is a homomorphism from V_{3} into W_{1}, as modules over A, then Θ(α_{3,1}) = ψ_{1} ∘ α_{3,1} ∘ ϕ_{2} (3.11.5) is a homomorphism from V_{2} into W_{2}, as modules over A. This defines a homomorphism (3.11.6) from Hom_{A}(V_{3}, V_{1}) into Hom_{A}(V_{2}, W_{2}), as modules over k. Equivalently, Θ = Ψ_{2,1} ∘ Φ_{2,1} = Φ_{2,2} ∘ Ψ_{3,1}. (3.11.7)

If ϕ_{2} ∘ ϕ_{1} = 0, (3.11.8) then Φ_{1,2} ∘ Φ_{2,2} = 0, (3.11.9) because Φ_{1,2} ∘ Φ_{2,2} is the same as the homomorphism from Hom_{A}(V_{3}, W_{2}) into Hom_{A}(V_{1}, W_{2}) corresponding to composition with ϕ_{2} ∘ ϕ_{1}. This implies that Φ_{2,1} ∘ Θ = 0, (3.11.10) by (3.11.7). This means that (3.11.11) Θ(Hom_{A}(V_{3}, W_{1})) ⊆ ker Φ_{1,2} when (3.11.8) holds.

Similarly, if ψ_{2} ∘ ψ_{1} = 0, (3.11.12) then Ψ_{2,2} ∘ Ψ_{2,1} = 0, (3.11.13) because Ψ_{2,2} ∘ Ψ_{2,1} is the same as the homomorphism from Hom_{A}(V_{2}, W_{1}) into Hom_{A}(V_{2}, W_{3}) corresponding to composition with ψ_{2} ∘ ψ_{1}. This implies that Ψ_{2,2} ∘ Θ = 0, (3.11.14) by (3.11.7). Thus (3.11.15) Θ(Hom_{A}(V_{3}, W_{1})) ⊆ ker Ψ_{2,2} when (3.11.12) holds.

Suppose from now on in this section that

V_{1} ⟶^{ϕ_{1}} V_{2} ⟶^{ϕ_{2}} V_{3} ⟶ 0 (3.11.16)

and

0 ⟶ W_{1} ⟶^{ψ_{1}} W_{2} ⟶^{ψ_{1}} W_{3} (3.11.17)

are exact sequences. This means that ϕ_{2} is surjective, ψ_{1} is injective, and

ϕ_{1}(V_{1}) = ker ϕ_{2}, ψ_{1}(W_{1}) = ker ψ_{2}. (3.11.18)

It is well known that

0 ⟶ Hom_{A}(V_{3}, V_{1}) ⟶^{Θ} Hom_{A}(V_{2}, W_{2}) (3.11.19)

⟶ Hom_{A}(V_{1}, W_{2}) ⨁ Hom_{A}(V_{2}, W_{3})

is an exact sequence in this case, using the mapping obtained from Φ_{1,2} and Ψ_{2,2} in the last step. This corresponds to part of Proposition 4.3a on p25 of [50], which is stated more abstractly and under slightly different conditions, as mentioned at the top of p26 of [50].

The exactness of (3.11.19) means that Θ is injective, and that

(3.11.20) Θ(Hom_{A}(V_{3}, W_{1})) = (ker Φ_{1,2}) ∩ (ker Ψ_{2,2}).

Note that (3.11.21) Θ(Hom_{A}(V_{3}, W_{1})) ⊆ (ker Φ_{1,2}) ∩ (ker Ψ_{2,2}),

by (3.11.11) and (3.11.15).

The statement and argument in [50] use the left exactness of Hom_{A}(·, ·) in each variable, as in Sections 2.1 and 2.3, to get the exactness of (3.11.19). We can also use arguments like those in Sections 2.1 and 2.3 more directly here. In particular, the injectivity of Θ can be obtained from the injectivity of Φ_{2,1} and Ψ_{2,1}, or from the injectivity of Φ_{2,2} and Ψ_{3,1}, by (3.11.7). The injectivity of these mappings follows from the surjectivity of ϕ_{2} and the injectivity of ψ_{1}, as before. The injectivity of Θ can be obtained directly from the surjectivity of ϕ_{2} and the injectivity of ψ_{1} as well, using the definition (3.11.5) of Θ.

Let α_{2,2} be a homomorphism from V_{2} into W_{2}, as modules over A, with

(3.11.22) α_{2,2} ∈ (ker Φ_{1,2}) ∩ (ker Ψ_{2,2}).

The condition that α_{2,2} ∈ ker Φ_{1,2} means that

α_{2,2} ∘ ϕ_{1} = Φ_{1,2}(α_{2,2}) = 0, (3.11.23)

which is the same as saying that

ker ϕ_{2} = ϕ_{1}(V_{1}) ⊆ ker α_{2,2}. (3.11.24)

This implies that there is a homomorphism α_{3,2} from V_{3} into W_{2}, as modules over A, such that Φ_{2,2}(α_{3,2}) = α_{3,2} ∘ ϕ_{2} = α_{2,2}, (3.11.25)

because ϕ_{2}(V_{2}) = V_{3}, by hypothesis.

Similarly, the condition that α_{2,2} ∈ ker Ψ_{2,2} means that

ψ_{2} ∘ α_{2,2} = Ψ_{2,2}(α_{2,2}) = 0, (3.11.26) which is the same as saying that

α_{2,2}(V_{2}) ⊆ ker ψ_{2} = ψ_{1}(W_{1}). (3.11.27)

This implies that

α_{3,2}(V_{3}) = α_{2,3}(ϕ_{2}(V_{2})) = α_{2,2}(V_{2}) ⊆ ψ_{1}(W_{1}). (3.11.28)

It follows that there is a homomorphism α_{3,1} from V_{3} into W_{1}, as modules over A, such that Ψ_{3,1}(α_{3,1}) = ψ_{1} ∘ α_{3,1} = α_{3,2}, (3.11.29)

because ψ_{1} is injective, by hypothesis. Combining this with (3.11.25), we obtain that Θ(α_{3,1}) = ψ_{1} ∘ α_{3,1} ∘ ϕ_{2} = α_{3,2} ∘ ϕ_{2} = α_{2,2}, (3.11.30)

as desired.

3.12 Finitely-generated submodules

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a left or right module over A. If V_{1}, V_{2} are submodules of V, then it is easy to see that

V_{1} + V_{2} = {v_{1} + v_{2} : v_{1} ∈ V_{1}, v_{2} ∈ V_{2}} (3.12.1)

is a submodule of V as well. If V_{1}, V_{2} are finitely generated as modules over A, then V_{1} + V_{2} is finitely generated as a module over A too.

The collection of all finitely generated submodules of V is partially ordered by inclusion, and in fact it is a directed set, by the remarks in the preceding paragraph. Note that every element of V is contained in a finitely-generated submodule of V, because the action of A on any element of V defines a submodule of V. Thus V is the union of all of its finitely-generated submodules, so that V may be considered as the direct limit of its finitely-generated submodules. This corresponds to part of Exercise 17 on p33 of [8].

Suppose now that V is a right module over A, let W be a left module over A, and let V ⨂_{A} W be a tensor product of V and W. Also let V_{0}, W_{0} be submodules of V, W, respectively, as modules over A, and let V_{0} ⨂ W_{0} be a tensor product of V_{0} and W_{0} over A. If v ∈ V and w ∈ W, then we let v ⊗ w be the corresponding element of V ⨂_{A} W, as usual. Similarly, if v ∈ V_{0} and w ∈ W_{0}, then let v ⊗_{0} w be the corresponding element of V_{0} ⨂_{A} W_{0}. As in Section 1.9, there is a unique homomorphism from V_{0} ⨂_{A} W_{0} into V ⨂_{A} W, as modules over k, with v ⊗_{0} w ↦ v ⊗ w (3.12.2)

for every v ∈ V_{0} and w ∈ W_{0}.

Let n be a positive integer, and suppose that v_{1}, …, v_{n} ∈ V and w_{1}, …, w_{n} in W satisfy

∑_{j=1}^{n} v_{j} ⊗ w_{j} = 0 (3.12.3)

in V ⨂_{A} W. If v_{1}, …, v_{n} ∈ V_{0} and w_{1}, …, w_{n} ∈ W_{0}, (3.12.4)

then

∑_{j=1}^{n} v_{j} ⊗_{0} w_{j} (3.12.5)

defines an element of V_{0} ⨂_{A} W_{0}. Of course, (3.12.3) holds when (3.12.5) is equal to 0 in V_{0} ⨂_{A} W_{0}.

In fact, there are finitely-generated submodules V_{0}, W_{0} of V, W, respectively, as modules over A, such that (3.12.4) holds and (3.12.5) is equal to 0 in V_{0} ⨂_{A} W_{0}, as in Corollary 2.13 on p25 of [8]. The proof uses the standard way of constructing tensor products as quotients. Thus (3.12.3) says that a certain formal expression involving v_{1}, …, v_{n} and w_{1}, …, w_{n} can be expressed in terms of finitely many elements of V and W in a suitable way. It suffices to use finitely-generated submodules V_{0}, W_{0} of V, W, respectively, such that (3.12.4) holds, and which contain the finitely many elements of V and W just mentioned, to get that (3.12.5) is equal to 0 in V_{0} ⨂_{A} W_{0}.

Alternatively, V ⨂_{A} W may be obtained as a direct limit of tensor products of finitely-generated submodules of V and W, as in Section 3.8. This uses the fact that V and W may be obtained as direct limits of finitely-generated submodules, as before. One can use this to get that there are finitely-generated submodules V_{0}, W_{0} of V, W, respectively, such that (3.12.4) holds and (3.12.5) is equal to 0 in V_{0} ⨂_{A} W_{0}, as in Section 3.2.

One could obtain the same conclusion using direct limits in each factor separately, as in Section 3.6. That is, one can first obtain V ⊗_{A} W as a direct limit of tensor products of finitely-generated submodules of V with W, as in Section 3.6. This implies that there is a finitely-generated submodule V_{0} of V such that v_{1}, …, v_{n} ∈ V_{0} and (3.12.5) is equal to 0 in V_{0} ⨂_{A} W_{0}, with W_{0} = W. Similarly, one can obtain V_{0} ⨂_{A} W as a direct limit of tensor products of V_{0} with finitely-generated submodules of W. This implies that there is a finitelygenerated submodule W_{0} of W such that (3.12.4) holds, and (3.12.5) is equal to 0 in V_{0} ⨂_{A} W_{0}.

3.13 An interesting homomorphism

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, W be both left or both right modules over A, and let V_{0}, W_{0} be submodules of V, W, respectively. Suppose that ϕ is a homomorphism from V into W, as modules over A, with ϕ(V_{0}) ⊆ W_{0}. (3.13.1)

Thus the restriction ϕ_{0} of ϕ to V_{0} may be considered as a homomorphism from V_{0} into W_{0}, as modules over A. Note that

ker ϕ_{0} = (ker ϕ) ∩ V_{0}. (3.13.2)

Of course, the quotients V/V_{0}, W/W_{0} may be defined as modules over A in the usual way. Let q_{V}, q_{W} be the natural quotient mappings from V, W onto V/V_{0}, W/W_{0}, respectively. Because of (3.13.1), there is a unique homomorphism ϕ_{q} from V/V_{0} into W/W_{0}, as modules over A, such that

ϕ_{q} ∘ q_{V} = q_{W} ∘ ϕ, (3.13.3)

as homomorphisms from V into W/W_{0}. In particular,

q_{V} (ker ϕ) ⊆ ker ϕ_{q}. (3.13.4)

Because ϕ(V) is a submodule of W, the quotient

W/ϕ(V) (3.13.5)

is defined as a module over A. This is known as the cokernel of ϕ, as on p19 of [8], and p3 of [50]. Let q_{ϕ} be the natural quotient mapping from W onto W/ϕ(V). Similarly, W_{0}/ϕ_{0}(V_{0}) = W_{0}/ϕ(V_{0}) (3.13.6)

is the cokernel of ϕ_{0}, as a homomorphism from V_{0} into W_{0}, and we let q_{ϕ_{0}} be the natural quotient mapping from W_{0} onto W_{0}/ϕ_{0}(V_{0}).

Suppose that v ∈ V and

q_{V} (v) ∈ ker ϕ_{q}. (3.13.7)

This implies that q_{W} (ϕ(v)) = ϕ_{q}(q_{V} (v)) = 0, (3.13.8)

so that ϕ(v) ∈ W_{0}. (3.13.9)

We would like to put Δ(q_{V} (v)) = q_{ϕ_{0}}(ϕ(v)), (3.13.10)

which is an element of W_{0}/ϕ_{0}(V_{0}). If u ∈ V and u − v ∈ V_{0}, so that q_{V} (u) = q_{V} (v), then ϕ(u) − ϕ(v) = ϕ(u − v) ∈ ϕ_{0}(V_{0}), (3.13.11)

and thus q_{ϕ_{0}}(ϕ(u)) = q_{ϕ_{0}}(ϕ(v)). This shows that the right side of (3.13.10) does not depend on the choice of v.

It is easy to see that Δ defines a homomorphism from ker ϕ_{q} into W_{0}/ϕ_{0}(V_{0}), as modules over A. Let us check that

ker Δ = q_{V} (ker ϕ). (3.13.12)

If v ∈ ker ϕ, then (3.13.7) holds, by (3.13.4), and (3.13.10) is equal to 0. This shows that q_{V} (ker ϕ) is contained in the kernel of Δ. If v ∈ V satisfies (3.13.7), and (3.13.10) is equal to 0, then

ϕ(v) ∈ ϕ_{0}(V_{0}) = ϕ(V_{0}). (3.13.13)

This means that there is a u ∈ V_{0} such that ϕ(v) = ϕ(u), and thus v −u ∈ ker ϕ. It follows that q_{V} (v) = q_{V} (v − u) ∈ q_{V} (ker ϕ), (3.13.14)

which shows that the kernel of Δ is contained in q_{V} (ker ϕ).

Let us verify that

Δ(ker ϕ_{q}) = q_{ϕ_{0}}(ϕ(V) ∩ W_{0}). (3.13.15)

The fact that Δ maps ker ϕ_{q} into the right side follows from the definition of Δ. If v ∈ V and ϕ(v) ∈ W_{0}, then

ϕ_{q}(q_{V} (v)) = q_{W} (ϕ(v)) = 0, (3.13.16)

so that q_{V} (v) ∈ ker ϕ_{q}. This means that Δ(q_{V} (v)) is as in (3.13.10), so that the right side of (3.13.15) is contained in the image of Δ.

There is a natural homomorphism

(3.13.17) from W_{0}/ϕ_{0}(V_{0}) into W/ϕ(V),

as modules over A, obtained from the obvious inclusion mapping from W_{0} into W. This also uses the fact that ϕ_{0}(V_{0}) = ϕ(V_{0}) is contained in ϕ(V). It is easy to see that the kernel of this homomorphism is equal to the right side of (3.13.15).

The quotient (W/W_{0})/ϕ_{q}(V/V_{0}) (3.13.18)

is defined as a module over A, because ϕ_{q}(V/V_{0}) is a submodule of W/W_{0}. Of course, ϕ_{q}(V/V_{0}) = ϕ_{q}(q_{V} (V)) = q_{W} (ϕ(V)), (3.13.19)

using (3.13.3) in the second step. Thus (3.13.18) is the same as

(W/W_{0})/q_{W} (ϕ(V)). (3.13.20)

There is a natural homomorphism

(3.13.21) from W/ϕ(V) onto (W/W_{0})/q_{W} (ϕ(V)),

as modules over A, which is induced by q_{W}. The kernel of this homomorphism is equal to q_{ϕ}(W_{0}). (3.13.22)

This is the same as the image of W_{0}/ϕ_{0}(V) in W/ϕ(V) under the natural homomorphism as in (3.13.17).

The remarks in this section correspond to Proposition 2.10 and its proof on p23 of [8], and are related to Lemmas 3.2 and 3.3 on p40 of [50].

3.14 An abstract construction

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. If V is a left module over A, then we would like to find a left module D(V) over V that contains V as a submodule, and has the following property:

(3.14.1) if I is a left ideal in A, and ϕ is a homomorphism from I into V, then there is a w ∈ D(V) such that ϕ(a) = a · w for every a ∈ I.

Here I may be considered as a left module over A, and ϕ is supposed to be a homomorphism from I into V, as left modules over A. This corresponds to the first part of the proof of Theorem 3.3 on p9 of [50].

Let L_{A} be the collection of all left ideals in A, and for each I ∈ L_{A}, let Hom_{A}(I, V) be the space of all homomorphisms from I into V, as left modules over A. This is a module over k, and we let

(3.14.2) Φ_{0}(I) be a subset of Hom_{A}(I, V) that generates Hom_{A}(I, V),

as a module over k. Put

Φ_{0} = ⋃_{I∈L_{A}}({I} × Φ_{0}(I)), (3.14.3)

and let Z_{0} be a free left module over A with a basis consisting of elements denoted z_{(I,ϕ)}, with (I, ϕ) ∈ Φ_{0}. Thus

Z_{1} = V ⨁ Z_{0} (3.14.4)

is a left module over A.

Let (I, ϕ) ∈ Φ_{0} be given. If x ∈ I, then ϕ(x) ∈ V, and

(ϕ(x), −x z_{(I,ϕ)}) (3.14.5)

is an element of Z_{1}. If a ∈ A, then a x ∈ I, and

a · (ϕ(x), −x z_{(I,ϕ)}) = (a · ϕ(x), −a x z_{(I,ϕ)}) = (ϕ(a x), −a x z_{(I,ϕ)}). (3.14.6)

This implies that the collection of elements of Z_{1} of the form (3.14.5) with x ∈ I is a submodule of Z_{1}, as a left module over A.

Let Y be the submodule of Z_{1}, as a left module over A, generated by elements of the form (3.14.5), with (I, ϕ) ∈ Φ_{0} and x ∈ I. We would like to take

D(V) = Z_{1}/Y, (3.14.7)

as a left module over A.

Suppose that v ∈ V and (v, 0) ∈ Y. (3.14.8)

This means that there are finitely many distinct elements (I_{j}, ϕ_{j}), 1 ≤ j ≤ n, of Φ_{0}, and x_{j} ∈ I_{j} for each j = 1, …, n, such that

(v, 0) = ∑_{j=1}^{n}(ϕ_{j}(x_{j}), −x_{j} z_{(I_{j},ϕ_{j})}). (3.14.9)

It follows that

∑_{j=1}^{n} ϕ_{j}(x_{j}) = v (3.14.10)

and

∑_{j=1}^{n} x_{j} z_{(I_{j},ϕ_{j})} = 0 (3.14.11)

in Z_{0}. Because the (I_{j}, ϕ_{j})’s are distinct elements of Φ_{0}, we get that x_{j} = 0 for each j = 1, …, n. This implies that v = 0, by (3.14.10).

Consider the mapping from V into D(V) that sends v ∈ V to the image of (v, 0) under the quotient mapping from Z_{1} onto D(V). This is a homomorphism from V into D(V), as left modules over A, which is injective, by the remarks in the preceding paragraph. Thus we may identify V with its image in D(V) under this mapping.

Let (I, ϕ) ∈ Φ_{0} be given, so that (0, z_{(I,ϕ)}) is an element of Z_{1}. Let w be the image of (0, z_{(I,ϕ)}) under the quotient mapping from Z_{1} onto D(V). If a ∈ I, then ϕ(a) ∈ V is identified with the image of (ϕ(a), 0) in the quotient, as before. This is the same as the image of

(0, a z_{(I,ϕ)}) = a · (0, z_{(I,ϕ)}) (3.14.12)

in the quotient, by construction. Thus ϕ(a) is identified with a · w in D(V).

This shows that (3.14.1) holds when ϕ ∈ Φ_{0}(I). If ϕ is any homomorphism from I into V, as left modules over A, then ϕ can be expressed as a linear combination of elements of Φ_{0}(I) with coefficients in k. Using this, one can get the analogous statement for ϕ, by reducing to the previous case.

3.15 Another abstract construction

Let us continue with the same notation and hypotheses as in the previous section. We would like to show that there is an injective left module over A that contains V as a submodule, as in Theorem 3.3 on p9 of [50].

Let B be an infinite set whose cardinality is strictly larger than the cardinality of A, and which is as small as possible with these two properties. Suppose that B is well ordered by ⪯. We would like B to be “minimal” as a well-ordered set with this cardinality, in the sense that for each β ∈ B, the cardinality of

{α ∈ B : α ⪯ β, α ≠ β} (3.15.1) is strictly less than the cardinality of B. If B does not have this property already, then there is a smallest β ∈ B such that (3.15.1) has the same cardinality as B. In this case, we can replace B with (3.15.1), using the restriction of ⪯ to this set.

If β ∈ B, then we would like to define a left module Q_{β}(V) over A as follows. If β is the smallest element of B, then we take Q_{β}(V) = V. Otherwise, (3.15.1) is not empty, and we suppose that Q_{α}(V) has been defined for every element α of this set. Suppose for the moment that (3.15.1) has a maximal element α, which means that β is the minimal element of B with α ⪯ β and α ≠ β. Under these conditions, we take

Q_{β}(V) = D(Q_{α}(V)). (3.15.2)

If (3.15.1) has no maximal element, then we take

Q_{β}(V) = ⋃{Q_{α}(V) : α ∈ B, α ⪯ β, α ≠ β}. (3.15.3)

More precisely, if α_{1}, α_{2} ∈ B, α_{1} ⪯ α_{2}, and Q_{α_{1}}(V), Q_{α_{2}}(V) have already been defined, then Q_{α_{1}}(V) should be a submodule of Q_{α_{2}}(V). This implies that (3.15.3) is a left module over A too, that contains Q_{α}(V) as a submodule when α is an element of (3.15.1). Thus we can define Q_{β}(V) for every β ∈ B, and we put

Q(V) = ⋃_{β∈B} Q_{β}(V). (3.15.4)

This is a left module over A that contains Q_{β}(V) as a submodule for every β ∈ B, as before.

In particular, Q(V) contains V as a submodule, and we would like to show that Q(V) is injective as a left module over A. To do this, let I be a left ideal in A, and let ϕ be a homomorphism from I into Q(V), as left modules over A. If x ∈ I, then we can choose α(x) ∈ B such that

ϕ(x) ∈ Q_{α(x)}(V). (3.15.5)

One can check that the cardinality of

⋃_{x∈I}{α ∈ B : α ⪯ α(x)} (3.15.6)

is strictly less than the cardinality of B. More precisely, for each x ∈ I, the cardinality of {α ∈ B : α ⪯ α(x)} (3.15.7)

is strictly less than the cardinality of B, by construction. If I has only finitely many elements, then one can choose x ∈ I so that α(x) is maximal, and use this condition on (3.15.7). If I is an infinite set, then A is an infinite set, and the cardinality of (3.15.7) is less than or equal to the cardinality of A for every x ∈ I. This implies that the cardinality of (3.15.6) is less than or equal to the cardinality of I × A, which is equal to the cardinality of A.

It follows that (3.15.6) is a proper subset of B. If β ∈ B is not in (3.15.6), then it is easy to see that α(x) ⪯ β (3.15.8)

for every x ∈ I. This implies that

ϕ(I) ⊆ Q_{β}(V). (3.15.9)

Observe that B has no maximal element. Let β_{1} be the minimal element of B such that β ⪯ β_{1} and β ≠ β_{1}. Thus Q_{β_{1}}(V) = D(Q_{β}(V)), by construction. It follows that there is a w ∈ Q_{β_{1}}(V) ⊆ Q(V) (3.15.10)

such that ϕ(a) = a · w (3.15.11)

for every a ∈ I, as in the previous section. This implies that Q(V) is injective as a left module over A, as in Section 2.8.

Some associative algebras

4.1 Tensor products of algebras

Let k be a commutative ring with a multiplicative identity element, and let A_{1}, A_{2} be associative algebras over k. In particular, A_{1} and A_{2} are modules over k, and we take A = A_{1} ⨂_{k} A_{2} to be a tensor product of A_{1} and A_{2}, as modules over k. We would like to define multiplication on A in such a way that

(a_{1} ⊗ a_{2}) (b_{1} ⊗ b_{2}) = (a_{1} b_{1}) ⊗ (a_{2} b_{2}) (4.1.1)

for every a_{1}, b_{1} ∈ A_{1} and a_{2}, b_{2} ∈ A_{2}.

Note that (a_{1}, a_{2}, b_{1}, b_{2}) ↦ (a_{1} b_{1}) ⊗ (a_{2} b_{2}) (4.1.2)

is a multilinear mapping from A_{1} × A_{2} × A_{1} × A_{2} into A, which is to say that it is linear over k in each variable. If tensor products of finitely many modules over k are defined in terms of multilinear mappings, then it follows that there is a unique homomorphism from a tensor product

A_{1} ⨂_{k}A_{2} ⨂_{k}A_{1} ⨂_{k}A_{2} (4.1.3)

into A, as modules over k, with

a_{1} ⊗ a_{2} ⊗ b_{1} ⊗ b_{2} ↦ (a_{1} b_{1}) ⊗ (a_{2} b_{2}) (4.1.4)

for every a_{1}, b_{1} ∈ A_{1} and a_{2}, b_{2} ∈ A_{2}.

As in Section 1.4, there is a natural isomorphism between (4.1.3) and a tensor product

(A_{1} ⨂_{k}A_{2}) ⨂_{k}(A_{1} ⨂_{k}A_{2}) = A ⨂_{k}A, (4.1.5)

where a_{1}⊗a_{2}⊗b_{1}⊗b_{2} corresponds to (a_{1}⊗a_{2})⊗(b_{1}⊗b_{2}) for every a_{1}, b_{1} ∈ A_{1} and a_{2}, b_{2} ∈ A_{2}. Thus the module homomorphism from (4.1.3) into A mentioned in the preceding paragraph corresponds to a unique homomorphism from (4.1.5) into A, as modules over k, such that

(a_{1} ⊗ a_{2}) ⊗ (b_{1} ⊗ b_{2}) ↦ (a_{1} b_{1}) ⊗ (a_{2} b_{2}) (4.1.6)

for every a_{1}, b_{1} ∈ A_{1} and a_{2}, b_{2} ∈ A_{2}.

Of course, A ⨂_{k} A comes with a mapping from A×A that is bilinear over k. The composition of this mapping with the module homomorphism from (4.1.5) into A just mentioned is a bilinear mapping from A × A into A that can be used to define multiplication on A, and which satisfies (4.1.1), by construction.

Alternatively, if b_{1} ∈ A_{1} and b_{2} ∈ A_{2}, then

(a_{1}, a_{2}) ↦ (a_{1} b_{1}) ⊗ (a_{2} b_{2}) (4.1.7)

defines a mapping from A_{1} × A_{2} into A that is bilinear over k. This leads to a unique homomorphism from A = A_{1} ⨂_{k} A_{2} into itself, as a module over k, such that a_{1} ⊗ a_{2} ↦ (a_{1} b_{1}) ⊗ (a_{2} b_{2}) (4.1.8)

for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. It is easy to see that this homomorphism depends linearly on b_{1} and b_{2}, by uniqueness.

If a ∈ A, then the value of the homomorphism just mentioned at a defines a mapping from A_{1} × A_{2} into A, as a function of b_{1} ∈ A_{1} and b_{2} ∈ A_{2}, that is bilinear over k. This leads to a unique homomorphism from A into itself, as a module over k, that sends b_{1} ⊗ b_{2} to the value of the previous homomorphism depending on b_{1} and b_{2} at a for each b_{1} ∈ A_{1} and b_{2} ∈ A_{2}. This homomorphism depends linearly on a, by uniqueness.

This defines a mapping from A × A into A that is bilinear over k and can be used to define multiplication on A, where (4.1.1) holds by construction. One can check that multiplication on A is associative, because multiplication is associative on A_{1} and A_{2}, by hypothesis. Similarly, if multiplication is commutative on A_{1} and A_{2}, then multiplication is commutative on A.

If A_{1} and A_{2} have multiplicative identity elements e_{1} and e_{2}, respectively, then one can verify that e_{1} ⊗ e_{2} is the multiplicative identity element in A. In this case, a_{1} ↦ a_{1} ⊗ e_{2} (4.1.9)

and a_{2} ↦ e_{1} ⊗ a_{2} (4.1.10)

are algebra homomorphisms from A_{1} and A_{2} into A, respectively. Note that

(a_{1} ⊗ e_{2}) (e_{1} ⊗ a_{2}) = (e_{1} ⊗ a_{2}) (a_{1} ⊗ e_{2}) = a_{1} ⊗ a_{2} (4.1.11)

for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}.

Let C be another associative algebra over k, and let ϕ_{1}, ϕ_{2} be algebra homomorphisms from A_{1}, A_{2} into C, respectively. Suppose that

ϕ_{1}(a_{1}) ϕ_{2}(a_{2}) = ϕ_{2}(a_{2}) ϕ_{1}(a_{1}) (4.1.12) for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. Observe that

(a_{1}, a_{2}) ↦ ϕ_{1}(a_{1}) ϕ_{2}(a_{2}) (4.1.13)

is bilinear over k as a mapping from A_{1} × A_{2} into C. This leads to a unique homomorphism ϕ from A into C, as modules over k, such that

ϕ(a_{1} ⊗ a_{2}) = ϕ_{1}(a_{1}) ⊗ ϕ_{2}(a_{2}) (4.1.14)

for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. One can check that ϕ is an algebra homomorphism from A into C under these conditions.

4.2 Modules over tensor products

Let k be a commutative ring with a multiplicative identity element, and let A_{1}, A_{2} be associative algebras over k, with multiplicative identity elements e_{1}, e_{2}, respectively. Also let V be a module over k. Suppose for the moment that V is a left module over each of A_{1} and A_{2}, and that the actions of A_{1} and A_{2} on V commute with each other. This means that

a_{1} ·_{A_{1}} (a_{2} ·_{A_{2}} v) = a_{2} ·_{A_{2}} (a_{1} ·_{A_{1}} v) (4.2.1)

for every a_{1} ∈ A_{1}, a_{2} ∈ A_{2}, and v ∈ V, where ·_{A_{1}} and ·_{A_{2}} refer to the actions of A_{1} and A_{2} on V.

If v ∈ V, then the mapping from (a_{1}, a_{2}) ∈ A_{1} ×A_{2} to (4.2.1) is bilinear over k. Let A = A_{1} ⨂_{k} A_{2} be a tensor product of A_{1} and A_{2}, as modules over k. It follows that there is a unique homomorphism from A into V, as modules over k, that maps a_{1} ⊗ a_{2} to (4.2.1) for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. If a ∈ A, then let a · v = a ·_{A} v be the image of a under this mapping. Using this notation, we have that (a_{1} ⊗ a_{2}) ·_{A} v = a_{1} ·_{A_{1}} (a_{2} ·_{A_{2}} v) = a_{2} ·_{A_{2}} (a_{1} ·_{A_{1}} v) (4.2.2)

for every a_{1} ∈ A_{1} and a_{2} ∈ A_{2}.

Of course, one can do this for every v ∈ V, so that a ·_{A} v is defined as an element of V for every a ∈ A and v ∈ V. Remember that A may be considered as an associative algebra over k, as in the previous section. One can check that this makes V into a left module over A, as on p163 of [50].

Conversely, if V is a left module over A, then one can get commuting actions of A_{1} and A_{2} on V, as on p163 of [50]. More precisely, if a_{1} ∈ A_{1}, a_{2} ∈ A_{2}, and v ∈ V, then the actions of a_{1} and a_{2} on v are defined to be the actions of a_{1} ⊗ e_{2} and e_{1} ⊗ a_{2} on V, using the given action of A on V.

There are analogous statements for commuting right actions of A_{1} and A_{2} on V, as on p163 of [50].

Suppose now that V is a left module over A_{1} and a right module over A_{2}, and that the actions of A_{1} and A_{2} on V commute with each other. Thus

a_{1} ·_{A_{1}} (v ·_{A_{2}} a_{2}) = (a_{1} ·_{A_{1}} v) ·_{A_{2}} a_{2} (4.2.3) for every a_{1} ∈ A_{1}, a_{2} ∈ A_{2}, and v ∈ V. Let A_{2}^{op} be the opposite algebra associated to A_{2}, as in Section 1.11. Remember that we may use a_{2}^{op} to indicate that a_{2} ∈ A_{2} is being considered as an element of A_{2}^{op}. We may consider V as a left module over A_{2}^{op}, with

a_{2}^{op} ·_{A_{2}^{op}} v = v ·_{A_{2}} a_{2} (4.2.4)

for every a_{2} ∈ A_{2} and v ∈ V, as before.

Under these conditions, we may consider V to be a left module over a tensor product A_{1} ⨂_{k} A_{2}^{op} of A_{1} and A_{2}^{op}, as on p163 of [50] again. Similarly, we may consider V to be a right module over a tensor product of A_{1}^{op} and A_{2}, as in [50].

4.3 Formal power series

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. Also let n be a positive integer, and let

T_{1}, …, T_{n} (4.3.1)

be n distinct commuting indeterminates. As on p93 of [114], we normally try to use upper-case letters for indeterminates, and lower-case letters for elements of rings or modules.

By a multi-index of length n we mean an n-tuple α = (α_{1}, …, α_{n}) of nonnegative integers, and we put

|α| = α_{1} + ⋯ + α_{n}. (4.3.2)

The corresponding formal monomial

T^{α} = T_{1}^{α_{1}} ⋯ T_{n}^{α_{n}} (4.3.3)

in T_{1}, …, T_{n} has degree |α|. We use Z_{+} for the set of all positive integers, so that (Z_{+} ∪ {0})^{n} is the set of all multi-indices of length n.

A formal power series in T_{1}, …, T_{n} with coefficients in V can be expressed as

f(T) = f(T_{1}, …, T_{n}) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} T^{α}, (4.3.4)

where f_{α} ∈ V (4.3.5)

for every multi-index α. The space

V [[T_{1}, …, T_{n}]] (4.3.6)

of all such formal power series can be defined as the space

c((Z_{+} ∪ {0})^{n}, V) (4.3.7) of all V -valued functions on (Z_{+} ∪ {0})^{n}. This is a module over k with respect to pointwise addition and scalar multiplication of V -valued functions on (Z_{+} ∪ {0})^{n}, which corresponds to termwise addition and scalar multiplication of formal power series as in (4.3.4).

The space V [T_{1}, …, T_{n}] (4.3.8)

of formal polynomials in T_{1}, …, T_{n} with coefficients in V may be defined as the subset of V [[T_{1}, …, T_{n}]] consisting of formal power series as in (4.3.4) such that

(4.3.9) f_{α} = 0 for all but finitely many multi-indices α.

More precisely, this may be defined as the space

c_{00}((Z_{+} ∪ {0})^{n}, V) (4.3.10)

of V -valued functions on (Z_{+} ∪ {0})^{n} with finite support. This is a submodule of V [[T_{1}, …, T_{n}]], as a module over k. We may identify V with the submodule of V [T_{1}, …, T_{n}] consisting of f(T) as in (4.3.4) such that

f_{α} = 0 when α ≠ 0. (4.3.11)

Let A be an associative algebra over k, let f(T) ∈ A[[T_{1}, …, T_{n}]] be as in (4.3.4), and let

g(T) = ∑_{β∈(Z_{+}∪{0})^{n}} g_{β} T^{β} (4.3.12)

be another element of A[[T_{1}, …, T_{n}]]. If γ is a multi-index of length n, then put

h_{γ} = ∑_{{α,β∈(Z_{+}∪\{0\})^{n} \atop α+β=γ}} f_{α} g_{β}. (4.3.13)

Note that the sum on the right has only finitely many terms, and thus defines an element of A. This means that

h(T) = ∑_{γ∈(Z_{+}∪{0})^{n}} h_{γ} T^{γ} (4.3.14)

defines an element of A[[T_{1}, …, T_{n}]], and we put

f(T) g(T) = h(T). (4.3.15)

One can check that

(4.3.16) A[[T_{1}, …, T_{n}]] is an associative algebra over k

with respect to this definition of multiplication. If multiplication on A is commutative, then (4.3.17) A[[T_{1}, …, T_{n}]] is a commutative algebra too. It is easy to see that

(4.3.18) A[T_{1}, …, T_{n}] is a subalgebra of A[[T_{1}, …, T_{n}]],

which contains A as a subalgebra. If A has a multiplicative identity element e, then the corresponding formal power series is the multiplicative identity element in A[[T_{1}, …, T_{n}]]. In particular, k[[T_{1}, …, T_{n}]] is a commutative associative algebra over k with a multiplicative identity element.

If f(T) ∈ k[[T_{1}, …, T_{n}]] is as in (4.3.4), and g(T) ∈ V [[T_{1}, …, T_{n}]] is as in (4.3.12), then (4.3.13) defines an element of V for each multi-index γ, using scalar multiplication on V. Under these conditions, we can define h(T) as an element of V [[T_{1}, …, T_{n}]] as in (4.3.14), so that f(T) g(T) can be defined as in (4.3.15). One can verify that

(4.3.19) V [[T_{1}, …, T_{n}]] is a module over k[[T_{1}, …, T_{n}]]

in this way. If f(T) ∈ k[T_{1}, …, T_{n}] and g(T) ∈ V [T_{1}, …, T_{n}], then h(T) is an element of V [T_{1}, …, T_{n}], which makes

(4.3.20) V [T_{1}, …, T_{n}] a module over k[T_{1}, …, T_{n}].

Similarly, let A be an associative algebra over k with a multiplicative identity element e, and suppose that V is a left or right module over A. As before, the action of A on V can be extended to an action of A[[T_{1}, …, T_{n}]] on V [[T_{1}, …, T_{n}]], so that

(4.3.21) V [[T_{1}, …, T_{n}]] becomes a left or right module over A[[T_{1}, …, T_{n}]],

as appropriate. One may also consider

(4.3.22) V [T_{1}, …, T_{n}] as a left or right module over A[T_{1}, …, T_{n}],

as appropriate, in the same way.

Consider the mapping from A[[T_{1}, …, T_{n}]] onto A defined by

f(T) ↦ f_{0}, (4.3.23)

where f(T) is as in (4.3.4). It is easy to see that this defines a homomorphism from A[[T_{1}, …, T_{n}]] onto A, as algebras over k.

4.4 Polynomial functions

Let k be a commutative ring with a multiplicative identity element, and let n be a positive integer. If V is a module over k, then let V^{n} be the space of ntuples of elements of V, which is a module over k with respect to coordinatewise addition and scalar multiplication.

If t = (t_{1}, …, t_{n}) ∈ k^{n} and α is a multi-index of length n, then t^{α} is defined as an element of k by t^{α} = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}}, (4.4.1) where t_{j}^{α_{j}} is interpreted as being the multiplicative identity element 1 in k when α_{j} = 0, as usual. If β is another multi-index of length n, then

t^{α+β} = t^{α} t^{β}. (4.4.2)

Let T_{1}, …, T_{n} be n distinct commuting indeterminates, and let V be a module over k again. Also let f(T) be a formal polynomial in T_{1}, …, T_{n} with coefficients in V, as in (4.3.4). If t ∈ k^{n}, then

f(t) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} t^{α} (4.4.3)

defines an element of V, where all but finitely many terms in the sum on the right are equal to 0, by hypothesis. Of course,

f(T) ↦ f(t) (4.4.4)

is a homomorphism from V [T_{1}, …, T_{n}] into V, as modules over k.

Let A be an associative algebra over k, so that A[T_{1}, …, T_{n}] is an associative algebra over k too, as in the previous section. If f(T), g(T) ∈ A[T_{1}, …, T_{n}], h(T) = f(T) g(T), and t ∈ k^{n}, then one can check that

h(t) = f(t) g(t). (4.4.5)

Let V be a module over k again, and remember that V [T_{1}, …, T_{n}] may be considered as a module over k[T_{1}, …, T_{n}]. Let f(T) ∈ k[T_{1}, …, T_{n}] and g(T) ∈ V [T_{1}, …, T_{n}] be given, so that h(T) = f(T) g(T) ∈ V [T_{1}, …, T_{n}]. If t ∈ k^{n}, then f(t) is defined as an element of k, g(t) and h(t) are defined as elements of V, and one can verify that (4.4.5) holds.

Let A be an associative algebra over k with a multiplicative identity element e, and suppose that V is a left or right module over A. If one of f(T), g(T) is in A[T_{1}, …, T_{n}] and the other is in V [T_{1}, …, T_{n}], as appropriate, then h(T) = f(T) g(T) ∈ V [T_{1}, …, T_{n}], as in the previous section. If t ∈ k^{n}, then f(t), g(t) are defined as elements of A or V, as appropriate, h(t) ∈ V, and (4.4.5) holds.

Suppose that a = (a_{1}, …, a_{n}) ∈ A^{n} has commuting coordinates, so that

a_{j} a_{l} = a_{l} a_{j} (4.4.6)

for every j, l = 1, …, n, which holds automatically when n = 1. If α is a multi-index of length n, then

a^{α} = a_{1}^{α_{1}} ⋯ a_{n}^{α_{n}} (4.4.7)

defines an element of A, where a_{j}^{α_{j}} is interpreted as being equal to e when α_{j} = 0. If β is another multi-index of length n, then

a^{α+β} = a^{α} a^{β}. (4.4.8)

Let f(T) be a formal polynomial in T_{1}, …, T_{n} with coefficients in k, as in (4.3.4) again. Under these conditions, f(a) is defined as an element of A by

f(a) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} a^{α}, (4.4.9)

where all but finitely many terms in the sum on the right are equal to 0, by hypothesis. If g(T) ∈ k[T_{1}, …, T_{n}] and h(T) = f(T) g(T), then one can check that h(a) = f(a) g(a). (4.4.10)

Thus f(T) ↦ f(a) defines an algebra homomorphism from k[T_{1}, …, T_{n}] into A.

4.5 Invertibility in A[[T_{1}, …, T_{n}]]

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let T_{1}, …, T_{n} be n distinct commuting indeterminates. Also let V be a module over k, let f(T) ∈ V [[T_{1}, …, T_{n}]] be as in (4.3.4), and let N be a nonnegative integer. Let us say that f(T) vanishes to order N if

f_{α} = 0 (4.5.1)

for every multi-index α of length n with |α| ≤ N. One may consider this to hold trivially when N = −1.

Suppose that f(T) ∈ k[[T_{1}, …, T_{n}]] and g(T) ∈ V [[T_{1}, …, T_{n}]] vanish to order N_{1}, N_{2} ≥ −1, respectively. Under these conditions, one can check that

(4.5.2) f(T) g(T) vanishes to order N_{1} + N_{2} + 1.

Let A be an associative algebra over k with a multiplicative identity element e. If f(T), g(T) ∈ A[[T_{1}, …, T_{n}]] vanish to order N_{1}, N_{2} ≥ −1, respectively. then (4.5.2) holds, as before.

Suppose that a(T) ∈ A[[T_{1}, …, T_{n}]] vanishes to order 0, and let j be a nonnegative integer. Observe that

(4.5.3) a(T)^{j} vanishes to order j − 1.

Let l be a nonnegative integer, and consider

∑_{j=0}^{l} a(T)^{j}. (4.5.4)

If α is a multi-index of length n, then the coefficient of T^{α} in (4.5.4) does not depend on l when l ≥ |α|, (4.5.5) because of (4.5.3). In this case, we can define

∑_{j=0}^{∞} a(T)^{j} (4.5.6)

as an element of A[[T_{1}, …, T_{n}]] by saying that for each multi-index α of length n, the coefficient of T^{α} in (4.5.6) is the same as in (4.5.4) when (4.5.5) holds.

It is easy to see that

(e − a(T)) ∑_{j=0}^{l} a(T)^{j} = ( ∑_{j=0}^{l} a(T)^{j}) (e − a(T)) = e − a(T)^{l+1} (4.5.7)

for every l ≥ 0, by a standard argument. One can use this to check that

(e − a(T)) ∑_{j=0}^{∞} a(T)^{j} = ( ∑_{j=0}^{∞} a(T)^{j}) (e − a(T)) = e. (4.5.8)

This shows that e − a(T) is invertible in A[[T_{1}, …, T_{n}]], with inverse equal to (4.5.6).

Let f(T) ∈ A[[T_{1}, …, T_{n}]] be as in (4.3.4), and suppose that f_{0} is invertible in A. This implies that f(T) can be expressed as

f(T) = f_{0} (e − a(T)), (4.5.9)

where a(T) ∈ A[[T_{1}, …, T_{n}]] vanishes to order 0. It follows that f(T) is invertible in A[[T_{1}, …, T_{n}]], by the remarks in the preceding paragraph.

Conversely, if f(T) is invertible in A[[T_{1}, …, T_{n}]], then f_{0} is invertible in A. This follows from the fact that f(T) ↦ f_{0} is an algebra homomorphism from A[[T_{1}, …, T_{n}]] onto A, as in Section 4.3.

4.6 Partial derivatives

Let n be a positive integer, let α be a multi-index of length n, and let l be an integer with 1 ≤ l ≤ n. Under these conditions, we define the multi-index α(l) of length n by

(4.6.1)  α_{j}(l) = α_{j}  when j ≠ l
= α_{l} − 1 when j = l and α_{l} ≥ 1
= 0  when j = l and α_{l} = 0.

Similarly, the multi-index α^{+}(l) of length n is defined by

(4.6.2)  α_{j}^{+}(l) = α_{j}  when j ≠ l
= α_{l} + 1 when j = l.

Let k be a commutative ring with a multiplicative identity element, let V be a module over k, and let T_{1}, …, T_{n} be n distinct commuting indeterminates. If f(T) ∈ V [[T_{1}, …, T_{n}]] is as in (4.3.4), then the formal partial derivative of f(T) in T_{l} can be defined as an element of V [[T_{1}, …, T_{n}]] by

∂_{l}f(T) = \frac{∂}{∂T_{l}} f(T) = ∑_{α∈(Z_{+}∪\{0\})^{n}}(α_{l} + 1) · f_{α^{+}(l)} T^{α}. (4.6.3)

This is essentially the same as

∑_{α∈(Z_{+}∪\{0\})^{n}} α_{l} · f_{α} T^{α(l)} = ∑_{{α∈(Z_{+}∪\{0\})^{n} \atop α_{l}≥1}} α_{l} · f_{α} T^{α(l)}. (4.6.4)

Clearly f(T) ↦ ∂_{l}f(T) is a homomorphism from V [[T_{1}, …, T_{n}]] into itself, as a module over k, which maps V [T_{1}, …, T_{n}] into itself. One can verify that

∂_{l}(∂_{m}f(T)) = ∂_{m}(∂_{l}f(T)) (4.6.5)

for every l, m = 1, …, n and f(T) ∈ V [[T_{1}, …, n]].

Let A be an associative algebra over k, so that A[[T_{1}, …, T_{n}]] is an associative algebra over k too, as in Section 4.3. If f(T), g(T) ∈ A[[T_{1}, …, T_{n}]], then one can check that

∂_{l}(f(T) g(T)) = (∂_{l}f(T)) g(T) + f(T) ∂_{l}g(T) (4.6.6)

for every l = 1, …, n. If f(T) ∈ k[[T_{1}, …, T_{n}]] and g(T) ∈ V [[T_{1}, …, T_{n}]], then f(T) g(T) ∈ V [[T_{1}, …, T_{n}]], as in Section 4.3, and satisfies (4.6.6). If A has a multiplicative identity element e, and V is a left or right module over A, then V [[T_{1}, …, T_{n}]] is a left or right module over A too, as before. One can verify that the analogue of (4.6.6) holds in this case as well.

Let t ∈ k^{n} be given, and suppose that u ∈ k^{n} satisfies

u_{j} u_{l} = 0 (4.6.7)

for each j, l = 1, …, n. Let α be a multi-index of length n, so that t^{α} and (t + u)^{α} are defined as elements of k as in Section 4.4. It is easy to see that

(t_{l} + u_{l})^{α_{l}} = t_{l}^{α_{l}} + α_{l} · t_{l}^{α_{l}−1} u_{l} (4.6.8)

for each l = 1, …, n with α_{l} ≥ 1, because u_{l}^{2} = 0. This implies that

(t_{l} + u_{l})^{α_{l}} = t_{l}^{α_{l}} + α_{l} · t_{l}^{α_{l}(l)} u_{l} (4.6.9)

for every l = 1, …, n, which is automatic when α_{l} = 0. It follows that

(t + u)^{α} = t^{α} + ∑_{l=1}^{n} α_{l} · t^{α(l)} u_{l}. (4.6.10)

If V is a module over k and f(T) ∈ V [T_{1}, …, T_{n}], then we get that

f(t + u) = f(t) + ∑_{l=1}^{n}(∂_{l}f)(t) u_{l}. (4.6.11)

Here f(t), f(t + u), and (∂_{l}f)(t) are defined as elements of V as in Section 4.4.

Let A be an associative algebra over k with a multiplicative identity element e, and suppose that a ∈ A^{n} has commuting coordinates. Also let u be an element of A^{n} that satisfies (4.6.7), and whose coordinates commute with the coordinates of a, so that a_{j} u_{l} = u_{l} a_{j} (4.6.12)

for every j, l = 1, …, n. Note that a + u has commuting coordinates, because u has commuting coordinates. Let α be a multi-index of length n again, so that a^{α} and (a + u)^{α} are defined as elements of A. As in (4.6.9), we have that

(a_{l} + u_{l})^{α_{l}} = a_{l}^{α_{l}} + α_{l} · a_{l}^{α_{l}(l)} u_{l} (4.6.13)

for each l = 1, …, n. This means that

(a + u)^{α} = a^{α} + ∑_{l=1}^{n} α_{l} · a^{α(l)} u_{l}, (4.6.14)

as before. If f(T) ∈ k[T_{1}, …, T_{n}], then it follows that

f(a + u) = f(a) + ∑_{l=1}^{n}(∂_{l}f)(a) u_{l}, (4.6.15)

where f(a), f(a + u), and (∂_{l}f)(a) are defined as elements of A as in Section 4.4.

4.7 Algebras of module homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. The space Hom_{k}(V, V) of homomorphisms from V into itself, as a module over k, is an associative algebra over k, with respect to composition of mappings. Of course, the identity mapping on V is the multiplicative identity element in Hom_{k}(V, V).

Let A be an associative algebra over k with a multiplicative identity element e_{A}. If V is a left module over A, then the space Hom_{A}(V, V) of homomorphisms from V into itself, as a left module over A, is a subalgebra of Hom_{k}(V, V), as an algebra over k. Similarly, if V is a right module over A, then the space Hom_{A}(V, V) of homomorphisms from V into itself, as a right module over A, is a subalgebra of Hom_{k}(V, V), as an algebra over k. Note that the identity mapping on V is an element of Hom_{A}(V, V) in both cases.

A representation of A on a module V over k may be defined as an algebra homomorphism from A into Hom_{k}(V, V) that sends e_{A} to the identity mapping on V. Thus a representation of A on V corresponds exactly to an action of A on V that makes V a left module over A. Similarly, an opposite algebra homomorphism from A into Hom_{k}(V, V) that sends e_{A} to the identity mapping on V corresponds exactly to an action of A on V that makes V a right module over A.

Let n be a positive integer, and let A^{n} be the space of n-tuples of elements of A. This is a module over k with respect to coordinatewise addition and scalar multiplication, and both a left and right module over A with respect to left and right multiplication by elements of A coordinatewise on A^{n}. Note that these actions of A on A^{n} on the left and the right commute with each other, by associativity.

Let α = (α_{j,l}) be an n × n matrix with entries in A. If x ∈ A^{n}, then let T_{α}^{L}(x) and T_{α}^{R}(x) be the elements of A^{n} whose jth coordinates are given by

(T_{α}^{L}(x))_{j} = ∑_{l=1}^{n} α_{j,l} x_{l} (4.7.1)

and

(T_{α}^{R}(x))_{j} = ∑_{l=1}^{n} x_{l} α_{j,l}, (4.7.2)

respectively, for every j = 1, …, n. These define homomorphisms from A^{n} into itself as a module over k, which are the same when A is commutative. More precisely,

(4.7.3) T_{α}^{L} is a homomorphism from A^{n} into itself, as a right module over A,

and

(4.7.4) T_{α}^{R} is a homomorphism from A^{n} into itself, as a left module over A.

The superscripts L and R here are intended to indicate that the components of x are multiplied on the left and right, respectively, by the entries of α. If n = 1, then these reduce to the left and right multiplication operators on A corresponding to an element of A.

Equivalently, let u_{1}, …, u_{n} be the standard basis for A^{n} as a left and right module over A, so that

(4.7.5)  (u_{j})_{l} = e_{A} when j = l
= 0 when j ≠ l.

Using (4.7.1) and (4.7.2), we get that

T_{α}^{L}(u_{l}) = T_{α}^{R}(u_{l}) = ∑_{j=1}^{n} α_{j,l} u_{l} = ∑_{j=1}^{n} u_{j} α_{j,l} (4.7.6)

for each l = 1, …, n. One can check that every homomorphism from A^{n} into itself, as a right module over A, can be expressed as T_{α}^{L} for a unique matrix α. Similarly, every homomorphism from A^{n}, as a left module over itself, can be expressed as T_{α}^{R} for a unique matrix α.

Remember that the space M_{n}(A) of n × n matrices with entries in A is an associative algebra over k with respect to matrix multiplication. Let I_{n} be the identity matrix in M_{n}(A), whose diagonal entries are equal to e_{A}, and whose other entries are equal to 0. This is the multiplicative identity element in M_{n}(A). It is easy to see that T_{I_{n}}^{L} = T_{I_{n}}^{R} (4.7.7)

is the identity mapping on A^{n}.

If β is another n × n matrix with entries in A, then the product α β may be defined as an n × n matrix with entries in A in the usual way. It is easy to see that T_{α}^{L} ∘ T_{β}^{L} = T_{α β}^{L} (4.7.8)

as mappings from A^{n} into itself. This means that

α ↦ T_{α}^{L} (4.7.9)

defines an isomorphism from M_{n}(A) onto the algebra of homomorphisms from A^{n} onto itself, as a right module over A.

Note that

(T_{β^{t}}^{R}(x))_{l} = ∑_{r=1}^{n} x_{r} β_{r,l} (4.7.10)

for every x ∈ A^{n} and r = 1, …, n, where β^{t} is the transpose of β. Thus

(T_{α^{t}}^{R}(T_{β^{t}}^{R}(x)))_{j} = ∑_{l=1}^{n} ∑_{r=1}^{n} x_{r} β_{r,l} α_{l,j} (4.7.11)

for each j = 1, …, n. This implies that

T_{α^{t}}^{R} ∘ T_{β^{t}}^{R} = T_{(β α)^{t}}^{R}, (4.7.12)

as mappings from A^{n} into itself. It follows that

α ↦ T_{α^{t}}^{R} (4.7.13)

defines an opposite algebra isomorphism from M_{n}(A) onto the algebra of homomorphisms from A^{n} into itself, as a left module over A.

Remember that the transpose defines an opposite algebra isomorphism from M_{n}(A) onto the algebra M_{n}(A^{op}) of n×n matrices with elements in the opposite algebra A^{op} associated to A, as in Section 1.11. In fact, we have that

α ↦ T_{α}^{R} (4.7.14)

defines an isomorphism from the algebra M_{n}(A^{op}) of n×n matrices with entries in the opposite algebra A^{op} of A onto the algebra of homomorphisms from A^{n} into itself, as a left module over A. This is equivalent to the statement in the preceding paragraph, and it could also be verified directly from the definitions.

4.8 Modules over semigroups

Let k be a commutative ring with a multiplicative identity element, and let Σ be a semigroup, with the semigroup operation expressed multiplicatively, and with an identity element e_{Σ}. Also let V be a module over k. Suppose that for every x ∈ Σ and v ∈ V, x · v is defined as an element of V, and that

v ↦ x · v (4.8.1)

is a homomorphism from V into itself, as a module over k. Suppose too that

x · (y · v) = (x y) · v (4.8.2)

for every x, y ∈ Σ and v ∈ V, and that

e_{Σ} · v = v (4.8.3)

for every v ∈ V. Under these conditions, V is said to be a left module over Σ.

Equivalently, the action of Σ on V on the left corresponds to a mapping from Σ into the algebra Hom_{k}(V, V) of all homomorphisms from V into itself, as a module over k. The condition (4.8.2) says that this mapping is a homomorphism from Σ into Hom_{k}(V, V), as a semigroup with respect to composition of mappings. Similarly, (4.8.3) says that this mapping sends e_{Σ} to the identity mapping on V. A mapping from Σ into Hom_{k}(V, V) with these properties is also known as a representation of Σ on V.

Suppose now that for every x ∈ Σ and v ∈ V, v · x is defined as an element of V, and that v ↦ v · x (4.8.4)

is a homomorphism from V into itself, as a module over k. Suppose in addition that (v · x) · y = v · (x y) (4.8.5)

for every x, y ∈ Σ and v ∈ V, and that

v · e_{Σ} = v (4.8.6)

for every v ∈ V. In this case, V is said to be a right module over Σ.

The opposite semigroup Σ^{op} associated to Σ is defined to be the same as Σ as a set, with the product of x, y ∈ Σ^{op} equal to the product y x of y and x in Σ. If x ∈ Σ, then it may be helpful to use x^{op} to indicate that x is being considered as an element of Σ^{op}, as in Section 1.11. Thus multiplication in Σ^{op} is given by

x^{op} y^{op} = (y x)^{op}, (4.8.7)

as before. Note that e_{Σ} is the identity element in Σ^{op} as well. A homomorphism from Σ^{op} into another semigroup may be called an opposite semigroup homomorphism from Σ into the other semigroup.

A mapping from Σ into Hom_{k}(V, V) can be used to define an action of Σ on V on the right, as in (4.8.4). Observe that (4.8.5) is the same as saying that this mapping is an opposite semigroup homomorphism, with respect to composition of mappings on Hom_{k}(V, V). Of course, (4.8.6) is the same as saying that this mapping sends e_{Σ} to the identity mapping on V. This means that a right module over Σ corresponds to a representation of Σ^{op}. A left or right module over Σ may be considered as a right or left module over Σ^{op}, respectively.

4.9 Semigroup algebras

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let Σ be a semigroup, with the semigroup operation expressed multiplicatively, and with an identity element e_{Σ}. The corresponding semigroup algebra A(Σ) of Σ over A basically consists of finite formal sums of the form

a_{1} x_{1} + ⋯ + a_{n} x_{n}, (4.9.1)

where a_{j} ∈ A and x_{j} ∈ Σ for every j = 1, …, n.

More precisely, A(Σ) can be defined as the space c_{00}(Σ, A) of A-valued functions on Σ with finite support. This is a module over k with respect to pointwise addition and scalar multiplication, and both a left and right module over A with respect to pointwise multiplication on the left and right by elements of A. An element x of Σ may be identified with the A-valued function on Σ equal to e_{A} at x, and equal to 0 at every other element of Σ.

Multiplication on Σ can be extended to A(Σ) in such a way that it is bilinear over k, and (a x) (b y) = (a b) (x y) (4.9.2)

for every a, b ∈ A and x, y ∈ Σ, as on p148 of [50]. It is easy to see that A(Σ) is an associative algebra over k with respect to this definition of multiplication.

The element of A(Σ) corresponding to e_{Σ} is the multiplicative identity element of A(Σ). If we identify a ∈ A with a e_{Σ} ∈ A(Σ), then A corresponds to a subalgebra of A(Σ).

Let V be a module over k, and suppose for the moment that V is a left module over A(Σ). In particular, this means that V is a left module over A, because A corresponds to a subalgebra of A(Σ). If x ∈ Σ, then v ↦ x · v defines a homomorphism from V into itself, as a module over k, and as a left module over A, by identifying x with an element of A(Σ) as before. This makes V a left module over Σ, as in the previous section. The condition that the actions of elements of Σ be homomorphisms from V into itself, as a left module over A, means that the actions of Σ and A on V commute with each other.

Conversely, suppose that V is a left module over Σ and A, and that the actions of Σ and A commute with each other. Under these conditions, one can define an action of A(Σ) on V, so that V becomes a left module over A(Σ), as an associative algebra over k. This corresponds to some remarks on p149 of [50].

Similarly, if V is a right module over A(Σ), then V is a right module over Σ and A, and the actions of Σ and A on V commute with each other. Conversely, if V is a right module over Σ and A, and the actions of Σ and A commute with each other, then one can define an action of A(Σ) on V, so that V becomes a right module over A(Σ), as an associative algebra over k.

Suppose that V is a left module over A, so that the space Hom_{A}(V, V) of homomorphisms from V into itself, as a left module over A, is a subalgebra of Hom_{k}(V, V), as an algebra over k. An action of Σ on V by elements of Hom_{A}(V, V) that makes V a left module over Σ corresponds exactly to a homomorphism from Σ into Hom_{A}(V, V), as a semigroup with respect to composition, that sends e_{Σ} to the identity mapping on V.

Similarly, if V is a right module over A, then the space Hom_{A}(V, V) of homomorphisms from V into itself, as a right module over A, is a subalgebra of Hom_{k}(V, V). An action of Σ on A by elements of Hom_{A}(V, V) that makes V a right module over Σ corresponds exactly to an opposite semigroup homomorphism from Σ into Hom_{A}(V, V) that sends e_{Σ} to the identity mapping on V.

Let B be another associative algebra over k, with a multiplicative identity element e_{B}. Suppose that ϕ_{A} is a homomorphism from A into B, as algebras over k, such that ϕ_{A}(e_{A}) = e_{B}. Let ϕ_{Σ} be a homomorphism from Σ into B, as a semigroup with respect to multiplication, such that ϕ_{Σ}(e_{Σ}) = e_{B}. Suppose also that ϕ_{A}(a) ϕ_{Σ}(x) = ϕ_{Σ}(x) ϕ_{A}(a) (4.9.3)

for every a ∈ A and x ∈ Σ. Under these conditions, there is a unique homomorphism ϕ_{A(Σ)} from A(Σ) into B, as algebras over k, such that

ϕ_{A(Σ)}(a x) = ϕ_{A}(a) ϕ_{Σ}(x) (4.9.4)

for every a ∈ A and x ∈ Σ.

Note that A(Σ) satisfies the conditions on B mentioned in the preceding paragraph. This uses the natural mappings from A and Σ into A(Σ) as ϕ_{A} and ϕ_{Σ}, respectively.

4.10 Free semigroups

Let E be a nonempty set, and for each positive integer n, let E^{n} be the nth Cartesian power of E, consisting of n-tuples of elements of E. We may consider the elements of E^{n} as formal products of n elements of E, which may be expressed as strings of n elements of E. We take E^{0} to be a set with a single element, denoted e_{Σ}, representing the empty string.

Thus

Σ = Σ(E) = ⋃_{n=0}^{∞} E^{n} (4.10.1)

consists of all finite formal products of elements of E, or finite strings of elements of E, including the empty string. This is a semigroup with respect to formal products, which corresponds to concatenation of finite strings of elements of E.

By construction, e_{Σ} is the identity element in Σ. This is the free semigroup generated by E.

Let Σ_{1} be a semigroup with identity element e_{Σ_{1}}. Any mapping from E into Σ_{1} has a unique extension to a semigroup homomorphism from Σ into Σ_{1} that sends e_{Σ} to e_{Σ_{1}}.

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k, with a multiplicative identity element e_{A}. One can use Σ to get the corresponding semigroup algebra A(Σ) over A, as in the previous section. This is mentioned on p148 of [50], when E is finite. In this case, A(Σ) corresponds to the free ring mentioned on p146 of [50].

Let B be another associative algebra over k, with a multiplicative identity element e_{B}. Also let ϕ_{A} be a homomorphism from A into B, as algebras over k, with ϕ_{A}(e_{A}) = e_{B}. Suppose that ϕ_{E} is a mapping from E into B such that

ϕ_{A}(a) ϕ_{E}(x) = ϕ_{E}(x) ϕ_{A}(a) (4.10.2)

for every a ∈ A and x ∈ E. Let ϕ_{Σ} be the unique extension of ϕ_{E} to a homomorphism from Σ into B, as a semigroup with respect to multiplication, such that ϕ_{Σ}(e_{Σ}) = e_{B}. It is easy to see that (4.9.3) holds in this case. This implies that there is a unique homomorphism ϕ_{A(Σ)} from A(Σ) into B, as algebras over k, that satisfies (4.9.4). As before, A(Σ) satisfies these conditions on B, with respect to the natural mappings from A and E into A(Σ).

4.11 Tensor algebras using bimodules

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a module over k that is a bimodule over A, which is to say that A is both a left and right module over A, and that the actions of A on V on the left and right commute with each other.

Let V ⨂_{A} V be a tensor product of V with itself over A. This is a module over k, and in fact a left and right module over A, using the action of A on the left on the left factor of V, and the action of A on the right on the the right factor of V. Note that the actions of A on V ⨂_{A} V on the left and on the right commute with each other.

Put T^{1}V = V, and T^{2}V = V ⨂_{A} V. Similarly, if n ≥ 3 is an integer, then we take T^{n}V to be a tensor product of n V ’s over A. The order of the n factors of V is important here, but the way that the n-fold tensor product is arranged into products of pairs does not really matter, up to natural isomorphisms, as in Section 1.12.

More precisely, T^{n}V is a left module over A, using the action of A on the left on the left-most factor of V. Similarly, T^{n}V is a right module over A, using the action of A on the right on the right-most factor of A. These actions of A on T^{n}V on the left and right commute with each other.

If n_{1} and n_{2} are positive integers, then a tensor product (T^{n_{1}}V) ⨂_{A}(T^{n_{2}}V) of T^{n_{1}}V and T^{n_{2}}V over A is a left and right module over A, using the action of A on the left on T^{n_{1}}V, and the action of A on the right on T^{n_{2}}V. These actions of A on (T^{n_{1}}V) ⨂_{A}(T^{n_{2}}V) on the left and the right commute with each other, and (T^{n_{1}}V) ⨂_{A}(T^{n_{2}}V) is isomorphic to T^{n_{1}+n_{2}}V in a natural way, as a bimodule over A.

The natural bilinear mapping from (T^{n_{1}}V)×(T^{n_{2}}V) into (T^{n_{1}}V) ⨂_{A}(T^{n_{2}}V) leads to a mapping

(4.11.1) from (T^{n_{1}}V) × (T^{n_{2}}V) into T^{n_{1}+n_{2}}V

that is bilinear over k. If u_{1} ∈ T^{n_{1}}V and u_{2} ∈ T^{n_{2}}V, then let u_{1} u_{2} be the image of (u_{1}, u_{2}) into T^{n_{1}+n_{2}}V under this mapping. If a ∈ A, then

(u_{1} · a) u_{2} = u_{1} (a · u_{2}), (4.11.2)

by construction. We also have that

(a · u_{1}) u_{2} = a · (u_{1} u_{2}) (4.11.3)

and u_{1} (u_{2} · a) = (u_{1} u_{2}) · a. (4.11.4)

Let us take T^{0}V = A, which may be considered as a bimodule over A too. The mapping as in (4.11.1) mentioned earlier can be extended to the cases where n_{1} = 0 or n_{2} = 0, using the action of A on T^{n_{2}}V on the left when n_{1} = 0, the action of A on T^{n_{1}}V on the right when n_{2} = 0, and multiplication on A when n_{1} = n_{2} = 0. This extension satisfies (4.11.2), (4.11.3), and (4.11.4) when n_{1} = 0 or n_{2} = 0 as well.

Let n_{1}, n_{2}, and n_{3} be nonnegative integers, and let u_{j} ∈ T^{n_{j}}V, j = 1, 2, 3, be given. One can check that

(u_{1} u_{2}) u_{3} = u_{1} (u_{2} u_{3}) (4.11.5)

in T^{n_{1}+n_{2}+n_{3}}V. This uses the natural isomorphisms between

((T^{n_{1}}V) ⨂_{A}(T^{n_{2}}V)) ⨂_{A}(T^{n_{3}}V), (4.11.6)

(T^{n_{1}}V) ⨂_{A}((T^{n_{2}}V) ⨂_{A}(T^{n_{3}}V)), (4.11.7)

and T^{n_{1}+n_{2}+n_{3}}V, as in Section 1.12.

Put

T V = ⨁_{n=0}^{∞} T^{n}V, (4.11.8)

which may be considered initially as a module over k, and a bimodule over A. There is a natural mapping from (T V) × (T V) into T V, using the mappings as in (4.11.1) for every n_{1}, n_{2} ≥ 0. If u, w ∈ T V, then let u w be the image of (u, w) in T V under this mapping. This makes T V an associative algebra over k.

Let us identify A = T^{0}V with the corresponding subset of T V, which is a subalgebra of T V. Multiplication of elements of T V by elements of A on the left or right is the same as the action of A on T V on the left and right, respectively, as a bimodule over A. In particular, e_{A} is the multiplicative identity element in T V.

Let E be a nonempty set, and suppose that V is freely generated, as a bimodule over A, by E. Thus V corresponds to the direct sum of a family of copies of A indexed by E, as a bimodule over A. If n is a positive integer, then we can take T^{n}V to be freely generated by E^{n}, as a bimodule over A. This also works with n = 0, where E^{0} is interpreted as consisting of a single element, as in the previous section. If Σ is the free semigroup generated by E, then T V corresponds to the semigroup algebra A(Σ).

4.12 Tensor algebras and homomorphisms

Let us continue with the same notation and hypotheses as in the previous section. Also let B be an associative algebra over k, with a multiplicative identity element e_{B}. Suppose that ϕ_{A} is a homomorphism from A into B, as algebras over k, with ϕ_{A}(e_{A}) = e_{B}. Using this, B may be considered as a bimodule over A.

Let ϕ_{V} be a homomorphism from V into B, as modules over k, and as bimodules over A. This means that

ϕ_{V} (a · v) = ϕ_{A}(a) ϕ_{V} (v) (4.12.1)

and ϕ_{V} (v · a) = ϕ_{V} (v) ϕ_{A}(a) (4.12.2)

for every a ∈ A and v ∈ V. If v_{1}, v_{2} ∈ V and a ∈ A, then it follows that

ϕ_{V} (v_{1} · a) ϕ_{V} (v_{2}) = ϕ_{V} (v_{1}) ϕ_{A}(a) ϕ_{V} (v_{2}) = ϕ_{V} (v_{1}) ϕ_{V} (a · v_{2}). (4.12.3)

Of course, (v_{1}, v_{2}) ↦ ϕ_{V} (v_{1}) ϕ_{V} (v_{2}) (4.12.4)

defines a mapping from V × V into B that is bilinear over k. Using (4.12.3), we get that there is a unique homomorphism ϕ_{T^{2}V} from T^{2}V into B, as modules over k, such that ϕ_{T^{2}V} (v_{1} ⊗ v_{2}) = ϕ_{V} (v_{1}) ϕ_{V} (v_{2}) (4.12.5)

for every v_{1}, v_{2} ∈ V. More precisely, ϕ_{T^{2}V} is a homomorphism from T^{2}V into B, as bimodules over A.

Similarly, we can get a natural homomorphism

(4.12.6) ϕ_{T^{n}V} from T^{n}V into B

for every n ≥ 0, as modules over k, and as bimodules over A. We use ϕ_{A} when n = 0, and ϕ_{V} when n = 1. If n_{1}, n_{2} are nonnegative integers and u_{j} ∈ T^{n_{j}}V for j = 1, 2, then

ϕ_{T^{n_{1}+n_{2}} V} (u_{1} u_{2}) = ϕ_{T^{n_{1}} V} (u_{1}) ϕ_{T^{n_{2}} V} (u_{2}). (4.12.7)

Let ϕ_{T V} be the mapping from T V into B that corresponds to ϕ_{T^{n}V} on T^{n}V for every n ≥ 0. This is a homomorphism from T V into B, as algebras over k.

Note that T V is generated, as an algebra over k, by A and V = T^{1}V, considered as subsets of T V. This implies that any algebra homomorphism from T V into B is uniquely determined by its restriction to A and V. In particular, ϕ_{T V} is uniquely determined by ϕ_{A} and ϕ_{V}.

If ϕ is any algebra homomorphism from T V into B, then we can take ϕ_{A} to be the restriction of ϕ to A, and ϕ_{V} to be the restriction of ϕ to V = T^{1}V.

Observe that T V satisfies the conditions on B here, using the natural mappings from A and V into T V.

4.13 Local rings

Let A be a ring, with a nonzero multiplicative identity element e_{A}. Put

(4.13.1) I_{0} = {a ∈ A : a does not have a left inverse in A}.

We say that A is a local ring if

(4.13.2) I_{0} is a left ideal in A.

This is the same as the condition (LC) on p147 in [50].

Suppose that A is a local ring. If I is a proper left ideal in A, then

I ⊆ I_{0}, (4.13.3)

as in Proposition 2.1 on p147 of [50]. Indeed, if I is a proper left ideal in A, then no element of I can have a left inverse in A, so that (4.13.3) holds.

Suppose for the sake of a contradiction that a ∈ I_{0} and b ∈ A satisfy

a b = e_{A}. (4.13.4)

This implies that (e_{A} − b a) b = b − b a b = b − b = 0. (4.13.5)

Note that b a ∈ I_{0}, because I_{0} is a left ideal in A. It follows that e_{A} − b a ∉ I_{0}, because e_{A} ∉ I_{0}. Thus e_{A} − b b has a left inverse c in A. However, this means that b = c (e_{a} − b a) b = 0, (4.13.6)

which is a contradiction. This shows that

(4.13.7) no element of I_{0} has a right inverse in A,

as in Proposition 2.1 on p147 of [50].

If b ∈ A, then I_{0} b is a left ideal in A, because I_{0} is a left ideal in A. We also have that e_{A} ∉ I_{0} b, as in the preceding paragraph, so that I_{0} b ≠ A. This implies that I_{0} b ⊆ I_{0}, (4.13.8) as in (4.13.3). This means that

(4.13.9) I_{0} is a right ideal in A

too, as in Proposition 2.1 on p147 of [50].

Let a ∈ A \ I_{0} be given, so that a has a left inverse b ∈ A. This means that a is a right inverse of b, so that b ∉ I_{0}. It follows that b has a left inverse c ∈ A. This implies that c = c b a = a, (4.13.10)

so that a b = c b = e_{A}. (4.13.11)

Thus (4.13.12) every element of A \ I_{0} has a two-sided inverse in A,

as in Proposition 2.1 on p147 of [50].

Using the remarks in the previous paragraphs, we get that

(4.13.13) I_{0} = {a ∈ A : a does not have a right inverse in A}.

This implies that I_{0} contains every proper right ideal in A, as before. This is another part of Proposition 2.1 on p147 of [50].

Because I_{0} is a two-sided ideal in A, the quotient A/I_{0} is defined as a ring. Of course, this ring is nontrivial, because I_{0} ≠ A. In fact, A/I_{0} is a division ring, which is to say that every nonzero element of A/I_{0} has a multiplicative inverse in A/I_{0}, as in Proposition 2.1 on p147 of [50].

One might consider the previous definition of a local ring as being the definition of a “left local ring”, and define a notion of “right local ring” analogously. The earlier remarks show that left local rings are right local rings, and the converse holds similarly. Thus one may simply define local rings in this way, as on p148 of [50].

Suppose for the moment that A is commutative. In this case, the property of being a local ring is often defined in terms of the uniqueness of a maximal proper ideal in A, as on p4 of [8]. If A is a local ring in the sense of (4.13.2), then I_{0} is the unique maximal ideal in A. Conversely, if A has a unique maximal proper ideal, then A is a local ring in the sense considered in this section. Indeed, every non-invertible element of A generates a proper ideal in A. There are also well-known arguments for showing that every proper ideal in A is contained in a maximal proper ideal, using Zorn’s lemma or Hausdorff’s maximality principle. If there is a unique maximal proper ideal in A, then it consists exactly of the noninvertible elements of A.

4.14 Local rings and power series

Let A be any ring with a nonzero multiplicative identity element e_{A} again, let n be a positive integer, and let T_{1}, …, T_{n} be n distinct commuting indeterminates. Consider the corresponding ring A[[T_{1}, …, T_{n}]] of formal power series in T_{1}, …, T_{n} with coefficients in A, as in Section 4.3. Remember that A may be identified with the subring of A[[T_{1}, …, T_{n}]] consisting of formal power series for which the coefficient of T^{α} is equal to 0 when α ≠ 0.

We have also seen that the mapping from an element of A[[T_{1}, …, T_{n}]] to its coefficient of T^{0} defines a ring homomorphism from A[[T_{1}, …, T_{n}]] onto A. If a ∈ A has a left or right inverse in A[[T_{1}, …, T_{n}]], then it follows that a has a left or right inverse in A, as appropriate.

Let f(T) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} T^{α} be an element of A[[T_{1}, …, T_{n}]]. If f(T) has a left or right inverse in A[[T_{1}, …, T_{n}]], then f_{0} has a left or right inverse in A, as appropriate, because of the homomorphism mentioned in the preceding paragraph. Conversely, if f_{0} has a left or right inverse in A, then f(T) has a left or right inverse in A[[T_{1}, …, T_{n}]], as appropriate. This follows from the invertibility of the elements of A[[T_{1}, …, T_{n}]] for which the coefficient of T^{0} is e_{A}, as in Section 4.5.

Let I_{0} be as in (4.13.1), and consider

{f(T) ∈ A[[T_{1}, …, T_{n}]] : f_{0} ∈ I_{0}}. (4.14.1)

This is the same as

(4.14.2) {f(T) ∈ A[[T_{1}, …, T_{n}]] : f(T) does not have a left inverse in A[[T_{1}, …, T_{n}]]},

by the remarks in the preceding paragraph. It is easy to see that (4.14.1) is a left ideal in A[[T_{1}, …, T_{n}]] if and only if I_{0} is a left ideal in A. This implies that

(4.14.3) A[[T_{1}, …, T_{n}]] is a local ring

if and only if A is a local ring, as in Exercise 6 on p159 of [50]. In particular, (4.14.3) holds when A is a division ring, as on p148 of [50].

4.15 Exterior rings

Let A be a ring with a nonzero multiplicative identity element, let n be a positive integer, and let x_{1}, …, x_{n} be n distinct indeterminates. We would like to consider the corresponding exterior or Grassmann ring E_{A}(x_{1}, …, x_{n}) in x_{1}, …, x_{n} with coefficients in A.

Let I = {j_{1}, …, j_{r}} be a subset of {1, …, n}, where

1 ≤ j_{1} < j_{2} < ⋯ < j_{r} ≤ n. (4.15.1)

Put x_{I} = x_{j_{1}} ∧ x_{j_{2}} ∧ ⋯ ∧ x_{j_{r}}, (4.15.2)

which may be considered as a formal expression, for the moment. We can define E_{A}(x_{1}, …, x_{n}) initially as the space of formal sums of the form

∑_{I⊆{1,...,n}} a_{I} x_{I}, (4.15.3)

where a_{I} ∈ A for each I ⊆ {1, …, n}. More precisely, this can be defined as the set of all A-valued functions on the set of all subsets of {1, …, n}, which are identified with formal sums as in (4.15.3). If I = ∅, then we may identify x_{I} with e_{A}, so that a ∈ A is identified with a x_{I}.

Of course, E_{A}(x_{1}, …, x_{n}) is a left and right module over A, with respect to termwise addition and multiplication on the left or right by elements of A. Note that E_{A}(x_{1}, …, x_{n}) is freely generated by x_{I}, I ⊆ {1, …, n}, as a left or right module over A. If I_{1}, I_{2} ⊆ {1, …, n}, then we would like to define the product of x_{I_{1}} and x_{I_{2}}. Put

x_{I_{1}} ∧ x_{I_{2}} = 0 when I_{1} ∩ I_{2} ≠ ∅. (4.15.4)

Suppose now that I_{1} and I_{2} are disjoint sets, with r_{1}, r_{2} elements, respectively. Let σ be the permutation on I_{1} ∪ I_{2} that sends the elements of I_{1} to the first r_{1} elements of I_{1} ∪ I_{2}, in order, and send I_{2} to the next r_{2} elements of I_{1} ∪ I_{2}, in order. This can be extended to a permutation on {1, …, n}, by sending any element of {1, …, n} \ (I_{1} ∪ I_{2}) to itself. Put

(4.15.5) x_{I_{1}} ∧ x_{I_{2}} = x_{I_{1}∪I_{2}} when σ is an even permutation,
= −x_{I_{1}∪I_{2}} when σ is an odd permutation.

We can use this to define multiplication on E_{A}(x_{1}, …, x_{n}) in an obvious way, where x_{I} commutes with every element of A for each I ⊆ {1, …, n}. It is well known that E_{A}(x_{1}, …, x_{n}) a ring with respect to this definition of multiplication, which contains A as a subring, and with e_{A} as the multiplicative identity element in E_{A}(x_{1}, …, x_{n}). If A is an algebra over a commutative ring k with a multiplicative identity element, then E_{A}(x_{1}, …, x_{n}) is an algebra over k as well.

Let V be a left or right module over A, and let E_{V} (x_{1}, …, x_{n}) be the space of formal sums

∑_{I⊆{1,...,n}} v_{I} x_{I}, (4.15.6)

with v_{I} ∈ V for each I ⊆ {1, …, n}. As before, this can be defined more precisely as the space of V -valued functions on the set of all subsets if {1, …, n}, which are identified with formal sums as in (4.15.6). This is a left or right module over A too, as appropriate, with respect to termwise addition, and where A acts termwise on the coefficients v_{I} in (4.15.6).

In fact, E_{V} (x_{1}, …, x_{n}) is a left or right module over E_{A}(x_{1}, …, x_{n}), as appropriate. If I_{0} ⊆ {1, …, n}, then we put

x_{I_{0}} · ( ∑_{I⊆{1,...,n}} v_{I} x_{I}) = ∑_{I⊆{1,...,n}} v_{I} x_{I_{0}} ∧ x_{I} (4.15.7)

when V is a left module over A, and

( ∑_{I⊆{1,...,n}} v_{I} x_{I}) · x_{I_{0}} = ∑_{I⊆{1,...,n}} v_{I} x_{I} ∧ x_{I_{0}} (4.15.8)

when V is a right module over A. One can use this and the action of A on V to define the action of E_{A}(x_{1}, …, x_{n}) on E_{V} (x_{1}, …, x_{n}) on the left or right, as appropriate, in an obvious way.

Part II

Differentiation and complexes Differentiation and dual numbers

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, W be modules over k that are either both left modules over A, or both right modules over A. If ϕ is a homomorphism from V into W, as modules over A, then (5.0.1) the cokernel of ϕ is W/ϕ(V),

as on p19 of [8] and p3 of [50]. Similarly,

(5.0.2) the coimage of ϕ is V/ ker ϕ.

Note that ϕ induces an isomorphism from its coimage onto its image ϕ(V), as modules over A.

5.1 Modules with differentiation

Let k and A be as before, and let V be a module over k that is a left or right module over A. Supose that d_{V} is a homomorphism from V into itself, as a module over A, such that d_{V} ∘ d_{V} = 0. (5.1.1)

Under these conditions, V is said to be a module with differentiation, as on p53 of [50].

Put Z(V) = ker d_{V} , B(V) = d_{V} (V), (5.1.2)

which are submodules of V, as a module over A. Note that

B(V) ⊆ Z(V), (5.1.3)

by (5.1.1).

Similarly, put Z′(V) = V/d_{V} (V) (5.1.4)

and B′(V) = V/ ker d_{V} , (5.1.5)

which are also modules over A, as quotients of V. Equivalently, Z′(V) is the cokernel of d_{V}, and B′(V) is the coimage of d_{V}, as on p53 of [50].

The natural quotient mapping from V onto B′(V) can be expressed as the composition of the natural quotient mapping from V onto Z′(V) with a unique homomorphism (5.1.6) from Z′(V) onto B′(V),

as modules over A, because d_{V} (V) ⊆ ker d_{V}. We also have that d_{V} can be expressed as the composition of the natural quotient mapping from V onto B′(V) with a unique isomorphism

(5.1.7) δ from B′(V) onto B(V),

as modules over A, as on p53 of [50].

Thus d_{V} can be factored into a composition of module homomorphisms,

V ⟶ Z′(V) ⟶ B′(V) ⟶^{δ} B(V) ⟶ Z(V) ⟶ V, (5.1.8)

as on p53 of [50]. The first step uses the natural quotient mapping, and the second step uses the mapping obtained from the natural quotient mapping as in (5.1.6). The last two steps use the natural inclusion mappings.

This leads to a homomorphism

(5.1.9)  d̃_{V} from Z′(V) into Z(V),

as modules over A, as on p54 of [50]. More precisely, d̃_{V} is obtained by composing the second, third, and fourth homomorphisms indicated in (5.1.8). Equivalently, d̃_{V} is induced by d_{V} in an obvious way.

Observe that d̃_{V} (Z′(V)) = B(V), (5.1.10)

so that (5.1.11) the cokernel of d̃_{V} is equal to Z(V)/B(V).

The kernel of d̃_{V} is the same as the kernel of the mapping in (5.1.6), because the third and fourth homomorphisms indicated in (5.1.8) are injections. This means that ker d̃_{V} = ker d_{V} /d_{V} (V) = Z(V)/B(V), (5.1.12)

as on p54 of [50]. Put H(V) = Z(V)/B(V), (5.1.13)

which is the homology module of V with respect to d_{V}, as on p54 of [50]. This may be considered as a submodule of Z′(V), as a module over A.

If W is any left or right module over A, then W may be considered as a module with differentiation, by taking d_{W} = 0. In this case,

Z(W) = Z′(W) = H(W) = W, B(W) = B′(W) = {0}. (5.1.14)

If V is any module with differentiation, then Z(V), Z′(V), H(V), B(V), and B′(V) may be considered as modules with differentiation equal to 0, as on p55 of [50].

5.2 Mappings and differentiation

Let k be a commutative ring with a multiplicative identity element again, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that (V, d_{V}), (W, d_{W}) are both left modules or both right modules over A, with differentiation. Let ϕ be a homomorphism from V into W, as modules over A. We say that ϕ is a homomorphism from V into W as modules with differentiation if d_{W} ∘ ϕ = ϕ ∘ d_{V} , (5.2.1)

as on p54 of [50]. One may refer to ϕ as a map or mapping from V into W, as modules with differentiation, as well, as in [50].

In this case, we get that

ϕ(Z(V)) ⊆ Z(W) (5.2.2)

and ϕ(B(V)) ⊆ B(W). (5.2.3)

We also obtain induced homomorphisms

(5.2.4) ϕ_{Z′} from Z′(V) into Z′(W)

and (5.2.5) ϕ_{B′} from B′(V) into B′(W),

as modules over A. Note that

ϕ_{Z′}(H(V)) ⊆ H(W), (5.2.6)

because of (5.2.2). The restriction of ϕ_{Z′} to H(V) may be denoted ϕ_{H}, considered as a homomorphism from H(V) into H(W), as modules over A.

Remember that d̃_{V} is the homomorphism from Z′(V) into Z(V) induced by d_{V}, and let d̃_{W} be the analogous homomorphism for W. It is easy to see that

ϕ ∘ d̃_{V} = d̃_{W} ∘ ϕ_{Z′}, (5.2.7)

as homomorphisms from Z′(V) into Z(W). This corresponds to the commutativity of the diagram at the bottom of p54 of [50].

If ϕ is injective, then it is easy to see that

ϕ(Z(V)) = ϕ(V) ∩ Z(W). (5.2.8)

Similarly, we get that (5.2.9) ϕ_{B′} is injective.

Suppose for the moment that

ϕ(V) = W. (5.2.10)

Observe that ϕ_{Z′}(Z′(V)) = Z′(W) (5.2.11)

and ϕ_{B′}(B′(V)) = B′(W). (5.2.12)

One can check that ϕ(B(V)) = B(W). (5.2.13)

Let ψ be another homomorphism from V into W, as modules over A, and as modules with differentiation. A homomorphism σ from V into W, as modules over A, is said to be a homotopy between ϕ and ψ if

d_{W} ∘ σ + σ ∘ d_{V} = ϕ − ψ, (5.2.14)

as on p54 of [50]. This implies that

(ϕ − ψ)(Z(V)) ⊆ B(W). (5.2.15)

It follows that ϕ_{H} = ψ_{H}, (5.2.16)

as on p54 of [50].

Let (W_{1}, d_{W_{1}}) be another left or right module over A with differentiation, depending on whether V, W are left or right modules over A. Also let ϕ_{1} be a homomorphism from W into W_{1}, as modules over A with differentiation. This implies that ϕ_{1} ∘ ϕ is a homomorphism from V into W_{1}, as modules over A with differentiation. One can verify that

(ϕ_{1} ∘ ϕ)_{Z′} = (ϕ_{1})_{Z′} ∘ ϕ_{Z′} (5.2.17)

and (ϕ_{1} ∘ ϕ)_{B′} = (ϕ_{1})_{B′} ∘ ϕ_{B′}. (5.2.18)

In particular, (ϕ_{1} ∘ ϕ)_{H} = (ϕ_{1})_{H} ∘ ϕ_{H}. (5.2.19)

Note that d_{V} (ker ϕ) ⊆ ker ϕ, (5.2.20)

by (5.2.1). Similarly, d_{W} (ϕ(V)) ⊆ ϕ(V). (5.2.21)

5.3 Submodules and quotients

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a left or right module over A with differentiation, and let V_{0} be a submodule of V, as a module over A. Let us say that V_{0} is a submodule of V as a module with differentiation if

d_{V} (V_{0}) ⊆ V_{0}. (5.3.1)

In this case, V_{0} is a module with differentiation too, with d_{V_{0}} equal to the restriction of d_{V} to V_{0}. Of course, the natural inclusion mapping from V_{0} into V is a homomorphism from V_{0} into V, as modules with differentiation.

Let q_{0} be the natural quotient mapping from V onto V/V_{0}. Note that q_{0} ∘d_{V} is equal to 0 on V_{0}, by (5.3.1), so that there is a unique homomorphism d_{V/V_{0}} from V/V_{0} into itself, as a module over A, such that

d_{V/V_{0}} ∘ q_{0} = q_{0} ∘ d_{V} . (5.3.2)

It is easy to see that d_{V/V_{0}} ∘ d_{V/V_{0}} = 0, (5.3.3)

so that V/V_{0} is a module with differentiation with respect to d_{V/V_{0}}. By construction, q_{0} is a homomorphism from V onto V/V_{0}, as modules with differentiation.

Observe that

Z(V) ∩ (ker q_{0}) = Z(V) ∩ V_{0} = Z(V_{0}). (5.3.4)

This is basically the same as (5.2.8), and it corresponds to the exactness of the bottom row of the diagram on the bottom of p54 of [50].

Remember that q_{0}(B(V)) = B(V/V_{0}), (5.3.5)

as in (5.2.13). Suppose that v ∈ V satisfies

q_{0}(v) ∈ B(V/V_{0}). (5.3.6)

This means that there is a u ∈ V such that

q_{0}(v) = q_{0}(d_{V} (u)), (5.3.7)

by (5.3.5). Equivalently, there is a v_{0} ∈ V_{0} such that

v − d_{V} (u) = v_{0}. (5.3.8)

It follows that

(5.3.9) the images of v and v_{0} under the natural quotient mapping from V onto Z′(V) are the same.

5.3. SUBMODULES AND QUOTIENTS 119

This corresponds to the exactness of the top row of the diagram on the bottom of p54 of [50]. Similarly, if we also have that v ∈ Z(V), (5.3.10) then v_{0} ∈ Z(V_{0}), (5.3.11) by (5.3.8). This corresponds to the exactness of the first part of the sequence in (4) on p55 of [50].

There is a natural homomorphism (5.3.12) Δ from H(V/V_{0}) into H(V_{0}), as modules over A, as on p55 of [50]. To see this, let an element of H(V/V_{0}) = Z(V/V_{0})/B(V/V_{0}) (5.3.13) be given. This can be represented by the image of q_{0}(v) ∈ Z(V/V_{0}) in this quotient, where v ∈ V. Observe that q_{0}(d_{V} (v)) = d_{V/V_{0}}(q_{0}(v)) = 0, (5.3.14) so that d_{V} (v) ∈ V_{0}. More precisely, d_{V} (v) ∈ Z(V_{0}). (5.3.15)

We would like to take Δ of the image of q_{0}(v) in (5.3.13) to be the image of d_{V} (v) in H(V_{0}) = Z(V_{0})/B(V_{0}). (5.3.16) To show that this is well defined, suppose that v′ ∈ V and the image of q_{0}(v′) in (5.3.13) is the same as the image of q_{0}(v), so that q_{0}(v − v′) = q_{0}(v) − q_{0}(v′) ∈ B(V/V_{0}). (5.3.17) This means that there is a u ∈ V such that q_{0}(v − v′) = d_{V/V_{0}}(q_{0}(u)) = q_{0}(d_{V} (u)), (5.3.18) and thus v − v′ − d_{V} (u) ∈ V_{0}. (5.3.19) It follows that d_{V} (v) − d_{V} (v′) = d_{V} (v − v′ − d_{V} (u)) ∈ B(V_{0}), (5.3.20) so that the images of d_{V} (v) and d_{V} (v′) in (5.3.16) are the same. It is easy to see that Δ is a homomorphism from H(V/V_{0}) into H(V_{0}), as modules over A.

Let (q_{0})_{Z′} be the homomorphism from Z′(V) into Z′(V/V_{0}), as modules over A, induced by q_{0}, as in the previous section. The restriction of (q_{0})_{Z′} to H(V) is denoted (q_{0})_{H}, and maps H(V) into H(V/V_{0}), as before. Let us check that

Δ ∘ (q_{0})_{H} = 0. (5.3.21)

If v ∈ Z(V), so that d_{V} (v) = 0, then Δ of the image of q_{0}(v) in H(V/V_{0}) is equal to 0, by construction. This means that Δ ∘ (q_{0})_{H} of the image of v in H(V) = Z(V)/B(v) is equal to 0, as desired.

Suppose that v ∈ V, q_{0}(v) ∈ Z(V/V_{0}), and that Δ maps the image of q_{0}(v) in H(V/V_{0}) to 0 in H(V_{0}). This means that

d_{V} (v) ∈ B(V_{0}), (5.3.22)

so that d_{V} (v) = d_{V} (u) for some u ∈ V_{0}. It follows that

q_{0}(v) = q_{0}(v − u) and v − u ∈ Z(V). (5.3.23)

Thus the image of q_{0}(v) in H(V/V_{0}) is the same as the image of q_{0}(v − u) in H(V/V_{0}), and the image of q_{0}(v − u) in H(V/V_{0}) is the same as (q_{0})_{H} of the image of v − u in H(V). This shows that

(q_{0})_{H}(H(V)) = ker Δ, (5.3.24)

which corresponds to the exactness of part of the sequence (4) on p55 of [50].

Let ι_{0} be the natural inclusion mapping from V_{0} into V. This leads to an induced homomorphism (ι_{0})_{Z′} from Z′(V_{0}) into Z′(V), as in the previous section. The restriction of (ι_{0})_{Z′} to H(V_{0}) is denoted (ι_{0})_{H}, as usual, and maps H(V_{0}) into H(V). It is easy to see that

(ι_{0})_{H} ∘ Δ = 0, (5.3.25)

by construction. This is because the element of Z(V_{0}) in (5.3.15) is automatically in B(V).

Suppose that w ∈ Z(V_{0}), and that the image of w in H(V_{0}) is in the kernel of (ι_{0})_{H}. This means that there is a v ∈ V such that

w = d_{V} (v). (5.3.26)

This implies that

d_{V/V_{0}}(q_{0}(v)) = q_{0}(d_{V} (v)) = q_{0}(w) = 0, (5.3.27)

so that q_{0}(v) ∈ Z(V/V_{0}). It follows that the image of w in H(V_{0}) is the same as Δ of the image of q_{0}(v) in H(V/V_{0}). Thus

ker(ι_{0})_{H} = Δ(H(V/V_{0})), (5.3.28)

which corresponds to the exactness of another part of the sequence (4) on p55 of [50].

5.4 Dual numbers

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let d be a formal symbol, which is taken to commute with all elements of k and A, and satisfy d^{2} = 0. The algebra A[d] of dual numbers associated to A consists of expressions of the form a_{1} + a_{2} d, with a_{1}, a_{2} ∈ A, as on p56 of [50].

More precisely, A[d] is free as a left and right module over A, with generators e_{A} and d. If a_{1}, a_{2}, b_{1}, b_{2} ∈ A, then we put

(a_{1} + a_{2} d) (b_{1} + b_{2} d) = a_{1} b_{1} + (a_{1} b_{2} + a_{2} b_{1}) d. (5.4.1)

One can check that this makes A[d] into an associative algebra over k that contains A as a subalgebra, and with e_{A} as the multiplicative identity element.

A left or right module (V, d_{V}) over A with differentiation corresponds exactly to a left or right module over A[d], as appropriate, as on p56 of [50]. In this case, d_{V} corresponds to the action of d on V, as a module over A[d]. Similarly, a homomorphism between left or right modules over A with differentiation corresponds exactly to a homomorphism between left or right modules over A[d], as appropriate.

Remember that any module V over A may be considered as a module with differentiation, by taking d_{V} = 0. Equivalently, V may be considered as a module over A[d], where the action of d on V is equal to 0. In particular, A may be considered as a left and right module over A[d] in this way, as on p56 of [50].

Let (V, d_{V}) be a left module over A with differentiation, which may be considered as a left module over A[d]. If v ∈ V, then

a ↦ a · v (5.4.2)

defines a homomorphism from A into V, as left modules over A. Every homomorphism from A into V, as left modules over A, corresponds to a unique v ∈ V in this way. If v ∈ Z(V), then (5.4.2) defines a homomorphism from A into V, as left modules over A with differentiation, or equivalently as left modules over A[d]. Conversely, if (5.4.2) defines a homomorphism from A into V as left modules over A with differentiation, or equivalently as left modules over A[d], then one can check that v ∈ Z(V).

This defines an isomorphism

(5.4.3) between Hom_{A[d]}(A, V) and Z(V),

as modules over k, as on p56 of [50]. More precisely, Hom_{A[d]}(A, V) may be considered as a left module over A, by considering A as a right module over itself. In fact, we have an isomorphism as in (5.4.3) as left modules over A. This may also be considered as an isomorphism between left modules over A with differentiation, or left modules over A[d], with differentiation equal to 0, which corresponds to considering A as a right module over itself with differentiation equal to 0. There are analogous statements for right modules, as usual.

Let us check that

A ⨂_{A[d]}V = Z′(V) = V/d_{V} (V), (5.4.4)

as on p56 of [50]. More precisely, this means that Z′(V) satisfies the requirements of a tensor product, initially as a module over k.

Let W be a module over k, and let b be a mapping from A × V into W that is bilinear over k. We may be interested in situations where

b(a · α, v) = b(a, α · v) (5.4.5)

for every a ∈ A, α ∈ A[d], and v ∈ V. This happens if and only if

b(a a_{1}, v) = b(a, a_{1} · v) (5.4.6)

and 0 = b(a · d, v) = b(a, d · v) = b(a, d_{V} (v)) (5.4.7)

for every a, a_{1} ∈ A and v ∈ V.

There is an obvious mapping from A × V into Z′(V), defined by

(5.4.8) (a, v) ↦ the image of a · v in Z′(V) = V/d_{V} (V).

It is easy to see that this mapping satisfies (5.4.6), (5.4.7), and thus (5.4.5).

Let W be a module over k again, and let b be a mapping from A × V into W that is bilinear over k and satisfies (5.4.5). Observe that

b(a, v) = b(e_{A}, a · v) (5.4.9)

and b(e_{A}, d_{V} (v)) = 0 (5.4.10)

for every a ∈ A and v ∈ V. Because of (5.4.10), there is a unique homomorphism c from Z′(V) into W such that

(5.4.11) the composition of the natural quotient mapping from V onto Z′(V) with c

is the same as v ↦ b(e_{A}, v). (5.4.12)

It follows that

(5.4.13) b is the same as the composition of (5.4.8) with c,

by (5.4.9). It is easy to see that c is uniquely determined by this property as well.

This shows that Z′(V) satisfies the requirements of a tensor product of A and V over A[d], as a module over k, and using the mapping (5.4.8) from A × V into Z′(V). Such a tensor product is also a left module over A in a natural way, because A is a left module over itself. It is easy to see that this corresponds to Z′(V) as a left module over A, as a quotientof V.

Similarly, if V is a right module over A with differentiation, and thus a right module over A[d], then Z′(V) satisfies the requirements of a tensor product of V and A over A[d], as a module over k. Such a tensor product is a right module over A in a natural way, because A is a right module over itself, which corresponds to Z′(V) as a right module over A, as a quotient of V.

5.5 Easy differentiation

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let W_{1}, W_{2} be modules over k that are both left or both right modules over A, and let ψ_{1,2} be a homomorphism from W_{1} into W_{2}, as modules over A. Put

W_{1,2} = W_{1} × W_{2}, (5.5.1)

considered as a left or right module over A, as appropriate, where the module operations are defined coordinatewise. Equivalently, W_{1,2} is the direct sum of W_{1} and W_{2}, as a module over A.

If w_{1} ∈ W_{1} and w_{2} ∈ W_{2}, then put

d_{W_{1,2}}((w_{1}, w_{2})) = (0, ψ_{1,2}(w_{1})). (5.5.2)

This defines a homomorphism from W_{1,2} into itself, as a module over A, with

d_{W_{1,2}} ∘ d_{W_{1,2}} = 0. (5.5.3)

This makes W_{1,2} a module with differentiation.

Observe that Z(W_{1,2}) = (ker ψ_{1,2}) × W_{2} (5.5.4)

and B(W_{1,2}) = {0} × ψ_{1,2}(W_{1}). (5.5.5)

It follows that

(5.5.6) Z′(W_{1,2}) is isomorphic to W_{1} × (W_{2}/ψ_{1,2}(W_{1}))

and (5.5.7) B′(W_{1,2}) is isomorphic to (W_{1}/(ker ψ_{1,2})) × {0},

as modules over A. Similarly,

(5.5.8) H(W_{1,2}) is isomorphic to (ker ψ_{1,2}) × (W_{2}/ψ_{1,2}(W_{1})),

as a module over A.

Let W be a left or right module over A, and put

W^{x} = W × W, (5.5.9) following the notation on p57 of [50]. This is the same as W_{1,2}, with W_{1} = W_{2} = W. More precisely, W^{x} is a module with differentiation, with

d_{W^{x}}((w_{1}, w_{2})) = (0, w_{1}) (5.5.10)

for every w_{1}, w_{2} ∈ W, as on p57 of [50]. This is the same as d_{W_{1,2}}, with ψ_{1,2} equal to the identity mapping on W.

In this case, Z(W^{x}) = B(W^{x}) = {0} × W. (5.5.11)

Similarly, Z′(W^{x}) = B′(W^{x}) = W^{x}/({0} × W), (5.5.12)

which is isomorphic to W in an obvious way, as a module over A. Of course,

H(W^{x}) = {0}. (5.5.13)

Let d and A[d] be as in the previous section, and let η be the obvious inclusion mapping from A into A[d], as on p57 of [50]. If W is a left module over A, then

(η)W = A[d] ⨂_{A}W (5.5.14)

is a left module over A[d], which is the covariant η-extension of W, as in Section 2.9. Similarly, (η)W = Hom_{A}(A[d], W) (5.5.15)

is a left module over A[d], which is the contravariant η-extension of W, as in Section 2.11.

Consider the mapping

(w_{1}, w_{2}) ↦ e_{A} ⊗ w_{1} + (e_{A} d) ⊗ w_{2} (5.5.16)

from W^{x} into (5.5.14). This defines an isomorphism from W^{x} onto (5.5.14), as modules over k, and left modules over A. More precisely, this defines an isomorphism from W^{x} onto (5.5.14), as left modules over A[d]. This corresponds to a remark on p57 of [50].

If w_{1}, w_{2} ∈ W, then

a_{1} + a_{2} d ↦ a_{1} · w_{2} + a_{2} · w_{1} (5.5.17)

defines a homomorphism from A[d] into W, as left modules over A. Thus

(w_{1}, w_{2}) ↦ (5.5.17) (5.5.18)

defines a mapping from W^{x} into (5.5.15). It is easy to see that this is an isomorphism from W^{x} onto (5.5.15) as modules over k, and left modules over A.

If w_{1}, w_{2} ∈ W, then the analogue of (5.5.17) for d_{W^{x}}((w_{1}, w_{2})) = (0, w_{1}) is

a_{1} + a_{2} d ↦ a_{1} · w_{1}. (5.5.19)

5.6. SOME ISOMORPHISMS WITH W^{X} 125

Equivalently, this is the image of d_{W^{x}}((w_{1}, w_{2})) in (5.5.15) under the mapping (5.5.18). If a_{1}, a_{2} ∈ A, then

(a_{1} + a_{2} d) d = a_{1} d (5.5.20)

is mapped to a_{1} · w_{1} (5.5.21)

by (5.5.17). This implies that (5.5.18) defines an isomorphism from W^{x} onto (5.5.15), as left modules over A[d]. This corresponds to another remark on p57 of [50].

Of course, there are analogous statements when W is a right module over A.

5.6 Some isomorphisms with W^{x}

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra with a multiplicative identity element e_{A}. Also let d and A[d] be as in Section 5.4, and let η be the obvious inclusion mapping from A into A[d], as before. Suppose that (V, d_{V}) is a left module over A with differentiation, which may be considered as a left module over A[d]. Let σ be a homomorphism from V into itself, as a module over A, and suppose that

(5.6.1) d_{V} ∘ σ + σ ∘ d_{V} is the identity mapping on V.

Equivalently, this means that σ is a homotopy between the identity mapping and the zero mapping on V, as homomorphisms from V into itself as a module with differentiation, as in Section 5.2.

If v ∈ Z(V), then (5.6.1) implies that

v = d_{V} (σ(v)) ∈ B(V). (5.6.2)

This means that B(V) = Z(V) (5.6.3)

in this case. Put W = B(V), (5.6.4)

which is a left module over A. Thus W^{x} may be defined as in the previous section.

Consider the mapping ϕ from V into W^{x} defined by

ϕ(v) = (d_{V} (v), d_{V} (σ(v))) (5.6.5)

for every v ∈ V. Note that this is a homomorphism from V into W^{x}, as modules over A. If v ∈ V, then

ϕ(d_{V} (v)) = (d_{V} (d_{V} (v)), d_{V} (σ(d_{V} (v))))
(5.6.6)  = (0, d_{V} (v) − d_{V} (d_{V} (σ(v)))) = (0, d_{V} (v)) = d_{W^{x}}(ϕ(v)),

using (5.6.1) in the second step. This shows that ϕ is a homomorphism from V into W^{x}, as modules with differentation. If ϕ(v) = 0, then

v = σ(d_{V} (v)) + d_{V} (σ(v)) = 0, (5.6.7)

using (5.6.1) in the second step again.

Let w ∈ W be given, and observe that

d_{V} (σ(w)) = w, (5.6.8)

by (5.6.1). We also have that

(5.6.9)  d_{V} (σ(σ(w))) = σ(w) − σ(d_{V} (σ(w)))
= σ(w) − σ(w) + σ(σ(d_{V} (w))) = 0,

using (5.6.1) in the first and second steps. It follows that

ϕ(w) = (0, w) (5.6.10)

and ϕ(σ(w)) = (w, 0). (5.6.11)

This implies that ϕ(V) = W^{x}. This shows that ϕ is an isomorphism from V onto W^{x}, as modules with differentiation.

Using this, we get that

(5.6.12) V is isomorphic to (5.5.14)

and (5.6.13) V is isomorphic to (5.5.15),

as modules over A[d]. This corresponds to the fact that (b) implies (c), (c’) in Proposition 2.3 on p57 of [50].

5.7 Some related properties of (V, d_{V})

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let d and A[d] be as in Section 5.4, and let η be the obvious inclusion mapping from A into A[d], as in the previous sections. Suppose that (V, d_{V}) is a left module over A with differentiation, which may be considered as a left module over A[d], as before.

If there is a left module W over A such that (5.6.12) holds, then

(5.7.1) V is η-projective,

as in Section 2.14. Similarly, if there is a left module W over A such that (5.6.13) holds, then (5.7.2) V is η-injective, 5.7. SOME RELATED PROPERTIES OF (V, D_{V}) 127

as in Section 2.15. These statements correspond to (c) implies (a) and (c’) implies (a’) in Proposition 2.3 on p57 of [50].

Suppose that (5.7.1) holds, and let us show that there is a homomorphism σ from V into itself, as a module over A, that satisfies (5.6.1). This corresponds to the fact that (a) implies (b) in Proposition 2.3 on p57 of [50]. Let us use V (A) to refer to V as a left module over A, so that

(η)V (A) = A[d] ⨂_{A}V (A), (5.7.3)

as in Section 2.9. This is isomorphic to V (A)^{x} in a natural way, as left modules over A[d], as in Section 5.5.

There is a natural homomorphism from (5.7.3) into V, as left modules over A[d], obtained from the action of A[d] on V on the left, as in Section 2.12. If we identify (5.7.3) with V (A)^{x}, as in the preceding paragraph, then this mapping corresponds to the mapping f from V (A)^{x} into V defined by

f((v_{1}, v_{2})) = v_{1} + d_{V} (v_{2}) (5.7.4)

for every v_{1}, v_{2} ∈ V. Our hypothesis (5.7.1) is the same as saying that V (A)^{x} corresponds to the direct sum of the kernel of f and another submodule of V (A)^{x}, as a left module over A[d], as in Section 2.12. This means that there is a homomorphism g from V into V (A)^{x}, as left modules over A[d], such that

(5.7.5) f ∘ g is the identity mapping on V.

We can express g as g(v) = (τ(v), σ(v)) (5.7.6)

for every v ∈ V, where σ, τ are mappings from V into itself, by definition of V (A)^{x}. More precisely, σ and τ are homomorphisms from V into itself, as a left module over A, because g is a homomorphism from V into V (A)^{x} as left modules over A[d], and thus over A. The condition that g be a homomorphism from V into V (A)^{x} as left modules over A[d] implies that

(0, τ(v)) = d_{V (A)^{x}}(g(v)) = g(d_{V} (v)) = (τ(d_{V} (v)), σ(d_{V} (v))) (5.7.7)

for every v ∈ V. We also have that

v = f(g(v)) = τ(v) + d_{V} (σ(v)) (5.7.8)

for every v ∈ V, by (5.7.5). It is easy to obtain (5.6.1) from (5.7.7) and (5.7.8), as desired.

Suppose now that (5.7.2) holds, and let us check that there is a homomorphism σ from V into itself, as a module over A, that satisfies (5.6.1). This corresponds to (a’) implies (b) in Proposition 2.3 on p57 of [50]. Remember that (η)V (A) = Hom_{A}(A[d], V (A)), (5.7.9) as in Section 2.11. This is isomorphic to V (A)^{x} in a natural way, as left modules over A[d], as in Section 5.5 again.

There is a natural homomorphism from V into (5.7.9), as left modules over A[d], obtained from the action of A[d] on V on the left, as in Section 2.13. Using the identification of (5.7.9) with V (A)^{x} mentioned in the preceding paragraph, we get that this mapping corresponds to the mapping f from V into V (A)^{x} defined by f(v) = (d_{V} (v), v) (5.7.10)

for every v ∈ V.

As in Section 2.13, (5.7.2) means that V (A)^{x} corresponds to the direct sum of f(V) and another submodule of V (A)^{x}, as a left module over A[d]. This implies that there is a homomorphism g from V (A)^{x} into V, as left modules over A[d], such that

(5.7.11) g ∘ f is the identity mapping on V.

We may express g as g((v_{1}, v_{2})) = σ(v_{1}) + τ(v_{2}) (5.7.12)

for some homomorphisms σ, τ from V into itself, as a left module over A, and every v_{1}, v_{2} ∈ V.

Because g is a homomorphism from V (A)^{x} into V as left modules over A[d], we have that

d_{V} (σ(v_{1})) + d_{V} (τ(v_{2})) = d_{V} (g((v_{1}, v_{2})))
(5.7.13)  = g(d_{V (A)^{x}}((v_{1}, v_{2}))) = g((0, v_{1})) = τ(v_{1})

for every v_{1}, v_{2} ∈ V. We also have that

σ(d_{V} (v)) + τ(v) = v (5.7.14)

for every v ∈ V, by (5.7.11). One can get (5.6.1) by taking v_{1} = v and v_{2} = 0 in (5.7.13), and combining the result with (5.7.14).

If V satisfies (5.7.1) or (5.7.2), then it follows that

H(V) = {0}, (5.7.15)

by (5.6.3). This corresponds to Corollary 2.4 on p58 of [50].

5.8 Some more properties of (V, d_{V})

Let us continue with the same notation and hypotheses as at the beginning of the previous section.

Suppose for the moment that

(5.8.1) V is projective as a left module over A[d].

5.8. SOME MORE PROPERTIES OF (V, D_{V}) 129

This automatically implies that V is η-projective, as in Section 2.12. Put W = B(V), considered as a left module over A, as in Section 5.6. The remarks in the previous two sections imply that

(5.8.2) V is isomorphic to W^{x}, as a left module over A[d].

Equivalently,

(5.8.3) V is isomorphic to _{(η)}W, as a left module over A[d].

Using (5.8.1), we also get that

(5.8.4) V is projective as a left module over A,

as in Section 2.12. More precisely, this uses the fact that A[d] is projective as a left module over A, by construction. Note that V is isomorphic to W^{x}, as a left module over A, by (5.8.2). Of course, W^{x} is isomorphic to the direct sum of two copies of W, as a left module over A, by construction. It follows that

(5.8.5) W is projective as a left module over A.

Conversely, suppose that there is a left module W over A such that (5.8.3) and (5.8.5) hold. Using (5.8.3), we get that V is η-projective, as in Section 2.14. As before, (5.8.3) is equivalent to (5.8.2), which implies that V is isomorphic to W^{x}, as a left module over A. Of course, W^{x} is projective as a left module over A, by (5.8.5). This means that (5.8.4) holds. It follows that (5.8.1) holds, because V is η-projective, as in Section 2.12. This corresponds to the first part of Proposition 2.5 on p58 of [50].

Similarly, suppose for the moment that

(5.8.6) V is injective as a left module over A[d].

This implies that V is η-injective, as in Section 2.13. If W = B(V), as a left module over A again, then (5.8.2) holds, as in the previous two sections. This is equivalent to saying that

(5.8.7) V is isomorphic to ^{(η)}W, as a left module over A[d],

as before.

Note that A[d] is projective as a right module over A. Using this and (5.8.6), we obtain that (5.8.8) V is injective as a left module over A,

as in Section 2.13. As before, (5.8.2) implies that V is isomorphic to W^{x}, as a left module over A. It follows that

(5.8.9) W is injective as a left module over A,

because W^{x} is isomorphic to the direct sum of two copies of W, as a left module over A, by construction.

Conversely, suppose that (5.8.7) and (5.8.9) hold for some left module W over A. As in Section 2.15, (5.8.7) implies that V is η-injective. Because (5.8.7) is equivalent to (5.8.2), as before, we get that V is isomorphic to W^{x}, as a left module over A. Note that W^{x} is injective as a left module over A, by (5.8.9). Thus (5.8.8) holds. This implies that (5.8.6) holds, because V is η-injective, as in Section 2.13. This corresponds to the second part of Proposition 2.5 on p58 of [50].

5.9 Graded modules

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a left or right module over A. Of course, Z denotes the set of integers.

As on p58 of [50], a grading of V is defined by a family of submodules V^{j}, j ∈ Z, such that V corresponds to the direct sum of the V^{j}’s, as a module over A. One may refer to this as a Z-grading of V, to be more precise.

In this case, if v ∈ V, then v can be expressed in a unique way as

v = ∑_{j=−∞}^{∞} v^{j}, (5.9.1)

where v^{j} ∈ V^{j} for each j, and v^{j} = 0 for all but finitely many j. If j ∈ Z, then v^{j} is called the homogeneous component of v of degree j, as in [50]. Every element of V^{j} is said to be homogeneous of degree j, so that 0 is homogeneous of every degree.

If V^{j} = {0} when j < 0, then V is said to be positive as a graded module, as on p58 of [50]. Similarly, if V^{j} = {0} when j > 0, then V is said to be negative as a graded module.

It is sometimes convenient to put

V_{j} = V^{−j} (5.9.2)

for each j ∈ Z, particularly when V is negative as a graded module. Let U be a submodule of V, as a module over A. Thus

U^{j} = U ∩ V^{j} (5.9.3)

is a submodule of U, as a module over A, for every j ∈ Z. If every element of U can be expressed as the sum of elements of finitely many U^{j}’s, then U is said to be homogeneous, as a submodule of V, as on p58 of [50]. This means that U corresponds to the direct sum of the U^{j}’s, j ∈ Z, as a module over A, so that U is graded too.

Of course, the quotient V/U is a module over A as well. If j ∈ Z, then the natural quotient mapping leads to a homomorphism

(5.9.4) from V^{j}/U^{j} onto (V^{j} + U)/U, as modules over A. More precisely, the kernel of this homomorphism is trivial, by definition of U^{j}, and so we get an isomorphism as in (5.9.4). Note that every element of V/U may be expressed as the sum of elements of finitely many of the submodules (V^{j} + U)/U, because every element of V can be expressed as the sum of elements of finitely many V^{j}’s.

If U is a homogeneous submodule of V, then it is easy to see that V/U corresponds to the direct sum of the submodules (V^{j} + U)/U, j ∈ Z. This implies that V/U is graded as a module over A, with

(V/U)^{j} = (V^{j} + U)/U (5.9.5)

for each j ∈ Z, as on p58 of [50]. It is sometimes convenient to identify V/U with the direct sum of V^{j}/U^{j}, j ∈ Z, under these conditions.

Let W be another left or right module over A, depending on whether V is a left or right module over A, and suppose that W is graded too. A homomorphism ϕ from V into W, as modules over A, is said to have degree m ∈ Z if

ϕ(V^{j}) ⊆ W^{j+m} (5.9.6)

for every j ∈ Z, as on p58 of [50]. If j ∈ Z, then the restriction of ϕ to V^{j}, considered as a mapping from V^{j} into W^{j+m}, is denoted ϕ^{j}, and called the jth component of ϕ, as in [50].

Observe that

(5.9.7) ker ϕ is a homogeneous submodule of V

and (5.9.8) ϕ(V) is a homogeneous submodule of W

in this case. More precisely, if j ∈ Z, then

(5.9.9) (ker ϕ)^{j} = (ker ϕ) ∩ V^{j} = ker ϕ^{j}

and (ϕ(V))^{j} = ϕ(V) ∩ W^{j} = ϕ^{j−m}(V^{j−m}). (5.9.10)

Thus the coimage V/ ker ϕ and cokernel W/ϕ(V) of ϕ may be considered as graded modules over A, as on p58 of [50]. If j ∈ Z, then

(5.9.11) (V/ ker ϕ)^{j} is isomorphic to V^{j}/(ker ϕ)^{j} = V^{j}/ ker ϕ^{j}

and

(5.9.12) (W/ϕ(V))^{j} is isomorphic to W^{j}/(ϕ(V))^{j} = W^{j}/ϕ^{j−m}(V^{j−m}),

as modules over A, as before.

Note that the isomorphism from V/ ker ϕ onto ϕ(V), as modules over A, induced by ϕ has degree m. As a result, this should not normally be used to identify the coimage of ϕ with ϕ(V), as mentioned on p58 of [50].

If U is a homogeneous submodule of V, then the natural quotient mapping from V onto V/U has degree 0, with respect to the grading on V/U defined earlier.

Let Z be another left or right module over A, depending on whether V, W are left or right modules over A, and suppose that Z is graded as well. Also let ϕ be a homomorphism from V into W of degree m again, and let ψ be a homomorphism from W into Z, as modules over A, or degree n ∈ Z. Under these conditions, (5.9.13) ψ ∘ ϕ has degree m + n,

as a homomorphism from V into Z.

5.10 Complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. A left or right module (V, d_{V}) with differentiation is said to be a complex if V is graded and d_{V} has degree 1, as on p58 of [50].

Equivalently, suppose that V is a graded left or right module over A, and that for each j ∈ Z, d_{V}^{j} is a homomorphism from V^{j} into V^{j+1}, as modules over A. This leads to a unique homomorphism d_{V} from V into itself, as a module over A, of degree 1, with jth component equal to d_{V}^{j} for each j ∈ Z. The condition that d_{V} ∘ d_{V} = 0 on V is the same as saying that

d_{V}^{j+1} ∘ d_{V}^{j} = 0 (5.10.1)

on V_{j} for each j ∈ Z, as on p58 of [50].

The term “cochain complex” is often used for a complex in this sense, as mentioned on p58 of [50]. Similarly, a “chain complex” may be obtained by lowering indices, as in (5.9.2).

If (V, d_{V}) is a complex, then Z(V) = ker d_{V} and B(V) = d_{V} (V) are homogeneous submodules of V, as in the previous section. If j ∈ Z, then

(5.10.2) Z(V)^{j} = (ker d_{V})^{j} = (ker d_{V}) ∩ V^{j} = ker d_{V}^{j}

and B(V)^{j} = (d_{V} (V))^{j} = d_{V} (V) ∩ V^{j} = d_{V}^{j−1}(V^{j−1}), (5.10.3)

as in the previous section.

This leads to natural gradings on Z′(V) = V/d_{V} (V), B′(V) = V/ ker d_{V}, and H(V) = Z(V)/B(V), as before. If j ∈ Z, then

(5.10.4) Z′(V)^{j} = (V/d_{V} (V))^{j} is isomorphic to V^{j}/(d_{V} (V))^{j} = V^{j}/d_{V}^{j−1}(V^{j−1})

and

(5.10.5) B′(V)^{j} = (V/ ker d_{V})^{j} is isomorphic to V^{j}/(ker d_{V})^{j} = V^{j}/ ker d_{V}^{j} , as modules over A. Similarly,

(5.10.6) H(V)^{j} = (Z(V)/B(V))^{j} is isomorphic to Z(V)^{j}/B(V)^{j} = (ker d_{V}^{j})/d_{V}^{j−1}(V^{j−1}),

as modules over A. Note that

(5.10.7) H(V) is homogeneous as a submodule of Z′(V).

Remember that d̃_{V} is the homomorphism from Z′(V) into Z(V), as modules over A, induced by d_{V} in the obvous way. It is easy to see that d̃_{V} has degree 1, because of the same property of d_{V}.

Let U be a submodule of V, as a module over A with differentiation. Thus U may be considered as a module over A with differentiation too, with d_{U} equal to the restriction of d_{V} to U. Suppose that U is also homogeneous as a submodule of V, so that U is graded as well. Of course, d_{U} has degree 1 on U, so that (U, d_{U}) is a complex.

As in Section 5.3, V/U is a module over A with differentiation, where d_{V/U} is induced on V/U by d_{V} in the usual way. Similarly, V/U is graded in a natural way, because U is homogeneous, as in the previous section. It is easy to see that d_{V/U} has degree 1 on V/U, so that (V/U, d_{V/U}) is a complex too.

5.11 Maps between complexes

Let k be a commutative ring with a mutliplcative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (V, d_{V}), (W, d_{W}) be both left or both right modules over A with differentiation that are complexes. Let ϕ be a homomorphism from V into W, as modules over A with differentiation. If ϕ has degree 0, then ϕ is said to be a map from V into W as complexes, as on p59 of [50]. Note that d_{W} ∘ ϕ = ϕ ∘ d_{V} is the same as saying that d_{W}^{j} ∘ ϕ^{j} = ϕ^{j+1} ∘ d_{V}^{j} (5.11.1)

for every integer j.

As in Section 5.2, we have that ϕ(Z(V)) ⊆ Z(W), or equivalently

ϕ^{j}(Z(V)^{j}) ⊆ Z(W)^{j} (5.11.2)

for each j. Similarly, ϕ(B(V)) ⊆ B(W), which means that

ϕ^{j}(B(V)^{j}) ⊆ B(W)^{j} (5.11.3)

for every j.

Remember that ϕ_{Z′}, ϕ_{B′} are the homomorphisms from Z′(V), B′(V) into Z′(W), B′(W), respectively, induced by ϕ. It is easy to see that ϕ_{Z′} and ϕ_{B′} have degree 0. Thus the restriction ϕ_{H} of ϕ_{Z′} to H(V) has degree 0, as a homomorphism into H(W). We also have that

ϕ^{j+1} ∘ d̃_{V}^{j} = d̃_{W}^{j} ∘ ϕ_{Z′}^{j} (5.11.4)

134 CHAPTER 5. DIFFERENTIATION AND DUAL NUMBERS for each j, because of the analogous statement for ϕ in Section 5.2. This corresponds to some remarks on p59 of [50].

Let ψ be another map from V into W, as complexes. Suppose that σ is a homomorphism from V into W, as modules over A, of degree −1. If d_{W} ∘ σ + σ ∘ d_{V} = ϕ − ψ, (5.11.5) then σ is said to be a homotopy between ϕ and ψ, as maps between complexes, as on p59 of [50]. Equivalently, this means that d_{W}^{j−1} ∘ σ^{j} + σ^{j+1} ∘ d_{V}^{j} = ϕ^{j} − ψ^{j} (5.11.6) for every j. This implies that (ϕ^{j} − ψ^{j})(Z(V)^{j}) ⊆ B(W)^{j} (5.11.7) for each j, so that ϕ_{H}^{j} = ψ_{H}^{j} (5.11.8) for each j, as before.

Note that ϕ is injective on V if and only if (5.11.9) ϕ^{j} is injective on V^{j} for each j. In this case, ϕ maps Z(V) onto the intersection of ϕ(V) and Z(W), as in Section 5.2. Equivalently, ϕ^{j}(Z(V)^{j}) = ϕ^{j}(V^{j}) ∩ Z(W)^{j} (5.11.10) for each j. It follows that ϕ_{B′} is injective on B′(V), as before, which means that (5.11.11) ϕ_{B′}^{j} is injective on B′(V)^{j} for each j.

Similarly, ϕ maps V onto W if and only if ϕ^{j}(V^{j}) = W^{j} (5.11.12) for each j. Under these conditions, ϕ_{Z′} is surjective, which is the same as saying that ϕ_{Z′}^{j}(Z′(V)^{j}) = Z′(W)^{j} (5.11.13) for each j. We also have that ϕ_{B′} is surjective, or equivalently ϕ_{B′}^{j}(B′(V)^{j}) = B′(W)^{j} (5.11.14) for every j.

The surjectivity of ϕ implies that ϕ maps B(V) onto B(W), as in Section 5.2. This means that ϕ^{j}(B(V)^{j}) = B(W)^{j} (5.11.15) for each j.

If ϕ is any map from V into W as complexes, then the kernel of ϕ is a submodule of V, as a module over A with differentiation, and ker ϕ is homogeneous as a submodule of V. Similarly, ϕ(V) is a submodule of W, as a module over A with differentiation, and ϕ(V) is homogeneous as a submodule of W. This means that ker ϕ and ϕ(V) are complexes, with respect to the restriction of d_{V} to ker ϕ, and the restriction of d_{W} to ϕ(V), respectively.

5.12 Some related homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (V, d_{V}) be a left or right module over A with differentiation that is a complex. Suppose that U is a submodule of V, as a module over A with differentiation, so that U is a module with differentiation too, with d_{U} equal to the restriction of d_{V} to U. Suppose that U is homogeneous as a submodule of V as well, so that (U, d_{U}) is a complex. Note that the natural inclusion mapping from U into V is a map between complexes.

Remember that V/U is a module over A with differentiation, where d_{V/U} is induced by d_{V} in the usual way, and that V/U is a complex. The natural quotient mapping q from V onto V/U is a map between complexes.

Observe that

Z(V)^{j} ∩ (ker q^{j}) = Z(V)^{j} ∩ U^{j} = Z(U)^{j} (5.12.1)

for each j. This corresponds to a remark in Section 5.3, and is basically the same as (5.11.10). This is also related to some remarks on p59 of [50].

As in (5.11.15), we have that

q^{j}(B(V)^{j}) = B(V/U)^{j} (5.12.2)

for every integer j. If v^{j} ∈ V^{j} for some j and

q^{j}(v^{j}) ∈ B(V/U)^{j}, (5.12.3)

then there is a w^{j−1} ∈ V^{j−1} such that

q^{j}(v^{j}) = q^{j}(d_{V}^{j−1}(w^{j−1})), (5.12.4)

by (5.12.2). This means that there is a u^{j} ∈ U^{j} such that

v^{j} − d_{V}^{j−1}(w^{j−1}) = u^{j}. (5.12.5)

This implies that

(5.12.6) the images of v^{j} and u^{j} in Z′(V)^{j}, under the natural quotient mapping from V onto Z′(V), are the same.

This corresponds to a remark in Section 5.3, and is related to some remarks on p59 of [50].

If we also have that v^{j} ∈ Z(V)^{j}, (5.12.7)

then we get that u^{j} ∈ Z(U)^{j}, (5.12.8)

by (5.12.5). This corresponds to another remark in Section 5.3, and is related to some remarks on p59 of [50].

Remember that there is a natural homomorphism Δ from H(V/U) into H(U), as modules over A, as in Section 5.3. In fact, this homomorphism has degree 1. Indeed, let an integer j and an element of H(V/U)^{j} be given. This element can be represented as the image of

q^{j}(v^{j}) ∈ Z(V/U)^{j} (5.12.9)

under the natural quotient mapping from Z(V/U) onto H(V/U), where v^{j} ∈ V^{j}. Using (5.12.9), we get that

q^{j+1}(d_{V}^{j} (v^{j})) = d_{V/U}^{j}(q^{j}(v^{j})) = 0, (5.12.10)

so that d_{V}^{j} (v^{j}) ∈ U^{j+1}.

More precisely, d_{V}^{j} (v^{j}) ∈ Z(U)^{j+1}. (5.12.11)

Remember that Δ of the image of q^{j}(v^{j}) in H(V/U)^{j} is equal to the image of d_{V}^{j} (v^{j}) in H(U), under the natural quotient mapping from Z(U) onto H(U). It follows that Δ of the image of q^{j}(v^{j}) in H(V/U) is an element of H(U)^{j+1}, because of (5.12.11). This means that

Δ(H(V/U)^{j}) ⊆ H(U)^{j+1}, (5.12.12)

as desired.

Let q_{Z′} be the homomorphism from Z′(V) into Z′(V/U) induced by q, as usual. The restriction q_{H} of q_{Z′} to H(V) maps H(V) into H(V/U). Remember that Δ ∘ q_{H} = 0, as in Section 5.3. This implies that

Δ^{j} ∘ q_{H}^{j} = 0 (5.12.13)

for every j.

Let an integer j and v^{j} ∈ V^{j} be given, with q^{j}(v^{j}) ∈ Z(V/U)^{j}, and suppose that Δ^{j} maps the image of q^{j}(v^{j}) in H(V/U)^{j} to 0 in H(U)^{j+1}. This means that d_{V}^{j} (v^{j}) ∈ B(U)^{j+1}, (5.12.14)

which is to say that d_{V}^{j} (v^{j}) = d_{V}^{j} (u^{j}) for some u^{j} ∈ U^{j}. Thus

q^{j}(v^{j}) = q^{j}(v^{j} − u^{j}) and v^{j} − u^{j} ∈ Z(V)^{j}. (5.12.15)

This implies that q_{H}^{j} of the image of v^{j} − u^{j} in H(V)^{j} is the same as the image of q^{j}(v^{j}) in H(V/U)^{j}. It follows that

q_{H}^{j}(H(V)^{j}) = ker Δ^{j}, (5.12.16)

as in Section 5.3, and which is related to some remarks on p59 of [50].

Let ι be the natural inclusion mapping from U into V, and let ι_{Z′} be the induced mapping from Z′(U) into Z′(V), as usual. The restriction of ι_{Z′} to H(U) is denoted ι_{H}, as before, and maps H(U) into H(V). Remember that ι_{H} ∘ Δ = 0, as in Section 5.3, so that

ι_{H}^{j+1} ∘ Δ^{j} = 0 (5.12.17)

for every j.

Let an integer j and w^{j+1} ∈ Z(U)^{j+1} be given, and suppose that the image of w^{j+1} in H(U)^{j+1} is in the kernel of ι_{H}^{j+1}. This implies that there is a v^{j} ∈ V^{j} such that w^{j+1} = d_{V}^{j} (v^{j}). (5.12.18)

It follows that

d_{V/U}^{j}(q^{j}(v^{j})) = q^{j+1}(d_{V}^{j} (v^{j})) = q^{j+1}(w^{j+1}) = 0, (5.12.19)

so that q^{j}(v^{j}) ∈ Z(V/U)^{j}. This means that the image of w^{j+1} in H(U)^{j+1} is the same as Δ^{j} of the image of q^{j}(v^{j}) in H^{j}(V/U). This shows that

ker ι_{H}^{j+1} = Δ^{j}(H(V/U)^{j}), (5.12.20)

as in Section 5.3, and which is related to some remarks on p59 of [50].

5.13 Double gradings

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V be a left or right module over A. A double grading or bi-grading of V is defined by a family of submodules V^{j,l}, j, l ∈ Z, such that V corresponds to the direct sum of the V^{j,l}’s, as a module over A, as on p60 of [50]. One may also refer to this as a Z^{2}-grading of V, as a module over A. The elements of V^{j,l} are said to be bihomogeneous of bidegree (j, l).

If r ∈ Z, then put

V^{r} = ∑_{j+l=r} V^{j,l}, (5.13.1)

which is the subset of V consisting of finite sums of elements of V^{j,l}, with j, l ∈ Z and j + l = r. This is a submodule of V, which corresponds to the direct sum of the V^{j,l}’s with j + l = r, as a module over A. Note that V corresponds to the direct sum of the V^{r}’s, r ∈ Z, as a module over A. This is the associated grading of V by the given double grading, as on p60 of [50]. A bihomogeneous element of V of bidegree (j, l) is thus homogeneous of degree j + l with respect to the associated grading.

If V^{j,l} = {0} when j < 0 or l < 0, then V is said to be positive as a doubly-graded module, as on p60 of [50]. Similarly, if V^{j,l} = {0} when j > 0 or l > 0, then V is said to be negative as a doubly-graded module. As before, it is sometimes convenient to put

V_{j,l} = V^{−j,−l} (5.13.2)

for every j, l ∈ Z, particularly when V is negative as a doubly-graded module. Let U be a submodule of V, as a module over A, so that

U^{j,l} = U ∩ V^{j,l} (5.13.3)

is a submodule of U for every j, l ∈ Z. If every element of U can be expressed as the sum of elements of finitely many U^{j,l}’s, then U is said to be bihomogeneous, as a submodule of V, as on p60 of [50]. In this case, U corresponds to the direct sum of the U^{j,l}’s, j, l ∈ Z, as a module over A, so that U is doubly-graded as well. Note that this implies that U is a homogeneous submodule of V with respect to the associated grading.

If j, l ∈ Z, then the natural quotient mapping from V onto V/U leads to a homomorphism (5.13.4) from V^{j,l}/U^{j,l} onto (V^{j+l} + U)/U,

as modules over A. The kernel of this homomorphism is trivial, by the definition of U^{j,l}, so that we get an isomorphism as in (5.13.4). Every element of V/U can be expressed as the sum of elements of finitely many of the submodules (V^{j,l} +U)/U, because every element of V can be expressed as the sum of finitely many V^{j,l}’s.

If U is a bihomogeneous submodule of V, then one can check that V/U corresponds to the direct sum of the submodules (V^{j,l} + U)/U, j, l ∈ Z. This means that V/U is doubly-graded as a module over A, with

(V/U)^{j,l} = (V^{j,l} + U)/U (5.13.5)

for every j, l ∈ Z. In this case, it is sometimes convenient to identify V/U with the direct sum of V^{j,l}/U^{j,l}, as on p60 of [50].

Let W be another doubly-graded left or right module over A, depending on whether V is a left or right module over A. Also let ϕ be a homomorphism from V into W, as modules over A, and let p, q be integers. We say that ϕ has bidegree (p, q) if ϕ(V^{j,l}) ⊆ W^{j+p,l+q} (5.13.6)

for every j, l ∈ Z, as on p60 of [50]. Under these conditions, if j, l ∈ Z, then the restriction of ϕ to V^{j,l}, considered as a mapping from V^{j,l} into W^{j+p,l+q}, is denoted ϕ^{j,l}, and called the (p, q)-component of ϕ, as in [50]. A homomorphism of bidegree (p, q) from V into W may be considered as a homomorphism of degree p + q with respect to the associated gradings on V, W.

In this case, it is easy to see that

(5.13.7) ker ϕ is a bihomogeneous submodule of V

and (5.13.8) ϕ(V) is a bihomogeneous submodule of W.

If j, l ∈ Z, then (5.13.9) (ker ϕ)^{j,l} = (ker ϕ) ∩ V^{j,l} = ker ϕ^{j,l}

and (ϕ(V))^{j,l} = ϕ(V) ∩ W^{j,l} = ϕ^{j−p,l−q}(V^{j−p,l−q}). (5.13.10)

The coimage V/ ker ϕ and cokernel W/ϕ(V) of ϕ may be considered as doubly-graded modules over A. If j, l ∈ Z, then

(5.13.11) (V/ ker ϕ)^{j,l} is isomorphic to V^{j,l}/(ker ϕ)^{j,l} = V^{j,l}/ ker ϕ^{j,l}

and

(5.13.12) (W/ϕ(V))^{j,l} is isomorphic to W^{j,l}/ϕ(V)^{j,l} = W^{j,l}/ϕ^{j−p,l−q}(V^{j−p,l−q}),

as modules over A. The isomorphism from V/ ker ϕ onto ϕ(V), as modules over A, induced by ϕ has bidegree (p, q) as well. This is related to some remarks on p60 of [50].

If U is a bihomogeneous submodule of V, then the natural quotient mapping from V onto V/U has bidegree (0, 0), with respect to the double grading on V/U defined previously.

Let Z be another doubly-graded left or right module over A, depending on whether V, W are left or right modules over A. If ψ is a homomorphism from W into Z, as modules over A, of bidegree (p′, q′), then

(5.13.13) ψ ∘ ϕ has bidegree (p + p′, q + q′),

as a homomorphism from V into Z.

One may also consider n-graded modules over A for any positive integer n, as on p62 of [50].

5.14 Double complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a doubly-graded left or right module over A.

Suppose that d_{1} = d_{V,1} and d_{2} = d_{V,2} are two differentiation operators on V, which is to say that d_{1}, d_{2} are homomorphisms from V into itself, as a module over A, such that d_{1} ∘ d_{1} = d_{2} ∘ d_{2} = 0. (5.14.1)

Suppose as well that d_{1} has bidegree (1, 0) on V, d_{2} has bidegree (0, 1), and that d_{1}, d_{2} anticommute on V, so that

d_{2} ∘ d_{1} + d_{1} ∘ d_{2} = 0. (5.14.2)

Under these conditions, V is said to be a double complex with respect to d_{1}, d_{2}, as on p60 of [50].

If j, l ∈ Z, then we get homomorphisms d_{1}^{j,l}, d_{2}^{j,l} from V^{j,l} into V^{j+1,l}, V^{j,l+1}, respectively, as modules over A, as in the previous section. As before, (5.14.1) is the same as saying that d_{1}^{j+1,l} ∘ d_{1}^{j,l} = 0 (5.14.3)

nad d_{2}^{j,l+1} ∘ d_{2}^{j,l} = 0 (5.14.4)

on V^{j,l} for every j, l ∈ Z. Similarly, (5.14.2) is the same as asking that

d_{2}^{j+1,l} ∘ d_{1}^{j,l} + d_{1}^{j,l+1} ∘ d_{2}^{j,l} = 0 (5.14.5)

on V^{j,l} for every j, l ∈ Z.

Remember that V may be considered as a singly-graded module over A, with V^{r} as in (5.13.1) for every r ∈ Z. Put

d = d_{V} = d_{1} + d_{2}, (5.14.6)

which is a homomorphism from V into itself of degree 1, as a singly-graded module over A. Observe that d ∘ d = 0 on V, by (5.14.1) and (5.14.2). Thus V is a single complex with respect to d, as on p61 of [50]. One calls d the total differentiation operator on V, as in [50]. If j, l ∈ Z, then the restriction of d to V^{j,l} is equal to d_{1}^{j,l} + d_{2}^{j,l}. (5.14.7)

In particular, d(V^{j,l}) ⊆ V^{j+1,l} + V^{j,l+1}. (5.14.8)

Conversely, suppose that d is a homomorphism from V into itself of degree 0, as a singly-graded module over A, such that d∘d = 0 and (5.14.8) holds for every j, l ∈ Z. If j, l ∈ Z, then there are unique homomorphisms d_{1}^{j,l}, d_{2}^{j,l} from V^{j,l} into V^{j+1,l}, V^{j,l+1}, respectively, as modules over A, such that the restriction of d to V^{j,l} is equal to (5.14.7). One can check that (5.14.3), (5.14.4), and (5.14.5) hold for every j, l ∈ Z in this case. Let d_{1}, d_{2} be the homomorphisms from V into itself whose restrictions to V^{j,l} are equal to d_{1}^{j,l}, d_{2}^{j,l}, respectively, for every j, l ∈ Z. It follows that V is a double complex with respect to d_{1}, d_{2}, and that d is the corresponding total differentiation operator, as on p61 of [50].

If V is a double complex, then the corresponding modules Z(V), B(V), Z′(V), B′(V), and H(V) are defined by considering V as a single complex, as before. More precisely, these are singly-graded modules over A.

If n is any positive integer, then one may consider n-complexes over A, as on p62 of [50].

5.15 Maps between double complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with multiplicative identity element e_{A}. Also let V and W be both left or both right modules over A that are double complexes, with differentiation operators d_{V,1}, d_{V,2} and d_{W,1}, d_{W,2}, respectively.

Suppose that ϕ is a homomorphism from V into W, as modules over A, of bidegree (0, 0). If d_{W,1} ∘ ϕ = ϕ ∘ d_{V,1} (5.15.1)

and d_{W,2} ∘ ϕ = ϕ ∘ d_{V,2}, (5.15.2)

then ϕ is said to be a map from V into W, as double complexes, as on p61 of [50]. In this case, it is easy to see that ϕ is a map from V into W, considered as single complexes, as in the previous section.

Note that (5.15.1) is the same as saying that

d_{W,1}^{j,l} ∘ ϕ^{j,l} = ϕ^{j+1,l} ∘ d_{V,1}^{j,l} (5.15.3)

for every j, l ∈ Z. Similarly, (5.15.2) is the same as saying that

d_{W,2}^{j,l} ∘ ϕ^{j,l} = ϕ^{j,l+1} ∘ d_{V,2}^{j,l} (5.15.4)

for every j, l.

Let ϕ and ψ be maps from V into W, as double complexes. Also let σ_{1}, σ_{2} be homomorphisms from V into W, as modules over A, of bidegrees (−1, 0), (0, −1), respectively. Suppose that

d_{W,1} ∘ σ_{1} + σ_{1} ∘ d_{V,1} + d_{W,2} ∘ σ_{2} + σ_{2} ∘ d_{V,2} = ϕ − ψ. (5.15.5)

Suppose in addition that

σ_{1} ∘ d_{V,2} + d_{W,2} ∘ σ_{1} = 0 (5.15.6)

and σ_{2} ∘ d_{V,1} + d_{W,1} ∘ σ_{2} = 0. (5.15.7)

Under these conditions, (σ_{1}, σ_{2}) is said to define a homotopy between ϕ and ψ, as on p61 of [50].

Of course, (5.15.5) is the same as saying that

d_{W,1}^{j−1,l} ∘ σ_{1}^{j,l} + σ_{1}^{j+1,l} ∘ d_{V,1}^{j,l} + d_{W,2}^{j,l−1} ∘ σ_{2}^{j,l} + σ_{2}^{j,l+1} ∘ d_{V,2}^{j,l} (5.15.8)

= ϕ^{j,l} − ψ^{j,l}

for every j, l ∈ Z. Similarly, (5.15.6) is the same as saying that

σ_{1}^{j,l+1} ∘ d_{V,2}^{j,l} + d_{W,2}^{j−1,l} ∘ σ_{1}^{j,l} = 0 (5.15.9) for every j, l. We also have that (5.15.7) is the same as saying that

σ_{2}^{j+1,l} ∘ d_{V,1}^{j,l} + d_{W,1}^{j,l−1} ∘ σ_{2}^{j,l} = 0 (5.15.10)

for every j, l.

Put σ = σ_{1} + σ_{2}, (5.15.11)

which is a homomorphism from V into W of degree −1, as singly-graded modules over A. Let d_{V}, d_{W} be the total differentiation operators on V, W, respectively. One can check that d_{W} ∘ σ + σ ∘ d_{V} = ϕ − ψ, (5.15.12)

using (5.15.8), (5.15.9), and (5.15.10). This means that σ is a homotopy between ϕ and ψ, as maps from V into W, as single complexes over A, as on p62 of [50]. If j, l ∈ Z, then the restriction of σ to V^{j,l} is equal to

σ_{1}^{j,l} + σ_{2}^{j,l}, (5.15.13)

so that σ(V^{j,l}) ⊆ V^{j−1,l} + V^{j,l−1}, (5.15.14)

as in [50].

Conversely, suppose that σ is a homotopy between ϕ and ψ, as maps from V into W, as single complexes over A, that satisfies (5.15.14) for every j, l ∈ Z. In this case, if j, l ∈ Z, then there are unique homomorphisms σ_{1}^{j,l}, σ_{2}^{j,l} from V^{j,l} into V^{j−1,l}, V^{j,l−1}, respectively, as modules over A, such that the restriction of σ to V^{j,l} is equal to (5.15.13). This leads to homomorphisms σ_{1}, σ_{2} from V into W, as modules over A, of bidegrees (−1, 0), (0, −1), respectively, whose restrictions to V^{j,l} are equal to σ_{1}^{j,l}, σ_{2}^{j,l}, respectively, for each j, l ∈ Z. One can check that (5.15.8), (5.15.9), and (5.15.10) hold for every j, l ∈ Z, because of (5.15.12). This implies that (5.15.5), (5.15.6), and (5.15.7) hold, so that (σ_{1}, σ_{2}) is a homotopy between ϕ and ψ, as maps from V into W as double complexes, as on p62 of [50].

More on differentiation

6.1 Tensor products and gradings

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a graded right module over A, and let W be a graded left module over A.

Suppose that V ⨂_{A} W is a tensor product of V and W, as right and left modules over A, respectively. Thus V ⨂_{A} W is a module over k that is isomorphic to the direct sum of a family of tensor products V^{j} ⨂_{A} W^{l}, j, l ∈ Z_{+}. This means that V ⨂_{A} W is a doubly-graded module over k, with

(V ⨂_{A}W )^{j,l} = V^{j} ⨂_{A}W^{l} (6.1.1)

for every j, l ∈ Z, considered as a submodule of V ⨂_{A} W. This corresponds to a remark on p63 of [50].

If r ∈ Z, then put

(V ⨂_{A}W )^{r} = ∑_{j+l=r} (V ⨂_{A}W )^{j,l} = ∑_{j+l=r} V^{j} ⨂_{A}W^{l}, (6.1.2)

as in Section 5.13. This is the subset of V ⨂_{A} W consisting of finite sums of elements of (6.1.1), with j, l ∈ Z and j + l = r, as before. This is a submodule of V ⨂_{A} W, as a module over k, which corresponds to the direct sum of (6.1.1) with j + l = r, as a module over k. We also have that V ⨂_{A} W corresponds to the direct sum of (6.1.2) with r ∈ Z, as a module over k. This defines the grading on V ⨂_{A} W associated to the double grading defined in the preceding paragraph, as in Section 5.13.

Let V_{1} be another graded right module over A, let W be another graded left module over A, and let V_{1} ⨂_{A} W_{1} be a tensor product of V_{1} and W_{1} over A. This is a doubly-graded module over k, as before.

Let ϕ_{1}, ψ_{1} be homomorphisms from V, W into V_{1}, W_{1}, as modules over A, of degrees p_{1}, q_{1} ∈ Z, respectively. The corresponding mapping from V ⨂_{A} W into V_{1} ⨂_{A} W_{1} is defined a bit differently from before, without gradings. Namely, there is a unique homomorphism

(6.1.3) ϕ_{1} ⊗ ψ_{1} from V ⨂_{A}W into V_{1} ⨂_{A}W_{1},

as modules over k, such that for every j, l ∈ Z, v^{j} ∈ V^{j}, and w^{l} ∈ W_{l}, we have that (ϕ_{1} ⊗ ψ_{1})(v^{j} ⊗ w^{l}) = (−1)^{j q_{1}}(ϕ_{1}(v^{j}) ⊗ ψ_{1}(w^{l})), (6.1.4)

as on p64 of [50]. Note that

(6.1.5) ϕ_{1} ⊗ ψ_{1} has bidegree (p_{1}, q_{1}).

Equivalently, if j, l ∈ Z, then the restriction (ϕ_{1} ⊗ ψ_{1})^{j,l} of ϕ_{1} ⊗ ψ_{1} to (V ⨂_{A} W )^{j,l} is a homomorphism into (V_{1} ⨂_{A} W_{1})^{j+p_{1},l+q_{1}}, as modules over k. This homomorphism corresponds to

(6.1.6) (−1)^{j q_{1}} times the homomorphism from V^{j} ⨂_{A}W^{l} into

V_{1}^{j+p_{1}} ⨂_{A}W_{1}^{l+q_{1}}, as modules over k, associated to ϕ_{1}^{j} and ψ_{1}^{l}

in the usual way, as on p63 of [50].

Let V_{2} be a third graded right module over A, let W_{2} be a third graded left module over A, and let V_{2} ⨂_{A} W_{2} be a tensor product of V_{2} and W_{2} over A, which is a doubly-graded module over k. Also let ϕ_{2}, ψ_{2} be homomorphisms from V_{1}, W_{2} into V_{2}, W_{2}, as modules over A, of degrees p_{2}, q_{2} ∈ Z, respectively. This leads to a homomorphism ϕ_{2} ⊗ ψ_{2} from V_{1} ⨂_{A} W_{1} into V_{2} ⨂_{A} W_{2}, as modules over k, of bidegree (p_{2}, q_{2}), as before.

Similarly, ϕ_{2} ∘ ϕ_{1}, ψ_{2} ∘ ψ_{1} are homomorphisms from V, W into V_{2}, W_{2}, as modules over A, of degrees p_{1} + p_{2}, q_{1} + q_{2}, respectively. This leads to a homomorphism (ϕ_{2} ∘ ϕ_{1}) ⊗ (ψ_{2} ∘ ψ_{1}) from V ⨂_{A} W into V_{2} ⨂_{A} W_{2}, as modules over k, of bidegree (p_{1} + p_{2}, q_{1} + q_{2}). One can check that

(ϕ_{2} ∘ ϕ_{1}) ⊗ (ψ_{2} ∘ ψ_{1}) = (−1)^{p_{1} q_{2}} (ϕ_{2} ⊗ ψ_{2}) ∘ (ϕ_{1} ⊗ ψ_{1}), (6.1.7)

as on p63 of [50].

6.2 Tensor products of complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (V, d_{V}) be a graded right module over A with differentiation that is a complex, and let (W, d_{W}) be a graded left module over A with differentiation that is a complex. Suppose that V ⨂_{A} W is a tensor product of V and W over A, which is doubly-graded as a module over k, as in the previous section.

The identity mappings I_{V}, I_{W} on V, W are homomorphisms from V, W into themselves, respectively, as modules over A, of degree 0. Let

δ_{1} = d_{V} ⊗ I_{W} (6.2.1) and δ_{2} = I_{V} ⊗ d_{W} (6.2.2)

be the homomorphisms from V ⨂_{A} W into itself, as a module over k, defined as in the previous section. If j, l ∈ Z, v^{j} ∈ V_{j}, and w^{l} ∈ W^{l}, then

δ_{1}(v^{j} ⊗ w^{l}) = d_{V} (v^{j}) ⊗ w^{l} (6.2.3)

and δ_{2}(v^{j} ⊗ w^{l}) = (−1)^{j} (v^{j} ⊗ d_{W} (w^{l})), (6.2.4)

and δ_{1}, δ_{2} are uniquely determined by these properties. Of course, δ_{1} has bidegree (1, 0), and δ_{2} has bidegree (0, 1).

Observe that δ_{1} ∘ δ_{1} = δ_{2} ∘ δ_{2} = 0 (6.2.5)

on V ⨂_{A} W, because d_{V} ∘ d_{V} = 0 on V and d_{W} ∘ d_{W} = 0 on W. We also have that δ_{1} ∘ δ_{2} + δ_{2} ∘ δ_{1} = 0 (6.2.6)

on V ⨂_{A} W, by (6.1.7). This shows that

(6.2.7) V ⨂_{A}W is a double complex with respect to δ_{1} and δ_{2},

as on p63f of [50].

Let (V_{0}, d_{V_{0}}) be another graded right module over A with differentiation that is a complex, let (W_{0}, d_{W_{0}}) be another graded left module over A with differentiation that is a complex, and let V_{0} ⨂_{A} W_{0} be a tensor product of V_{0} and W_{0} over A. This is a double complex with respect to the corresponding differentiation operators δ_{1,0} and δ_{2,0}, as before.

Let ϕ, ψ be mappings from V, W into V_{0}, W_{0}, respectively, as complexes. In this case, ϕ ⊗ ψ is the same as the mapping from V ⨂_{A} W into V_{0} ⨂_{A} W_{0} defined without using gradings, because ψ has degree 0, by hypothesis. One can check that (6.2.8) ϕ ⊗ ψ is a map from V ⨂_{A}W into V_{0} ⨂_{A}W_{0},

as double complexes, as on p63 of [50].

Let ϕ_{0}, ψ_{0} be another pair of mappings from V, W into V_{0}, W_{0}, respectively, as complexes. Suppose that σ, τ are homotopies between ϕ, ψ and ϕ_{0}, ψ_{0}, respectively. Thus σ, τ are homomorphisms from V, W into V_{0}, W_{0}, respectively, as modules over A, and with degree −1, such that

d_{V_{0}} ∘ σ + σ ∘ d_{V} = ϕ − ϕ_{0} (6.2.9)

and d_{W_{0}} ∘ τ + τ ∘ d_{W} = ψ − ψ_{0}. (6.2.10)

Put σ_{1} = σ ⊗ ψ (6.2.11)

146 CHAPTER 6. MORE ON DIFFERENTIATION and σ_{2} = ϕ_{0} ⊗ τ, (6.2.12) which are homomorphisms from V ⨂_{A} W into V_{0} ⨂_{A} W_{0}, as modules over k, with bidegrees (−1, 0) and (0, −1), respectively.

Observe that

σ_{1} ∘ δ_{2} = (σ ⊗ ψ) ∘ (I_{V} ⊗ d_{W}) = σ ⊗ (ψ ∘ d_{W}) = σ ⊗ (d_{W_{0}} ∘ ψ)
(6.2.13)  = −(I_{V_{0}} ⊗ d_{W_{0}}) ∘ (σ ⊗ ψ) = −δ_{2,0} ∘ σ_{1},

by (6.1.7). Similarly,

σ_{2} ∘ δ_{1} = (ϕ_{0} ⊗ τ) ∘ (d_{V} ⊗ I_{W}) = −(ϕ_{0} ∘ d_{V}) ⊗ τ = −(d_{V_{0}} ∘ ϕ_{0}) ⊗ τ
(6.2.14)  = −(d_{V_{0}} ⊗ I_{W_{0}}) ∘ (ϕ_{0} ⊗ τ) = −δ_{1,0} ∘ σ_{2}.

We also have that δ_{1,0} ∘ σ_{1} = (d_{V_{0}} ⊗ I_{W_{0}}) ∘ (σ ⊗ ψ) = (d_{V_{0}} ∘ σ) ⊗ ψ (6.2.15) and σ_{1} ∘ δ_{1} = (σ ⊗ ψ) ∘ (d_{V} ⊗ I_{W}) = (σ ∘ d_{V}) ⊗ ψ. (6.2.16) This implies that δ_{1,0} ∘ σ_{1} + σ_{1} ∘ δ_{1} = (d_{V_{0}} ∘ σ + σ ∘ d_{V}) ⊗ ψ = (ϕ − ϕ_{0}) ⊗ ψ. (6.2.17)

Similarly, δ_{2,0} ∘ σ_{2} = (I_{V_{0}} ⊗ d_{W_{0}}) ∘ (ϕ_{0} ⊗ τ) = ϕ_{0} ⊗ (d_{W_{0}} ∘ τ) (6.2.18) and σ_{2} ∘ δ_{2} = (ϕ_{0} ⊗ τ) ∘ (I_{V} ⊗ d_{W}) = ϕ_{0} ⊗ (τ ∘ d_{W}). (6.2.19) It follows that δ_{2,0} ∘ σ_{2} + σ_{2} ∘ δ_{2} = ϕ_{0} ⊗ (d_{W_{0}} ∘ τ + τ ∘ d_{W}) = ϕ_{0} ⊗ (ψ − ψ_{0}). (6.2.20)

Combining (6.2.17) and (6.2.20), we obtain that δ_{1,0} ∘ σ_{1} + σ_{1} ∘ δ_{1} + δ_{2,0} ∘ σ_{2} + σ_{2} ∘ δ_{2} (6.2.21)

= (ϕ − ϕ_{0}) ⊗ ψ + ϕ_{0} ⊗ (ψ − ψ_{0}) = ϕ ⊗ ψ − ϕ_{0} ⊗ ψ_{0}.

This shows that (6.2.22) (σ_{1}, σ_{2}) defines a homotopy between ϕ ⊗ ψ and ϕ_{0} ⊗ ψ_{0}, as maps from V ⨂_{A} W into V_{0} ⨂_{A} W_{0}, as double complexes. This corresponds to a remark on p63 of [50].

6.3 Gradings and homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose for the moment that V, W are both left or both right modules over A, so that the corresponding space Hom_{A}(V, W) of homomorphisms from V into W, as modules over A, is a module over k.

Let V_{1}, W_{1} be left or right modules over A, depending on whether V, W are left or right modules over A. Also let ϕ_{1} be a homomorphism from V_{1} into V, and let ψ_{1} be a homomorphism from W into W_{1}, as modules over A. If α ∈ Hom(V, W), then put

(Hom(ϕ_{1}, ψ_{1}))(α) = ψ_{1} ∘ α ∘ ϕ_{1}, (6.3.1)

as on p20 of [50]. Note that Hom(ϕ_{1}, ψ_{1}) defines a homomorphism

(6.3.2) from Hom_{A}(V, W) into Hom_{A}(V_{1}, W_{1}),

as modules over k, as in [50].

Let V_{2}, W_{2} be left or right modules over A as well, depending on whether V, W are left or right modules over A. Suppose that ϕ_{2} is a homomorphism from V_{2} into V_{1}, and that ψ_{2} is a homomorphism from W_{1} into W_{2}, as modules over A. Thus Hom(ϕ_{2}, ψ_{2}) defines a homomorphism

(6.3.3) from Hom_{A}(V_{1}, W_{1}) into Hom_{A}(V_{2}, W_{2}),

as modules over k, as before.

Under these conditions, ϕ_{1} ∘ ϕ_{2} is a homomorphism from V_{2} into V, and ψ_{2} ∘ ψ_{1} is a homomorphism from W into W_{2}, as modules over A. This means that Hom(ϕ_{1} ∘ ϕ_{2}, ψ_{2} ∘ ψ_{1}) defines a homomorphism

(6.3.4) from Hom_{A}(V, W) into Hom_{A}(V_{2}, W_{2}),

as usual. One can check that

(6.3.5) Hom(ϕ_{1} ∘ ϕ_{2}, ψ_{2} ∘ ψ_{1}) = Hom(ϕ_{2}, ψ_{2}) ∘ Hom(ϕ_{1}, ψ_{1}).

Suppose now that V, W are graded modules over A. We would like to define Hom_{A}^{gr}(V, W) as a doubly-graded module over k, with

(Hom_{A}^{gr}(V, W))^{j,l} = Hom_{A}(V^{−j}, W^{l}) (6.3.6)

for every j, l ∈ Z, as on p62 of [50]. We can simply take Hom_{A}^{gr}(V, W) to be the direct sum of (6.3.6), as a module over k, but it is sometimes convenient to realize it in a slightly different way.

Let |V |, |W | be the underlying modules over A corresponding to V, W, respectively, without gradings, as on p63 of [50]. If α ∈ Hom_{A}^{gr}(V, W) and j, l ∈ Z, then let α^{j,l} be the corresponding element of (6.3.6), as usual. This leads to a unique homomorphism from |V | into |W |, that we shall also denote by α, with

α = ∑_{l=−∞}^{∞} α^{j,l} on V^{−j} (6.3.7)

for every j ∈ Z. More precisely, α^{j,l} = 0 for all but finitely many (j, l) ∈ Z^{2}, by hypothesis. This implies that all but finitely many terms in the sum on the right side of (6.3.7) are equal to 0 for any j, and that all of the terms in the sum are equal to 0 for all but finitely many j.

If j, l ∈ Z, then α^{j,l} is uniquely determined by this homomorphism from |V | into |W |, by restricting the homomorphism to V^{−j}, and taking the component of this restriction that maps into W^{l}. This defines an injective homomorphism

(6.3.8) from Hom_{A}^{gr}(V, W) into Hom_{A}(|V |, |W |),

as modules over k, and we may identify Hom_{A}^{gr}(V, W) with the corresponding submodule of Hom_{A}(|V |, |W |), as a module over k. Note that Hom_{A}^{gr}(V, W) normally corresponds to a proper submodule of Hom_{A}(|V |, |W |), as on p63 of [50].

If r ∈ Z, then put

(Hom_{A}^{gr}(V, W))^{r} = ∑_{j+l=r} (Hom_{A}^{gr}(V, W))^{j,l}, (6.3.9)

as in Section 5.13. This is a submodule of Hom_{A}^{gr}(V, W), as a module over k, which corresponds to the direct sum of (6.3.6) over all j, l ∈ Z with j + l = r. As before, Hom_{A}^{gr}(V, W) corresponds to the direct sum of (6.3.9) over r ∈ Z, as a module over k, which defines the grading on Hom_{A}^{gr}(V, W) associated to the double grading already defined.

Let M^{r}(V, W) be the space of homomorphisms from V into W, as modules over A, of degree r, as in Exercise 5 on p73 of [50]. This is a submodule of Hom_{A}(|V |, |W |), as a module over k. Observe that

(Hom_{A}^{gr}(V, W))^{r} = (Hom_{A}^{gr}(V, W)) ∩ M^{r}(V, W). (6.3.10)

Put

M(V, W) = ∑_{r=−∞}^{∞} M^{r}(V, W), (6.3.11)

as in Exercise 5 on p73 of [50]. This is a submodule of Hom_{A}(|V |, |W |), as a module over k, which corresponds to the direct sum of M^{r}(V, W), r ∈ Z. We also have that Hom_{A}^{gr}(V, W) ⊆ M(V, W), (6.3.12)

by (6.3.10).

6.4. INDUCED HOMOMORPHISMS ON HOM_{A}^{GR}(V, W) 149

6.4 Induced homomorphisms on Hom_{A}^{gr}(V, W)

Let us continue with the same notation and hypotheses as in the previous section. Let V_{1}, W_{1} be left or right graded modules over A, depending on whether V, W are left or right modules over A. Also let ϕ_{1} be a homomorphism from V_{1} into V, and let ψ_{1} be a homomorphism from W into W_{1}, as modules over A, of degrees p_{1}, q_{1} ∈ Z, respectively. The corresponding homomorphism Hom^{gr}(ϕ_{1}, ψ_{1}) (6.4.1) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{1}, W_{1}),

as modules over k, is defined a bit differently from before, without gradings. More precisely, (6.4.2) Hom^{gr}(ϕ_{1}, ψ_{1}) has bidegree (p_{1}, q_{1}).

If j, l ∈ Z, then Hom^{gr}(ϕ_{1}, ψ_{1}) should map

(6.4.3) (Hom_{A}^{gr}(V, W))^{j,l} into (Hom_{A}^{gr}(V_{1}, W_{1}))^{j+p_{1},l+q_{1}}.

This corresponds to a mapping

(6.4.4) from Hom_{A}(V^{−j}, W^{l}) into Hom_{A}(V_{1}^{−j−p_{1}}, W_{1}^{l+q_{1}}),

as in (6.3.6). Note that

ϕ_{1}^{−j−p_{1}}(V_{1}^{−j−p_{1}}) ⊆ V^{−j} (6.4.5)

and ψ_{1}^{l}(W^{l}) ⊆ W_{1}^{l+q_{1}}, (6.4.6)

by hypothesis. Thus Hom(ϕ_{1}^{−j−p_{1}}, ψ_{1}^{l}) defines a homomorphism as in (6.4.4), as modules over k.

We would like to take

(6.4.7) the restriction of Hom^{gr}(ϕ_{1}, ψ_{1}) to (Hom_{A}^{gr}(V, W))^{j,l} to be

the mapping corresponding to (−1)^{j q_{1}} Hom(ϕ_{1}^{−j−p_{1}}, ψ_{1}^{l}),

as on p63 of [50]. Of course, there is a unique homomorphism as in (6.4.1), as modules over k, with this property.

Let V_{2}, W_{2} be another pair of left or right graded modules over A, depending on whether V, W are left or right modules over A. Also let ϕ_{2} be a homomorphism from V_{2} into V_{1}, and let ψ_{2} be a homomorphism from W_{1} into W_{2}, as modules over A, of degrees p_{2}, q_{2} ∈ Z, respectively. This leads to a homomorphism Hom^{gr}(ϕ_{2}, ψ_{2})

(6.4.8) from Hom_{A}^{gr}(V_{1}, W_{1}) into Hom_{A}^{gr}(V_{2}, W_{2}),

as modules over k, as before. Of course, this homomorphism has bidegree (p_{2}, q_{2}).

It follows that ϕ_{1} ∘ ϕ_{2} is a homomorphism from V_{2} into V, and that ψ_{2} ∘ ψ_{1} is a homomorphism from W into W_{2}, as modules over A, of degrees p_{1} + p_{2}, q_{1} + q_{2}, respectively. Thus we get a homomorphism Hom^{gr}(ϕ_{1} ∘ ϕ_{2}, ψ_{2} ∘ ψ_{1})

(6.4.9) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{2}, W_{2}),

as modules over k, of bidegree (p_{1} + p_{2}, q_{1} + q_{2}). One can verify that

(6.4.10) Hom^{gr}(ϕ_{1} ∘ ϕ_{2}, ψ_{2} ∘ ψ_{1}) = (−1)^{p_{1} q_{2}} Hom^{gr}(ϕ_{2}, ψ_{2}) ∘ Hom^{gr}(ϕ_{1}, ψ_{1}),

as on p63 of [50]. More precisely, it suffices to check that this holds on (6.3.6) for each j, l ∈ Z. This uses (6.3.5), and the way that these homomorphisms are defined.

6.5 Complexes and homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (V, d_{V}) and (W, d_{W}) be both left or both right graded modules over A with differentiation that are complexes. Thus Hom_{A}^{gr}(V, W) is a doubly-graded module over k, as in Section 6.3.

The identity mappings I_{V}, I_{W} on V, W are homomorphisms from V, W into themselves, respectively, of degree 0, as before. Let

δ_{1} = Hom^{gr}(d_{V} , I_{W}) (6.5.1)

and δ_{2} = Hom^{gr}(I_{V} , d_{W}) (6.5.2)

be the homomorphisms from Hom_{A}^{gr}(V, W) into itself, as a module over k, defined as in the previous section. Remember that δ_{1} has bidegree (1, 0), and δ_{2} has bidegree (0, 1).

It is easy to see that δ_{1} ∘ δ_{1} = δ_{2} ∘ δ_{2} = 0 (6.5.3)

on Hom_{A}^{gr}(V, W), because d_{V} ∘ d_{V} = 0 on V and d_{W} ∘ d_{W} = 0 on W, and using (6.4.10). One can also use (6.4.10) to get that

δ_{1} ∘ δ_{2} + δ_{2} ∘ δ_{1} = 0 (6.5.4)

on Hom_{A}^{gr}(V, W). This means that

(6.5.5) Hom_{A}^{gr}(V, W) is a double complex with respect to δ_{1} and δ_{2},

as on p63 of [50].

Let (V_{0}, d_{V_{0}}), (W_{0}, d_{W_{0}}) be another pair of left or right graded modules over A with differentiation that are complexes, depending on whether V, W are left or right modules over A. As before, Hom_{A}^{gr}(V_{0}, W_{0}) is a doubly-graded module over k, and we let δ_{1,0}, δ_{2,0} be the corresponding differentiation operators on it.

Let ϕ be a map from V_{0} into V, and let ψ be a map from W into W_{0}, as complexes. This leads to a homomorphism Hom^{gr}(ϕ, ψ) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W_{0}), as modules over k, of bidegree (0, 0), as in the previous section. This is the same as the homomorphism Hom(ϕ, ψ) defined as in Section 6.3 without using gradings, restricted to Hom_{A}^{gr}(V, W), because ψ has degree 0. One can verify that

(6.5.6) Hom^{gr}(ϕ, ψ) is a map from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W_{0}),

as double complexes, as on p63 of [50].

Let ϕ_{0} be another map from V_{0} into V, and let ψ_{0} be another map from W into W_{0}, as complexes. Also let σ, τ be homotopies between ϕ, ψ and ϕ_{0}, ψ_{0}, respectively. This means that σ is a homomorphism from V_{0} into V, as modules over A, with degree −1, such that

d_{V} ∘ σ + σ ∘ d_{V_{0}} = ϕ − ϕ_{0}. (6.5.7)

Similarly, τ is a homomorphism from W into W_{0}, as modules over A, of degree −1, with d_{W_{0}} ∘ τ + τ ∘ d_{W} = ψ − ψ_{0}. (6.5.8)

Note that σ_{1} = Hom^{gr}(σ, ψ) (6.5.9)

and σ_{2} = Hom^{gr}(ϕ_{0}, τ) (6.5.10)

are homomorphisms from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W_{0}), as modules over k, of bidegrees (−1, 0) and (0, −1), respectively.

It is easy to see that

(6.5.11) σ_{1} ∘ δ_{2} = Hom^{gr}(σ, ψ) ∘ Hom^{gr}(I_{V} , d_{W})
= Hom^{gr}(σ, ψ ∘ d_{W}) = Hom^{gr}(σ, d_{W_{0}} ∘ ψ)
= −Hom^{gr}(I_{V_{0}}, d_{W_{0}}) ∘ Hom^{gr}(σ, ψ) = −δ_{2,0} ∘ σ_{1},

using (6.4.10). Similarly,

(6.5.12) σ_{2} ∘ δ_{1} = Hom^{gr}(ϕ_{0}, τ) ∘ Hom^{gr}(d_{V} , I_{W})
= −Hom^{gr}(d_{V} ∘ ϕ_{0}, τ) = −Hom^{gr}(ϕ_{0} ∘ d_{V_{0}}, τ)
= −Hom^{gr}(d_{V_{0}}, I_{W_{0}}) ∘ Hom^{gr}(ϕ_{0}, τ) = −δ_{1,0} ∘ σ_{2}.

Observe that

(6.5.13) δ_{1,0} ∘ σ_{1} = Hom^{gr}(d_{V_{0}}, I_{W_{0}}) ∘ Hom^{gr}(σ, ψ) = Hom^{gr}(σ ∘ d_{V_{0}}, ψ)

and

(6.5.14) σ_{1} ∘ δ_{1} = Hom^{gr}(σ, ψ) ∘ Hom^{gr}(d_{V} , I_{W}) = Hom^{gr}(d_{V} ∘ σ, ψ).

This shows that

(6.5.15)  δ_{1,0} ∘ σ_{1} + σ_{1} ∘ δ_{1} = Hom^{gr}(d_{V_{0}} ∘ σ + σ ∘ d_{V} , ψ)
= Hom^{gr}(ϕ − ϕ_{0}, ψ).

Similarly,

(6.5.16) δ_{2,0} ∘ σ_{2} = Hom^{gr}(I_{V_{0}}, d_{W_{0}}) ∘ Hom^{gr}(ϕ_{0}, τ) = Hom^{gr}(ϕ_{0}, d_{W_{0}} ∘ τ)

and

(6.5.17) σ_{2} ∘ δ_{2} = Hom^{gr}(ϕ_{0}, τ) ∘ Hom^{gr}(I_{V} , d_{W}) = Hom^{gr}(ϕ_{0}, τ ∘ d_{W}).

This implies that

(6.5.18)  δ_{2,0} ∘ σ_{2} + σ_{2} ∘ δ_{2} = Hom^{gr}(ϕ_{0}, d_{W_{0}} ∘ τ + τ ∘ d_{W})
= Hom^{gr}(ϕ_{0}, ψ − ϕ_{0}).

It follows that

δ_{1,0} ∘ σ_{1} + σ_{1} ∘ δ_{1} + δ_{2,0} ∘ σ_{2} + σ_{2} ∘ δ_{2} (6.5.19)

= Hom^{gr}(ϕ − ϕ_{0}, ψ) + Hom^{gr}(ϕ_{0}, ψ − ψ_{0})
= Hom^{gr}(ϕ, ψ) + Hom^{gr}(ϕ_{0}, ψ_{0}).

This means that

(6.5.20) (σ_{1}, σ_{2}) defines a homotopy between Hom^{gr}(ϕ, ψ) and Hom^{gr}(ϕ_{0}, ψ_{0}),

as maps from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W_{0}), as double complexes. This corresponds to a remark on p63 of [50], as before.

6.6 Maps of any degree

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (V, d_{V}), (W, d_{W}) be both left or both right graded modules over A with differentiation that are complexes.

Suppose that ϕ is a homomorphism from V into W, as modules over A, of degree r ∈ Z such that d_{W} ∘ ϕ = (−1)^{r} ϕ ∘ d_{V} . (6.6.1)

Under these conditions, ϕ is considered to be a map of degree r from V into W, as complexes, as in Exercise 1 on p72 of [50]. If r = 0, then this reduces to the definition of a map between complexes in Section 5.11. Of course, (6.6.1) is the same as saying that d_{W}^{j+r} ∘ ϕ^{j} = (−1)^{r} ϕ^{j+1} ∘ d_{V}^{j} (6.6.2) for each integer j.

It is easy to see that ϕ(Z(V)) ⊆ Z(W) in this case, as in Section 5.2. More precisely, we have that ϕ^{j}(Z(V)^{j}) ⊆ Z(W)^{j+r} (6.6.3)

for every integer j, as in Section 5.11. Similarly, ϕ(B(V)) ⊆ B(W), and in fact

ϕ^{j}(B(V)^{j}) ⊆ B(W)^{j+r} (6.6.4)

for every j.

As in Section 5.2, ϕ induces homomorphisms ϕ_{Z′}, ϕ_{B′} from Z′(V), B′(V) into Z′(W), B′(W), respectively, as modules over A. We also have that ϕ_{Z′} maps H(V) into H(W), and we let ϕ_{H} be the restriction of ϕ_{Z′} to H, as before. Note that ϕ_{Z′}, ϕ_{B′}, and ϕ_{H} have degree r, as in Section 5.11.

Remember that d̃_{V}, d̃_{W} are the homomorphisms from Z′(V), Z′(W) into Z(V), Z(W) induced by d_{V}, d_{W}, respectively, as in Section 5.1. One can check that ϕ ∘ d̃_{V} = (−1)^{r} d̃_{W} ∘ ϕ_{Z′}, (6.6.5)

as in Section 5.2. Equivalently, this means that

ϕ^{j+1} ∘ d̃_{V}^{j} = (−1)^{r} d̃_{W}^{j+r} ∘ ϕ_{Z′}^{j} (6.6.6)

for every j, as in Section 5.11.

Let ψ be another map of degree r from V into W, as complexes, and let σ be a homomorphism from V into W, as modules over A, of degree r − 1. If

d_{W} ∘ σ + (−1)^{r} σ ∘ d_{V} = ϕ − ψ, (6.6.7)

then σ is said to be a homotopy between ϕ and ψ, as maps of degree r between complexes, as in Exercise 1 on p72 of [50]. This reduces to the definition of a homotopy between maps between complexes when r = 0, as in Section 5.11. This is the same as saying that

d_{W}^{j+r−1} ∘ σ^{j} + (−1)^{r} σ^{j+1} ∘ d_{V}^{j} = ϕ^{j} − ψ^{j} (6.6.8)

for every j, as usual. This implies that

(ϕ^{j} − ψ^{j})(Z(V)^{j}) ⊆ B(W)^{j+r} (6.6.9)

for every j, so that ϕ_{H}^{j} = ψ_{H}^{j} for each j, as before.

Let (W_{1}, d_{W_{1}}) be another left or right graded module over A with differentiation that is a complex, depending on whether V, W are left or right modules over A. Also let ϕ_{1} be a map of degree r_{1} ∈ Z from W into W_{1}, as complexes. Thus ϕ_{1} ∘ ϕ is a homomorphism from V into W_{1}, as modules over A, of degree r + r_{1}. Observe that

d_{W_{1}} ∘ ϕ_{1} ∘ ϕ = (−1)^{r_{1}} ϕ_{1} ∘ d_{W} ∘ ϕ = (−1)^{r+r_{1}} ϕ_{1} ∘ ϕ ∘ d_{V} , (6.6.10) so that ϕ_{1} ∘ ϕ is a map of degree r + r_{1} from V into W_{1}, as complexes. This corresponds to part of Exercise 1 on p72f of [50].

Let ψ_{1} be another map of degree r_{1} from W into W_{1}, as complexes, and let σ_{1} be a homotopy between ϕ_{1} and ψ_{1}, as maps of degree r_{1} between complexes. Put τ = σ_{1} ∘ ϕ + (−1)^{r_{1}} ψ_{1} ∘ σ, (6.6.11)

where ϕ, ψ, and σ are as before. This is a homomorphism from V into W_{1}, as modules over A, of degree r + r_{1} − 1. We would like to verify that

(6.6.12) τ is a homotopy between ϕ_{1} ∘ ϕ and ψ_{1} ∘ ψ,

as maps of degree r + r_{1} between complexes, as in Exercise 1 on p72f of [50]. More precisely,

(6.6.13) σ_{1} ∘ ϕ is a homotopy between ϕ_{1} ∘ ϕ and ψ_{1} ∘ ϕ,

and (6.6.14) (−1)^{r_{1}} ψ_{1} ∘ σ is a homotopy between ψ_{1} ∘ ϕ and ψ_{1} ∘ ψ,

as maps of degree r + r_{1} between complexes. Of course, these are homomorphisms from V into W_{1}, as modules over A, of degree r + r_{1} − 1, as before. Observe that

d_{W_{1}} ∘ σ_{1} ∘ ϕ + (−1)^{r+r_{1}} σ_{1} ∘ ϕ ∘ d_{V} (6.6.15)

= d_{W_{1}} ∘ σ_{1} ∘ ϕ + (−1)^{r_{1}} σ_{1} ∘ d_{W} ∘ ϕ
= (ϕ_{1} − ψ_{1}) ∘ ϕ,

which implies (6.6.13). Similarly,

(−1)^{r_{1}} d_{W_{1}} ∘ ψ_{1} ∘ σ + (−1)^{r+r_{1}} (−1)^{r_{1}} ψ_{1} ∘ σ ∘ d_{V} (6.6.16)

= ψ_{1} ∘ d_{W_{1}} ∘ σ + (−1)^{r} ψ_{1} ∘ σ ∘ d_{V}
= ψ_{1} ∘ (ϕ − ψ),

which implies (6.6.14). Combining these two equations, we get that

d_{W_{1}} ∘ τ + (−1)^{r+r_{1}} τ ∘ d_{V} = ϕ_{1} ∘ ϕ − ψ_{1} ∘ ψ, (6.6.17)

which implies (6.6.12).

6.7 Double complexes and any bidegree

Let k be a commutative ring with a mutliplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, W be both left or both right modules over A that are double complexes, with differentiation operators d_{V,1}, d_{V,2} and d_{W,1}, d_{W,2}, respectively.

Suppose that ϕ is a homomorphism from V into W, as modules over A, of bidegree (p, q) for some p, q ∈ Z. If

d_{W,1} ∘ ϕ = (−1)^{p+q} ϕ ∘ d_{V,1} (6.7.1)

and d_{W,2} ∘ ϕ = (−1)^{p+q} ϕ ∘ d_{V,2}, (6.7.2)

then ϕ is considered to be a map of bidegree (p, q) from V into W, as double complexes, as in Exercise 2 on p73 of [50]. This reduces to the definition of a map between double complexes in Section 5.15 when p, q = 0. As usual, (6.7.1) and (6.7.2) are the same as saying that

d_{W,1}^{j+p,l+q} ∘ ϕ^{j,l} = (−1)^{p+q} ϕ^{j+1,l} ∘ d_{V_{1}} (6.7.3)

and d_{W,2}^{j+p,l+q} ∘ ϕ^{j,l} = (−1)^{p+q} ϕ^{j,l+1} ∘ d_{V,2} (6.7.4)

for all integers j, l.

Remember that V, W may be considered as single complexes, using the total differentiation operators

d_{V} = d_{V,1} + d_{V,2}, d_{W} = d_{W,1} + d_{W,2}, (6.7.5)

as in Section 5.14. If ϕ is a map of bidegree (p, q) from V into W, as double complexes, then it is easy to see that ϕ is a map of degree p + q from V into W, as single complexes, as in the previous section.

Let ψ be another map of bidegree (p, q) from V into W, as double complexes, and let σ_{1}, σ_{2} be homomomorphisms from V into W, as modules over A, of bidegrees (p − 1, q), (p, q − 1), respectively. Suppose that

d_{W,1} ∘ σ_{1} + (−1)^{p+q} σ_{1} ∘ d_{V,1} + d_{W,2} ∘ σ_{2} + (−1)^{p+q} σ_{2} ∘ d_{V,2} (6.7.6)

= ϕ − ψ.

Suppose also that (−1)^{p+q} σ_{1} ∘ d_{V,2} + d_{W,2} ∘ σ_{1} = 0 (6.7.7)

and (−1)^{p+q} σ_{2} ∘ d_{V,1} + d_{W,1} ∘ σ_{2} = 0. (6.7.8)

In this case, (σ_{1}, σ_{2}) defines a homotopy between ϕ and ψ, as maps of bidegree (p, q) between double complexes, as in Exercise 2 on p73 of [50]. This reduces to the definition of a homotopy between maps between double complexes in Section 5.15 when p = q = 0.

Note that (6.7.6) is the same as saying that

d_{W,1}^{j+p−1,l+q} ∘ σ_{1}^{j,l} + (−1)^{p+q} σ_{1}^{j+1,l} ∘ d_{V,1}^{j,l} (6.7.9)

+d_{W,2}^{j+p,l+q−1} ∘ σ_{2}^{j,l} + (−1)^{p+q} σ_{2}^{j,l+1} ∘ d_{V,2}^{j,l}
= ϕ^{j,l} − ψ^{j,l}

for all integers j, l. Similarly, (6.7.7) is the same as saying that

(−1)^{p+q} σ_{1}^{j,l+1} ∘ d_{V,2}^{j,l} + d_{W,2}^{j+p−1,l+q} ∘ σ_{1}^{j,l} = 0 (6.7.10)

for every j, l, and (6.7.8) is the same as saying that

(−1)^{p+q} σ_{2}^{j+1,l} ∘ d_{V,1}^{j,l} + d_{W,1}^{j+p,l+q−1} ∘ σ_{2}^{j,l} = 0 (6.7.11)

for every j, l.

Put σ = σ_{1} + σ_{2}, (6.7.12)

which is a homomorphism from V into W of degree p + q − 1, as singly-graded modules over A. One can verify that

d_{W} ∘ σ + (−1)^{p+q} σ ∘ d_{V} = ϕ − ψ, (6.7.13)

using (6.7.6), (6.7.7), and (6.7.8). This implies that σ is a homotopy between ϕ and ψ, as maps of degree p + q from V into W as single complexes over A, as in the previous section. This reduces to the analogous statement in Section 5.15 when p = q = 0.

Let W̃ be another left or right module over A that is a double complex, depending on whether V, W are left or right modules over A, and let d_{_{W} ,1},

d_{_{W} ,2} be the corresponding differentiation operators on W̃. Also let ϕ̃ be a map

of bidegree (p̃, q̃) from W into W̃, as double complexes, for some p̃, q̃ ∈ Z. It is easy to see that ϕ̃ ∘ ϕ is a map of bidegree (p + p̃, q + q̃) from V into W̃, as double complexes.

Let ψ̃ be another map of bidegree (p̃, q̃) from W into W̃, as double complexes, and let (σ̃_{1}, σ̃_{2}) be a homotopy between ϕ̃ and ψ̃, as maps of bidegree (p̃, q̃) between double complexes. One can check that

(6.7.14) (σ̃_{1} ∘ ϕ, σ̃_{2} ∘ ϕ) is a homotopy between ϕ̃ ∘ ϕ and ψ̃ ∘ ϕ,

as maps of bidegree (p + p̃, q + q̃) between double complexes. Similarly,

(6.7.15) ((−1)^{_{p}+_{q}}ψ̃ ∘ σ_{1}, (−1)^{_{p}+_{q}}ψ̃ ∘ σ_{2}) is a homotopy between

ψ̃ ∘ ϕ and ψ̃ ∘ ψ,

as maps of bidegree (p + p̃, q + q̃) between double complexes. This is analogous to the corresponding statements in the previous section.

Put

τ_{j} = σ̃_{j} ∘ ϕ + (−1)^{_{p}+_{q}}ψ̃ ∘ σ_{j}, (6.7.16)

for j = 1, 2. The remarks in the preceding paragraph imply that

(6.7.17) (τ_{1}, τ_{2}) is a homotopy between ϕ̃ ∘ ϕ and ψ̃ ∘ ψ,

as maps of bidegree (p + p̃, q + q̃) between double complexes. This corresponds to part of Exercise 2 on p73 of [50].

6.8 Tensor products and any degrees

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that (V, d_{V}), (V_{0}, d_{V_{0}}) are graded right modules over A with differentiation that are complexes, and that (W, d_{W}), (W_{0}, d_{W_{0}}) are graded left modules over A with differentiation that are complexes. Let V ⨂_{A} W, V_{0} ⨂_{A} W_{0} be tensor products of V, W and V_{0}, W_{0} over A, respectively. These are double complexes over k with respect to the corresponding differentiation operators δ_{1}, δ_{2} and δ_{1,0}, δ_{2,0}, respectively, discussed in Section 6.2.

Let ϕ, ψ be maps of degrees p, q ∈ Z from V, W into V_{0}, W_{0}, respectively, as complexes. This leads to a homomorphism ϕ⊗ψ from V ⨂_{A} W into V_{0} ⨂_{A} W_{0}, as modules over k, of bidegree (p, q), as in Section 6.1. Let us check that

(6.8.1) ϕ ⊗ ψ is a map of bidegree (p, q) from V ⨂_{A}W into V_{0} ⨂_{A}W_{0},

as double complexes. This is part of Exercise 2 on p73 of [50], and the analogous statement for p = q = 0 was mentioned in Section 6.2.

We shall use (6.1.7) repeatedly, as before. Observe that

δ_{1,0} ∘ (ϕ ⊗ ψ) = (d_{V_{0}} ⊗ I_{W_{0}}) ∘ (ϕ ⊗ ψ) = (d_{V_{0}} ∘ ϕ) ⊗ ψ
(6.8.2)  = (−1)^{p} (ϕ ∘ d_{V}) ⊗ ψ = (−1)^{p+q} (ϕ ⊗ ψ) ∘ (d_{V} ⊗ I_{W})
= (−1)^{p+q} (ϕ ⊗ ψ) ∘ δ_{1}.

Similarly,

δ_{2,0} ∘ (ϕ ⊗ ψ) = (I_{V_{0}} ⊗ d_{W_{0}}) ∘ (ϕ ⊗ ψ) = (−1)^{p} ϕ ⊗ (d_{W_{0}} ∘ ψ)
(6.8.3)  = (−1)^{p+q} ϕ ⊗ (ψ ∘ d_{W}) = (−1)^{p+q} (ϕ ⊗ ψ) ∘ (I_{V} ⊗ d_{W})
= (−1)^{p+q} (ϕ ⊗ ψ) ∘ δ_{2},

as desired.

Let ϕ_{0}, ψ_{0} be another pair of mappings of degrees p, q from V, W into V_{0}, W_{0}, respectively, as complexes. Also let σ, τ be homotopies between ϕ, ψ and ϕ_{0}, ψ_{0}, as maps of degrees p, q between complexes, respectively. This means that σ, τ are homomorphisms from V, W into V_{0}, W_{0}, as modules over A, with degrees p − 1, q − 1, respectively, such that

d_{V_{0}} ∘ σ + (−1)^{p} σ ∘ d_{V} = ϕ − ϕ_{0} (6.8.4)

and d_{W_{0}} ∘ τ + (−1)^{q} τ ∘ d_{W} = ψ − ψ_{0}, (6.8.5)

as in Section 6.6. Put σ_{1} = σ ⊗ ψ (6.8.6)

and σ_{2} = (−1)^{p} ϕ_{0} ⊗ τ. (6.8.7)

These are homomorphisms from V ⨂_{A} W into V_{0} ⨂_{A} W_{0}, as modules over k, of bidegrees (p − 1, q) and (p, q − 1), respectively.

The basic properties of σ_{1} and σ_{2} are analogous to those in Section 6.2. First,

σ_{1} ∘ δ_{2} = (σ ⊗ ψ) ∘ (I_{V} ⊗ d_{W}) = σ ⊗ (ψ ∘ d_{W})
(6.8.8)  = (−1)^{q} σ ⊗ (d_{W_{0}} ∘ ψ) = (−1)^{p−1+q} (I_{V_{0}} ⊗ d_{W_{0}}) ∘ (σ ⊗ ψ)
= −(−1)^{p+q} δ_{2,0} ∘ σ_{1}.

Similarly,

σ_{2} ∘ δ_{1} = (−1)^{p} (ϕ_{0} ⊗ τ) ∘ (d_{V} ⊗ I_{W}) = (−1)^{p+q−1} (ϕ_{0} ∘ d_{V}) ⊗ τ
(6.8.9)  = (−1)^{q−1} (d_{V_{0}} ∘ ϕ_{0}) ⊗ τ = (−1)^{q−1} (d_{V_{0}} ⊗ I_{W_{0}}) ∘ (ϕ_{0} ⊗ τ)
= −(−1)^{p+q} δ_{1,0} ∘ σ_{2}.

Next, δ_{1,0} ∘ σ_{1} = (d_{V_{0}} ⊗ I_{W_{0}}) ∘ (σ ⊗ ψ) = (d_{V_{0}} ∘ σ) ⊗ ψ (6.8.10)

and σ_{1} ∘ δ_{1} = (σ ⊗ ψ) ∘ (d_{V} ⊗ I_{W}) = (−1)^{q} (σ ∘ d_{V}) ⊗ ψ. (6.8.11)

It follows that

δ_{1,0} ∘ σ_{1} + (−1)^{p+q} σ_{1} ∘ δ_{1} = (d_{V_{0}} ∘ σ) ⊗ ψ + (−1)^{p} (σ ∘ d_{V}) ⊗ ψ
(6.8.12)  = (d_{V_{0}} ∘ σ + (−1)^{p} σ ∘ d_{V}) ⊗ ψ = (ϕ − ϕ_{0}) ⊗ ψ.

Similarly,

δ_{2,0} ∘ σ_{2} = (−1)^{p} (I_{V_{0}} ⊗ d_{W_{0}}) ∘ (ϕ_{0} ⊗ τ) = ϕ_{0} ⊗ (d_{W_{0}} ∘ τ) (6.8.13)

and σ_{2} ∘ δ_{2} = (−1)^{p} (ϕ_{0} ⊗ τ) ∘ (I_{V} ⊗ d_{W}) = (−1)^{p} ϕ_{0} ⊗ (τ ∘ d_{W}). (6.8.14)

This implies that

δ_{2,0} ∘ σ_{2} + (−1)^{p+q} σ_{2} ∘ δ_{2} = ϕ_{0} ⊗ (d_{W_{0}} ∘ τ) + (−1)^{q} ϕ_{0} ⊗ (τ ∘ d_{W})
(6.8.15)  = ϕ_{0} ⊗ (d_{W_{0}} ∘ τ + (−1)^{q} τ ∘ d_{W})
= ϕ_{0} ⊗ (ψ − ψ_{0}).

We can combine (6.8.12) and (6.8.15) to get that

δ_{1,0} ∘ σ_{1} + (−1)^{p+q} σ_{1} ∘ δ_{1} + δ_{2,0} ∘ σ_{2} + (−1)^{p+q} σ_{2} ∘ δ_{2} (6.8.16)

= ϕ ⊗ ψ − ϕ_{0} ⊗ ψ_{0}.

This shows that

(6.8.17) (σ_{1}, σ_{2}) is a homotopy between ϕ ⊗ ψ and ϕ_{0} ⊗ ψ_{0},

as maps of bidegree (p, q) from V ⨂_{A} W into V_{0} ⨂_{A} W_{0}, as double complexes. This is another part of Exercise 2 on p73 of [50].

6.9 Homomorphisms and any degrees

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that (V, d_{V}), (V_{0}, d_{V_{0}}), (W, d_{W}), and (W_{0}, d_{W_{0}}) are all left or all right graded modules over A with differentiation that are complexes. Thus Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V_{0}, W_{0}) are double complexes over k with respect to the corresponding differentiation operators δ_{1}, δ_{2} and δ_{1,0}, δ_{2,0} discussed in Section 6.5.

Let ϕ be a map of degree p ∈ Z from V_{0} into V, and let ψ be a map of degree q ∈ Z from W into W_{0}, as complexes. This leads to a homomorphism Hom^{gr}(ϕ, ψ) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W_{0}), as modules over k, of bidegree (p, q), as in Section 6.4. We would like to verify that

(6.9.1) Hom^{gr}(ϕ, ψ) is a map of bidegree (p, q)

from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W_{0}),

as double complexes. This is part of Exercise 2 on p73 of [50], and the analogous statement for p = q = 0 was mentioned in Section 6.5.

We shall use (6.4.10) repeatedly in this section, as before. In particular, we have that

δ_{1,0} ∘ Hom^{gr}(ϕ, ψ) = Hom^{gr}(d_{V_{0}}, I_{W_{0}}) ∘ Hom^{gr}(ϕ, ψ)
(6.9.2)  = Hom^{gr}(ϕ ∘ d_{V_{0}}, ψ) = (−1)^{p} Hom^{gr}(d_{V} ∘ ϕ, ψ)
= (−1)^{p+q} Hom^{gr}(ϕ, ψ) ∘ Hom^{gr}(d_{V} , I_{W})
= (−1)^{p+q} Hom^{gr}(ϕ, ψ) ∘ δ_{1}.

Similarly,

δ_{2,0} ∘ Hom^{gr}(ϕ, ψ) = Hom^{gr}(I_{V_{0}}, d_{W_{0}}) ∘ Hom^{gr}(ϕ, ψ)
(6.9.3)  = (−1)^{p} Hom^{gr}(ϕ, d_{W_{0}} ∘ ψ) = (−1)^{p+q} Hom^{gr}(ϕ, ψ ∘ d_{W})
= (−1)^{p+q} Hom^{gr}(ϕ, ψ) ∘ Hom^{gr}(I_{V} , d_{W})
= (−1)^{p+q} Hom^{gr}(ϕ, ψ) ∘ δ_{2}.

Let ϕ_{0} be another map of degree p from V_{0} into V, and let ψ_{0} be another map of degree q from W into W_{0}, as complexes. Suppose that σ, τ are homotopies between ϕ, ψ and ϕ_{0}, ψ_{0}, as maps of degree p, q between complexes, respectively. Thus σ is a homomorphism from V_{0} into V, as modules over A, with degree p−1, such that d_{V} ∘ σ + (−1)^{p} σ ∘ d_{V_{0}} = ϕ − ϕ_{0}. (6.9.4)

Similarly, τ is a homomorphism from W into W_{0}, as modules over A, with degree q − 1, such that

d_{W_{0}} ∘ τ + (−1)^{q} τ ∘ d_{W} = ψ − ψ_{0}. (6.9.5)

Put σ_{1} = (−1)^{p} Hom^{gr}(σ, ψ) (6.9.6) and σ_{2} = (−1)^{p} Hom^{gr}(ϕ_{0}, τ). (6.9.7)

These are homomorphisms from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W_{0}), as modules over k, of bidegrees (p−1, q) and (p, q −1), respectively. The basic properties of these homomorphisms are analogous to those in Section 6.5. Observe first that

σ_{1} ∘ δ_{2} = (−1)^{p} Hom^{gr}(σ, ψ) ∘ Hom^{gr}(I_{V} , d_{W})
(6.9.8)  = (−1)^{p} Hom^{gr}(σ, ψ ∘ d_{W}) = (−1)^{p+q} Hom^{gr}(σ, d_{W_{0}} ∘ ψ)
= (−1)^{q−1} Hom^{gr}(I_{V_{0}}, d_{W_{0}}) ∘ Hom^{gr}(σ, ψ) = −(−1)^{p+q} δ_{2,0} ∘ σ_{1}.

Similarly,

σ_{2} ∘ δ_{1} = (−1)^{p} Hom^{gr}(ϕ_{0}, τ) ∘ Hom^{gr}(d_{V} , I_{W})
= (−1)^{p+q−1} Hom^{gr}(d_{V} ∘ ϕ_{0}, τ) = (−1)^{q−1} Hom^{gr}(ϕ_{0} ∘ d_{V_{0}}, τ)
= (−1)^{q−1} Hom^{gr}(d_{V_{0}}, I_{W_{0}}) ∘ Hom^{gr}(ϕ_{0}, τ) = −(−1)^{p+q} δ_{1,0} ∘ σ_{2}.

We also have that

(6.9.9)  δ_{1,0} ∘ σ_{1} = (−1)^{p} Hom^{gr}(d_{V_{0}}, I_{W_{0}}) ∘ Hom^{gr}(σ, ψ)
= (−1)^{p} Hom^{gr}(σ ∘ d_{V_{0}}, ψ)

and

(6.9.10)  σ_{1} ∘ δ_{1} = (−1)^{p} Hom^{gr}(σ, ψ) ∘ Hom^{gr}(d_{V} , I_{W})
= (−1)^{p+q} Hom^{gr}(d_{V} ∘ σ, ψ).

This implies that

δ_{1,0} ∘ σ_{1} + (−1)^{p+q} σ_{1} ∘ δ_{1} = Hom^{gr}((−1)^{p} σ ∘ d_{V_{0}} + d_{V} ∘ σ, ψ)
(6.9.11)  = Hom^{gr}(ϕ − ϕ_{0}, ψ).

Similarly,

(6.9.12)  δ_{2,0} ∘ σ_{2} = (−1)^{p} Hom^{gr}(I_{V_{0}}, d_{W_{0}}) ∘ Hom^{gr}(ϕ_{0}, τ)
= Hom^{gr}(ϕ_{0}, d_{W_{0}} ∘ τ)

and

(6.9.13)  σ_{2} ∘ δ_{2} = (−1)^{p} Hom^{gr}(ϕ_{0}, τ) ∘ Hom^{gr}(I_{V} , d_{W})
= (−1)^{p} Hom^{gr}(ϕ_{0}, τ ∘ d_{W}).

It follows that

δ_{2,0} ∘ σ_{2} + (−1)^{p+q} σ_{2} ∘ δ_{2} = Hom^{gr}(ϕ_{0}, d_{W_{0}} ∘ τ + (−1)^{q} τ ∘ d_{W})
(6.9.14)  = Hom^{gr}(ϕ_{0}, ψ − ψ_{0}).

6.10. THE COMPLEX M(V, W) 161

Combining (6.9.11) and (6.9.14), we get that

δ_{1,0} ∘ σ_{1} + (−1)^{p+q} σ_{1} ∘ δ_{1} + δ_{2,0} ∘ σ_{2} + (−1)^{p+2} σ_{2} ∘ δ_{2} (6.9.15)

= Hom^{gr}(ϕ, ψ) + Hom^{gr}(ϕ_{0}, ψ_{0}).

This shows that

(6.9.16) (σ_{1}, σ_{2}) is a homotopy between Hom^{gr}(ϕ, ψ) and Hom^{gr}(ϕ_{0}, ψ_{0}),

as maps of bidegree (p, q) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W_{0}), as double complexes. This is another part of Exercise 2 on p73 of [50].

6.10 The complex M(V, W)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (V, d_{V}), (W, d_{W}) be both left or both right modules over A with differentiation that are complexes. As in Section 6.3, we let |V |, |W | be the underlying modules over A corresponding to V, W, without gradings.

If r ∈ Z, then M^{r}(V, W) is the space of homomorphisms from V into W, as modules over A, of degree r, as in Section 6.3. This is a submodule of Hom_{A}(|V |, |W |), as a module over k, and M(V, W) is the submodule of Hom_{A}(V, W) spanned by the M^{r}(V, W)’s, r ∈ Z, as before. More precisely, M(V, W) corresponds to the direct sum of the M^{r}(V, W)’s, r ∈ Z, as a module over k, so that M(V, W) may be considered as a graded module over k.

If ϕ ∈ M^{r}(V, W) for some r ∈ Z, then put

d_{1}^{r}(ϕ) = ϕ ∘ d_{V} (6.10.1)

and d_{2}^{r}(ϕ) = (−1)^{r+1} d_{W} ∘ ϕ, (6.10.2)

which are elements of M^{r+1}(V, W). These define unique homomorphisms d_{1}, d_{2} from M(V, W) into itself, as a module over k, of degree 1, whose restrictions to M^{r}(V, W) are equal to d_{1}^{r}, d_{2}^{r}, respectively, for each r ∈ Z. Observe that

d_{1} ∘ d_{1} = d_{2} ∘ d_{2} = 0 (6.10.3)

on M(V, W), because d_{V} ∘ d_{V} = 0 on V and d_{W} ∘ d_{W} = 0 on W, by hypothesis. One can also verify that d_{1} ∘ d_{2} + d_{2} ∘ d_{1} = 0 (6.10.4)

on M(V, W).

Put d = d_{1} + d_{2}, (6.10.5)

which is a homomorphism from M(V, W) into itself, as a module over k, of degree 1. It is easy to see that d ∘ d = 0 (6.10.6) on M(V, W). This means that M(V, W) is a module with differentiation with respect to d, and in fact a complex. This corresponds to part of Exrcise 5 on p73 of [50].

If r ∈ Z, then let Map^{r}(V, W) be the space of maps of degree r from V into W, as complexes. This is a submodule of M^{r}(V, W), as a module over k. It is easy to see that Z(M(V, W))^{r} = Map^{r}(V, W), (6.10.7)

where the left side is defined by considering M(V, W) as a complex with respect to d. This is part of Exercise 5 on p73f of [50].

Let Map_{0}^{r}(V, W) be the subset of Map^{r}(V, W) consisting of maps of degree r that are homotopic to 0, as maps of degree r between complexes. This is a submodule of Map^{r}(V, W), as a module over k. One can check that

B(M(V, W))^{r} = Map_{0}^{r}(V, W), (6.10.8)

where the left side is defined by considering M(V, W) as a complex with respect to d again. This is another part of Exercise 5 on p73f of [50].

Let Hom_{A}^{gr}(V, W) be the doubly-graded module over k defined in Section 6.3. Remember that Hom_{A}^{gr}(V, W) corresponds to a submodule of M(V, W), as a module over k, and that the single grading on Hom_{A}^{gr}(V, W) obtained from the double grading is compatible with the grading on M(V, W). We also have that Hom_{A}^{gr}(V, W) is a double complex with respect to the homomorphisms δ_{1}, δ_{2} defined in Section 6.5.

One can verify that δ_{1} is the same as the restriction of d_{1} to Hom_{A}^{gr}(V, W). More precisely, if j, l ∈ Z, then the restriction of δ_{1} to ( Hom_{A}^{gr}(V, W))^{j,l} is defined by composition with d_{V}, which is the same as d_{1}. Similarly,

(6.10.9) the restriction of δ_{2} to (Hom_{A}^{gr}(V, W))^{j,l} is (−1)^{j} times the mapping defined by composition with d_{W} .

However,

(6.10.10) the restriction of d_{2} to (Hom_{A}^{gr}(V, W))^{j,l} is (−1)^{j+l+1} times the mapping defined by composition with d_{W} ,

because ( Hom_{A}^{gr}(V, W))^{j,l} corresponds to a submodule of M^{r}(V, W), with r = j + l.

6.11 Tensor products and homology

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (V, d_{V}) be a graded right module over A with differentiation that is a complex, and let (W, d_{W}) be a graded left module over A with differentiation that is a complex. Suppose that V ⨂_{A} W is a tensor product of V and W over A, which is a double complex over k with respect to the double grading and differentiation operators δ_{1}, δ_{2} defined in Sections 6.1 and 6.2, respectively.

Remember that V ⨂_{A} W may be considered as a single complex over k, with respect to the single grading obtained from the double grading, and the corresponding total differentiation operator

δ = δ_{1} + δ_{2}, (6.11.1)

as in Section 5.14. In particular,

Z(V ⨂_{A}W ), B(V ⨂_{A}W ), and H(V ⨂_{A}W ) (6.11.2)

are defined as singly-graded modules over k in the usual way, by considering V ⨂_{A} W as a single complex.

Let j, l ∈ Z be given. Of course,

(v^{j}, w^{l}) ↦ v^{j} ⊗ w^{l} (6.11.3)

defines a mapping from V^{j} ×W^{l} into V^{j} ⨂_{A} W^{l} = (V ⨂_{A} W )^{j,l} that is bilinear over k and satisfies (v^{j} · a) ⊗ w^{l} = v^{j} ⊗ (a · w^{l}) (6.11.4)

for every a ∈ A, v^{j} ∈ V^{j}, and w^{l} ∈ W^{l}. Observe that

δ_{1}(v^{j} ⊗ w^{l}) = 0 (6.11.5)

when d_{V} (v^{j}) = 0, and δ_{2}(v^{j} ⊗ w^{l}) = 0 (6.11.6)

when d_{W} (w^{l}) = 0. This means that

δ(v^{j} ⊗ w^{l}) = 0 (6.11.7)

when d_{V} (v^{j}), d_{W} (w^{l}) = 0.

Equivalently, v^{j} ⊗ w^{l} ∈ Z(V ⨂_{A}W )^{j+l} (6.11.8)

when v^{j} ∈ Z(V)^{j} and w^{l} ∈ Z(W)^{l}. Let Z(V) ⨂_{A} Z(W) be a tensor product of Z(V) and Z(W) over A, which is a doubly-graded module over k. In particular,

(Z(V) ⨂_{A}Z(W))^{j,l} = Z(V)^{j} ⨂_{A}Z(W)^{l} (6.11.9)

is a tensor product of Z(V)^{j} and Z(W)^{l} over A. The restriction of (6.11.3) to Z(V)^{j} × Z(W)^{l} is a bilinear mapping over k into (V ⨂_{W} W )^{j+l} that satisfies (6.11.4). This leads to a homomorphism

(6.11.10) from Z(V)^{j} ⨂_{A}Z(W)^{l} into Z(V ⨂_{A}W )^{j+l},

as modules over k, in the usual way.

This defines a homomorphism

(6.11.11) from Z(V) ⨂_{A}Z(W) into Z(V ⨂_{A}W ),

as modules over k. This homomorphism has degree 0, with respect to the single grading on Z(V) ⨂_{A} Z(W) obtained from the double grading mentioned earlier.

This leads to a homomorphism

(6.11.12) η from Z(V) ⨂_{A}Z(W) into H(V ⨂_{A}W ),

as modules over k, by composing the homomorphism as in (6.11.11) with the natural quotient mapping

(6.11.13) from Z(V ⨂_{A}W ) onto H(V ⨂_{A}W ).

This corresponds to the homomorphism η in the diagram (1) on p64 of [50], under slightly different conditions. This homomorphism has degree 0 too, with respect to the single grading on Z(V) ⨂_{A} Z(W) obtained from the double grading. More precisely, the restriction of this homomorphism to (6.11.9) is the homomorphism

(6.11.14) from Z(V)^{j} ⨂_{A}Z(W)^{l} into H(V ⨂_{A}W )^{j+l},

as modules over k, obtained by composing the homomorphism as in (6.11.10) with the natural quotient mapping

(6.11.15) from Z(V ⨂_{A}W )^{j+l} onto H(V ⨂_{A}W )^{j+l}.

Similarly, one can check that

v^{j} ⊗ w^{l} ∈ B(V ⨂_{A}W )^{j+l} (6.11.16)

when v^{j} ∈ B(V)^{j} and w^{l} ∈ Z(W)^{l}, (6.11.17)

and when v^{j} ∈ Z(V)^{j} and w^{l} ∈ B(W)^{l}. (6.11.18)

This means that

(6.11.19) v^{j} ⊗ w^{l} is mapped to 0 in H(V ⨂_{A}W )^{j+l}

by the homomorphism in (6.11.14)

when either (6.11.17) or (6.11.18) holds.

Consider the mapping

6.12. SOME HOMOMORPHISMS RELATED TO α 165

(6.11.20) from Z(V)^{j} × Z(W)^{l} into H(V ⨂_{A}W )^{j+l}, with

(6.11.21)  (v^{j}, w^{l}) ↦ the image of v^{j} ⊗ w^{l} under

the quotient mapping (6.11.15)

for every v^{j} ∈ V^{j} and w^{l} ∈ W^{l}. This mapping is equal to 0 on B(V)^{j} × Z(W)^{l}, and on Z(V)^{j} ×B(W)^{l}, as in the preceding paragraph. If v^{j} ∈ V^{j} and w^{l} ∈ W^{l}, then it follows the image of (v^{j}, w^{l}) under this mapping only depends on the images of v^{j}, w^{l} under the natural quotient mappings from Z(V)^{j}, Z(W)^{l} onto H(V)^{j}, H(W)^{l}, respectively. This leads to a mapping

(6.11.22) from H(V)^{j} × H(W)^{l} into H(V ⨂_{A}W )^{j+l},

that is bilinear over k.

Remember that H(V) is a graded right module over A, and that H(W) is a graded left module over A. Let H(V) ⨂_{A} H(W) be a tensor product of H(V) and H(W) over A, which is a doubly-graded module over k. Thus

(H(V) ⨂_{A}H(W))^{j,l} = H(V)^{j} ⨂_{A}H(W)^{l} (6.11.23)

is a tensor product of H(V)^{j} and H(W)^{l} over A. Using (6.11.4), we get that the mapping in (6.11.22) satisfies the analogous property with respect to the actions of A on H(V)^{j} on the right and on H(W)^{l} on the left. This leads to a homomorphism

(6.11.24) from H(V)^{j} ⨂_{A}H(W)^{l} into H(V ⨂_{A}W )^{j+l},

as modules over k.

This defines a homomorphism

(6.11.25) α from H(V) ⨂_{A}H(W) into H(V ⨂_{A}W ),

as modules over k. This homomorphism has degree 0, with respect to the single grading on H(V) ⨂_{A} H(W) obtained from the double grading. This corresponds to the homomorphism in Proposition 6.1 on p64 of [50], under slightly different conditions.

6.12 Some homomorphisms related to α

Let us continue with the same notation and hypotheses as in the previous section.

Using the natural quotient mappings from Z(V), Z(W) onto H(V), H(W), respectively, we get a homomorphism

(6.12.1) ξ from Z(V) ⨂_{A}Z(W) onto H(V) ⨂_{A}H(W), as modules over k. This corresponds to the homomorphism ξ in the diagram (1) on p64 of [50], under slightly different conditions. More precisely, the surjectivity of ξ follows from the surjectivity of the quotient mappings, as in Section 1.9. Note that ξ has bidegee (0, 0) with respect to the usual double gradings on Z(V) ⨂_{A} Z(W) and H(V) ⨂_{A} H(W), because the quotient mappings have degree 0. In particular, ξ has degree 0 with respect to the corresponding single gradings.

By construction, α ∘ ξ = η, (6.12.2)

where η is as in (6.11.12). This corresponds to part of Proposition 6.1 on p64 of [50], and in fact α is uniquely determined by this property.

Remember that Z′(V) = V/d_{V} (V), Z′(W) = W/d_{W} (W), as in Section 5.1. Let Z′(V) ⨂_{A} Z′(W) be a tensor product of Z′(V) and Z′(W) over A, which is a doubly-graded module over k. Using the natural quotient mappings from V, W onto Z′(V), Z′(W), respectively, we get a homomorphism

(6.12.3) from V ⨂_{A}W onto Z′(V) ⨂_{A}Z′(W),

as modules over k. This homomorphism has bidegree (0, 0) with respect to the induced double gradings, and thus degree 0 with respect to the associated single gradings.

The natural inclusion mappings from Z(V), Z(W) into V, W, respectively, lead to a natural homomorphism

(6.12.4) from Z(V) ⨂_{A}Z(W) into V ⨂_{A}W,

as modules over k. This homomorphism has bidegree (0, 0) with respect to the appropriate double gradings, and thus degree 0 with respect to the associated single gradings. We can compose this homomorphism with the one in (6.12.3) to get a natural homomorphism

(6.12.5) from Z(V) ⨂_{A}Z(W) into Z′(V) ⨂_{A}Z′(W),

as modules over k. This homomorphism ha bidegree (0, 0) with respect to the usual double gradings, and thus degree 0 with respect to the associated single gradings.

Alternatively, the restrictions of the natural quotient mappings from V, W onto Z′(V), Z′(W) to Z(V), Z(W) define natural homomorphisms

(6.12.6) from Z(V), Z(W) into Z′(V), Z′(W),

respectively, as modules over A. The homomorphism in (6.12.5) is the same as the one obtained from these homomorphisms in the usual way.

Of course, H(V) = Z(V)/d_{V} (V), H(W) = Z(W)/d_{W} (W) are homogeneous submodules of Z′(V), Z′(W), as graded modules over A, as in Section 5.10. Using the corresponding inclusion mappings, we get a natural homomorphism

(6.12.7) τ from H(V) ⨂_{A}H(W) into Z′(V) ⨂_{A}Z′(W), as modules over k. This corresponds to the homomorphism τ in the diagram (1) on p64 of [50], under slightly different conditions. This homomorphism has bidegree (0, 0) with respect to the appropriate double gradings, and thus degree 0 with respect to the associated single gradings.

The homomorphisms as in (6.12.6) are the same as the compositions of the natural quotient mappings from Z(V), Z(W) onto H(V), H(W) with the natural inclusion mappings into Z′(V), Z′(W), respectively. This implies that

(6.12.8) τ ∘ ξ is the same as the homomorphism as in (6.12.5).

More precisely, the homomorphism as in (6.12.4)

(6.12.9) maps Z(V) ⨂_{A}Z(W) into Z(V ⨂_{A}W ),

as in the previous section. Equivalently, the homomorphism as in (6.12.4) is essentially the same as the one as in (6.11.11), because Z(V ⨂_{A} W ) is a submodule of V ⨂_{A} W.

The restriction of the homomorphism as in (6.12.3) to Z(V ⨂_{A} W ) defines a homomorphism

(6.12.10) from Z(V ⨂_{A}W ) into Z′(V) ⨂_{A}Z′(W),

as modules over k. This homomorphism has degree 0 with respect to the corresponding single gradings.

One can check that

the homomorphism as in (6.12.10)

(6.12.11)

is equal to 0 on B(V ⨂_{A}W ).

This leads to a homomorphism

(6.12.12) ζ from H(V ⨂_{A}W ) into Z′(V) ⨂_{A}Z′(W),

as modules over k. This corresponds to the homomorphism ζ in the diagram (1) on p64 of [50], under slightly different conditions. This homomorphism has degree 0 with respect to the appropriate single gradings.

It is easy to see that

(6.12.13) ζ ∘ η is the same as the homomorphism as in (6.12.5),

where η is as in (6.11.12), by construction. This means that

τ ∘ ξ = ζ ∘ η, (6.12.14)

by (6.12.8). This corresponds to the commutativity of the diagram (1) on p64 of [50], under slightly different conditions.

Observe that ζ ∘ α ∘ ξ = ζ ∘ η = τ ∘ ξ, (6.12.15) using (6.12.2) in the first step. This implies that

ζ ∘ α = τ, (6.12.16)

because ξ is surjective, as in (6.12.1). This corresponds to part of Proposition 6.1 on p64 of [50], under slightly different conditions.

6.13 Homology and Hom_{A}^{gr}(V, W)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (V, d_{V}) and (W, d_{W}) be both left or both right graded modules over A with differentiation that are complexes. Remember that Hom_{A}^{gr}(V, W) may be defined as a doubly-graded module over k as in Section 6.3, and as a double complex with respect to the differentiation operators δ_{1}, δ_{2} discussed in Section 6.5. It follows that Hom_{A}^{gr}(V, W) may be considered as a single complex over k, with respect to the single grading obtained from the double grading, and the total differentiation operator δ = δ_{1} + δ_{2}. (6.13.1)

Using this,

(6.13.2) Z(Hom_{A}^{gr}(V, W)), B(Hom_{A}^{gr}(V, W)), and H(Hom_{A}^{gr}(V, W))

may be defined as singly-graded modules over k in the usual way.

Let j, l ∈ Z and

ϕ^{j,l} ∈ (Hom_{A}^{gr}(V, W))^{j,l} = Hom_{A}(V^{−j}, W^{l}) (6.13.3)

be given. Of course, ϕ^{j,l} may be considered as an element of Hom_{A}^{gr}(V, W) as well. By construction, δ_{1}(ϕ^{j,l}) = ϕ^{j,l} ∘ d_{V}^{−j−1}, (6.13.4)

which is an element of Hom_{A}^{gr}(V, W)^{j+1,l}, and

δ_{2}(ϕ^{j,l}) = (−1)^{j} d_{W}^{l} ∘ ϕ^{j,l}, (6.13.5)

which is an element of Hom_{A}^{gr}(V, W)^{j,l+1}. This means that

δ_{1}(ϕ^{j−1,l}) = ϕ^{j−1,l} ∘ d_{V}^{−j} (6.13.6)

and δ_{2}(ϕ^{j,l−1}) = (−1)^{j} d_{W}^{l−1} ∘ ϕ^{j,l−1} (6.13.7)

are elements of ( Hom_{A} gr(V, W))^{j,l}.

Let ϕ ∈ Hom_{A}^{gr}(V, W) be given, and let ϕ^{j,l} be the component of ϕ in ( Hom_{A}^{gr}(V, W))^{j,l}, as usual. Observe that

(δ_{1}(ϕ))^{j,l} = ϕ^{j−1,l} ∘ d_{V}^{−j} (6.13.8)

6.13. HOMOLOGY AND HOM_{A}^{GR}(V, W) 169

and (δ_{2}(ϕ))^{j,l} = (−1)^{j} d_{W}^{l−1} ∘ ϕ^{j,l−1}, (6.13.9) as in the preceding paragraph. Thus (δ(ϕ))^{j,l} = ϕ^{j−1,l} ∘ d_{V}^{−j} + (−1)^{j} d_{W}^{l−1} ∘ ϕ^{j,l−1}. (6.13.10)

It follows that ϕ ∈ Z( Hom_{A}^{gr}(V, W)) if and only if ϕ^{j−1,l} ∘ d_{V}^{−j} + (−1)^{j} d_{W}^{l−1} ∘ ϕ^{j,l−1} = 0 (6.13.11) for every j, l. Equivalently, this means that ϕ^{j−1,l+1} ∘ d_{V}^{−j} + (−1)^{j} d_{W}^{l} ∘ ϕ^{j,l} = 0 (6.13.12) for every j, l. This is also the same as saying that ϕ^{j,l} ∘ d_{V}^{−j−1} + (−1)^{j+1} d_{W}^{l−1} ∘ ϕ^{j+1,l−1} = 0 (6.13.13) for every j, l.

Let j, l ∈ Z be given, and remember that ϕ^{j,l} is a homomorphism from V^{−j} into W^{l}, as modules over A. If ϕ ∈ Z( Hom_{A}^{gr}(V, W)), then ϕ^{j,l}(Z(V)^{−j}) ⊆ Z(W)^{l}, (6.13.14) by (6.13.12). In this case, we also have that ϕ^{j,l}(B(V)^{−j}) ⊆ B(W)^{l}, (6.13.15) by (6.13.13). This implies that ϕ^{j,l} induces a homomorphism (6.13.16) from H(V)^{−j} into H(W)^{l}, as modules over A. Note that this induced homomorphism is equal to 0 for all but finitely many (j, l), because ϕ^{j,l} = 0 for all but finitely many (j, l), by the definition of Hom_{A}^{gr}(V, W).

Let Hom_{A}^{gr}(H(V), H(W)) be the doubly-graded module over k obtained from H(V), H(W) as singly-graded modules over A as in Section 6.3. Using the induced homomorphisms in (6.13.16) for all j, l, we get an element of Hom_{A}^{gr}(H(V), H(W)) from ϕ. This defines a homomorphism (6.13.17) from Z(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(H(V), H(W)), as modules over k. Remember that Z(Hom_{A}^{gr}(V, W)) is a singly-graded module over k, and that Hom_{A}^{gr}(H(V), H(W)) has a single grading induced by the double grading. It is easy to see that the homomorphism in (6.13.17) has degree 0 with respect to these single gradings.

Suppose for the moment that ϕ ∈ B( Hom_{A}^{gr}(V, W)), so that ϕ = δ(ψ) (6.13.18)

170 CHAPTER 6. MORE ON DIFFERENTIATION for some ψ ∈ Hom_{A}^{gr}(V, W). This means that ϕ^{j,l} = (δ(ψ))^{j,l} = ψ^{j−1,l} ∘ d_{V}^{−j} + (−1)^{j} d_{W}^{l−1} ∘ ψ^{j,l−1} (6.13.19) for every j, l, as in (6.13.10). It follows that ϕ^{j,l}(Z(V)^{−j}) ⊆ B(W)^{l} (6.13.20) for every j, l. This implies that the induced homomorphism as in (6.13.16) is equal to 0 for every j, l. Equivalently, ϕ is mapped to 0 by the homomorphism as in (6.13.17).

This shows that the homomorphism as in (6.13.17) leads to a homomorphism (6.13.21) α′ from H(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(H(V), H(W)). This homomorphism has degree 0 with respect to the appropriate single gradings. This corresponds to the homomorphism in Proposition 6.1a on p65f of [50], under slightly different conditions.

6.14 Some properties of Hom_{A}^{gr}(Z′(V), Z(W))

We continue with the same notation and hypotheses as in the previous section. Remember that Z′(V) = V/d_{V} (V) and Z(W) may be considered as singlygraded modules over A, as in Section 5.10. Thus Hom_{A}^{gr}(Z′(V), Z(W)) may be defined as a doubly-graded module over k, as in Section 6.3.

Using the natural quotient mapping from V onto Z′(V), and the natural inclusion mapping from Z(W) into W, we get a natural homomorphism (6.14.1) from Hom_{A}^{gr}(Z′(V), Z(W)) into Hom_{A}^{gr}(V, W), as modules over k, as in Section 6.4. This homomorphism has bidegree (0, 0), because the quotient and inclusion mappings mentioned before have degree 0. This homomorphism is also injective.

Let ϕ ∈ Hom_{A}^{gr}(V, W) be given. Suppose that ϕ^{j,l}(V^{−j}) ⊆ Z(W)^{l} (6.14.2) for every j, l. Suppose too that ϕ^{j,l} = 0 on B(V)^{−j} (6.14.3) for every j, l. It is easy to see that ϕ is in the image of the homomorphism as in (6.14.1) under the conditions. Conversely, every element of the image of the homomorphism as in (6.14.1) has these two properties.

6.15. SOME PROPERTIES OF HOM_{A}^{GR}(Z(V), Z′(W)) 171

Observe that (6.14.3) is the same as saying that δ_{1}(ϕ^{j,l}) = 0 (6.14.4) in (Hom_{A}^{gr}(V, W))^{j+1,l}, by (6.13.4). Similarly, (6.14.2) is the same as saying that δ_{2}(ϕ^{j,l}) = 0 (6.14.5)

in Hom_{A}^{gr}(V, W)^{j,l+1}, by (6.13.5). It follows that (6.14.3) holds for every j, l if and only if δ_{1}(ϕ) = 0. (6.14.6)

Similarly, (6.14.2) holds for every j, l if and only if

δ_{2}(ϕ) = 0. (6.14.7)

In particular, the natural homomorphism as in (6.14.1) may be considered as a homomorphism

(6.14.8) from Hom^{gr}(Z′(V), Z(W)) into Z(Hom_{A}^{gr}(V, W)),

as modules over k. This homomorphism has degree 0, with respect to the appropriate single gradings. This leads to a natural homomorphism

(6.14.9) η from Hom_{A}^{gr}(Z′(V), Z(W)) into H(Hom_{A}^{gr}(V, W)),

as modules over k, by composing the homomorphism as in (6.14.8) with the natural quotient mapping from Z( Hom_{A}^{gr}(V, W)) onto H(Hom_{A}^{gr}(V, W)). This homomorphism has degree 0 with respect to the appropriate single gradings as well. This corresponds to the homomorphism η in the diagram (1) on p64 of [50], under slightly different consitions.

There is also a natural homomorphism

(6.14.10) ξ from Hom_{A}^{gr}(Z′(V), Z(W)) into Hom_{A}^{gr}(H(V), H(W)),

as modules over k. This uses the natural inclusion mapping from H(V) into Z′(V), and the natural quotient mapping from Z(W) onto H(W), as in Section 6.4 again. This homomorphism has bidegree (0, 0), because these inclusion and quotient mappings have degree 0. This corresponds to the homomorphism ξ in the diagram (1) on p64 of [50], under slightly different conditions.

One can check that α′ ∘ η = ξ, (6.14.11)

as homomorphisms from Hom_{A}^{gr}(Z′(V), Z(W)) into Hom_{A}^{gr}(H(V), H(W)). To see this, observe first that α′ ∘ η is the same as the composition of the homomorphisms as in (6.14.8) and (6.13.17). It is easy to see that this composition is the same as ξ. This corresponds to part of Proposition 6.1a on p65f of [50], under slightly different conditions.

6.15 Some properties of Hom_{A}^{gr}(Z(V), Z′(W))

Let us continue with the same notation and hypotheses as in the previous two sections. As before, Z(V) and Z′(W) = W/d_{W} (W) may be considered as singly-graded modules over A, so that Hom_{A}^{gr}(Z(V), Z′(W)) may be defined as a doubly-graded module over k.

We can use the natural inclusion mapping from Z(V) into V, and the natural quotient mapping from W onto Z′(W), to get a natural homomorphism

(6.15.1) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(Z(V), Z′(W)),

as modules over k, as in Section 6.4. This homomorphism has bidegree (0, 0), because the inclusion and quotient maps being used have degree 0.

Of course, there are natural homomorphisms

(6.15.2) from Z(V), Z(W) into Z′(V), Z′(W),

respectively, as modules over A. These homomorphisms are obtained by composing the natural inclusion mappings from Z(V), Z(W) into V, W with the natural quotient mappings from V, W onto Z′(V), Z′(W), respectively. Note that these homomorphisms have degree 0.

Using these homomorphisms, we get a natural homomorphism

(6.15.3) from Hom_{A}^{gr}(Z′(V), Z(W)) into Hom_{A}^{gr}(Z(V), Z′(W)),

as modules over k, as in Section 6.4 again. This homomorphism has bidegree (0, 0), as before. This homomorphism is the same as the composition of the homomorphism as in (6.14.1) with the homomorphism as in (6.15.1).

If ϕ ∈ B(Hom_{A}^{gr}(V, W)), then ϕ maps Z(V)^{−j} into B(W)^{l} for every j, l, as in (6.13.20). This means that ϕ is mapped to 0 in Hom_{A}^{gr}(Z(V), Z′(W)) by the homomorphism as in (6.15.1). In particular, this leads to a natural homomorphism

(6.15.4) ζ from H(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(Z(V), Z′(W)),

as modules over k. This homomorphism has degree 0 with respect to the appropriate single gradings. This corresponds to the homomorphism ζ in the diagram (1) on p64 of [50], under slightly different conditions.

Remember that η is the homomorphism as in (6.14.9). It is easy to see that

(6.15.5) ζ ∘ η is the same as the homomorphism as in (6.15.3).

Using the natural quotient mapping from Z(V) onto H(V), and the natural inclusion mapping from H(W) into Z′(W), we get a natural homomorphism

(6.15.6) τ from Hom_{A}^{gr}(H(V), H(W)) into Hom_{A}^{gr}(Z(V), Z′(W)),

as modules over k, as in Section 6.4. Note that

(6.15.7) τ is injective,

as has bidegree (0, 0), as usual. This corresponds to the homomorphism τ in diagram (1) on p64 of [50], under slightly different conditions.

Remember that ξ is the homomorphism in (6.14.10). It is easy to see that

(6.15.8) τ ∘ ξ is the same as the homomorphism as in (6.15.3).

It follows that τ ∘ ξ = ζ ∘ η, (6.15.9)

by (6.15.5). This corresponds to the commutativity of the diagram (1) on p64 of [50], under slightly different conditions.

Remember that α′ is the homomorphism as in (6.13.21). One can check that

τ ∘ α′ = ζ, (6.15.10)

as homomorphisms from H( Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(Z(V), Z′(W)). In fact, α′ is uniquely determined by this property, because τ is injective. This corresponds to part of Proposition 6.1a on p65f of [50], under slightly different conditions. Note that (6.14.11) could also be obtained from (6.15.10), by composing both sides with η, and using (6.15.9).

More on differentiation, 2

7.1 Some splitting and α

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. As in Section 6.11, we let (V, d_{V}) be a graded right module over A with differentiation that is a complex, and (W, d_{W}) be a graded left module over A with differentiation that is a complex. We also let V ⨂_{A} W be a tensor product of V and W over A, which is a double complex over k with respect to the double grading and differentiation operators δ_{1}, δ_{2} defined in Sections 6.1 and 6.2. Thus V ⨂_{A} W may be considered as a single complex over k too, with respect to the single grading obtained from the double grading, and the total differentiation operator δ = δ_{1} + δ_{2}, as in Section 5.14.

Remember that

Z′(V) = V/d_{V} (V), Z′(W) = W/d_{W} (W) (7.1.1)

are graded modules over A, and that H(V), H(W) are homogeneous submodules of Z′(V), Z′(W), respectively, as in Section 5.10. In this section, we suppose that

(7.1.2) Z′(V) corresponds to the direct sum of H(V)

and another homogeneous submodule of Z′(V),

as a right module over A, and similarly that

(7.1.3) Z′(W) corresponds to the direct sum of H(W)

and another homogeneous submodule of Z′(W),

as a left module over A. This means that Z′(V), Z′(W) correspond to the direct sums of H(V), H(W) and other submodules, respectively, as graded modules over A.

7.1. SOME SPLITTING AND α 175

Equivalently,

(7.1.4) there are homomorphisms from Z′(V), Z′(W) onto H(V), H(W), as modules over A, that are equal to the identity mappings on H(V), H(W), respectively, and that have degree 0.

We can compose the natural quotient mappings from V, W onto Z′(V), Z′(W), respectively, with the homomorphisms as in (7.1.4) to get homomorphisms

(7.1.5) β, γ from V, W onto H(V), H(W),

respectively, as modules over A, of degree 0.

Of course, the natural quotient mappings from V, W onto Z′(V), Z′(W) are equal to 0 on d_{V} (V), d_{W} (W), respectively. This means that the compositions of these quotient mappings with d_{V}, d_{W}, respectively, are equal to 0. It follows that β ∘ d_{V} , γ ∘ d_{W} = 0. (7.1.6)

Let H(V) ⨂_{A} H(W) be a tensor product of H(V) and H(W), which is a doubly-graded module over k. Using β and γ, we get a homomorphism β ⊗ γ from V ⨂_{A} W onto H(V) ⨂_{A} H(W), as modules over k, of bidegree (0, 0), as in Section 6.1. It is easy to see that

(β ⊗ γ) ∘ δ_{1} = (β ⊗ γ) ∘ δ_{2} = 0, (7.1.7)

using (7.1.6). This implies that

(β ⊗ γ) ∘ δ = 0. (7.1.8)

This leads to a homomorphism

(7.1.9) from H(V ⨂_{A}W ) into H(V) ⨂_{A}H(W),

as modules over k, in a natural way. More precisely, the composition of the natural quotient mapping from Z(V ⨂_{A} W ) onto H(V ⨂_{A} W ) with this homomorphism is the same as the restriction of β⊗γ to Z(V ⨂_{A} W ). This homomorphism has degree 0, with respect to the usual single grading on H(V ⨂_{A} W ), and the single grading on H(V) ⨂_{A} H(W) obtained from the double grading in the usual way.

Let α be the homormophism from H(V) ⨂_{A} H(W) into H(V ⨂_{A} W ) discussed in Section 6.11. One can check that

(7.1.10) the composition of α with the homomorphism as in (7.1.9)

is the identity mapping on H(V) ⨂_{A}H(W).

This implies that

(7.1.11) α is injective on H(V) ⨂_{A}H(W).

176 CHAPTER 7. MORE ON DIFFERENTIATION, 2 We also get that (7.1.12) H(V ⨂_{A}W ) corresponds to the direct sum of

α(H(V) ⨂_{A}H(W)) and another homogeneous

submodule of H(V ⨂_{A}W ),

as a singly-graded module over k. This corresponds to Proposition 6.2 on p66 of [50], under slightly different conditions.

7.2 More splitting and α

Let us return to the same notation and hypotheses as at the beginning of the previous section. Put V_{1} = B(V) = d_{V} (V), W_{1} = B(W) = d_{W} (W), (7.2.1) which are homogeneous submodules of V, W, respectively. Suppose that (7.2.2) V_{2}, V_{3} are homogeneous submodules of V and (7.2.3) W_{2}, W_{3} are homogeneous submodules of W, with the following two properties. First, (7.2.4) V corresponds to the direct sum of V_{1}, V_{2}, and V_{3}, as a right module over A, and (7.2.5) W corresponds to the direct sum of W_{1}, W_{2}, and W_{3}, as a left module over A. Second, Z(V) = V_{1} + V_{2} (7.2.6) and Z(W) = W_{1} + W_{2}. (7.2.7)

More precisely, Z(V) and Z(W) correspond to the direct sums of V_{1}, V_{2} and W_{1}, W_{2}, respectively, as graded modules over A. It follows that the restrictions of the natural quotient mappings from Z(V), Z(W) onto H(V), H(W) to V_{2}, W_{2} are isomorphisms (7.2.8) from V_{2}, W_{2} onto H(V), H(W), respectively, as graded modules over A.

7.2. MORE SPLITTING AND α 177

Similarly, the restrictions of the natural quotient mappings from V, W onto Z′(V), Z′(W) to V_{2} + V_{3}, W_{2} + W_{3} are isomorphisms

(7.2.9) from V_{2} + V_{3}, W_{2} + W_{3} onto Z′(V), Z′(W),

respectively, as modules over A. The images of V_{2}, W_{2} under these quotient mappings are the same as the images of Z(V), Z(W), by hypothesis. Thus these images correspond to H(V), H(W), respectively. Of course, V_{2} + V_{3} and W_{2} + W_{3} correspond to the direct sums of V_{2}, V_{3} and W_{2}, W_{3}, respectively, as modules over A. It follows that (7.1.2) and (7.1.3) hold under these conditions.

Let V_{a} ⨂_{A} W_{b} be a tensor product of V_{a} and W_{b} over A for each a, b = 1, 2, 3. Of course, V ⨂_{A} W is isomorphic to the direct sum of V_{a} ⨂_{A} W_{b}, a, b = 1, 2, 3, as a doubly-graded module over k. Thus we may identify V_{a} ⨂_{A} W_{b} with a bihomogeneous submodule of V ⨂_{A} W, as a doubly-graded module over k, for each a, b = 1, 2, 3.

If a = 1, 2, or 3, then we may take V_{a} ⨂_{A} Z(W) to be the bihomogeneous submodule of V ⨂_{A} W corresponding to the direct sum of V_{a} ⨂_{A} W_{b}, b = 1, 2. Similarly, if b = 1, 2, or 3, then we may take Z(V) ⨂_{A} W_{b} to be the bihomogeneous submodule of V ⨂_{A} W corresponding to the direct sum of V_{a} ⨂_{A} W_{b}, a = 1, 2. We may take Z(V) ⨂_{A} Z(W) to be the bihomogeneous submodule of V ⨂_{A} W corresponding to the direct sum of V_{a} ⨂_{A} W_{b}, a, b = 1, 2, as well.

The restrictions of d_{V}, d_{W} to V_{3}, W_{3} are one-to-one mappings onto V_{1}, W_{1}, respectively. If a = 1, 2, or 3, then it follows that

(7.2.10) the restriction of δ_{2} to V_{a} ⨂_{A}W_{3}

is a one-to-one mapping onto V_{a} ⨂_{A}W_{1}.

If a = 1 or 2, then δ_{1} = 0 on V_{a} ⨂_{A} W_{3}, so that δ = δ_{2} on V_{a} ⨂_{A} W_{3}, and thus

(7.2.11) the restriction of δ to V_{a} ⨂_{A}W_{3}

is a one-to-one mapping onto V_{a} ⨂_{A}W_{1}.

Similarly, if b = 1, 2, or 3, then

(7.2.12) the restriction of δ_{1} to V_{3} ⨂_{A}W_{b}

is a one-to-one mapping onto V_{1} ⨂_{A}W_{b}.

If b = 1 or 2, then δ_{2} = 0 on V_{3} ⨂_{A} W_{b}, so that δ = δ_{1} on V_{3} ⨂_{A} W_{b}, and

(7.2.13) the restriction of δ to V_{3} ⨂_{A}W_{b}

is a one-to-one mapping onto V_{1} ⨂_{A}W_{b}.

Observe that

(7.2.14) the restriction of δ to V_{3} ⨂_{A}W_{3} is a one-to-one

mapping into (V_{1} ⨂_{A}W_{3}) + (V_{3} ⨂_{A}W_{1}),

178 CHAPTER 7. MORE ON DIFFERENTIATION, 2 by (7.2.10) and (7.2.12), with a = b = 3. Of course, Z(V) ⨂_{A}Z(W) ⊆ Z(V ⨂_{A}W ). (7.2.15) One can check that Z(V ⨂_{A}W ) ⊆ (Z(V) ⨂_{A}Z(W)) + (V_{1} ⨂_{A}W_{3}) + (V_{3} ⨂_{A}W_{1}), (7.2.16) using the earlier remarks.

It is easy to see that

B(V ⨂_{A}W ) = (V_{1} ⨂_{A}W_{1}) + (V_{2} ⨂_{A}W_{1}) + (V_{1} ⨂_{A}W_{2})

+δ(V_{3} ⨂_{A}W_{3}), (7.2.17) using (7.2.11) and (7.2.13). Note that δ(V_{3} ⨂_{A}W_{3}) ⊆ Z(V ⨂_{A}W ) ∩ ((V_{1} ⨂_{A}W_{3}) + (V_{3} ⨂_{A}W_{1})). (7.2.18) We would like to verify that δ(V_{3} ⨂_{A}W_{3}) = Z(V ⨂_{A}W ) ∩ ((V_{1} ⨂_{A}W_{3}) + (V_{3} ⨂_{A}W_{1})). (7.2.19)

Every element of V_{1} ⨂_{A} W_{3} can be expressed as δ_{1}(y) for a unique y in V_{3} ⨂_{A} W_{3}, by (7.2.12), with b = 3. Similarly, every element of V_{3} ⨂_{A} W_{1} can be expressed as δ_{2}(z) for a unique z ∈ V_{3} ⨂ W_{3}, by (7.2.10), with a = 3. We also have that δ(δ_{1}(y) + δ_{2}(z)) = δ_{2}(δ_{1}(y)) + δ_{1}(δ_{2}(z)). (7.2.20) This is equal to 0 when y = z, as in Section 6.2.

Conversely, if (7.2.20) is equal to 0, then one can check that y = z, using (7.2.10) and (7.2.12). This implies that (7.2.19) holds.

It follows that Z(V ⨂_{A}W ) = (Z(V) ⨂_{A}Z(W)) + δ(V_{3} ⨂_{A}W_{3}). (7.2.21) Using this, we get that (7.2.22) H(V ⨂_{A}W ) is isomorphic to V_{2} ⨂_{A}W_{2}, as graded modules over k. Equivalently, (7.2.23) H(V ⨂_{A}W ) is isomorphic to H(V) ⨂_{A}H(W), as graded modules over k. More precisely, (7.2.24) α is an isomorphism from H(V) ⨂_{A}H(W) onto H(V ⨂_{A}W ), as graded modules over k under these conditions, where α is as in Section 6.11. This corresponds to Proposition 7.4 on p70 of [50].

7.3. SOME SPLITTING AND α′ 179

7.3 Some splitting and α′

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. As in Section 6.13, we let (V, d_{V}) and (W, d_{W}) be both left or both right graded modules over A with differentiation that are complexes. Thus Hom_{A}^{gr}(V, W) may be defined as a doubly-graded module over k, and in fact as a double complex with respect to the appropriate differentiation operators δ_{1}, δ_{2}, as in Sections 6.3 and 6.5. This implies that Hom_{A}^{gr}(V, W) is a single complex over k with respect to the single grading obtained from the double grading and the total differentation operator δ = δ_{1} + δ_{2}, as before.

Suppose that Z′(V) satisfies (7.1.2), as a left or right module over A, as appropriate. In this section, we also ask that

(7.3.1) Z(W) corresponds to the direct sum of B(W)

and another homogeneous submodule of Z(W),

as a module over A. Equivalently, this means that Z(W) corresponds to the direct sum of B(W) and another submodule, as a graded module over A.

As before, the condition on Z′(V) is the same as saying that there is a homomorphism from Z′(V) onto H(V), as modules over A, that is equal to the identity mapping on H(V), and has degree 0. This leads to a homomorphism

(7.3.2) β from V onto H(V),

as modules over A, with degree 0, by composing the natural quotient mapping from V onto Z′(V) with the previous homomorphism from Z′(V) onto H(V). Note that β ∘ d_{V} = 0. (7.3.3)

Using (7.3.1), we get that

(7.3.4) there is a homomorphism from H(W) into Z(W), as modules over A, of degree 0, whose composition with the natural quotient mapping from Z(W) onto H(W) is the identity mapping on H(W).

More precisely, Z(W) corresponds to the direct sum of B(W) and the image of H(W) under this homomorphism, as a graded module over A. Let γ̃ be the homomorphism as in (7.3.4), considered as a homomorphism

(7.3.5) from H(W) into W.

Alternatively, γ̃ may be considered as the composition of the homomorphism in (7.3.4) with the natural inclusion mapping from Z(W) into W. Of course,

d_{W} ∘ γ̃ = 0, (7.3.6)

180 CHAPTER 7. MORE ON DIFFERENTIATION, 2

because γ̃(H(W)) ⊆ Z(W).

Note that Hom_{A}^{gr}(H(V), H(W)) may be defined as a doubly-graded module over k, as in Section 6.3. As in Section 6.4, we can use β and γ̃ to get a homomorphism (7.3.7) Hom^{gr}(β, γ̃) from Hom_{A}^{gr}(H(V), H(W)) into Hom_{A}^{gr}(V, W), as modules over k, of bidegree (0, 0).

One can check that δ_{1} ∘ Hom^{gr}(β, γ̃) = 0, (7.3.8) using (7.3.3). Similarly, δ_{2} ∘ Hom^{gr}(β, γ̃) = 0, (7.3.9) because of (7.3.6). This implies that δ ∘ Hom^{gr}(β, γ̃) = 0. (7.3.10) Equivalently, (7.3.11) Hom^{gr}(β, γ̃)(Hom_{A}^{gr}(H(V), H(W))) ⊆ Z(Hom_{A}^{gr}(V, W)).

This leads to a homomorphism (7.3.12) from Hom_{A}^{gr}(H(V), H(W)) into H(Hom_{A}^{gr}(V, W)), as modules over k, by composing Hom^{gr}(β, γ̃) with the natural quotient mapping (7.3.13) from Z(Hom_{A}^{gr}(V, W)) onto H(Hom_{A}^{gr}(V, W)). This homomorphism has degree 0, with respect to the single grading associated to the double grading on Hom_{A}^{gr}(H(V), H(W)), and the appropriate single grading on H( Hom_{A}^{gr}(V, W)).

Let α′ be the homomorphism (7.3.14) from H(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(H(V), H(W)) discussed in Section 6.13. One can check that (7.3.15) the composition of the homomorphism as in (7.3.12) with α′

is the identity mapping on Hom_{A}^{gr}(H(V), H(W)). It follows that

α′(H(Hom_{A}^{gr}(V, W))) = Hom_{A}^{gr}(H(V), H(W)). (7.3.16) We also get that (7.3.17) H(Hom_{A}^{gr}(V, W)) corresponds to the direct sum of ker α′

and another homogeneous submodule of H(Hom_{A}^{gr}(V, W)),

as a singly-graded module over k. This corresponds to Proposition 6.2a on p66 of [50], under slightly different conditions.

7.4. MORE SPLITTING AND α′ 181

7.4 More splitting and α′

We return to the same notation and hypotheses as at the beginning of the previous section. As in Section 7.2, we would like to consider stronger splitting conditions on V and W now.

As before, we put V_{1} = B(V), W_{1} = B(W), (7.4.1)

which are homogeneous submodules of V, W, respectively. We suppose again that V_{2}, V_{3} are homogeneous submodules of V, and that V corresponds to the direct sum of V_{1}, V_{2}, and V_{3}, as a module over A. Similarly, we suppose that W_{2}, W_{3} are homogeneous submodules of W, and that W corresponds to the direct sum of W_{1}, W_{2}, and W_{3}, as a module over A. We also ask that

Z(V) = V_{1} + V_{2}, Z(W) = W_{1} + W_{2}, (7.4.2)

as before, so that Z(V) and Z(W) correspond to the direct sums of V_{1}, V_{2} and W_{1}, W_{2}, respectively, as graded modules over A. This implies that the restrictions of the natural quotient mappings from Z(V), Z(W) onto H(V), H(W) to V_{2}, W_{2} are isomorphisms onto H(V), H(W), respectively, as graded modules over A, as before.

It follows that Z′(V) corresponds to the direct sum of H(V) and another homogeneous submodules of Z′(V), as a module over A, as in Section 7.2. Of course, (7.3.1) holds, by hypothesis. This means that V and W satisfy the conditions mentioned in the previous section.

Note that Hom_{A}^{gr}(V_{a}, W_{b}) may be defined as a doubly-graded module over k as in Section 6.3 for each a, b = 1, 2, 3. It is easy to see that

(7.4.3) Hom_{A}^{gr}(V, W) is isomorphic to the direct sum of Hom_{A}^{gr}(V_{a}, W_{b}), a, b = 1, 2, 3,

as a doubly-graded module over k, in a natural way. We shall use this to identify Hom_{A}^{gr}(V_{a}, W_{b}) with a bihomogeneous submodule of Hom_{A}^{gr}(V, W), as a doubly-graded module over k, for each a, b = 1, 2, 3.

Similarly, if a = 1, 2, or 3, then

(7.4.4) Hom_{A}^{gr}(V_{a}, Z(W)) is isomorphic to the direct sum of Hom_{A}^{gr}(V_{a}, W_{b}), b = 1, 2,

as a doubly-graded module over k in a natural way. If b = 1, 2, or 3, then

(7.4.5) Hom_{A}^{gr}(Z(V), W_{b}) is isomorphic to the direct sum of Hom_{A}^{gr}(V_{a}, W_{b}), a = 1, 2,

as a doubly-graded module over k in a natural way. We also have that

(7.4.6) Hom_{A}^{gr}(Z(V), Z(W)) is isomorphic to the direct sum of Hom_{A}^{gr}(V_{a}, W_{b}), a, b = 1, 2, as a doubly-graded module over k in a natural way. These modules may be identified with bihomogeneous submodules of Hom_{A}^{gr}(V, W), as a doubly-graded module over k, as before.

If a = 2 or 3, then one can check that

δ_{1} = 0 on Hom_{A}^{gr}(V_{a}, W_{b}) (7.4.7)

for each b = 1, 2, 3. This uses the way that Hom_{A}^{gr}(V_{a}, W_{b}) is identified with a submodule of Hom_{A}^{gr}(V, W), so that an element of Hom_{A}^{gr}(V_{a}, W_{b}) corresponds to an element of Hom_{A}^{gr}(V, W) whose components in Hom_{A}^{gr}(V_{a′}, W_{b′}) are equal to 0 when a ≠ a′ or b ≠ b′. This also uses the fact that V_{1} = d_{V} (V), by construction. Of course,

δ_{2} = 0 on Hom_{A}^{gr}(V_{a}, W_{b}) (7.4.8)

for each a = 1, 2, 3 and b = 1, 2, because d_{W} = 0 on W_{1}, W_{2}.

The restrictions of d_{V}, d_{W} to V_{3}, W_{3} are one-to-one mappings onto V_{1}, W_{1}, respectively, as in Section 7.2. If a = 1, 2, or 3, then it follows that

(7.4.9) the restriction of δ_{2} to Hom_{A}^{gr}(V_{a}, W_{3})

is a one-to-one mapping onto Hom_{A}^{gr}(V_{a}, W_{1}).

If a = 2 or 3, then δ_{1} = 0 on Hom_{A}^{gr}(V_{a}, W_{3}), as in (7.4.7), so that δ = δ_{2} on Hom_{A}^{gr}(V_{a}, W_{3}). This implies that

(7.4.10) the restriction of δ to Hom_{A}^{gr}(V_{a}, W_{3})

is a one-to-one mapping onto Hom_{A}^{gr}(V_{a}, W_{1})

when a = 2 or 3.

If b = 1, 2, or 3, then it is easy to see that

(7.4.11) the restriction of δ_{1} to Hom_{A}^{gr}(V_{1}, W_{b})

is a one-to-one mapping onto Hom_{A}^{gr}(V_{3}, W_{b}).

If b = 1 or 2, then δ_{2} = 0 on Hom_{A}^{gr}(V_{1}, W_{b}), so that δ = δ_{1} on Hom_{A}^{gr}(V_{1}, W_{b}), and thus

(7.4.12) the restriction of δ to Hom_{A}^{gr}(V_{1}, W_{b})

is a one-to-one mapping onto Hom_{A}^{gr}(V_{3}, W_{b}).

Note that

(7.4.13) the restriction of δ to Hom_{A}^{gr}(V_{1}, W_{3}) is a one-to-one mapping into Hom_{A}^{gr}(V_{3}, W_{3}) + Hom_{A}^{gr}(V_{1}, W_{1}),

by (7.4.9) with a = 1, and (7.4.11) with b = 3.

Clearly δ = 0 on Hom_{A}^{gr}(V_{a}, W_{b}) (7.4.14)

7.4. MORE SPLITTING AND α′ 183 when a = 2 or 3 and b = 1 or 2, by (7.4.7) and (7.4.8). This means that (7.4.15) Hom_{A}^{gr}(V_{2}, Z(W)) + Hom_{A}^{gr}(V_{3}, Z(W)) ⊆ Z(Hom_{A}^{gr}(V, W)).

Observe that (7.4.16) δ(Hom_{A}^{gr}(V_{1}, W_{1})) = δ(Hom_{A}^{gr}(V_{3}, W_{3})) = Hom_{A}^{gr}(V_{3}, W_{1}), by (7.4.10) with a = 3, and (7.4.12) with b = 1. One can verify that

Z(Hom_{A}^{gr}(V, W)) ⊆ Hom_{A}^{gr}(V_{2}, Z(W)) + Hom_{A}^{gr}(V_{3}, Z(W))

+Hom_{A}^{gr}(V_{1}, W_{1}) + Hom_{A}^{gr}(V_{3}, W_{3}). (7.4.17) This uses (7.4.10) with a = 2, (7.4.12) with b = 2, and (7.4.13).

We also have that

B(Hom_{A}^{gr}(V, W)) = Hom_{A}^{gr}(V_{2}, W_{1}) + Hom_{A}^{gr}(V_{3}, W_{1}) + Hom_{A}^{gr}(V_{3}, W_{2})

+δ(Hom_{A}^{gr}(V_{1}, W_{3})), (7.4.18) by (7.4.10), (7.4.12), and (7.4.14). Of course,

δ(Hom_{A}^{gr}(V_{1}, W_{3}))

(7.4.19) ⊆ Z(Hom_{A}^{gr}(V, W)) ∩ (Hom_{A}^{gr}(V_{1}, W_{1}) + Hom_{A}^{gr}(V_{3}, W_{3})),

by (7.4.13). We would like to check that

δ(Hom_{A}^{gr}(V_{1}, W_{3}))

(7.4.20) = Z(Hom_{A}^{gr}(V, W)) ∩ (Hom_{A}^{gr}(V_{1}, W_{1}) + Hom_{A}^{gr}(V_{3}, W_{3})).

Every element of Hom_{A}^{gr}(V_{1}, W_{1}) can be expressed as δ_{2}(ϕ) (7.4.21) for a unique ϕ ∈ Hom_{A}^{gr}(V_{1}, W_{3}), by (7.4.9), with a = 1. Similarly, every element of Hom_{A}^{gr}(V_{3}, W_{3}) can be expressed as δ_{1}(ψ) (7.4.22) for a unique ψ ∈ Hom_{A}^{gr}(V_{1}, W_{3}), by (7.4.11) with b = 3. Note that δ(δ_{2}(ϕ) + δ_{1}(ψ)) = δ_{1}(δ_{2}(ϕ)) + δ_{2}(δ_{1}(ψ)). (7.4.23) This is equal to 0 when ϕ = ψ, as in Section 6.5. Conversely, if (7.4.23) is equal to 0, then one can check that ϕ = ψ, using (7.4.9) and (7.4.11).

This implies (7.4.20). It follows that

Z(Hom_{A}^{gr}(V, W)) = Hom_{A}^{gr}(V_{2}, Z(W)) + Hom_{A}^{gr}(V_{3}, Z(W))

+δ(Hom_{A}^{gr}(V_{1}, W_{3})), (7.4.24) by (7.4.15) and (7.4.17).

Combining this with (7.4.18), we obtain that

(7.4.25) H(Hom_{A}^{gr}(V, W)) is isomorphic to Hom_{A}^{gr}(V_{2}, W_{2}),

as graded modules over k. This means that

(7.4.26) H(Hom_{A}^{gr}(V, W)) is isomorphic to Hom_{A}^{gr}(H(V), H(W)),

as graded modules over k. In fact,

(7.4.27) α′ is an isomorphism from H(Hom_{A}^{gr}(V, W))

onto Hom_{A}^{gr}(H(V), H(W))

as graded modules over k in this case, where α′ is as in Section 6.13. This corresponds to Proposition 7.4 on p70 of [50] again.

7.5 Transforming modules and complexes

We have seen a couple of basic ways in which pairs of singly-graded modules or complexes can be combined to get doubly-graded modules or complexes, as in the previous chapter. This is discussed more broadly in Section 5 of Chapter IV of [50], for combining r singly-graded modules of complexes into r-graded modules and r-tuple complexes for any r. In particular, one can take r = 1, and consider transformations of singly-graded modules or complexes into other singly-graded modules or complexes. We shall consider some instances of this here, and in later sections.

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a right module over A, let W be a left module over A, and let V ⨂_{A} W be a tensor product of V and W over A. In this section, we shall normally consider W as fixed, and consider V ⨂_{A} W as a transform of V. Of course, one could just as well consider V ⨂_{A} W as a transform of W, with V fixed.

If V is a graded module over A, then V ⨂_{A} W is graded as a module over k in a natural way. More precisely, if V^{j} ⨂_{A} W is a tensor product of V^{j} with W over A for each j, then V ⨂_{A} W is isomorphic to the direct sum of V^{j} ⨂_{A} W, j ∈ Z. Thus we can identify V^{j} ⨂_{A} W with a submodule of V ⨂_{A} W for each j, as a module over k, and take

(V ⨂_{A}W )^{j} = V^{j} ⨂_{A}W. (7.5.1)

This corresponds to some remarks on p62 of [50], with r = 1.

We may consider W as a graded module over A, with

(7.5.2)  W^{l} = W when l = 0
= {0} when l ≠ 0,

as on p75 of [50]. If V is a graded module over A, then V ⨂_{A} W may be considered as a doubly-graded module over k, as in Section 6.1, with

(7.5.3) (V ⨂_{A}W )^{j,l} = V^{j} ⨂_{A}W^{l} = V^{j} ⨂_{A}W when l = 0
= {0}  when l ≠ 0.

In this case, the single grading on V ⨂_{A} W associated to the double grading is the same as the single grading in the preceding paragraph.

Suppose now that (V, d_{V}) is a right module over A with differentiation. Let

d = d_{V ⨂_{A}W} (7.5.4)

be the homomorphism from V ⨂_{A}W into itself, as a module over k, obtained from d_{V} and the identity mapping on W in the usual way. Thus

d(v ⊗ w) = d_{V} (v) ⊗ w (7.5.5)

for every v ∈ V and w ∈ W, and d is uniquely determined by this property. It is easy to see that d ∘ d = 0 on V ⨂_{A}W, (7.5.6)

so that V ⨂_{A} W is a module with differentiation with respect to d.

Suppose that (V, d_{V}) is in fact a graded right module over A with differentiation that is a complex. We may consider W as a graded left module over A with differentiation that is a complex, with the grading as in (7.5.2), and differentiation operator d_{W} = 0, as on p75 of [50]. In this case, V ⨂_{A} W is a double complex, with differentiation operators

δ_{1} = d, δ_{2} = 0, (7.5.7)

as in Section 6.2. This means that V ⨂_{A} W is a single complex with respect to the total differentiation operator

δ = δ_{1} + δ_{2} = d, (7.5.8)

and the single gradng associated to the double grading on V ⨂_{A} W, as in Section 5.14.

Alternatively, if V ⨂_{A} W is considered as a singly-graded module over k as in (7.5.1), then it is easy to see that d has degree 1. This makes V ⨂_{A} W into a single complex, as on p63 of [50]. It is easy to see that this is the same as considering V ⨂_{A} W as a single complex as in the preceding paragraph.

7.6 Induced mappings and homotopies

let V be a right module over A, let W be a left module over A, and let V ⨂_{A} W be a tensor product of V and W over A, as before.

Suppose for the moment that (V, d_{V}) is a right module over A with differentiation, and let d = d_{V ⨂_{A}W} be defined on V ⨂_{A} W as in (7.5.4). Let (V_{0}, d_{V_{0}}) be another right module over A with differentiation, and let V_{0} ⨂_{A} W be a tensor product of V_{0} and W over A. We can define

d_{0} = d_{V_{0} ⨂_{A}W} (7.6.1)

on V_{0} ⨂_{A} W in the same way as before, so that V_{0} ⨂_{A} W is a module with differentiation with respect to d_{0}.

Let ϕ be a homomorphism from V into V_{0}, as right modules over A with differentiation. This leads to a homomorphism Φ

(7.6.2) from V ⨂_{A}W into V_{0} ⨂_{A}W,

as modules over k, using the identity mapping on W. It is easy to see that Φ is a homomorphism as in (7.6.2), as modules with differentiation.

Let ψ be another homomorphism from V into V_{0}, as right modules over A with differentiation, and let Ψ be the corresponding homomorphism as in (7.6.2), as modules over k with differentiation. Suppose that σ is a homotopy between ϕ and ψ, so that σ is a homomorphism from V into V_{0}, as right modules over A, such that d_{V_{0}} ∘ σ + σ ∘ d_{V} = ϕ − ψ. (7.6.3)

Let Σ be the homomorphism as in (7.6.2), as modules over k, corresponding to σ and the identity mapping on W. Observe that

d_{0} ∘ Σ + Σ ∘ d = Φ − Ψ, (7.6.4)

so that Σ is a homotopy between Φ and Ψ.

Suppose now that V, V_{0} are graded right modules over A, so that V ⨂_{A} W, V_{0} ⨂_{A} W may be considered as graded modules over k, as in (7.5.1). Let ϕ be a homomorphism from V into V_{0}, as right modules over A, and let Φ be the homomorphism as in (7.6.2) corresponding to ϕ and the identity mapping on W, as usual. Suppose that ϕ has degree p ∈ Z, and note that Φ has degree p as well.

If we consider W as a graded module as in (7.5.2), then V ⨂_{A} W, V_{0} ⨂_{A} W may be considered as doubly-graded modules over k, as in (7.5.3). It is easy to see that Φ is the same as the homomorphism as in (7.6.2) of bidegree (p, 0) associated to ϕ and the identity mapping on W as in Section 6.1 in this case.

Now let (V, d_{V}), (V_{0}, d_{V_{0}}) be graded right modules over A with differentiation that are complexes, so that V ⨂_{A} W, V_{0} ⨂_{A} W may be considered as single complexes too, as in the previous section. Suppose that ϕ is a map of degree p from V into V_{0}, as complexes, as in Section 6.6. Let Φ be the homomorphism as in (7.6.2), as modules over k, corresponding to ϕ and the identity mapping 7.7. AN ANALOGUE OF α 187

on W, as usual. It is easy to see that Φ is a map of degree p as in (7.6.2), as complexes.

Let ψ be another map of degree p from V into V_{0}, as complexes, and let Ψ be the corresponding map of degree p as in (7.6.2), as complexes, as in the preceding paragraph. Suppose that σ is a homotopy between ϕ and ψ, as maps of degree p between complexes, as in Section 6.6. Let Σ be the homomorphism as in (7.6.2), as modules over k, corresponding to σ and the identity mapping on W, as before. One can check that Σ is a homotopy between Φ and Ψ, as maps of degree p between complexes.

As in the previous section, we may consider W as a complex, with d_{W} = 0. Using this, we may consider V ⨂_{A} W, V_{0} ⨂_{A} W as double complexes, as before. We may also consider Φ, Ψ as maps of bidegree (p, 0) as in (7.6.2), as double complexes, as in Section 6.8. Under these conditions, (Σ, 0) defines a homotopy between Φ and Ψ, as maps of bidegree (p, 0) as in (7.6.2), as double complexes, as in Section 6.8.

Of course, if p = 0, then the remarks in the preceding paragraph correspond to some in Section 6.2.

7.7 An analogue of α

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. As before, we let V be a right module over A, W be a left module over A, and V ⨂_{A} W be a tensor product of V and W over A. Suppose that (V, d_{V}) is a right module over A with differentiation again, so that V ⨂_{A} W is a module over k with differentiation operator d = d_{V ⨂_{A} W} as in (7.5.4).

If v ∈ Z(V) and w ∈ W, then

d(v ⊗ w) = d_{V} (v) ⊗ w = 0, (7.7.1)

so that v ⊗ w ∈ Z(V ⨂_{A} W ). Thus

(v, w) ↦ v ⊗ w (7.7.2)

defines a mapping from Z(V) × W into Z(V ⨂_{A} W ) that is bilinear over k. If a ∈ A, v ∈ Z(V), and w ∈ W, then the values of this mapping at (v · a, w) and (v, a · w) are the same. Let Z(V) ⨂_{A} W be a tensor product of Z(V) and W over A. This leads to a natural homomorphism

(7.7.3) from Z(V) ⨂_{A}W into Z(V ⨂_{A}W ),

as modules over k. Equivalently, there is a natural homomorphism

(7.7.4) from Z(V) ⨂_{A}W into V ⨂_{A}W,

as modules over k, associated to the obvious inclusion mapping from Z(V) into V and the identity mapping on W. One can check that this homomorphism maps Z(V) ⨂_{A} W into Z(V ⨂_{A} W ), because of (7.7.1), so that we get a homomorphism as in (7.7.3).

Similarly, if v ∈ B(V) and w ∈ W, then v = d_{V} (u) for some u ∈ V, so that

v ⊗ w = d_{V} (u) ⊗ w = d(u ⊗ w) (7.7.5)

is an element of B(V ⨂_{A} W ). If B(V) ⨂_{A} W is a tensor product of B(V) and W over A, then one can use this to get a natural homomorphism

(7.7.6) from B(V) ⨂_{A}W into B(V ⨂_{A}W ),

as modules over k, as before. Equivalently, there is a natural homomorphism

(7.7.7) from B(V) ⨂_{A}W into V ⨂_{A}W,

as modules over k, associated to the obvious inclusion mapping from B(V) into V and the identity mapping on W. This homomorphism maps B(V) ⨂_{A} W into B(V ⨂_{A} W ), because of (7.7.5), and thus may be considered as a homomorphism as in (7.7.6).

Of course, there is a natural homomorphism

(7.7.8) from B(V) ⨂_{A}W into Z(V) ⨂_{A}W,

obtained from the obvious inclusion mapping from B(V) into Z(V) and the identity mapping on W. The composition of this homomorphism with the one as in (7.7.3) is the same as the composition of the homomorphism as in (7.7.6) with the obvious inclusion mapping from B(V ⨂_{A} W ) into Z(V ⨂_{A} W ). Equivalently, the composition of the homomorphism as in (7.7.8) with the one as in (7.7.4) is the same as the homomorphism as in (7.7.7).

Consider the natural quotient mapping

(7.7.9) from Z(V ⨂_{A}W ) onto H(V ⨂_{A}W ).

The composition of the homomorphism as in (7.7.3) with this quotient mapping leads to a natural homomorphism

(7.7.10) η from Z(V) ⨂_{A}W into H(V ⨂_{A}W ),

as modules over k.

If v ∈ Z(V) and w ∈ W, then v ⊗ w ∈ Z(V ⨂_{A} W ), as before, which can be mapped into H(V ⨂_{A} W ) by the natural quotient mapping. It is easy to see that the image in H(V ⨂_{A} W ) depends only on w and the image of v under the natural quotient mapping from Z(V) onto H(V). This defines a mapping from H(V) × W into H(V ⨂_{A} W) that is bilinear over k, and has the usual compatibility property for the actions of A on H(V) on the right, and on W on the left. If H(V) ⨂_{A} W is a tensor product of H(V) and W over A, then we get a natural homomorphism

(7.7.11) α from H(V) ⨂_{A}W into H(V ⨂_{A}W ), 7.8. ANALOGUES OF τ, ζ 189

as modules over k.

Using the natural quotient mapping from Z(V) onto H(V) and the identity mapping on W, we get a homomorphism

(7.7.12) ξ from Z(V) ⨂_{A}W onto H(V) ⨂_{A}W,

as modules over k. The surjectivity of ξ follows from a remark in Section 1.9, as usual. It is easy to see that

α ∘ ξ = η, (7.7.13)

by construction. This determines α uniquely, because ξ is surjective.

7.8 Analogues of τ, ζ

Let us continue with the same notation and hypotheses as in the previous section.

Let Z′(V) ⨂_{A} W be a tensor product of Z′(V) = V/d_{V} (V) and W over A. We can use the natural quotient mapping from V onto Z′(V) and the identity mapping on W to get a homomorphism

(7.8.1) from V ⨂_{A}W onto Z′(V) ⨂_{A}W,

as modules over k. The composition of the homomorphism as in (7.7.4) with this one defines a homomorphism

(7.8.2) from Z(V) ⨂_{A}W into Z′(V) ⨂_{A}W,

as modules over k. Alternatively, the restriction to Z(V) of the natural quotient mapping from V onto Z′(V) defines a homomorphism

(7.8.3) from Z(V) into Z′(V),

as modules over k. The homomorphism as in (7.8.2) is the same as the one obtained from the homomorphism as in (7.8.3) and the identity mapping on W in the usual way.

We can use the inclusion mapping from H(V) into Z′(V) and the identity mapping on W to get a natural homomorphism

(7.8.4) τ from H(V) ⨂_{A}W into Z′(V) ⨂_{A}W,

as modules over k. The homomorphism as in (7.8.3) is the same as the composition of the natural quotient mapping from Z(V) onto H(V) with the natural inclusion mapping into Z′(V). This implies that

(7.8.5) τ ∘ ξ is the same as the homomorphism as in (7.8.2).

The restriction of the homomorphism as in (7.8.1) to Z(V ⨂_{A} W ) defines a homomorphism (7.8.6) from Z(V ⨂_{A}W ) into Z′(V) ⨂_{A}W,

as modules over k. It is easy to see that this homomorphism is equal to 0 on B(V ⨂_{A} W ). This leads to a homomorphism

(7.8.7) ζ from H(V ⨂_{A}W ) into Z′(V) ⨂_{A} W,

as modules over k.

Observe that

(7.8.8) ζ ∘ η is the same as the homomorphism as in (7.8.2),

by construction. Thus τ ∘ ξ = ζ ∘ η, (7.8.9)

by (7.8.5).

This implies that ζ ∘ α ∘ ξ = ζ ∘ η = τ ∘ ξ, (7.8.10)

by (7.7.13). It follows that ζ ∘ α = τ, (7.8.11)

because ξ is surjective, as in (7.7.12). This could also be verified more directly from the definitions.

Suppose now that (V, d_{V}) is a graded right module over A with differentiation that is a complex. Thus V ⨂_{A} W may be considered as a graded module over k, as in (7.5.1), and in fact as a complex with respect to d, as in Section 7.5. Similarly,

B(V) ⨂_{A}W, Z(V) ⨂_{A}W, H(V) ⨂_{A}W, and Z′(V) ⨂_{A}W (7.8.12)

may be considered as graded modules over k. In this case, it is easy to see that the various homomorphisms mentioned earlier have degree 0. The properties of these homomorphisms correspond to the commutativity of the diagram (1) and parts of Proposition 6.1 on p64 of [50], under slightly different conditions.

We may also consider W as a graded left module over A with differentiation that is a complex, with the grading as in (7.5.2) and d_{W} = 0, as in Section 7.5. This means that B(W) = {0}, and Z(W) = H(W) = Z′(W) = W. Under these conditions, the remarks in this and the previous section correspond to those in Sections 6.11 and 6.12.

7.9 Splitting Z′(V) and α

7.9. SPLITTING Z′(V) AND α 191 let V be a right module over A, let W be a left module over A, and let V ⨂_{A} W be a tensor product of V and W over A, as before. We suppose that (V, d_{V}) is a right module over A with differentiation again, so that V ⨂_{A} W is a module over k with differentiation operator d = d_{V ⨂_{A} W} as in (7.5.4).

Suppose that (7.9.1) Z′(V) corresponds to the direct sum

of H(V) and another submodule of Z′(V),

as a right module over A. Equivalently, this means that (7.9.2) there is a homomorphism from Z′(V) onto H(V), as modules

over A, that is equal to the identity mapping on H(V).

We can compose the natural quotient mapping from V onto Z′(V) = V/d_{V} (V) with this homomorphism to get a homomorphism (7.9.3) β from V onto H(V), as modules over A. Note that β ∘ d_{V} = 0 (7.9.4) on V.

Using β and the identity mapping on W, we get a homomorphism (7.9.5) from V ⨂_{A}W onto H(V) ⨂_{A}W, as modules over k. Remember that d is the homomorphism from V ⨂_{A} W into itself associated to d_{V} and the identity mapping on W. It is easy to see that (7.9.6) the composition of d with the homomorphism as in (7.9.5)

is equal to 0 on V ⨂_{A}W.

This implies that the homomorphism as in (7.9.5) is equal to 0 on B(V ⨂_{A} W ).

In particular, this leads to a homomorphism (7.9.7) from H(V ⨂_{A}W ) into H(V) ⨂_{A}W, as modules over k. Namely, the composition of the natural quotient mapping from Z(V ⨂_{A} W ) onto H(V ⨂_{A} W ) with this homomorphism is the same as the restriction of the homomorphism as in (7.9.5) to Z(V ⨂_{A} W ). One can verify that (7.9.8) the composition of α with the homomorphism as in (7.9.7)

is the identity mapping on H(V) ⨂_{A}W, where α is as in (7.7.11).

It follows that (7.9.9) α is injective on H(V) ⨂_{A}W.

This also implies that

(7.9.10) H(V ⨂_{A}W ) corresponds to the direct sum of

α(H(V) ⨂_{A}W ) and another submodule of H(V ⨂_{A}W ),

as a module over k.

If (V, d_{V}) is a complex, then V ⨂_{A} W may be considered as a complex with respect to d, as in Section 7.5. In this case, one can ask that (7.9.1) hold for Z′(V) as a graded right module over A, which is to say that the other submodule of Z′(V) is homogeneous too. This implies that the homomorphisms mentioned in the previous paragraphs have degree 0. It follows that the other submodule of H(V ⨂_{A} W ) mentioned in (7.9.10) is homogeneous as well. This corresponds to Proposition 6.2 on p66 of [50], under slightly different conditions.

We may also consider W as a complex, with grading as in (7.5.2) and d_{W} = 0, as before. This permits one to consider the remarks in this section as a simpler version of those in Section 7.1.

7.10 An injectivity condition and α

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. As usual, we let V be a right module over A, W be a left module over A, and V ⨂_{A} W be a tensor product of V and W over A. Suppose again that (V, d_{V}) is a right module over A with differentiation, so that V ⨂_{A} W is a module over k with differentiation operator d = d_{V ⨂_{A} W} as in (7.5.4). In this section, we shall consider an additional injectivity condition related to V and W, and its consequences for V ⨂_{A} W.

Let B(V) ⨂_{A} W be a tensor product of B(V) and W over A again. As before, there is a natural homomorphism

(7.10.1) from B(V) ⨂_{A}W into V ⨂_{A}W,

as modules over k, corresponding to the obvious inclusion mapping from B(V) into V and the identity mapping on W. More precisely, this homomorphism maps B(V) ⨂_{A} W into the submodule B(V ⨂_{A} W ) of V ⨂_{A} W, as in Section 7.7. In fact, one can check that

the homomorphism as in (7.10.1)

(7.10.2)

maps B(V) ⨂_{A}W onto B(V ⨂_{A}W ).

7.10. AN INJECTIVITY CONDITION AND α 193

Let us suppose now that

(7.10.3) the homomorphism as in (7.10.1) is injective on B(V) ⨂_{A}W.

There is a natural homomorphism

(7.10.4) from V ⨂_{A}W onto B(V) ⨂_{A}W,

as modules over k, corresponding to d_{V} as a mapping from V onto B(V), and the identity mapping on W. Observe that

(7.10.5) d is the same as the composition of the homomorphisms as in (7.10.4) and (7.10.1),

as a homomorphism from V ⨂_{A} W into itself, as a module over k. In particular,

the kernel of the homomorphism as in (7.10.4)

(7.10.6)

is contained in ker d.

Using (7.10.3), we get that

(7.10.7) the kernel of the homomorphism as in (7.10.4) is equal to ker d,

as submodules of B ⨂_{A} W.

Let Z(V) ⨂_{A} W be a tensor product of Z(V) and W over A again too. As usual, there is a natural homomorphism

(7.10.8) from Z(V) ⨂_{A}W into V ⨂_{A}W,

as modules over k, corresponding to the obvious inclusion mapping from Z(V) into V, and the identity mapping on W. This homomorphism maps Z(V) ⨂_{A} W into Z(V ⨂_{A} W ), as in Section 7.7.

Of course,

0 ⟶ Z(V) ⟶ V ⟶^{d_{V}} B(V) ⟶ 0 (7.10.9)

is an exact sequence of right modules over A, using the obvious inclusion mapping from Z(V) into V. It follows that

Z(V) ⨂_{A}W ⟶ V ⨂_{A}W ⟶ B(V) ⨂_{A}W ⟶ 0 (7.10.10)

is an exact sequence of modules over k, using the homomorphism as in (7.10.8) in the first step, and the homomorphism as in (7.10.4) in the second step, as in Section 2.5. This means that

(7.10.11) the homomorphism as in (7.10.8) maps Z(V) ⨂_{A}W

onto the kernel of the homomorphism as in (7.10.4).

Combining this with (7.10.7), we obtain that

the homomorphism as in (7.10.8)

(7.10.12)

maps Z(V) ⨂_{A}W onto Z(V ⨂_{A}W ),

under these conditions.

Remember that the homomorphism η from Z(V) ⨂_{A} W into H(V ⨂_{A} W ) as in (7.7.10) is defined by taking the composition of the homomorphism as in (7.10.8), considered as a homomorphism from Z(V) ⨂_{A} W into Z(V ⨂_{A} W ), with the natural quotient mapping from Z(V ⨂_{A} W ) onto H(V ⨂_{A} W ). Using (7.10.12), we get that

(7.10.13) η maps Z(V) ⨂_{A}W onto H(V ⨂_{A}W ).

Let H(V) ⨂_{A} W be a tensor product of H(V) and W over A again, and remember that α is the homomorphism from H(V) ⨂_{A} W into H(V ⨂_{A} W ) as in (7.7.11). As before, we let ξ be the homomorphism from Z(V) ⨂_{A} W onto H(A) ⨂_{A} W, as modules over k, corresponding to the natural quotient mapping from Z(V) onto H(V), and the identity mapping on W. We have seen that α ∘ ξ = η, as in (7.7.13), which determines α uniquely, because ξ is surjective. This implies that

(7.10.14) α maps H(V) ⨂_{A}W onto H(V ⨂_{A}W ),

because of (7.10.13). This is related to Theorem 7.2 on p68 of [50], and some remarks on p70 of [50].

Suppose that V corresponds to the direct sum of B(V) and another submodule of V, as a right module over A. This implies that V ⨂_{A} W is isomorphic to the direct sum of B(V) ⨂_{A} W and a tensor product of the other submodule of V with W over A, as modules over k. In particular, (7.10.3) holds in this case. This is related to Proposition 7.4 on p70 of [50].

7.11 Injectivity of α

Let us return to the same notation and hypotheses as at the beginning of the previous section. As usual, we let B(V) ⨂_{A} W, Z(V) ⨂_{A} W, and H(V) ⨂_{A} W be tensor products of B(V), Z(V), and H(V) with W over A, respectively. Although we do not ask that (7.10.3) hold for the moment, some of the remarks in the previous section can still be used here.

There is a natural homomorphism

(7.11.1) from B(V) ⨂_{A}W into Z(V) ⨂_{A}W,

as modules over k, corresponding to the obvious inclusion mapping from B(V) into Z(V), and the identity mapping on W. Remember that

the homomorphism as in (7.10.1) is the same as

(7.11.2)

7.11. INJECTIVITY OF α 195

the composition of the homomorphism as in (7.11.1)

with the homomorphism as in (7.10.8),

as in Section 7.7.

Clearly 0 ⟶ B(V) ⟶ Z(V) ⟶ H(V) ⟶ 0 (7.11.3)

is an exact sequence of right modules over A, using the appropriate inclusion and quotient mappings. This implies that

B(V) ⨂_{A}W ⟶ Z(V) ⨂_{A}W ⟶^{ξ} H(V) ⨂_{A}W ⟶ 0 (7.11.4)

is an exact sequence of modules over k, as in Section 2.5. This uses the homomorphism as in (7.11.1) in the first step, and ξ is the homomorphism corresponding to the natural quotient mapping from Z(V) onto H(V) and the identity mapping on W, as before.

Remember that the homomorphism as in (7.10.8) maps Z(V) ⨂_{A} W into Z(V ⨂_{A} W ). Observe that

(7.11.5) B(V ⨂_{A}W ) is contained in the image of Z(V) ⨂_{A}W

under the homomorphism as in (7.10.8),

by (7.10.2) and (7.11.2).

Remember that η is the composition of the homomorphism as in (7.10.8), considered as a homomorphism from Z(V) ⨂_{A} W into Z(V ⨂_{A} W ), with the natural quotient mapping from Z(V ⨂_{A} W) onto H(V ⨂_{A} W ). Thus

(7.11.6) ker η is the same as the inverse image of B(V ⨂_{A}W )

under the homomorphism as in (7.10.8).

In fact,

(7.11.7) ker η is the submodule of Z(V) ⨂_{A}W spanned by

the kernel of the homomorphism as in (7.10.8)

and the image of B(V) ⨂_{A}W

under the homomorphism as in (7.11.1),

because of (7.10.2) and (7.11.2).

The exactness of (7.11.4) implies that

(7.11.8) ker ξ is the same as the image of B(V) ⨂_{A}W

under the homomorphism as in (7.11.1).

This means that

(7.11.9) ker η is the submodule of Z(V) ⨂_{A}W spanned by

the kernel of the homomorphism as in (7.10.8) and ker ξ.

Remember that α is the homomorphism from H(V) ⨂_{A}W into H(V ⨂_{A}W ) as in (7.7.11). We have seen that α∘ξ = η, as in (7.7.13), and that α is uniquely determined by this property, because ξ is surjective. Observe that

(7.11.10) α is injective if and only if ker η = ker ξ.

This happens if and only if

the kernel of the homomorphism as in (7.10.8)

(7.11.11)

is contained in the image of B(V) ⨂_{A}W

under the homomorphism as in (7.11.1).

In particular, α is injective when

(7.11.12) the homomorphism as in (7.10.8) is injective on Z(V) ⨂_{A}W.

Note that (7.10.3) implies that

(7.11.13) the homomorphism as in (7.11.1) is injective on B(V) ⨂_{A}W.

We also have that (7.10.3) follows from (7.11.12) and (7.11.13).

Let Z′(V) ⨂_{A} W be a tensor product of Z′(V) = V/d_{V} (V) and W over A, and let τ be the natural homomorphism from H(V) ⨂_{A} W into Z′(V) ⨂_{A} W, as modules over k, corresponding to the obvious inclusion mapping from H(V) into Z′(V) and the identity mapping on W, as in Section 7.8. Remember that τ is equal to the composition of α with the homomorphism ζ defined in Section 7.8, as in (7.8.11). This implies that

(7.11.14) α is injective when τ is injective.

These criteria for the injectivity of α are related to Theorem 7.2 on p68 of [50], and some remarks on p70 of [50].

If V corresponds to the direct sum of Z(V) and another submodule, as a right module over A, then V ⨂_{A} W is isomorphic to the direct sum of Z(V) ⨂_{A} W and a tensor product of the other submodule of V with W over A, as modules over k. This implies that the homomorphism as in (7.10.8) is injective on Z(V) ⨂_{A} W, so that α is injective, as before. Similarly, if Z′(V) corresponds to the direct sum of H(V) and another submodule of Z′(V), as a right module over A, then α is injective, as in Section 7.9, which could also be obtained from the injectivity of τ in this case. This is related to Proposition 7.4 on p70 of [50].

7.12 Differentiation and Hom(V, ·)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, W be both left or both right modules over A.

7.12. DIFFERENTIATION AND HOM(V, ·) 197

Suppose for the moment that (W, d_{W}) is a module with differentiation. Let

d = d_{Hom_{A}(V,W)} (7.12.1)

be the homomorphism from Hom_{A}(V, W) into itself, as a module over k, defined by composition with d_{W}. Thus, if ϕ ∈ Hom_{A}(V, W), then we put

d(ϕ) = d_{W} ∘ ϕ. (7.12.2)

Observe that d ∘ d = 0 on Hom_{A}(V, W), (7.12.3)

so that Hom_{A}(V, W) is a module with differentiation with respect to d.

Suppose now that W is a graded module over A. We would like to define Hom_{A}^{gr}(V, W) as a graded module over k, with

(Hom_{A}^{gr}(V, W))^{l} = Hom_{A}(V, W^{l}) (7.12.4)

for every l ∈ Z, as on p62 of [50], with r = 1. Although we can simply take Hom_{A}^{gr}(V, W) to be the direct sum of (7.12.4) over l, it is also sometimes convenient to consider it as a submodule of related modules over k.

Let |W | be the underlying module over A corresponding to W, without a grading, as on p63 of [50]. If ϕ ∈ Hom_{A}^{gr}(V, W) and l ∈ Z, then let ϕ^{l} be the corresponding element of (7.12.4), as usual, so that ϕ^{l} = 0 for all but finitely many l. It follows that

∑_{l=−∞}^{∞} ϕ^{l} (7.12.5)

defines a homomorphism from V into |W |, as modules over A. It is easy to see that ϕ is uniquely determined by (7.12.5), and so we shall often identify ϕ with (7.12.5).

This defines an injective homomorphism

(7.12.6) from Hom_{A}^{gr}(V, W) into Hom_{A}(V, |W |),

as modules over k, and we may identify Hom_{A}^{gr}(V, W) with the corresponding submodule of Hom_{A}(V, |W |), as a module over k. Note that Hom_{A}^{gr}(V, W) may correspond to a proper submodule of Hom_{A}(V, |W |), as on p63 of [50]. If V is finitely generated as a module over A, then one can check that the homomorphism as in (7.12.6) maps Hom_{A}^{gr}(V, W) onto Hom_{A}(V, |W |).

Of course, |W | may be considered as a submodule of ∏_{l=−∞}^{∞} W_{l}, as a module over A, so that Hom_{A}(V, |W |) corresponds to a submodule of

Hom_{A}(V, ∏_{l=−∞}^{∞} W_{l}), (7.12.7)

as a module over k. Remember that there is a natural isomorphism from (7.12.7) onto

∏_{l=−∞}^{∞} Hom_{A}(V, W_{l}), (7.12.8)

as modules over k, as in Section 1.7. Thus Hom_{A}^{gr}(V, W) corresponds to the submodule of Hom_{A}(V, |W |), as a module over k, that correesponds to the submodule of (7.12.8) given by the direct sum.

As in Section 7.5, V may be considered as a graded module over A, with

(7.12.9)  V^{j} = V  when j = 0
= {0} when j ≠ 0.

Using this, Hom_{A}^{gr}(V, W) may be defined as a doubly-graded module over k, as in Section 6.3, with

(Hom_{A}^{gr}(V, W))^{j,l} = Hom_{A}(V^{−j}, W^{l}) = Hom_{A}(V, W^{l}) when j = 0
(7.12.10)  = {0}  when j ≠ 0.

The single grading on Hom_{A}^{gr}(V, W) associated to this double grading is the same as in (7.12.4).

Suppose that (W, d_{W}) is a graded module over A that is a complex. In particular, Hom_{A}^{gr}(V, W) may be defined as a graded module over k as in (7.12.4). We can define d = d_{Hom_{A}^{gr}(V,W)} (7.12.11)

on Hom_{A}^{gr}(V, W) as follows. If ϕ ∈ Hom_{A}^{gr}(V, W), then d(ϕ) may be defined as an element of Hom_{A}^{gr}(V, W) by putting

d(ϕ)^{l+1} = d_{W}^{l} ∘ ϕ^{l} (7.12.12)

for each l. It is easy to see that Hom_{A}^{gr}(V, W) is a complex with respect to d, as on p63 of [50].

The underlying module |W | over A corresponding to W may be considered as a module with differentiation with respect to d_{W}, so that d = d_{Hom_{A}(V,|W |)} may be defined on Hom_{A}(V, |W |) as in (7.12.1). If we consider Hom_{A}^{gr}(V, W) as a submodule of Hom_{A}(V, |W |), as a module over k, then the restriction of d on Hom_{A}(V, |W |) to Hom_{A}^{gr}(V, W) is the same as the definition of d on Hom_{A}^{gr}(V, W) in the preceding paragraph. Thus Hom_{A}^{gr}(V, W) may be considered as a submodule of Hom_{A}(V, |W |), as a module over k with differentiation.

We may consider V as a complex, using the grading in (7.12.9), and differentiation operator d_{V} = 0, as on p75 of [50]. If we consider Hom_{A}^{gr}(V, W) as a doubly-graded module over k as in (7.12.10), then Hom_{A}^{gr}(V, W) is a double complex, with δ_{1} = 0, δ_{2} = d = d_{Hom_{A}^{gr}(V,W)}, (7.12.13)

as in Section 6.5. This implies that Hom_{A}^{gr}(V, W) is a single complex with respect to the total differentiation operator

δ = δ_{1} + δ_{2} = d = d_{Hom_{A}^{gr}(V,W)}, (7.12.14)

and the single grading associated to the double grading on Hom_{A}^{gr}(V, W), as in Section 5.14. This is the same as considering Hom_{A}^{gr}(V, W) as a single complex as before.

7.13. INDUCED MAPPINGS AND HOM(V, ·) 199

7.13 Induced mappings and Hom(V, ·)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, W, and W_{0} be all left or all right modules over A.

Suppose that (W, d_{W}) and (W_{0}, d_{W_{0}}) are modules with differentiation. Let d = d_{Hom_{A}(V,W)} be defined on Hom_{A}(V, W) as in (7.12.1), and define

d_{0} = d_{Hom_{A}(V,W_{0})} (7.13.1)

on Hom_{A}(V, W_{0}) in the same way.

Let ϕ be a homomorphism from W into W_{0}, as modules over A. Consider the homomorphism Φ

(7.13.2) from Hom_{A}(V, W) into Hom_{A}(V, W_{0}),

as modules over k, defined by composing an element of Hom_{A}(V, W) with ϕ to get an element of Hom_{A}(V, W_{0}). Suppose now that ϕ is a homomorphism from W into W_{0}, as modules over A with differentiation. It is easy to see that Φ is a homomorphism as in (7.13.2), as modules over k with differentiation, in this case.

Let ψ be another homomorphism from W into W_{0}, as modules over A with differentiation, and let Ψ be the corresponding homomorphism as in (7.13.2), as modules over k with differentiation. Suppose that σ is a homotopy between ϕ and ψ, which means that σ is a homomorphism from W into W_{0}, as modules over A, such that d_{W_{0}} ∘ σ + σ ∘ d_{W} = ϕ − ψ. (7.13.3)

Let Σ be the homomorphism as in (7.13.2), as modules over k, that corresponds to σ as before. It is easy to see that

d_{0} ∘ Σ + Σ ∘ d = Φ − Ψ, (7.13.4)

so that Σ is a homotopy between Φ and Ψ.

Suppose now that W, W_{0} are graded modules over A, so that Hom_{A}^{gr}(V, W), Hom_{A}^{gr}(V, W_{0}) may be defined as graded modules over k as in (7.12.4). Let ϕ be a homomorphism from W into W_{0}, as modules over A, of degree p ∈ Z. This leads to a homomorphism Φ

(7.13.5) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V, W_{0}),

as modules over k, of degree p too. More precisely, if l ∈ Z, then Φ acts on (7.12.4) by composition with ϕ, as on p63 of [50].

Let |W |, |W_{0}| be the underlying modules over A corresponding to W, W_{0}, respectively, without gradings, as on p63 of [50]. Remember that Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V, W_{0}) may be considered as submodules of Hom_{A}(V, |W |) and Hom_{A}(V, |W_{0}|), respectively, as modules over k, as in the previous section. Using this, we have that the homomorphism Φ as in (7.13.5) is the same as the restriction of the homomorphism Φ

(7.13.6) from Hom_{A}(V, |W |) into Hom_{A}(V, |W_{0}|) as in (7.13.2) to Hom_{A}^{gr}(V, W).

If V is considered as a graded module over A as in (7.12.9), then we can define Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V, W_{0}) as doubly-graded modules over k, as in Section 6.3. In this case,

(7.13.7) the homomorphism Φ as in (7.13.5)

corresponds to Hom^{gr}(I_{V} , ϕ),

in the notation of Section 6.4, where I_{V} is the identity mapping on V.

Suppose that (W, d_{W}), (W_{0}, d_{W_{0}}) are graded modules over A with differentiation that are complexes, so that Hom_{A}^{gr}(V, W), Hom_{A}^{gr}(V, W_{0}) may be considered as single complexes, as in the previous section. Let ϕ be a map of degree p ∈ Z from W into W_{0}, as complexes, as in Section 6.6. If Φ is the corresponding homomorphism as in (7.13.5), then Φ is a map of degree p as in (7.13.5), as complexes.

Let ψ be another map of degree p from W into W_{0}, as complexes, and let Ψ be the corresponding map of degree p as in (7.13.5), as complexes. Also let σ be a homotopy between ϕ and ψ, as maps of degree p between complexes, as in Section 6.6. In particular, σ is a homomorphism from W into W_{0} of degree p − 1, as modules over A. Let Σ be the corresponding homomorphism as in (7.13.5), as modules over k, which has degree p − 1 as well. One can verify that Σ is a homotopy between ϕ and ψ, as maps of degree p between complexes.

Let us consider V as a complex, with d_{V} = 0, and graded as in (7.12.9). Thus we may consider Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V, W_{0}) as double complexes, as in the previous section. In this case, Φ and Ψ may be considered as maps of bidegree (0, p) as in (7.13.5), as double complexes, as in Section 6.9. We also get that (0, Σ) defines a homotopy between Φ and Ψ, as maps of bidegree (0, p) as in (7.13.5), as double complexes, as in Section 6.9. If p = 0, then these remarks correspond to some of those in Section 6.5.

7.14 An analogue of α′

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V and W be both left or both right modules over A, and suppose that (W, d_{W}) is a module over A with differentiation. Thus Hom_{A}(V, W) is a module over k with differentiation operator d = d_{Hom_{A}(V,W)} as in (7.12.1).

By definition, Z( Hom_{A}(V, W)) consists of ϕ ∈ Hom_{A}(V, W) such that

d(ϕ) = d_{W} ∘ ϕ = 0 (7.14.1)

on V. This is the same as saying that

ϕ(V) ⊆ Z(W), (7.14.2)

so that Z(Hom_{A}(V, W)) = Hom_{A}(V, Z(W)), (7.14.3)

7.14. AN ANALOGUE OF α′ 201

where the latter is considered as a submodule of Hom_{A}(V, W).

Similarly, ϕ ∈ B( Hom_{A}(V, W)) if and only if ϕ = d(ψ) = d_{W} ∘ ψ (7.14.4) for some ψ ∈ Hom_{A}(V, W). In this case, ϕ(V) = d_{W} (ψ(V)) ⊆ d_{W} (W) = B(W). (7.14.5) This means that B(Hom_{A}(V, W)) ⊆ Hom_{A}(V, B(W)), (7.14.6) where the latter is considered as a submodule of Hom_{A}(V, W).

There is a natural homomorphism (7.14.7) ξ from Hom_{A}(V, Z(W)) into Hom_{A}(V, H(W)), as modules over k, defined by sending an element of Hom_{A}(V, Z(W)) to its composition with the natural quotient mapping from Z(W) onto H(W). Equivalently, ξ may be considered as a homomorphism (7.14.8) from Z(Hom_{A}(V, W)) into Hom_{A}(V, H(W)), by (7.14.3). Note that ker ξ = Hom_{A}(V, B(W)), (7.14.9) considered as a submodule of (7.14.3). This leads to a natural homomorphism (7.14.10) α′ from H(Hom_{A}(V, W)) into Hom_{A}(V, H(W)), because of (7.14.6).

The natural quotient mapping from Z( Hom_{A}(V, W)) onto H( Hom_{A}(V, W)) may be considered as a homomorphism (7.14.11) η from Hom_{A}(V, Z(W)) onto H(Hom_{A}(V, W)), as modules over k, by (7.14.3). By construction, α′ ∘ η = ξ. (7.14.12) This determines α′ uniquely, because η is surjective.

There is a natural homomorphism (7.14.13) from Hom_{A}(V, W) into Hom_{A}(V, Z′(W)), as modules over k, which sends an element of Hom_{A}(V, W) to its composition with the natural quotient mapping from W onto Z′(W) = W/d_{W} (W). This leads to a natural homomorphism (7.14.14) from Hom_{A}(V, Z(W)) into Hom_{A}(V, Z′(W)), as modules over k, which is defined by restricting the homomorphism as in (7.14.13) to Hom_{A}(V, Z(W)), considered as a submodule of Hom_{A}(V, W). This homomorphism could also be described by saying that it sends an element of Hom_{A}(V, Z(W)) to its composition with the restriction to Z(W) of the natural quotient mapping from W onto Z′(W).

Of course, H(W) may be considered as a submodule of Z′(W), as a module over A, so that Hom_{A}(V, H(W)) may be considered as a submodule of Hom_{A}(V, Z′(W)), as a module over k. The inclusion mapping

(7.14.15) τ from Hom_{A}(V, H(W)) into Hom_{A}(V, Z′(W))

is an injective module homomorphism, and we get that

(7.14.16) τ ∘ ξ is the same as the homomorphism as in (7.14.14).

Note that

the kernel of the homomorphism as in (7.14.14)

(7.14.17)

is equal to Hom_{A}(V, B(W)),

considered as a submodule of Hom_{A}(V, Z(W)), as a module over k. This can be seen directly, or using (7.14.9) and (7.14.16). This leads to a natural homomorphism (7.14.18) ζ from H(Hom_{A}(V, W)) into Hom_{A}(V, Z′(W)),

because of (7.14.3) and (7.14.6).

It is easy to see that

(7.14.19) ζ ∘ η is the same as the homomorphism as in (7.14.14),

by construction. This means that

τ ∘ ξ = ζ ∘ η, (7.14.20)

by (7.14.16).

One can verify that τ ∘ α′ = ζ, (7.14.21)

as homomorphisms from H( Hom_{A}(V, W)) into Hom_{A}(V, Z′(W)). Of course, α′ is uniquely determined by this property, because τ is injective. Each of (7.14.12) and (7.14.21) can be obtained from the other using (7.14.20), by composing with η or τ, as appropriate.

7.15 Complexes and α′

Let us continue with the same notation and hypotheses as in the previous section, and suppose now that (W, d_{W}) is a graded module over A that is a complex. Thus Hom_{A}^{gr}(V, W) may be defined as a graded module over k as in (7.12.4), and as a complex, with differentiation operator d = d_{Hom_{A}^{gr}(V,W)} as in (7.12.11).

7.15. COMPLEXES AND α′ 203

The underlying module |W | over A corresponding to W may be considered as a module with differentiation with respect to d_{W}, so that Hom_{A}(V, |W |) is a module over k with differentiation operator d = d_{Hom_{A}(V,|W |)} defined as in (7.12.1). Remember that Hom_{A}^{gr}(V, W) may be considered as a submodule of Hom_{A}(V, |W |), as a module over k with differentiation, as in Section 7.12.

Note that (7.15.1) Hom_{A}^{gr}(V, B(W)), Hom_{A}^{gr}(V, Z(W)), Hom_{A}^{gr}(V, H(W)),

and Hom_{A}^{gr}(V, Z′(W))

may be defined as graded modules over k as in (7.12.4) too. More precisely, (7.15.2) Hom_{A}^{gr}(V, B(W)) and Hom_{A}^{gr}(V, Z(W))

are homogeneous submodules of Hom_{A}^{gr}(V, W),

and (7.15.3) Hom_{A}^{gr}(V, H(W)) is a homogeneous submodule

of Hom_{A}^{gr}(V, Z′(W)).

The remarks in the previous section can be applied to |W |, and we would like to consider the analogous statements for Hom_{A}^{gr}(V, ·) in place of Hom_{A}(V, ·). We have that Z(Hom_{A}(V, |W |)) = Hom_{A}(V, Z(|W |)), (7.15.4) as in (7.14.3), for instance, and similarly Z(Hom_{A}^{gr}(V, W)) = Hom_{A}^{gr}(V, Z(W)). (7.15.5) We also have that B(Hom_{A}(V, |W |)) ⊆ Hom_{A}(V, B(|W |)), (7.15.6) as in (7.14.6), and B(Hom_{A}^{gr}(V, W)) ⊆ Hom_{A}^{gr}(V, B(W)). (7.15.7)

One can check that (7.15.8) B(Hom_{A}^{gr}(V, W)) = Hom_{A}^{gr}(V, W) ∩ B(Hom_{A}(V, |W |)). Using this, (7.15.9) we may consider H(Hom_{A}^{gr}(V, W))

as a submodule of H(Hom_{A}(V, |W |)),

as a module over k. Similarly, (7.15.10) we may consider Z′(Hom_{A}^{gr}(V, W))

as a submodule of Z′(Hom_{A}(V, |W |)), 204 CHAPTER 7. MORE ON DIFFERENTIATION, 2 as a module over k.

There is a natural homomorphism (7.15.11) ξ from Hom_{A}(V, Z(|W |)) into Hom_{A}(V, H(|W |)), as modules over k, as in (7.14.7). This may be considered as a homomorphism (7.15.12) from Z(Hom_{A}(V, |W |)) into Hom_{A}(V, H(|W |)), by (7.15.4), as before. It is easy to see that (7.15.13) ξ maps Hom_{A}^{gr}(V, Z(W)) into Hom_{A}^{gr}(V, H(W)), and that (7.15.14) the restriction of ξ to Hom_{A}^{gr}(V, Z(W)) has degree 0.

Remember that ker ξ = Hom_{A}(V, B(|W |)), (7.15.15) as in (7.14.9). This implies that (7.15.16) the kernel of the restriction of ξ to Hom_{A}^{gr}(V, Z(W))

is Hom_{A}^{gr}(V, B(W)).

Using (7.15.15), we get a natural homomorphism (7.15.17) α′ from H(Hom_{A}(V, |W |)) into Hom_{A}(V, H(|W |)), as in (7.14.10). One can check that (7.15.18) α′ maps H(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(V, H(W)), and that (7.15.19) the restriction of α′ to H(Hom_{A}^{gr}(V, W)) has degree 0.

Because of (7.15.4), the natural quotient mapping from Z( Hom_{A}(V, |W |)) onto H( Hom_{A}(V, |W |)) may be considered as a homomorphism (7.15.20) η from Hom_{A}(V, Z(|W |)) onto H(Hom_{A}(V, |W |)), as modules over k, as in (7.14.11). One can verify that (7.15.21) η maps Hom_{A}^{gr}(V, Z(W)) onto H(Hom_{A}^{gr}(V, W)), and that (7.15.22) the restriction of η to Hom_{A}^{gr}(V, Z(W)) has degree 0.

7.15. COMPLEXES AND α′ 205

More precisely, the restriction of η to Hom_{A}^{gr}(V, Z(W)) corresponds to the natural quotient mapping from Z( Hom_{A}^{gr}(V, W)) onto H( Hom_{A}^{gr}(V, W)), because of (7.15.5) and (7.15.8). We also have that α′ ∘ η = ξ, as in (7.14.12). In particular, this holds on Hom_{A}^{gr}(V, Z(W)).

There is a natural homomorphism

(7.15.23) from Hom_{A}(V, |W |) into Hom_{A}(V, Z′(|W |)),

as modules over k, as in (7.14.13). This leads to a natural homomorphism

(7.15.24) from Hom_{A}(V, Z(|W |)) into Hom_{A}(V, Z′(|W |)),

as modules over k, as in (7.14.14), by restricting the homomorphism as in (7.15.23) to Hom_{A}(V, Z(|W |)). It is easy to see that

the homomorphism as in (7.15.23)

(7.15.25)

maps Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V, Z′(W)),

and that

the restriction of the homomorphism as in (7.15.23)

(7.15.26)

to Hom_{A}^{gr}(V, W) has degree 0.

Thus the restriction of the homomorphism as in (7.15.24) to Hom_{A}^{gr}(V, Z(W)) defines a natural homomorphism

(7.15.27) from Hom_{A}^{gr}(V, Z(W)) into Hom_{A}^{gr}(V, Z′(W))

of degree 0.

The inclusion mapping

(7.15.28) τ from Hom_{A}(V, H(|W |)) into Hom_{A}(V, Z′(|W |))

is an injective module homomorphism, as in (7.14.15). Of course, τ maps Hom_{A}^{gr}(V, H(W)) into Hom_{A}^{gr}(V, Z′(W)). Remember that τ ∘ ξ is the same as the homomorphism as in (7.15.24), as in (7.14.16).

Remember that the kernel of the homomorphism as in (7.15.24) is equal to Hom_{A}(V, B(|W |)), as in (7.14.17). This leads to a natural homomorphism

(7.15.29) ζ from H(Hom_{A}(V, |W |)) into Hom_{A}(V, Z′(|W |)),

as in (7.14.18). It is easy to see that

ζ(H(Hom_{A}^{gr}(V, W))) ⊆ Hom_{A}^{gr}(V, Z′(W)) (7.15.30)

and (7.15.31) the restriction of ζ to H(Hom_{A}^{gr}(V, W)) has degree 0,

because of the analogous properties of the homomorphism as in (7.15.27).

As in (7.14.19), ζ ∘ η is the same as the homomorphism as in (7.15.24). This means that τ ∘ ξ = ζ ∘ η, as in (7.14.20). In particular, this holds on Hom_{A}^{gr}(V, Z(W)).

Similarly, τ ∘ α′ = ζ on H( Hom_{A}(V, |W |)), as in (7.14.21). In particular, this holds on H( Hom_{A}^{gr}(V, W)).

These homomorphisms and their properties correspond to the commutative diagram (1) on p64 of [50], and parts of Proposition 6.1a on p65 of [50], under slightly different conditions.

Remember that V may be considered as a graded module over A with differentiation that is a complex, with grading as in (7.12.9) and d_{V} = 0, as in Section 7.12. Of course, B(V) = {0}, and Z(V) = H(V) = Z′(V) = V in this case. Using this, the remarks in this and the previous section correspond to those in Sections 6.13 – 6.15.

More on differentiation, 3

8.1 Splitting Z(W) and α′

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V and W be both left or both right modules over A, and suppose that (W, d_{W}) is a module over A with differentiation. Remember that Hom_{A}(V, W) is a module over k with differentiation operator d = d_{Hom_{A}(V,W)} as in (7.12.1).

Suppose that (8.1.1) Z(W) corresponds to the direct sum

of B(W) and another submodule of Z(W),

as a module over A. This means that (8.1.2) there is a homomorphism from H(W) into Z(W),

as modules over A, whose composition with the natural quotient mapping from Z(W) onto H(W) is the identity mapping on H(W).

Indeed, Z(W) corresponds to the direct sum of B(W) and the image of H(W) under this homomorphism in this case, as a module over A.

Let γ̃ be the homomorphism as in (8.1.2), considered as a homomorphism (8.1.3) from H(W) into W. Equivalently, γ̃ may be considered as the composition of the homomorphism in (8.1.2) with the natural inclusion mapping from Z(W) into W. Note that d_{W} ∘ γ̃ = 0, (8.1.4) because γ̃(H(W)) ⊆ Z(W).

208 CHAPTER 8. MORE ON DIFFERENTIATION, 3

Using γ̃, we get a homomorphism (8.1.5) from Hom_{A}(V, H(W)) into Hom_{A}(V, W), as modules over k. This homomorphism sends an element of Hom_{A}(V, H(W)) to its composition with γ̃. This may be considered as a homomorphism (8.1.6) from Hom_{A}(V, H(W)) into Z(Hom_{A}(V, W)), or equivalently into Hom_{A}(V, Z(W)).

This leads to a homomorphism (8.1.7) from Hom_{A}(V, H(W)) into H(Hom_{A}(V, W)), as modules over k, by composing the previous homomorphism as in (8.1.6) with the natural quotient mapping (8.1.8) from Z(Hom_{A}(V, W)) onto H(Hom_{A}(V, W)). Let α′ be the homomorphism from H( Hom_{A}(V, W)) into Hom_{A}(V, H(W)) defined in Section 7.14. One can verify that (8.1.9) the composition of the homomorphism as in (8.1.7) with α′

is the identity mapping on Hom_{A}(V, H(W)). This implies that

α′(H(Hom_{A}(V, W))) = Hom_{A}(V, H(W)). (8.1.10) We also obtain that (8.1.11) H(Hom_{A}(V, W)) corresponds to the direct sum of ker α′

and another submodule of H(Hom_{A}(V, W)),

as a module over k.

Suppose now that (W, d_{W}) is a graded module over A with differentiation that is a complex, and that (8.1.12) Z(W) corresponds to the direct sum of B(W)

and another homogeneous submodule of Z(W),

as a module over A. This means that Z(W) corresponds to the direct sum of B(W) and another submodule, as a graded module over A. It follows that (8.1.13) the homomorphism as in (8.1.2) has degree 0, so that (8.1.14) the homomorphism γ̃ as in (8.1.3) has degree 0.

8.1. SPLITTING Z(W) AND α′ 209

Remember that Hom_{A}^{gr}(V, W) may be defined as a graded module over k as in (7.12.4), and as a complex, with differentiation operator d = d_{Hom_{A}^{gr}(V,W)} as in (7.12.11). We may also consider the underlying module |W | over A corresponding to W, without a grading, as a module with differentiation with respect to d_{W}, so that Hom_{A}(V, |W |) is a module over k with differentiation operator d = d_{Hom_{A}(V,|W |)} defined as in (7.12.1). As in Section 7.12, we may consider Hom_{A}^{gr}(V, W) as a submodule of Hom_{A}(V, |W |), as a module over k with differentiation.

We can use γ̃ to get a homomorphism (8.1.15) from Hom_{A}(V, H(|W |)) into Hom_{A}(V, |W |), as modules over k, as in (8.1.5). This may be considered as a homomorphism (8.1.16) from Hom_{A}(V, H(|W |)) into Z(Hom_{A}(V, |W |)), as in (8.1.6), or equivalently into Hom_{A}(V, Z(|W |)). It is easy to see that

the homomorphism as in (8.1.15)

(8.1.17)

maps Hom_{A}^{gr}(V, H(W)) into Hom_{A}^{gr}(V, W).

The restriction of this homomorphism to Hom_{A}^{gr}(V, H(W)) may be considered as a homomorphism (8.1.18) from Hom_{A}^{gr}(V, H(W)) into Z(Hom_{A}^{gr}(V, W)), or equivalently into Hom_{A}^{gr}(V, Z(W)), as before. Note that (8.1.19) the homomorphism as in (8.1.18) has degree 0, because of (8.1.14).

As before, we get a homomorphism (8.1.20) from Hom_{A}(V, H(|W |)) into H(Hom_{A}(V, |W |)), as modules over k, by composing the homomorphism as in (8.1.16) with the natural quotient mapping (8.1.21) from Z(Hom_{A}(V, |W |)) onto H(Hom_{A}(V, |W |)). One can check that the homomorphism as in (8.1.20) maps Hom_{A}^{gr}(V, H(W)) into H( Hom_{A}^{gr}(V, W)). More precisely, the restriction of this homomorphism to Hom_{A}^{gr}(V, H(W)) is the same as the homomorphism (8.1.22) from Hom_{A}^{gr}(V, H(W)) into H(Hom_{A}^{gr}(V, W)) obtained by composing the homomorphism as in (8.1.18) with the natural quotient mapping (8.1.23) from Z(Hom_{A}^{gr}(V, W)) onto H(Hom_{A}^{gr}(V, W)).

210 CHAPTER 8. MORE ON DIFFERENTIATION, 3 It is easy to see that (8.1.24) the homomorphism as in (8.1.22) has degree 0, because of (8.1.19).

Let α′ be the homomorphism (8.1.25) from H(Hom_{A}(V, |W |)) into Hom_{A}(V, H(|W |)) discussed in Sections 7.14 and 7.15. As in (8.1.9), (8.1.26) the composition of the homomorphism as in (8.1.20) with α′

is the identity mapping on Hom_{A}(V, H(|W |)).

It follows that α′(H(Hom_{A}(V, |W |))) = Hom_{A}(V, H(|W |)) (8.1.27) and (8.1.28) H(Hom_{A}(V, |W |)) corresponds to the direct sum of ker α′

and another submodule of H(Hom_{A}(V, |W |)),

as a module over k, as in (8.1.10) and (8.1.11).

Under these conditions, α′ maps H( Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(V, H(W)), and the restriction of α′ to H( Hom_{A}^{gr}(V, W)) has degree 0, as in (7.15.18) and (7.15.19). Using (8.1.26), we get that

the composition of the homomorphism as in (8.1.22)

(8.1.29)

with the restriction of α′ to H(Hom_{A}^{gr}(V, W))
is the identity mapping on Hom_{A}^{gr}(V, H(W)).

In particular, this implies that α′(H(Hom_{A}^{gr}(V, W))) = Hom_{A}^{gr}(V, H(W)). (8.1.30)

Of course, (8.1.31) the kernel of the restriction of α′ to H(Hom_{A}^{gr}(V, W))

is a homogeneous submodule of H(Hom_{A}^{gr}(V, W)),

as a graded module over k, because the restriction of α′ to H( Hom_{A}^{gr}(V, W)) has degree 0. Similarly, (8.1.32) the image of Hom_{A}^{gr}(V, H(W))

under the homomorphism as in (8.1.22)
is a homogeneous submodule of H(Hom_{A}^{gr}(V, W)),

8.2. SURJECTIVITY CONDITIONS AND α′ 211

because of (8.1.24). In fact,

(8.1.33) H(Hom_{A}^{gr}(V, W)) corresponds to the direct sum of the homogeneous submodules in (8.1.31) and (8.1.32),

as a graded module over k, because of (8.1.29). This corresponds to Proposition 6.2a on p66 of [50], under slightly different conditions.

We may consider V as a complex, with the grading as in (7.12.9), and d_{V} = 0, as in Section 7.12. In this case, the previous remarks correspond to those in Section 7.3.

8.2 Surjectivity conditions and α′

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V and W be both left or both right modules over A again, and suppose that (W, d_{W}) is a module over A with differentiation. Thus Hom_{A}(V, W) is a module over k with differentiation operator d = d_{Hom_{A}(V,W)} as in (7.12.1), as before.

Remember that Z( Hom_{A}(V, W)) is the same as Hom_{A}(V, Z(W)), and that B( Hom_{A}(V, W)) is contained in Hom_{A}(V, B(W)), as in Section 7.14. Consider the condition that

B(Hom_{A}(V, W)) = Hom_{A}(V, B(W)), (8.2.1)

which is to say that d maps Hom_{A}(V, W) onto Hom_{A}(V, B(W)). In particular, this holds when V is projective as a module over A.

Alternatively, let W_{0} be a submodule of W, as a module over A. If

(8.2.2) W corresponds to the direct sum of Z(W) and W_{0},

as a module over A, then

(8.2.3) the restriction of d_{W} to W_{0} is a one-to-one mapping onto B(W).

Conversely, one can check that (8.2.3) implies (8.2.2). It is easy to see that (8.2.3) implies (8.2.1).

Let α′ be the homomorphism from H( Hom_{A}(V, W)) into Hom_{A}(V, H(W)) defined in Section 7.14. One can verify that the kernel of α′ is the same as the image of Hom_{A}(V, B(W)) under the natural quotient homomorphism from Z( Hom_{A}(V, W)) onto H( Hom_{A}(V, W)), by construction. In particular,

(8.2.4) α′ is injective on H(Hom_{A}(V, W)) if and only if (8.2.1) holds.

Remember that ξ is the natural homomorphism from Hom_{A}(V, Z(W)) into Hom_{A}(V, H(W)) defined by composing an element of Hom_{A}(V, Z(W)) with the natural quotient mapping from Z(W) onto H(W), as in Section 7.14. Observe that α′(H(Hom_{A}(V, W))) = ξ(Hom_{A}(V, Z(W))), (8.2.5) by construction. In particular,

α′(H(Hom_{A}(V, W))) = Hom_{A}(V, H(W)) (8.2.6)

if and only if ξ(Hom_{A}(V, Z(W))) = Hom_{A}(V, H(W)). (8.2.7)

If V is projective as a module over A, then (8.2.7) holds. This implies that (8.2.6) holds, as in the preceding paragraph.

If Z(W) corresponds to the direct sum of B(W) and another submodule, as a module over A, then one can check directly that (8.2.7) holds. This means that (8.2.6) holds, which could also be obtained as in (8.1.10).

8.3 Surjectivity conditions and complexes

Let us continue with the same notation and hypotheses as in the previous section, and suppose in addition that (W, d_{W}) is a graded module over A that is a complex. This means that Hom_{A}^{gr}(V, W) may be defined as a graded module over k as in (7.12.4), and as a complex, with differentiation operator d = d_{Hom_{A}^{gr}(V,W)} as in (7.12.11).

As in Section 7.15, Z( Hom_{A}^{gr}(V, W)) is the same as Hom_{A}^{gr}(V, Z(W)), and B( Hom_{A}^{gr}(V, W)) is contained in Hom_{A}^{gr}(V, B(W)). Consider the condition that

B(Hom_{A}^{gr}(V, W)) = Hom_{A}^{gr}(V, B(W)), (8.3.1)

which means that d maps Hom_{A}^{gr}(V, W) onto Hom_{A}^{gr}(V, B(W)). One can check that this holds when V is projective as a module over A, as before.

Let W_{0} be a homogeneous submodule of W, as a graded module over A. Note that

(8.3.2) W corresponds to the direct sum of Z(W) and W_{0},

as a module over A, if and only if this holds with for W as a graded module over A. Of course, the restriction of d_{W} to W_{0} is a homomorphism into B(W), as modules over A, of degree 1. As in the previous section, (8.3.2) holds if and only if

(8.3.3) the restriction of d_{W} to W_{0} is a one-to-one mapping onto B(W).

One can verify that (8.3.1) holds when (8.3.3) holds.

Remember that the underlying module |W | over A corresponding to W, without a grading, may be considered as a module over A with differentiation with respect to d_{W}, so that Hom_{A}(V, |W |) may be considered as a module over k with differentiation operator d = d_{Hom_{A}(V,W)} as in (7.12.1). We may also consider Hom_{A}^{gr}(V, W) as a submodule of Hom_{A}(V, |W |), as a module over k with differentiation, as in Section 7.12. Similarly, Hom_{A}^{gr}(V, B(W)) may be 8.3. SURJECTIVITY CONDITIONS AND COMPLEXES 213

considered as a submodule of Hom_{A}(V, B(|W |)), as a module over k, which may be considered as a submodule of Hom_{A}(V, |W |). It is easy to see that (8.3.4) Hom_{A}^{gr}(V, B(W)) = Hom_{A}^{gr}(V, W) ∩ Hom_{A}(V, B(|W |)), as submodules of Hom_{A}(V, |W |).

Let α′ be the homomorphism (8.3.5) from H(Hom_{A}(V, |W |)) into Hom_{A}(V, H(|W |)) defined in Sections 7.14 and 7.15. As in the previous section, the kernel of α′ is the same as the image of Hom_{A}(V, B(|W |)) under the natural quotient homomorphism (8.3.6) from Z(Hom_{A}(V, |W |)) onto H(Hom_{A}(V, |W |)). Remember that the restriction of α′ to H(Hom_{A}^{gr}(V, W)) is a homomorphism into Hom_{A}^{gr}(V, H(W)) of degree 0. as in Section 7.15. The kernel of the restriction of α′ to H( Hom_{A}^{gr}(V, W)) is the same as the image of Hom_{A}^{gr}(V, B(W)) under the natural quotient homomorphism (8.3.7) from Z(Hom_{A}^{gr}(V, W)) onto H(Hom_{A}^{gr}(V, W)), because of (8.3.4). It follows that (8.3.8) the restriction of α′ to H(Hom_{A}^{gr}(V, W)) is injective

if and only if (8.3.1) holds.

Let ξ be the natural homomorphism (8.3.9) from Hom_{A}(V, Z(|W |)) into Hom_{A}(V, H(|W |)) defined by composing an element of Hom_{A}(V, Z(|W |)) with the natural quotient mapping from Z(|W |) onto H(|W |), as in Sections 7.14 and 7.15. Thus α′(H(Hom_{A}(V, |W |))) = ξ(Hom_{A}(V, Z(|W |))), (8.3.10) as in (8.2.5). The restriction of ξ to Hom_{A}^{gr}(V, Z(W)) is a homomorphism into Hom_{A}^{gr}(V, H(W)) of degree 0, as in Section 7.15. We also have that α′(H(Hom_{A}^{gr}(V, W))) = ξ(Hom_{A}^{gr}(V, Z(W))), (8.3.11) by construction. This means that α′(H(Hom_{A}^{gr}(V, W))) = Hom_{A}^{gr}(V, H(W)) (8.3.12) if and only if ξ(Hom_{A}^{gr}(V, Z(W))) = Hom_{A}^{gr}(V, H(W)). (8.3.13)

One can verify that (8.3.13) holds when V is projective as a module over A. Thus (8.3.12) holds in this case.

Suppose that Z(W) corresponds to the direct sum of B(W) and another homogeneous submodule of Z(W), as a module over A, and thus as a graded module over A. One can check directly that (8.3.13) under these conditions, so that (8.3.12) holds. This could also be obtained as in (8.1.30).

These properties of α′ are related to Propositions 7.2 and 7.4 on p68, 70 of [50], and some remarks on p70 of [50].

8.4 Differentiation and Hom(·, W)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, W be both left or both right modules over A.

Suppose for the moment that (V, d_{V}) is a module with differentiation. Let

d = d_{Hom_{A}(V,W)} (8.4.1)

be the homomorphism from Hom_{A}(V, W) into itself, as a module over A, defined by composing d_{V} with elements of Hom_{A}(V, W). That is to say, if ϕ is an element of Hom_{A}(V, W), then we put

d(ϕ) = ϕ ∘ d_{V} . (8.4.2)

Clearly d ∘ d = 0 on Hom_{A}(V, W), (8.4.3)

so that Hom_{A}(V, W) is a module with differentiation with respect to d.

Suppose now that V is a graded module over A. We would like to define Hom_{A}^{gr}(V, W) as a graded module over k, with

(Hom_{A}^{gr}(V, W))^{j} = Hom_{A}(V^{−j}, W) (8.4.4)

for every j ∈ Z, as on p62 of [50], with r = 1. As in previous situations like this, we can simply take Hom_{A}^{gr}(V, W) to be the direct sum of (8.4.4) over j ∈ Z, as a module over k, but it is sometimes convenient to consider it as a submodule of related modules over k.

Let |V | be the underlying module over A corresponding to V, without a grading, as on p63 of [50]. If ϕ ∈ Hom_{A}^{gr}(V, W), then let ϕ^{j} be the corresponding element of (8.4.4) for each j ∈ Z, as usual, so that ϕ^{j} = 0 for all but finitely many j. This leads to a unique homomorphism from |V | into W, as modules over A, that we shall also denote by ϕ, with

ϕ = ϕ^{j} on V^{−j} (8.4.5)

for each j ∈ Z.

This defines an injective homomorphism

(8.4.6) from Hom_{A}^{gr}(V, W) into Hom_{A}(|V |, W), 8.4. DIFFERENTIATION AND HOM(·, W) 215

as modules over k, so that we may identify Hom_{A}^{gr}(V, W) with the corresponding submodule of Hom_{A}(|V |, W), as a module over k. More precisely, Hom_{A}^{gr}(V, W) corresponds to the submodule of Hom_{A}(V, W) consisting of the homomorphisms from |V | into W, as modules over A, whose restrictions to V^{−j} are equal to 0 for all but finitely many j ∈ Z. This is related to a remark on p63 of [50]. Remember that Hom_{A}(|V |, W) corresponds to a direct product of Hom_{A}(V^{−j}, W), j ∈ Z, as a module over k, as in Section 1.7.

Suppose for the moment that W is considered as a graded module over A, with

(8.4.7)  W^{l} = W when l = 0
= {0} when l ≠ 0,

as in Section 7.5. This permits us to consider Hom_{A}^{gr}(V, W) as a doubly-graded module over k, as in Section 6.3, with

(Hom_{A}^{gr}(V, W))^{j,l} = Hom_{A}(V^{−j}, W_{l}) = Hom_{A}(V^{−j}, W) when l = 0
(8.4.8)  = {0}  when l ≠ 0.

Note that the single grading associated to this double grading is the same as in (8.4.4).

Suppose for the rest of the section that (V, d_{V}) is a graded module over A that is a complex. Thus Hom_{A}^{gr}(V, W) may be defined as a graded module over k as in (8.4.4). Let us define

d = d_{Hom_{A}^{gr}(V,W)} (8.4.9)

on Hom_{A}^{gr}(V, W) as follows. If ϕ ∈ Hom_{A}^{gr}(V, W), then d(ϕ) is defined as an element of Hom_{A}^{gr}(V, W) by putting

d(ϕ)^{j+1} = ϕ^{j} ∘ d_{V}^{−j−1} (8.4.10)

for each j. One can check that Hom_{A}^{gr}(V, W) is a complex with respect to d, as on p63 of [50].

The underlying module |V | over A corresponding to V may be considered as a module with differentiation with respect to d_{V}, so that d = d_{Hom_{A}(|V |,W)} may be defined on Hom_{A}(|V |, W) as in (8.4.1). It is easy to see that the restriction of d on Hom_{A}(|V |, W) to Hom_{A}^{gr}(V, W), considered as a submodule of Hom_{A}(|V |, W), as a module over k, is the same as the definition of d on Hom_{A}^{gr}(V, W) in the preceding paragraph. This means that Hom_{A}^{gr}(V, W) may be considered as a submodule of Hom_{A}(|V |, W), as a module over k with differentiation.

Let us consider W as a complex, with the grading as in (8.4.7), and differentiation operator d_{W} = 0, as on p75 of [50]. This permits us to consider Hom_{A}^{gr}(V, W) as a double complex, with double grading as in (8.4.8), and

δ_{1} = d = d_{Hom_{A}^{gr}(V,W)}, δ_{2} = 0, (8.4.11) as in Section 6.5. It follows that Hom_{A}^{gr}(V, W) is a single complex with respect to the total differentiation operator

δ = δ_{1} + δ_{2} = d = d_{Hom_{A}^{gr}(V,W)}, (8.4.12)

and the single grading associated to the double grading on Hom_{A}^{gr}(V, W), as in Section 5.14. This is equivalent to considering Hom_{A}^{gr}(V, W) as a single complex as before.

8.5 Induced mappings and Hom(·, W)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, V_{0}, and W be all left or all right modules over A.

Suppose for the moment that (V, d_{V}) and (V_{0}, d_{V_{0}}) are modules with differentiation. Let d = d_{Hom_{A}(V,W)} be defined on Hom_{A}(V, W) as in (8.4.1), and let d_{0} = d_{Hom_{A}(V_{0},W)} (8.5.1)

be defined on Hom_{A}(V_{0}, W) in the same way.

Let ϕ be a homomorphism from V_{0} into V, as modules over A. Consider the homomorphism Φ

(8.5.2) from Hom_{A}(V, W) into Hom_{A}(V_{0}, W),

as modules over k, defined by composing ϕ with an element of Hom_{A}(V, W) to get an element of Hom_{A}(V_{0}, W). Suppose now that ϕ is a homomorphism from V_{0} into V, as modules over A with differentiation. In this case, one can check that Φ is a homomorphism as in (8.5.2), as modules over k with differentiation.

Let ψ be another homomorphism from V_{0} into V, as modules over A with differentiation, and let Ψ be the corresponding homomorphism as in (8.5.2), as modules over k with differentiation. Also let σ be a homotopy between ϕ and ψ, so that σ is a homomorphism from V_{0} into V, as modules over A, such that

d_{V} ∘ σ + σ ∘ d_{V_{0}} = ϕ − ψ. (8.5.3)

Suppose that Σ is the homomorphism as in (8.5.2), as modules over k, that corresponds to σ in the same way as before. One can check that

d_{0} ∘ Σ + Σ ∘ d = Φ − Ψ, (8.5.4)

so that Σ is a homotopy between Φ and Ψ.

Suppose now that V, V_{0} are graded modules over A, so that Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V_{0}, W) may be defined as graded modules over k as in (8.4.4). Let ϕ be a homomorphism from V_{0} into V, as modules over A, of degree p ∈ Z. This leads to a homomorphism Φ

(8.5.5) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(V_{0}, W), 8.5. INDUCED MAPPINGS AND HOM(·, W) 217

as modules over k, of degree p as well. More precisely, if j ∈ Z, then Φ acts on (8.4.4) by composing the restriction of ϕ to V_{0}^{−j−p} with an element of (8.4.4) to get an element of

(Hom_{A}^{gr}(V_{0}, W))^{j+p} = Hom_{A}(V_{0}^{−j−p}, W), (8.5.6)

as on p63 of [50].

Let |V |, |V_{0}| be the underlying modules over A corresponding to V, V_{0}, respectively, without gradings, as on p63 of [50]. We may consider Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V_{0}, W) as submodules of Hom_{A}(|V |, W) and Hom_{A}(|V_{0}|, W), respectively, as modules over k, as in the previous section. Using this identification, the homomorphism Φ as in (8.5.5) is the same as the restriction of the homomorphism Φ (8.5.7) from Hom_{A}(|V |, W) into Hom_{A}(|V_{0}|, W)

as in (8.5.2) to Hom_{A}^{gr}(V, W).

If we consider W as a graded module as in (8.4.7), then Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V_{0}, W) may be defined as doubly-graded modules over k, as in Section 6.3. Under these conditions,

(8.5.8) the homomorphism Φ as in (8.5.5)

corresponds to Hom^{gr}(ϕ, I_{W}).

This uses the notation in Section 6.4, with I_{W} being the identity mapping on W.

Suppose that (V, d_{V}), (V_{0}, d_{V_{0}}) are graded modules over A with differentiation that are complexes, so that Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V_{0}, W) may be considered as single complexes too, as in the previous section. Let ϕ be a map of degree p ∈ Z from V_{0} into V, as complexes, as in Section 6.6. If Φ is the corresponding homomorphism as in (8.5.5), then it is easy to see that Φ is a map of degree p as in (8.5.5), as complexes.

Let ψ be another map of degree p from V_{0} into V, as complexes, and let Ψ be the corresponding map of degree p as in (8.5.5), as complexes. Suppose that σ is a homotopy between ϕ and ψ, as maps of degree p between complexes, as in Section 6.6. Remember that this means in particular that σ is a homomorphism from V_{0} into V of degree p−1, as modules over A. Thus we get a homomorphism Σ as in (8.5.5), as modules over k, of degree p − 1. One can check that (−1)^{p} Σ is a homotopy between ϕ and ψ, as maps of degree p between complexes.

We may consider W as a complex, with d_{W} = 0, and graded as in (8.4.7). This permits us to consider Hom_{A}^{gr}(V, W) and Hom_{A}^{gr}(V_{0}, W) as double complexes, as in the previous section. We may also consider Φ and Ψ as maps of bidegree (p, 0) as in (8.5.5), as double complexes, as in Section 6.9. Under these conditions, ((−1)^{p} Σ, 0) defines a homotopy between Φ and Ψ, as maps of bidgree (p, 0) as in (8.5.5), as double complexes, as in Section 6.9. These remarks correspond to some of those in Section 6.5 when p = 0.

218 CHAPTER 8. MORE ON DIFFERENTIATION, 3

8.6 Another analogue of α′

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V and W be both left or both right modules over A, and suppose that (V, d_{V}) is a module over A with differentiation. This means that Hom_{A}(V, W) is a module over k with differentiation operator d = d_{Hom_{A}(V,W)} as in (8.4.1).

In this case, Z( Hom_{A}(V, W)) consists of ϕ ∈ Hom_{A}(V, W) such that d(ϕ) = ϕ ∘ d_{V} = 0 (8.6.1) on V. Equivalently, this means that B(V) ⊆ ker ϕ. (8.6.2)

Of course, if ϕ ∈ Hom_{A}(V, W), then the restriction of ϕ to Z(V) defines a homomorphism from Z(V) into W, as modules over A. If ϕ ∈ Z( Hom_{A}(V, W)), then the restriction of ϕ to Z(V) induces a homomorphism (8.6.3) from H(V) into W, as modules over A, because of (8.6.2). This defines a homomorphism (8.6.4) from Z(Hom_{A}(V, W)) into Hom_{A}(H(V), W), as modules over k.

Suppose for the moment that ϕ ∈ B( Hom_{A}(V, W)), so that ϕ = d(ψ) = ψ ∘ d_{V} (8.6.5) for some ψ ∈ Hom_{A}(V, W). This implies that Z(V) ⊆ ker ϕ. (8.6.6) This means that the induced homomorphism as in (8.6.3) is equal to 0. This is the same as saying that ϕ is mapped to 0 by the homomorphism as in (8.6.4).

Thus B( Hom_{A}(V, W)) is contained in the kernel of the homomorphism as in (8.6.4). It follows that the homomorphism as in (8.6.4) induces a homomorphism (8.6.7) α′ from H(Hom_{A}(V, W)) into Hom_{A}(H(V), W), as modules over k.

There is a natural homomorphism (8.6.8) from Hom_{A}(Z′(V), W) into Hom_{A}(V, W), as modules over k. This is defined by composing the natural quotient mapping from V onto Z′(V) = V/d_{V} (V) with an element of Hom_{A}(Z′(V), W) to get an element of Hom_{A}(V, W). Note that (8.6.9) the homomorphism as in (8.6.8) is injective.

8.6. ANOTHER ANALOGUE OF α′ 219

We also have that

the homomorphism as in (8.6.8)

(8.6.10)

maps Hom_{A}(Z′(V), W) onto Z(Hom_{A}(V, W)).

This leads to a natural homomorphism

(8.6.11) η from Hom_{A}(Z′(V), W) onto H(Hom_{A}(V, W)),

as modules over k. More precisely, η is the composition of the homomorphism as in (8.6.8), considered as a homomorphism onto Z( Hom_{A}(V, W)), with the natural quotient mapping from Z( Hom_{A}(V, W)) onto H( Hom_{A}(V, W)).

There is a natural homomorphism

(8.6.12) ξ from Hom_{A}(Z′(V), W) into Hom_{A}(H(V), W),

as modules over k, which sends an element of Hom_{A}(Z′(V), W) to its restriction to H(V), considered as a submodule of Z′(V). It is easy to see that

α′ ∘ η = ξ, (8.6.13)

as homomorphisms from Hom_{A}(Z′(V), W) into Hom_{A}(H(V), W). More precisely, α′ ∘ η is the same as the composition of the homomorphism as in (8.6.8), considered as a homomorphism onto Z( Hom_{A}(v, W)), with the homomorphism as in (8.6.4). This composition is the same as ξ, by construction. Note that α′ is uniquely determined by (8.6.13), because η is surjective.

There is a natural homomorphism

(8.6.14) from Hom_{A}(V, W) into Hom_{A}(Z(V), W),

as modules over k, which sends an element of Hom_{A}(V, W) to its restriction to Z(V). There is also a natural homomorphism

(8.6.15) from Z(V) into Z′(V),

as modules over A, which is the restriction to Z(V) of the natural quotient mapping from V onto Z′(V). Equivalently, this is the composition of the natural quotient mapping from Z(V) onto H(V) with the natural inclusion mapping from H(V) into Z′(V).

Using the homomorphism as in (8.6.15), we get a natural homomorphism

(8.6.16) from Hom_{A}(Z′(V), W) into Hom_{A}(Z(V), W),

as modules over k. This homomorphism sends an element of Hom_{A}(Z′(V), W) to its composition with the homomorphism as in (8.6.15). This is the same as the composition of the homomorphism as in (8.6.8) with the homomorphism as in (8.6.14).

If ϕ ∈ B( Hom_{A}(V, W)), then ϕ is mapped to 0 by the homomorphism as in (8.6.14), because of (8.6.6). Using this and the restriction of the homomorphism as in (8.6.14) to Z( Hom_{A}(V, W)), we get a natural homomorphism

(8.6.17) ζ from H(Hom_{A}(V, W)) into Hom_{A}(Z(V), W),

as modules over k. One can check that

(8.6.18) ζ ∘ η is the same as the homomorphism as in (8.6.16).

More precisely, ζ ∘ η is the same as the composition of the homomorphism as in (8.6.8) with the homomorphism as in (8.6.14), which is the same as the homomorphism as in (8.6.16).

Using the natural quotient mapping from Z(V) onto H(V), we get a natural homomorphism

(8.6.19) τ from Hom_{A}(H(V), W) into Hom_{A}(Z(V), W),

as modules over k. That is to say, τ sends an element of Hom_{A}(H(V), W) to its composition with the natural quotient mapping from Z(V) onto H(V). In particular, (8.6.20) τ is injective.

One can verify that

(8.6.21) τ ∘ ξ is the same as the homomorphism as in (8.6.16).

More precisely, τ ∘ ξ is the same as the composition of the homomorphism as in (8.6.8) with the homomorphism as in (8.6.14), which is the same as the homomorphism as in (8.6.16). This implies that

τ ∘ ξ = ζ ∘ η, (8.6.22)

by (8.6.18).

We also have that τ ∘ α′ = ζ, (8.6.23)

as homomorphisms from H( Hom_{A}(V, W)) into Hom_{A}(Z(V), W). This can be seen directly from the definitions. Alternatively, each of (8.6.13) and (8.6.23) can be obtained from the other using (8.6.22), by composing with η or τ, as appropriate. Of course, α′ is uniquely determined by (8.6.23), because τ is injective.

8.7 Another α′ and complexes

We continue with the same notation and hypotheses as in the previous section, and suppose now that (V, d_{V}) is a graded module over A that is a complex. This means that Hom_{A}^{gr}(V, W) may be defined as a graded module over k as in 8.7. ANOTHER α′ AND COMPLEXES 221

(8.4.4), and as a complex, with differentiation operator d = d_{Hom_{A}^{gr}(V,W)} as in (8.4.9). Similarly,

(8.7.1) Hom_{A}^{gr}(B(V), W), Hom_{A}^{gr}(Z(V), W), Hom_{A}^{gr}(H(V), W), and Hom_{A}^{gr}(Z′(V), W)

may be defined as graded modules over k as in (8.4.4).

The underlying module |V | over A corresponding to V may be considered as a module with differentiation operator d_{V}, and Hom_{A}(|V |, W) is a module over k with differentiation operator d = d_{Hom_{A}(|V |,W)} defined as in (8.4.1). We have also seen that Hom_{A}^{gr}(V, W) may be considered as a submodule of Hom_{A}(|V |, W), as a module over k with differentiation, as in Section 8.4.

Of course, B(|V |) and Z(|V |) are the same as B(V) and Z(V), respectively, without the gradings. Thus B(|V |) and Z(|V |) may be considered as the same as |B(V)| and |Z(V)|, respectively. In particular,

(8.7.2) Hom_{A}^{gr}(B(V), W) and Hom_{A}^{gr}(Z(V), W) may be considered as submodules of Hom_{A}(B(|V |), W) and Hom_{A}(Z(|V |), W),

as modules over k, respectively.

Similarly, H(|V |) and Z′(|V |) are the same as H(V) and Z′(V), without the gradings. This means that H(|V |) and Z′(|V |) may be considered as the same as |H(V)| and |Z′(V)|, respectively. It follows that

(8.7.3) Hom_{A}^{gr}(H(V), W) and Hom_{A}^{gr}(Z′(V), W) may be considered as submodules of Hom_{A}(H(|V |), W) and Hom_{A}(Z′(|V |), W),

as modules over k, respectively.

The remarks in the previous section can be applied to |V |, and we would like to consider analogous statements for Hom_{A}^{gr}(·, W) instead of Hom_{A}(·, W) here. Note that

(8.7.4) Z(Hom_{A}^{gr}(V, W)) = Z(Hom_{A}(|V |, W)) ∩ Hom_{A}^{gr}(V, W),

because Hom_{A}^{gr}(V, W) is a submodule of Hom_{A}(|V |, W), as a module over k with differentiation.

We also have that

(8.7.5) B(Hom_{A}^{gr}(V, W)) = B(Hom_{A}(|V |, W)) ∩ Hom_{A}^{gr}(V, W).

More precisely, the fact that the left side is contained in the right side follows from Hom_{A}^{gr}(V, W) being a submodule of Hom_{A}(|V |, W), as a module over k with differentiation. The opposite inclusion can be verified using the definition of Hom_{A}^{gr}(V, W).

Using (8.7.5),

(8.7.6) we may consider H(Hom_{A}^{gr}(V, W))

as a submodule of H(Hom_{A}(|V |, W)),

222 CHAPTER 8. MORE ON DIFFERENTIATION, 3

as a module over k. Similarly, (8.7.7) we may consider Z′(Hom_{A}^{gr}(V, W))

as a submodule of Z′(Hom_{A}(|V |, W)),

as a module over k.

There is a natural homomorphism (8.7.8) from Z(Hom_{A}(|V |, W)) into Hom_{A}(H(|V |), W), as modules over k, as in (8.6.4). One can check that the restriction of this homomorphism to Z( Hom_{A}^{gr}(V, W)) defines a homomorphism (8.7.9) from Z(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(H(V), W), as modules over k, of degree 0.

There is a natural homomorphism (8.7.10) α′ from H(Hom_{A}(|V |, W)) into Hom_{A}(H(|V |), W), as modules over k, as in (8.6.7). One can check that (8.7.11) α′ maps H(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(H(V), W), and that (8.7.12) the restriction of α′ to H(Hom_{A}^{gr}(V, W)) has degree 0.

There is an natural injective homomorphism (8.7.13) from Hom_{A}(Z′(|V |), W) into Hom_{A}(|V |, W), as modules over k, as in (8.6.8). More precisely, this homomorphism maps Hom_{A}(Z′(|V |), W) onto Z( Hom_{A}(|V |, W)), as in (8.6.10). One can check that this homomorphism (8.7.14) maps Hom_{A}^{gr}(Z′(V), W) onto Z(Hom_{A}^{gr}(V, W)), and that the restriction of this homomorphism to Hom_{A}^{gr}(Z′(V), W) has degree 0.

There is a natural homomorphism (8.7.15) η from Hom_{A}(Z′(|V |), W) onto H(Hom_{A}(|V |, W)), as modules over k, as in (8.6.11). One can verify that (8.7.16) η maps Hom_{A}^{gr}(Z′(V), W) onto H(Hom_{A}^{gr}(V, W)), and that (8.7.17) the restriction of η to Hom_{A}^{gr}(Z′(V), W) has degree 0.

8.7. ANOTHER α′ AND COMPLEXES 223

The restriction of η to Hom_{A}^{gr}(Z′(V), W) is the same as the composition of the restriction of the homomorphism as in (8.7.13) to Hom_{A}^{gr}(Z′(V), W), considered as a homomorphism onto Z( Hom_{A}^{gr}(V, W)), with the natural quotient mapping from Z( Hom_{A}^{gr}(V, W)) onto H( Hom_{A}^{gr}(V, W)).

There is a natural homomorphism (8.7.18) ξ from Hom_{A}(Z′(|V |), W) into Hom_{A}(H(|V |), W), as modules over k, as in (8.6.12). One can check that (8.7.19) ξ maps Hom_{A}^{gr}(Z′(V), W) into Hom_{A}^{gr}(H(V), W), and that (8.7.20) the restriction of ξ to Hom_{A}^{gr}(Z′(V), W) has degree 0. Note that α′ ∘ η = ξ on Hom_{A}(Z′(|V |), W), as in (8.6.13). In particular, this holds on Hom_{A}^{gr}(Z′(V), W). The restriction of α′ to H( Hom_{A}^{gr}(V, W)) is uniquely determined by this equality of homomorphisms on Hom_{A}^{gr}(Z′(V), W), because of (8.7.16).

There is a natural homomorphism (8.7.21) from Hom_{A}(|V |, W) into Hom_{A}(Z(|V |), W), as modules over k, which sends an element of Hom_{A}(|V |, W) to its restriction to Z(|V |), as in (8.6.14). The restriction of this homomorphism to Hom_{A}^{gr}(V, W) defines a natural homomorphism (8.7.22) from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(Z(V), W), as modules over k. More precisely, this homomorphism has degree 0.

There is a natural homomorphism (8.7.23) from Z(|V |) into Z′(|V |), as modules over A, as in (8.6.15). The restriction of this homomorphism to Z(V) defines a homomorphism (8.7.24) from Z(V) into Z′(V), as modules over A, of degree 0. This is the same as the restriction to Z(V) of the natural quotient mapping from V onto Z′(V), and the composition of the natural quotient mapping from Z(V) onto H(V) with the natural inclusion from H(V) into Z′(V).

There is a natural homomorphism (8.7.25) from Hom_{A}(Z′(|V |), W) into Hom_{A}(Z(|V |), W), as modules over k, as in (8.6.16). The restriction of this homomorphism to Hom_{A}^{gr}(Z′(V), W) defines a homomorphism (8.7.26) from Hom_{A}^{gr}(Z′(V), W) into Hom_{A}^{gr}(Z(V), W), 224 CHAPTER 8. MORE ON DIFFERENTIATION, 3 as modules over k, of degree 0.

There is a natural homomorphism (8.7.27) ζ from H(Hom_{A}(|V |, W)) into Hom_{A}(Z(|V |), W), as modules over k, as in (8.6.17). One can check that (8.7.28) ζ maps H(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(Z(V), W), and that (8.7.29) the restriction of ζ to H(Hom_{A}^{gr}(V, W)) has degree 0. Remember that ζ ∘ η is the same as the homomorphism as in (8.7.25).

There is a natural injective homomorphism (8.7.30) τ from Hom_{A}(H(|V |), W) into Hom_{A}(Z(|V |), W), as modules over k, as in (8.6.19). One can verify that (8.7.31) τ maps Hom_{A}^{gr}(H(V), W) into Hom_{A}^{gr}(Z(V), W), and that (8.7.32) the restriction of τ to Hom_{A}^{gr}(H(V), W) has degree 0. We also have that τ ∘ ξ is the same as the homomorphism as in (8.7.25), as before. This implies that τ ∘ ξ = ζ ∘ η, as in (8.6.22). In particular, this holds on Hom_{A}^{gr}(Z′(V), W).

Remember that τ ∘ α′ = ζ on H( Hom_{A}(|V |, W)), as in (8.6.23). In particular, this holds on H( Hom_{A}^{gr}(V, W)).

These homomorphisms and their properties correspond to the commutative diagram (1) on p64 of [50], and to parts of Proposition 6.1a on p65 of [50], under slightly different conditions.

We may consider W as a graded module over A with differentiation that is a complex, with grading as in (8.4.7), and d_{W} = 0, as in Section 8.4. In this case, B(W) = {0}, Z(W) = H(W) = Z′(W) = W, and the remarks in this and the previous section correspond to those in Sections 6.13 – 6.15.

8.8 Splitting Z′(V) and α′

Let k be a commutative ring with a mutliplciative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that V and W are both left or both right modules over A, and that (V, d_{V}) is a module over A with differentiation. Thus Hom_{A}(V, W) is a module over k with differentiation operator d = d_{Hom_{A}(V,W)} as in (8.4.1).

8.8. SPLITTING Z′(V) AND α′ 225

Remember that Z′(V) = V/d_{V} (V), and suppose that (8.8.1) Z′(V) corresponds to the direct sum

of H(V) and another submodule of Z′(V),

as a module over A. This means that (8.8.2) there is a homomorphism from Z′(V) onto H(V), as modules

over A, that is equal to the identity mapping on H(V).

We can compose the natural quotient mapping from V onto Z′(V) with this homomorphism to get a homomorphism (8.8.3) β from V onto H(V), as modules over A. Of course, β ∘ d_{V} = 0 (8.8.4) on V, by construction. Note that (8.8.5) the restriction of β to Z(V) is the same as

the natural quotient mapping from Z(V) onto H(V),

by (8.8.2).

We can use β to get a homomorphism (8.8.6) from Hom_{A}(H(V), W) into Hom_{A}(V, W), as modules over k. This homomorphism sends an element of Hom_{A}(H(V), W) to its composition with β, to get an element of Hom_{A}(V, W). The composition of this element of Hom_{A}(V, W) with d_{V} is equal to 0, because of (8.8.4). This means that we get an element of Z( Hom_{A}(V, W)), by the definition of d. Thus we get a homomorphism (8.8.7) from Hom_{A}(H(V), W) into Z(Hom_{A}(V, W)) in this way.

This leads to a homomorphism (8.8.8) from Hom_{A}(H(V), W) into H(Hom_{A}(V, W)), as modules over k, by composing the previous homomorphism as in (8.8.7) with the natural quotient mapping (8.8.9) from Z(Hom_{A}(V, W)) onto H(Hom(V, W)). Let α′ be the homomorphism from H( Hom_{A}(V, W)) into Hom_{A}(H(V), W) defined in Section 8.6. One can check that (8.8.10) the composition of the homomorphism as in (8.8.8) with α′

is the identity mapping on Hom_{A}(H(V), W).

More precisely, the composition of the homomorphism as in (8.8.8) with α′ is the same as the composition of the homomorphism as in (8.8.7) with the homomorphism as in (8.6.4), by construction. To get that this is the identity mapping on Hom_{A}(H(V), W), one can use (8.8.5).

Using this, we get that

α′(H(Hom_{A}(V, W))) = Hom_{A}(H(V), W). (8.8.11)

We can also use (8.8.10) to get that

(8.8.12) H(Hom_{A}(V, W)) corresponds to the direct sum of ker α′

and another submodule of H(Hom_{A}(V, W)),

as a module over k.

Let us suppose now that (V, d_{V}) is a graded module over A with differentiation that is a complex, and that

(8.8.13) Z′(V) corresponds to the direct sum of H(V)

and another homogeneous submodule of Z′(V),

as a module over A. Equivalently, this means that Z′(V) corresponds to the direct sum of H(V) and another submodule, as a graded module over A. This implies that (8.8.14) the homomorphism as in (8.8.2) has degree 0.

It follows that

(8.8.15) the homomorphism β as in (8.8.3) has degree 0.

We can define Hom_{A}^{gr}(V, W) as a graded module over k as in (8.4.4), and as a complex, with differentiation operator d = d_{Hom_{A}^{gr}(V,W)} as in (8.4.9). The underlying module |V | over A corresponding to V, without a grading, may be considered as a module with differentiation with respect to d_{V}, so that Hom_{A}(|V |, W) is a module over k with differentiation operator d = d_{Hom_{A}(|V |,W)} defined as in (8.4.1). Remember that Hom_{A}^{gr}(V, W) may be considered as a submodule of Hom_{A}(|V |, W), as a module over k with differentiation, as in Section 8.4.

We can use β to get a homomorphism

(8.8.16) from Hom_{A}(H(|V |), W) into Hom_{A}(|V |, W),

as modules over k, as in (8.8.6). This may be considered as a homomorphism

(8.8.17) from Hom_{A}(H(|V |), W) into Z(Hom_{A}(|V |, W)),

as modules over k, as in (8.8.7). One can check that

the homomorphism as in (8.8.16)

(8.8.18)

maps Hom_{A}^{gr}(H(V), W) into Hom_{A}^{gr}(V, W).

8.8. SPLITTING Z′(V) AND α′ 227

The restriction of this homomorphism to Hom_{A}^{gr}(H(V), W) may be considered as a homomorphism (8.8.19) from Hom_{A}^{gr}(H(V), W) into Z(Hom_{A}^{gr}(V, W)), as modules over k. Observe that (8.8.20) the homomorphism as in (8.8.19) has degree 0, because of (8.8.15).

As before, we can compose the homomorphism as in (8.8.17) with the natural quotient mapping (8.8.21) from Z(Hom_{A}(|V |, W)) onto H(Hom_{A}(|V |, W)) to get a homomorphism (8.8.22) from Hom_{A}(H(|V |), W) into H(Hom_{A}(|V |, W)), as modules over k. Note that this homomorphism maps Hom_{A}^{gr}(H(V), W) into H( Hom_{A}^{gr}(V, W)). More precisely, the restriction of this homomorphism to Hom_{A}^{gr}(H(V), W) is the same as the homomorphism (8.8.23) from Hom_{A}^{gr}(H(V), W) into H(Hom_{A}^{gr}(V, W)) obtained by composing the homomorphism as in (8.8.19) with the natural quotient mapping (8.8.24) from Z(Hom_{A}^{gr}(V, W)) onto H(Hom_{A}^{gr}(V, W)). It is easy to see that (8.8.25) the homomorphism as in (8.8.23) has degree 0, because of (8.8.20).

Let α′ be the homomorphism (8.8.26) from H(Hom_{A}(H(|V |), W)) into Hom_{A}(H(|V |), W) discussed in the previous two sections. Thus (8.8.27) the composition of the homomorphism as in (8.8.22) with α′

is the identity mapping on Hom_{A}(H(|V |), W),

as in (8.8.10). This implies that α′(H(Hom_{A}(|V |, W))) = Hom_{A}(H(|V |), W)) (8.8.28) and (8.8.29) H(Hom_{A}(|V |, W)) corresponds to the direct sum of ker α′

and another submodule of H(Hom_{A}(|V |, W)), as a module over k, as in (8.8.11) and (8.8.12).

Remember that α′ maps H(Hom_{A}^{gr}(V, W)) into Hom_{A}^{gr}(H(V), W), and the restriction of α′ to H(Hom_{A}^{gr}(V, W)) has degree 0, as in (8.7.11) and (8.7.12). We also have that

the composition of the homomorphism as in (8.8.23)

(8.8.30)

with the restriction of α′ to H(Hom_{A}^{gr}(V, W))
is the identity mapping on Hom_{A}^{gr}(H(V), W),

because of (8.8.27). This implies in particular that

α′(H(Hom_{A}^{gr}(V, W))) = Hom_{A}^{gr}(H(V), W). (8.8.31)

Note that

(8.8.32) the kernel of the restriction of α′ to H(Hom_{A}^{gr}(V, W))

is a homogeneous submodule of H(Hom_{A}(V, W)),

as a graded module over k, because the restriction of α′ to H(Hom_{A}^{gr}(V, W)) has degree 0. Similarly,

(8.8.33) the image of Hom_{A}^{gr}(H(V), W)

under the homomorphism as in (8.8.23)
is a homogeneous submodule of H(Hom_{A}^{gr}(V, W)),

(8.8.34)

because of (8.8.25). Using (8.8.30), we get that

(8.8.35) H(Hom_{A}^{gr}(V, W)) corresponds to the direct sum of the homogeneous submodules in (8.8.32) and (8.8.33),

as a graded module over k. This corresponds to Proposition 6.2a on p66 of [50], under slightly different conditions.

We may also consider W as a complex, with the grading as in (8.4.7), and d_{W} = 0, as in Section 8.4. Under these conditions, the previous remarks correspond to those in Section 7.3.

8.9 Additional conditions and α′

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V and W be both left or both right modules over A again, and suppose that (V, d_{V}) is a module over A with differentiation. This means that Hom_{A}(V, W) is a module over k with differentiation operator d = d_{Hom_{A}(V,W)} as in (8.4.1).

8.9. ADDITIONAL CONDITIONS AND α′ 229

Remember that

(8.9.1) Z(Hom_{A}(V, W)) = {ϕ ∈ Hom_{A}(V, W) : d(ϕ) = ϕ ∘ d_{V} = 0}
= {ϕ ∈ Hom_{A}(V, W) : B(V) ⊆ ker ϕ},

as in Section 8.6. We have also seen that

(8.9.2) B(Hom_{A}(V, W)) = {ϕ ∈ Hom_{A}(V, W) : ϕ = d(ψ) = ψ ∘ d_{V}
for some ψ ∈ Hom_{A}(V, W)}

is contained in {ϕ ∈ Hom_{A}(V, W) : Z(V) ⊆ ker ϕ}. (8.9.3)

Note that (8.9.3) is contained in (8.9.1). We may be interested in situations in which (8.9.4) B(Hom_{A}(V, W)) is equal to (8.9.3).

It is easy to see that (8.9.3) is equal to

(8.9.5) {ϕ ∈ Hom_{A}(V, W) : ϕ = ψ_{0} ∘ d_{V} for some ψ_{0} ∈ Hom_{A}(B(V), W)}.

If W is injective as a module over A, then this is the same as (8.9.2), so that (8.9.4) holds. If

(8.9.6) V corresponds to the direct sum of B(V)

and another submodule of V,

as a module over A, then (8.9.5) is the same as (8.9.2), so that (8.9.4) holds again.

Let α′ be the homomorphism from H( Hom_{A}(V, W)) into Hom_{A}(H(V), W), as modules over k, defined in Section 8.6. By construction, the composition of the natural quotient mapping

(8.9.7) from Z(Hom_{A}(V, W)) onto H(Hom_{A}(V, W))

with α′ is the same as the homomorphism

(8.9.8) from Z(Hom_{A}(V, W)) into Hom_{A}(H(V), W)

mentioned in (8.6.4). One can check that the kernel of the homomorphism as in (8.9.8) is equal to (8.9.3). This means that

(8.9.9) ker α′ is the same as the image of (8.9.3)

under the natural quotient mapping as in (8.9.7).

In particular, (8.9.10) α′ is injective if and only if (8.9.4) holds.

Remember that ξ is the natural homomorphism from Hom_{A}(Z′(V), W) into Hom_{A}(H(V), W) that sends an element of Hom_{A}(Z′(V), W) to its restriction to H(V), considered as a submodule of Z′(V), as in Section 8.6. We have seen that ξ is the same as the composition of the natural homomorphism η from Hom_{A}(Z′(V), W) onto H(Hom_{A}(V, W)) defined in Section 8.6 with α′. This implies that α′(H(Hom_{A}(V, W))) = ξ(Hom_{A}(Z′(V), W)). (8.9.11)

Thus α′(H(Hom_{A}(V, W))) = Hom_{A}(H(V), W) (8.9.12)

if and only if ξ(Hom_{A}(Z′(V), W)) = Hom_{A}(H(V), W). (8.9.13)

If W is injective as a module over A, then (8.9.13) holds. This implies that (8.9.12) holds, as before.

If Z′(V) corresponds to the direct sum of H(V) and another submodule of Z′(V), as a module over A, then it is easy to see that (8.9.13) holds. This implies that (8.9.12) holds, which could also be obtained as in (8.8.11).

8.10 Additional conditions and complexes

We continue with the same notation and hypotheses as in the previous section, and suppose in addition that (V, d_{V}) is a graded module over A that is a complex. Thus Hom_{A}^{gr}(V, W) may be defined as a graded module over k as in (8.4.4), and as a complex, with differentiation operator d = d_{Hom_{A}^{gr}(V,W)} as in (8.4.9). We may define Hom_{A}^{gr}(B(V), W), Hom_{A}^{gr}(H(V), W), and Hom_{A}^{gr}(Z′(V), W) as graded modules over k too, as in (8.4.4).

We may consider the underlying module |V | over A corresponding to V as a module with differentiation operator d_{V}, so that Hom_{A}(|V |, W) is a module over k with differentiation operator d = d_{Hom_{A}(|V |,W)} defined as in (8.4.1). Remember that Hom_{A}^{gr}(V, W) may be considered as a submodule of Hom_{A}(|V |, W), as a module over k with differentiation, as in Section 8.4.

As in Section 8.7, H(|V |) and Z′(|V |) are the same as H(V) and Z′(V), respectively, without the gradings, so that they are the same as |H(V)| and |Z′(V)|, respectively. This means that Hom_{A}^{gr}(H(V), W) and Hom_{A}^{gr}(Z′(V), W) may be considered as submodules of Hom_{A}(H(|V |), W) and Hom_{A}(Z′(|V |), W), respectively, as modules over k.

Remember that

(8.10.1) Z(Hom_{A}^{gr}(V, W)) = Z(Hom_{A}(|V |, W)) ∩ Hom_{A}^{gr}(V, W)

and (8.10.2) B(Hom_{A}^{gr}(V, W)) = B(Hom_{A}(|V |, W)) ∩ Hom_{A}^{gr}(V, W),

as in Section 8.7. The latter implies that H(Hom_{A}^{gr}(V, W)) may be considered as a submodule of H(Hom_{A}(|V |, W)), as a modules over k, as before.

Of course, Z( Hom_{A}^{gr}(V, W)) consists of the ϕ ∈ Hom_{A}^{gr}(V, W) such that

d(ϕ)^{j+1} = ϕ^{j} ∘ d_{V}^{−j−1} = 0 (8.10.3)

8.10. ADDITIONAL CONDITIONS AND COMPLEXES 231

for every j ∈ Z. Equivalently, this means that B(V)^{−j} ⊆ ker ϕ^{j} (8.10.4) for every j ∈ Z. Similarly, B( Hom_{A}^{gr}(V, W)) is the same as the set of ϕ in Hom_{A}^{gr}(V, W) for which there is a ψ ∈ Hom_{A}^{gr}(V, W) such that ϕ^{j+1} = d(ψ)^{j+1} = ψ^{j} ∘ d_{V}^{−j−1} (8.10.5) for every j ∈ Z. This implies that Z(V)^{−j−1} ⊆ ker ϕ^{j+1} (8.10.6) for every j ∈ Z.

Consider (8.10.7) {ϕ ∈ Hom_{A}^{gr}(V, W) : Z(V)^{−j} ⊆ ker ϕ^{j} for every j ∈ Z}, which is a homogeneous submodule of Z( Hom_{A}^{gr}(V, W)), as a module over k. This is the same as the intersection of (8.9.3) with Hom_{A}^{gr}(V, W). Note that B( Hom_{A}^{gr}(V, W)) is contained in (8.10.7). As in the previous section, we may be interested in situations in which (8.10.8) B(Hom_{A}^{gr}(V, W)) is equal to (8.10.7).

Observe that ϕ ∈ Hom_{A}^{gr}(V, W) is an element of (8.10.7) if and only if for each j ∈ Z there is a homomorphism ψ_{0,j} from B(V)^{−j} = d_{V}^{−j−1}(V^{−j−1}) into W, as modules over A, such that ϕ^{j+1} = ψ_{0,j} ∘ d_{V}^{−j−1}. (8.10.9) Of course, if ϕ^{j+1} = 0, then one can simply take ψ_{0,j} = 0. It follows that ϕ ∈ Hom_{A}^{gr}(V, W) is an element of (8.10.7) if and only if there is a ψ_{0} in Hom_{A}^{gr}(B(V), W) such that ϕ^{j+1} = ψ_{0}^{j} ∘ d_{V}^{−j−1} (8.10.10) for every j ∈ Z.

If W is injective as a module over A, then one can take ψ_{0,j} to be a homomorphism from V^{−j} into W, as modules over A, in (8.10.9). This permits one to take ψ_{0} ∈ Hom_{A}^{gr}(V, W) in (8.10.10), so that (8.10.8) holds.

Suppose for the moment that (8.10.11) V corresponds to the direct sum of B(V)

and another homogeneous submodule of V,

as a module over A. This means that V^{−j} corresponds to the direct sum of B(V)^{−j} and another submodule of V^{−j}, as a module over A, for each j ∈ Z. In this case, one can take ψ_{0,j} to be a homomorphism from V^{−j} into W, as 232 CHAPTER 8. MORE ON DIFFERENTIATION, 3

modules over A, in (8.10.9) again. This permits one to take ψ_{0} ∈ Hom_{A}^{gr}(V, W) in (8.10.10), so that (8.10.8) holds, as before.

The natural quotient mapping (8.10.12) from Z(Hom_{A}^{gr}(V, W)) onto H(Hom_{A}^{gr}(V, W)) may be identified with the restriction to (8.10.1) of the natural quotient mapping (8.10.13) from Z(Hom_{A}(|V |, W)) onto H(Hom_{A}(|V |, W)), because of (8.10.2). Let α′ be the homomorphism (8.10.14) from H(Hom_{A}(|V |, W)) into Hom_{A}(H(|V |), W) defined in Sections 8.6 and 8.7. One can verify that (8.10.15) the kernel of the restriction of α′ to H(Hom_{A}^{gr}(V, W))

is the same as the image of (8.10.7)

under the natural quotient mapping as in (8.10.12).

This is similar to (8.9.9), and can be obtained from it. It follows in particular that (8.10.16) the restriction of α′ to H(Hom_{A}^{gr}(V, W)) is injective

if and only if (8.10.8) holds.

Let ξ be the natural homomorphism (8.10.17) from Hom_{A}(Z′(|V |), W) into Hom_{A}(H(|V |), W) that sends an element of Hom_{A}(Z′(|V |), W) to its restriction to H(|V |), considered as a submodule of Z′(|V |), as in Sections 8.6 and 8.7. Note that α′(H(Hom_{A}(|V |, W))) = ξ(Hom_{A}(Z′(|V |), W)), (8.10.18) as in (8.9.11). The restriction of ξ to Hom_{A}^{gr}(V, W) is a homomorphism into Hom_{A}^{gr}(H(V), W) of degree 0, as in Section 8.7. One can check that α′(H(Hom_{A}^{gr}(V, W))) = ξ(Hom_{A}^{gr}(Z′(V), W)) (8.10.19) using the same type of argument as before. More precisely, let η be the natural homomorphism from Hom_{A}(Z′(|V |), W) onto H( Hom_{A}(|V |, W)) defined in Section 8.7. We have seen that α′ ∘ η = ξ on Hom_{A}(Z′(|V |), W), and that η maps Hom_{A}^{gr}(Z′(V), W) onto H(Hom_{A}^{gr}(V, W)), which implies (8.10.19). It follows that α′(H(Hom_{A}^{gr}(V, W))) = Hom_{A}^{gr}(H(V), W) (8.10.20) if and only if ξ(Hom_{A}^{gr}(Z′(V), W)) = Hom_{A}^{gr}(H(V), W). (8.10.21)

One can check that (8.10.21) holds when W is injective as a module over A. Thus (8.10.20) holds in this case.

Suppose now that Z′(V) corresponds to the direct sum of H(V) and another homogeneous submodule of Z′(V), as a module over A. This means that Z′(V)^{−j} corresponds to the direct sum of H(V)^{−j} and another submodule of Z′(V)^{−j}, as a module over A, for each j ∈ Z. It is easy to see that (8.10.21) holds under these conditions. This implies that (8.10.20) holds, as before. This could also be obtained as in (8.8.31).

The properties of α′ discussed in this section are related to Propositions 7.2 and 7.4 on p68, 70 of [50], and some remarks on p70 of [50].

8.11 Easy complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a graded left or right module over A, with

V^{j} = {0} when j ≠ 0, 1. (8.11.1)

Suppose that d_{V} is a homomorphism from V into itself, as a module over A, with degree 1, so that d_{V}^{j} = 0 when j ≠ 0. (8.11.2)

Under these conditions, d_{V} ∘d_{V} = 0 automatically, so that (V, d_{V}) is a complex. The underlying module with differentiation is the same as in Section 5.5.

In this case,

(8.11.3)  Z(V)^{j} = ker d_{V}^{0} when j = 0
= V^{1}  when j = 1
= {0}  otherwise.

Similarly,

(8.11.4)  B(V)^{j} = d_{V}^{0} (V^{0}) when j = 1
= {0}  otherwise.

Thus

(8.11.5)  H(V)^{j} = ker d_{V}^{0}  when j = 0
= V^{1}/d_{V}^{0} (V^{0}) when j = 1
= {0}  otherwise.

We also have that

(8.11.6)  Z′(V)^{j} = V^{0}  when j = 0
= V^{1}/d_{V}^{0} (V^{0}) when j = 1
= {0}  otherwise.

Similarly,

(8.11.7)  B′(V)^{j} = V^{0}/ ker d_{V}^{0} when j = 0
= {0}  otherwise.

Using this, the remarks in Section 3.13 may be considered as a particular case of those in Section 5.12. This corresponds to the remark after Proposition 2.10 on p23 of [8].

Lemmas 3.2 and 3.3 on p40 of [50] deal with many of the same properties as in Section 3.13, under broader conditions appropriate to the particular parts. Some of the statements in Section 5.3 could be obtained using this, as on p55 of [50].

8.12 Sums, products, and differentiation

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let {V_{α}}_{α∈I} be a nonempty family of all left or all right modules over A. Thus the direct sum ⨁_{α∈I} V_{α} and direct product ∏_{α∈I} V_{α} may be defined as left or right modules over A, as appropriate.

Suppose that (V_{α}, d_{V_{α}}) is a module with differentiation for each α ∈ I. We can define differentiation operators on ⨁_{α∈I} and ∏_{α∈I} V_{α} in the obvious way, using d_{V_{α}} in each coordinate. It is easy to see that ⨁_{α∈I} V_{α} and ∏_{α∈I} V_{α} become modules with differentiation in this way. More precisely, ⨁_{α∈I} V_{α} is a submodule of ∏_{α∈I} V_{α}, as a module with differentiation.

Observe that

Z( ⨁_{α∈I} V_{α}) = ⨁_{α∈I} Z(V_{α}) (8.12.1)

and

Z( ∏_{α∈I} V_{α}) = ∏_{α∈I} Z(V_{α}) (8.12.2)

under these conditions. Similarly,

B( ⨁_{α∈I} V_{α}) = ⨁_{α∈I} B(V_{α}) (8.12.3)

and

B( ∏_{α∈I} V_{α}) = ∏_{α∈I} B(V_{α}). (8.12.4)

We also have that

H( ⨁_{α∈I} V_{α}) = ⨁_{α∈I} H(V_{α}) (8.12.5)

and

H( ∏_{α∈I} V_{α}) = ∏_{α∈I} H(V_{α}). (8.12.6)

Similarly, we get that

Z′( ⨁_{α∈I} V_{α}) = ⨁_{α∈I} Z′(V_{α}) (8.12.7)

and

Z′( ∏_{α∈I} V_{α}) = ∏_{α∈I} Z′(V_{α}), (8.12.8)

as well as

B′( ⨁_{α∈I} V_{α}) = ⨁_{α∈I} B′(V_{α}) (8.12.9)

and

B′( ∏_{α∈I} V_{α}) = ∏_{α∈I} B′(V_{α}). (8.12.10)

This corresponds to Proposition 9.3 on p98 of [50].

Suppose now that V_{α} is a graded module over A for each α ∈ I. In this case, we can identify ⨁_{α∈I} V_{α} with

⨁_{j∈Z} ( ⨁_{α∈I} V_{α}^{j}) (8.12.11)

in a straightforward way. This makes ⨁_{α∈I} V_{α} a graded module too, with

( ⨁_{α∈I} V_{α})^{j} = ⨁_{α∈I} V_{α}^{j} (8.12.12)

for each j ∈ Z.

Put

V^{j} = ∏_{α∈I} V_{α}^{j} (8.12.13)

for each j ∈ Z, which is a left or right module over A, as appropriate. Thus

V = ⨁_{j∈Z} V^{j} (8.12.14)

is a graded module over A. Of course, (8.12.12) is a submodule of (8.12.13), as a module over A, for each j ∈ Z. This means that (8.12.11) may be considered as a homogeneous submodule of (8.12.14).

Note that (8.12.14) may be considered as a submodule of

∏_{j∈Z} V^{j}, (8.12.15)

as a module over A. As before, we can identify

∏_{α∈I} ( ∏_{j∈Z} V_{α}^{j}) (8.12.16)

with (8.12.15), as modules over A, in a straightforward way. Of course,

∏_{α∈I} V_{α} = ∏_{α∈I} ( ⨁_{j∈Z} V_{α}^{j}) (8.12.17)

may be considered as a submodule of (8.12.16), as a module over A. It is easy to see that (8.12.14) corresponds to a submodule of (8.12.17), with respect to the identification between (8.12.15) and (8.12.16) just mentioned. This is compatible with considering ⨁_{α∈I} V_{α} as a submodule of (8.12.16), as a module over A, by considering V_{α} as a submodule of ∏_{j∈I} V_{α}^{j} for each α ∈ I.

In the next section, we shall consider the case where (V_{α}, d_{V_{α}}) is a complex for each α ∈ I.

8.13 Sums, products, and complexes

Let k be a commutative ring with a multiplicative identity element again, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Let (U, d_{U}) be a graded left or right module over A with differentiation that this a complex. Put

W = ∏_{j∈Z} U^{j}, (8.13.1)

so that W is a left or right module over A, as appropriate, and U corresponds to a submodule of W, as a module over A.

If w ∈ W, then define d_{W} (w) ∈ W by saying that for each j ∈ Z, the jth coordinate of d_{W} (w) is equal to d_{U}^{j−1} of the (j − 1)th coordinate of w. More precisely, the (j − 1)th coordinate of w is in U^{j−1}, so that d_{U}^{j−1} sends it into U^{j}. One can check that this makes (W, d_{W}) a module over A with differentiation. We also have that U corresponds to a submodule of W, as a module with differentiation.

Suppose now that {(V_{α}, d_{V_{α}})}_{α∈I} is a nonempty family of all left or all right graded modules over A with differentiation that are complexes. If α ∈ I, then put

W_{α} = ∏_{j∈Z} V_{α}^{j}, (8.13.2)

which is a left or right module over A. We can define d_{W_{α}} on W_{α} as in the preceding paragraph, so that (W, d_{W_{α}}) is a module over A with differentiation. Note that V_{α} corresponds to a submodule of W_{α}, as a module with differentiation, as before.

Using this, we can define a differentation operator on

∏_{α∈I} W_{α}, (8.13.3)

to get a module over A with differentiation, as in the previous section. Of course, (8.13.3) is the same as (8.12.16), as a module over A. One can verify that (8.12.17) is a submodule of (8.13.3), as a module with differentiation. This corresponds to defining a differentiation operator on ∏_{α∈I} V_{α}, as discussed near the beginning of the previous section.

Similarly, one can check that the module V defined in (8.12.14) corresponds to a submodule of (8.13.3), as a module over A with differentiation, using the identification between (8.12.15) and (8.12.16) mentioned earlier. More precisely, the restriction of the differentiation operator to (8.12.14) has degree 1 with respect to the natural grading on V. This makes V a complex, as a graded module over A with differentiation.

It is easy to see that ⨁_{α∈I} V_{α} is a submodule of (8.13.3), as a module with differentiation. As before, the restriction of the differentiation operator to ⨁_{α∈I} V_{α} has degree 1 with respect to the grading as in (8.12.12), so that ⨁_{α∈I} V_{α} becomes a complex.

If j ∈ Z, then

Z( ⨁_{α∈I} V_{α})^{j} = ⨁_{α∈I} Z(V_{α})^{j} (8.13.4)

and

Z(V)^{j} = ∏_{α∈I} Z(V_{α})^{j}. (8.13.5)

Similarly,

B( ⨁_{α∈I} V_{α})^{j} = ⨁_{α∈I} B(V_{α})^{j} (8.13.6)

and

B(V)^{j} = ∏_{α∈I} B(V_{α})^{j}. (8.13.7)

Using this, we get that

H( ⨁_{α∈I} V_{α})^{j} = ⨁_{α∈I} H(V_{α})^{j} (8.13.8)

and

H(V)^{j} = ∏_{α∈I} H(V_{α})^{j}. (8.13.9)

We also get that

Z′( ⨁_{α∈I} V_{α})^{j} = ⨁_{α∈I} Z′(V_{α})^{j} (8.13.10)

and

Z′(V)^{j} = ∏_{α∈I} Z′(V_{α})^{j}, (8.13.11)

and similarly

B′( ⨁_{α∈I} V_{α})^{j} = ⨁_{α∈I} B′(V_{α})^{j} (8.13.12)

and

B′(V)^{j} = ∏_{α∈I} B′(V_{α})^{j}. (8.13.13)

This corresponds to Proposition 9.3 on p98 of [50] again.

8.14 Direct limits and differentiation

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (I, ⪯) be a nonempty pre-directed set, let (V_{j}, d_{j}) be a module over A with differentiation for each j ∈ I. More precisely, we suppose that either V_{j} is a left module over A with differentiation for every j ∈ I, or that V_{j} is a right module over A with differentiation for every j ∈ I.

Suppose that for every j, l ∈ I with j ⪯ l, we have a homomorphism ν_{j,l} from V_{j} into V_{l}, as modules over A with differentiation. As usual, we ask that ν_{j,j} be the identity mapping on V_{j} for every j ∈ I. If j, l, r ∈ I and j ⪯ l ⪯ r, then we also ask that ν_{l,r} ∘ ν_{j,l} = ν_{j,r}, (8.14.1)

as before. Under these conditions, we get a direct or inductive system of modules with differentiation over (I, ⪯).

Let lim_{⟶} V_{j} be the direct limit of the V_{j}’s, as a direct system of modules over A, as in Section 3.2. This is a left or right module over A, as appropriate. If l ∈ I, then we also have a homomorphism ν_{l} from V_{l} into lim_{⟶} V_{j}, as modules over A, as before. Remember that

ν_{l} = ν_{r} ∘ ν_{l,r} (8.14.2)

for every l, r ∈ I with l ⪯ r.

If j, l ∈ I and j ⪯ l, then

d_{l} ∘ ν_{j,l} = ν_{j,l} ∘ d_{j}, (8.14.3)

because ν_{j,l} is supposed to be a homomorphism from V_{j} into V_{l}, as modules with differentiation. Thus the family of d_{j}’s, j ∈ I, defines a homomorphism from the direct systems of V_{j}’s, j ∈ I, into itself, as in Section 3.4. This implies that there is unique homomorphism d = lim_{⟶} d_{j} from lim_{⟶} V_{j} into itself, as a module over A, such that d ∘ ν_{l} = ν_{l} ∘ d_{l} (8.14.4)

for every l ∈ I, as before.

Of course, d_{j} ∘ d_{j} = 0 on V_{j} for each j ∈ I, because (V_{j}, d_{j}) is a module with differentiation. This implies that

d ∘ d = 0 on lim_{⟶} V_{j}, (8.14.5)

as in Section 3.4. This means that lim_{⟶} V_{j} is a module with differentiation with respect to d.

If j, l ∈ I and j ⪯ l, then

ν_{j,l}(Z(V_{j})) ⊆ Z(V_{l}), (8.14.6)

because of (8.14.3). Let ν_{j,l}^{Z} be the restriction of ν_{j,l} to Z(V_{j}), considered as a mapping into Z(V_{l}). Of course, ν_{j,j}^{Z} is the identity mapping on Z(V_{j}) for each j ∈ I. If j, l, r ∈ I and j ⪯ l ⪯ r, then

ν_{l,r}^{Z} ∘ ν_{j,l}^{Z} = ν_{j,r}^{Z}, (8.14.7)

by (8.14.1).

Thus the family of modules Z(V_{j}) and homomorphisms ν_{j,l}^{Z} forms a direct system over (I, ⪯), so that the direct limit lim_{⟶} Z(V_{j}) can be defined as a module

over A in the usual way. If l ∈ I, then we get a homomorphism ν_{l}^{Z} from Z(V_{l}) into lim_{⟶} Z(V_{j}), as modules over A, as in Section 3.2. If r ∈ I and l ⪯ r, then

ν_{l}^{Z} = ν_{r}^{Z} ∘ ν_{l,r}^{Z}, (8.14.8)

as before.

The family of obvious inclusion mappings from Z(V_{j}) into V_{j}, j ∈ I, defines a homomorphism from the direct system of Z(V_{j})’s into the direct system of V_{j}’s, j ∈ I, as in Section 3.4. This leads to a natural homomorphism

(8.14.9) from lim_{⟶} Z(V_{j}) into lim_{⟶} V_{j},

as modules over A, as before. This homomorphism is characterized by the fact that for each l ∈ I, the composition of ν_{l}^{Z} with the homomorphism as in (8.14.9) is the same as the composition of the natural inclusion mapping from Z(V_{l}) into V_{l} with ν_{l}.

If j ∈ I, then the composition of the obvious inclusion mapping from Z(V_{j}) into V_{j} with d_{j} is equal to 0, by construction. It follows that

the composition of the homomorphism as in (8.14.9)

(8.14.10)

with d is equal to 0,

as in Section 3.4. This means that

the homomorphism as in (8.14.9)

(8.14.11)

maps lim_{⟶} Z(V_{j}) into Z( lim_{⟶} V_{j}).

One can check that

(8.14.12) the homomorphism as in (8.14.9) is injective, using the remarks in Section 3.4, or more directly. Similarly, one can verify that

the homomorphism as in (8.14.9)

(8.14.13)

maps lim_{⟶} Z(V_{j}) onto Z( lim_{⟶} V_{j}),

using the remarks in Section 3.4. This corresponds to part of Proposition 9.3* on p100 of [50].

Similarly, if j, l ∈ I and j ⪯ l, then

ν_{j,l}(B(V_{j})) ⊆ B(V_{l}), (8.14.14)

by (8.14.3). Let ν_{j,l}^{B} be the restriction of ν_{j,l} to B(V_{j}), considered as a mapping into B(V_{l}). Thus ν_{j,j}^{B} is the identity mapping on B(V_{j}) for each j ∈ I, and

ν_{l,r}^{B} ∘ ν_{j,l}^{B} = ν_{j,r}^{B} (8.14.15)

when j, l, r ∈ I satisfy j ⪯ l ⪯ r, by (8.14.1). This means that the family of modules B(V_{j}) and homomorphisms ν_{j,l}^{B} forms a direct system over (I, ⪯), so that the direct limit lim_{⟶} B(V_{j}) can be defined as a module over A. As before,

we get a homomorphism ν_{l}^{B} from B(V_{l}) into lim_{⟶} B(V_{j}), as modules over A, for each l ∈ I, with ν_{l}^{B} = ν_{r}^{B} ∘ ν_{l,r}^{B} (8.14.16)

when r ∈ I and l ⪯ r.

The family of obvious inclusion mappings from B(V_{j}) into Z(V_{j}), j ∈ I, defines a homomorphism from the direct system of B(V_{j})’s into the direct system of Z(V_{j})’s, j ∈ I, as in Section 3.4. This leads to a natural homomorphism

(8.14.17) from lim_{⟶} B(V_{j}) into lim_{⟶} Z(V_{j}),

as modules over A, which is characterized by the property that for each l ∈ I, the composition of ν_{l}^{B} with this homomorphism is the same as the composition of the natural inclusion mapping from B(V_{l}) into Z(V_{l}) with ν_{l}^{Z}. One could also use the family of obvious inclusion mappings from B(V_{j}) into V_{j}, j ∈ I, to get a homomorphism from the direct system of B(V_{j})’s into the direct system of V_{j}’s, j ∈ I, and thus a homomorphism

(8.14.18) from lim_{⟶} B(V_{j}) into lim_{⟶} V_{j},

as modules over A. This homomorphism is the same as the composition of the homomorphisms as in (8.14.17) and (8.14.9), as in Section 3.4.

If j ∈ I, then let d_{j}^{B} be d_{j}, considered as a mapping from V_{j} onto B(V_{j}). Of course, d_{l}^{B} ∘ ν_{j,l}^{B} = ν_{j,l}^{B} ∘ d_{j}^{B} (8.14.19)

when l ∈ I satisfies j ⪯ l, as in (8.14.3). This means that the family of d_{j}^{B}’s, j ∈ I, defines a homomorphism from the direct system of V_{j}’s into the direct system 8.14. DIRECT LIMITS AND DIFFERENTIATION 241 of B(V_{j})’s, j ∈ I, as in Section 3.4. Thus there is a unique homomorphism d^{B} = lim_{⟶} d_{j}^{B} from lim_{⟶} V_{j} into lim_{⟶} B(V_{j}), as modules over A, such that d^{B} ∘ ν_{l}^{B} = ν_{l}^{B} ∘ d_{l}^{B} (8.14.20) for every l ∈ I. More precisely,

d^{B}( lim_{⟶} V_{j}) = lim_{⟶} B(V_{j}), (8.14.21)

because d_{j}^{B}(V_{j}) = B(V_{j}) for every j ∈ I, as in Section 3.4.

The composition of d^{B} with the homomorphism as in (8.14.18) is the same as d, because d_{j} is the same as the composition of d_{j}^{B} with the obvious inclusion mapping from B(V_{j}) into V_{j} for each j ∈ I. It follows that (8.14.22) B( lim_{⟶} V_{j}) is the same as the image of lim_{⟶} B(V_{j})

under the homomorphism as in (8.14.18).

Note that (8.14.23) the homomorphism as in (8.14.17) is injective, as in Section 3.4. Similarly, (8.14.24) the homomorphism as in (8.14.18) is injective, which could also be obtained from (8.14.12) and (8.14.23).

If j, l ∈ I and j ⪯ l, then ν_{j,l} induces a homomorphism (8.14.25) ν_{j,l}^{Z′} from Z′(V_{j}) into Z′(V_{l}), as modules over A, as in Section 5.2. Clearly ν_{j,j}^{Z′} is the identity mapping on Z′(V_{j}) = V_{j}/d_{j}(V_{j}) for each j ∈ I, and ν_{l,r}^{Z′} ∘ ν_{j,l}^{Z′} = ν_{j,r}^{Z′} (8.14.26) for every j, l, r ∈ I with j ⪯ l ⪯ r, because of (8.14.1). Thus the family of modules Z′(V_{j}) and homomorphisms ν_{j,l}^{Z′} forms a direct system over (I, ⪯), so that the direct limit lim_{⟶} Z′(V_{j}) can be defined as a module over A. We also get a homomorphism ν_{l}^{Z′} from Z′(V_{l}) into lim_{⟶} Z′(V_{j}), as modules over A, for each l ∈ I, with ν_{l}^{Z′} = ν_{r}^{Z′} ∘ ν_{l,r}^{Z′} (8.14.27) when r ∈ I satisfies l ⪯ r, as usual.

Let q_{j}^{Z′} be the natural quotient mapping from V_{j} onto Z′(V_{j}) for each j ∈ I. If j, l ∈ I and j ⪯ l, then q_{l}^{Z′} ∘ ν_{j,l} = ν_{j,l}^{Z′} ∘ q_{j}^{Z′}, (8.14.28) by construction. This means that the family of quotient mappings q_{j}^{Z′}, j ∈ I, defines a homomorphism from the direct system of V_{j}’s into the direct system of Z′(V_{j})’s, j ∈ I, as in Section 3.4. This leads to a natural homomorphism

(8.14.29) q^{Z′} = lim_{⟶} q_{j}^{Z′} from lim_{⟶} V_{j} into lim_{⟶} Z′(V_{j}),

as modules over A, which is characterized by the property that

q^{Z′} ∘ ν_{l} = ν_{l}^{Z′} ∘ q_{l}^{Z′} (8.14.30)

for every l ∈ I. More precisely,

q^{Z′}( lim_{⟶} V_{j}) = lim_{⟶} Z′(V_{j}), (8.14.31)

because q_{j}^{Z′}(V_{j}) = Z′(V_{j}) for every j ∈ I, as in Section 3.4.

Of course, ker q_{j}^{Z′} = B(V_{j}) for each j ∈ I, by construction. This implies that the kernel of q^{Z′} is the same as the image of lim_{⟶} B(V_{j}) under the homomorphism as in (8.14.18), as in Section 3.4. It follows that

ker q^{Z′} = B( lim_{⟶} V_{j}), (8.14.32)

because of (8.14.22). This means that

(8.14.33) Z′( lim_{⟶} V_{j}) is isomorphic to lim_{⟶} Z′(V_{j}),

as modules over A, in such a way that q^{Z′} corresponds to the natural quotient mapping from lim_{⟶} V_{j} onto Z′( lim_{⟶} V_{j}). This corresponds to another part of Proposition 9.3* on p100 of [50].

If j, l ∈ I and j ⪯ l, then ν_{j,l} induces a homomorphism

(8.14.34) ν_{j,l}^{H} from H(V_{j}) into H(V_{l}),

as modules over A, as in Section 5.2. This is the same as the restriction of ν_{j,l}^{Z′} to H(V_{j}) = Z(V_{j})/d_{j}(V_{j}), considered as a mapping into H(V_{l}). As before, ν_{j,j}^{H} is the identity mapping on H(V_{j}) for each j ∈ I, and

ν_{l,r}^{H} ∘ ν_{j,l}^{H} = ν_{j,r}^{H} (8.14.35)

for every j, l, r ∈ I with j ⪯ l ⪯ r. This means that the family of modules H(V_{j}) and homomorphisms ν_{j,l}^{H} forms a direct system over (I, ⪯), so that the direct limit lim_{⟶} H(V_{j}) can be defined as a module over A in the usual way. If

l ∈ I, then we get a homomorphism ν_{l}^{H} from H(V_{l}) into lim_{⟶} H(V_{j}), as modules over A, with ν_{l}^{H} = ν_{r}^{H} ∘ ν_{l,r}^{H} (8.14.36)

for every r ∈ I with l ⪯ r, as before.

8.14. DIRECT LIMITS AND DIFFERENTIATION 243

Let q_{j}^{H} be the natural quotient mapping from Z(V_{j}) onto H(V_{j}) for each j ∈ I, which is the same as the restriction of q_{j}^{Z′} to Z(V_{j}), considered as a mapping into H(V_{j}). If j, l ∈ I and j ⪯ l, then q_{l}^{H} ∘ ν_{j,l}^{Z} = ν_{j,l}^{H} ∘ q_{j}^{H}, (8.14.37) as mappings from Z(V_{j}) into H(V_{l}). Thus the family of quotient mappings q_{j}^{H}, j ∈ I, defines a homomorphism from the direct system of Z(V_{j})’s into the direct system of H(V_{j})’s, j ∈ I, as in Section 3.4. This leads to a natural homomorphism (8.14.38) q^{H} = lim_{⟶} q_{j}^{H} from lim_{⟶} Z(V_{j}) into lim_{⟶} H(V_{j}), as modules over A, which is characterized by the property that q^{H} ∘ ν_{l}^{Z} = ν_{l}^{H} ∘ q_{l}^{H} (8.14.39) for every l ∈ I. In fact,

q^{H}( lim_{⟶} Z(V_{j})) = lim_{⟶} H(V_{j}), (8.14.40)

because q_{j}^{H}(Z(V_{j})) = H(V_{j}) for every j ∈ I, as in Section 3.4.

Observe that (8.14.41) ker q^{H} is the same as the image of lim_{⟶} B(V_{j})

under the homomorphism as in (8.14.17),

because ker q_{j}^{H} = B(V_{j}) for each j ∈ I, as in Section 3.4. Remember that the homomorphism as in (8.14.9) defines an isomorphism from lim_{⟶} Z(V_{j}) onto Z( lim_{⟶} V_{j}), as modules over A, as in (8.14.12) and (8.14.13). We have also seen that the composition of the homomorphisms as in (8.14.17) and (8.14.9) is the same as the homomorphism as in (8.14.18). It follows that (8.14.42) the image of ker q^{H} under the homomorphism as in (8.14.9)

is equal to B( lim_{⟶} V_{j}),

because of (8.14.22).

Thus q^{H} corresponds to a homomorphism (8.14.43) from Z( lim_{⟶} V_{j}) onto lim_{⟶} H(V_{j}), as modules over A, with kernel equal to B( lim_{⟶} V_{j}). This shows that (8.14.44) H( lim_{⟶} V_{j}) is isomorphic to lim_{⟶} H(V_{j}), as modules over A, where the natural quotient mapping from Z( lim_{⟶} V_{j}) onto

H( lim_{⟶} V_{j}) corresponds to the homomorphism as in (8.14.43). This corresponds to part of Proposition 9.3* on p100 of [50] again.

Let us mention the analogous statements for B′(V_{j}) = V_{j}/ ker d_{j}, for the sake of completeness. If j, l ∈ I and j ⪯ l, then ν_{j,l} induces a homomorphism

(8.14.45) ν_{j,l}^{B′} from B′(V_{j}) into B′(V_{l}),

as modules over A, as in Section 5.2. This is the identity mapping on B′(V_{j}) when j = l, and ν_{l,r}^{B′} ∘ ν_{j,l}^{B′} = ν_{j,r}^{B′} (8.14.46) for every j, l, r ∈ I with j ⪯ l ⪯ r, by (8.14.1). Thus the family of modules B′(V_{j}) and homomorphisms ν_{j,l}^{B′} forms a direct system over (I, ⪯), so that the direct limit lim_{⟶} B′(V_{j}) can be defined as a module over A. If l ∈ I, then we also

get a homomorphism ν_{l}^{B′} from B′(V_{l}) into lim_{⟶} B′(V_{j}), as modules over A, with

ν_{l}^{B′} = ν_{r}^{B′} ∘ ν_{l,r}^{B′} (8.14.47)

for every r ∈ I with l ⪯ r.

Let q_{j}^{B′} be the natural quotient mapping from V_{j} onto B′(V_{j}) for each j ∈ I. If j, l ∈ I and j ⪯ l, then

q_{l}^{B′} ∘ ν_{j,l} = ν_{j,l}^{B′} ∘ q_{j}^{B′}, (8.14.48)

as mappings from V_{j} into B′(V_{l}). It follows that the family of quotient mappings q_{j}^{B′} defines a homomorphism from the direct system of V_{j}’s into the direct system of B′(V_{j})’s, j ∈ I, as in Section 3.4. This leads to a natural homomorphism

(8.14.49) q^{B′} = lim_{⟶} q_{j}^{B′} from lim_{⟶} V_{j} into lim_{⟶} B′(V_{j}),

as modules over A, which is characterized by the property that

q^{B′} ∘ ν_{l} = ν_{l}^{B′} ∘ q_{l}^{B′} (8.14.50)

for every l ∈ I. We also have that

q^{B′}( lim_{⟶} V_{j}) = lim_{⟶} B′(V_{j}), (8.14.51)

because q_{j}^{B′}(V_{j}) = B′(V_{j}) for every j ∈ I, as in Section 3.4.

The kernel of q^{B′} is the same as the image of lim_{⟶} Z(V_{j}) under the homomor-

phism as in (8.14.9), because ker q_{j}^{B′} = Z(V_{j}) for each j ∈ I, as in Section 3.4. This means that

ker q^{B′} = Z( lim_{⟶} V_{j}), (8.14.52)

because of (8.14.13). It follows that

(8.14.53) B′( lim_{⟶} V_{j}) is isomorphic to lim_{⟶} B′(V_{j}),

as modules over A, in such a way that q^{B′} corresponds to the natural quotient mapping from lim_{⟶} V_{j} onto B′( lim_{⟶} V_{j}).

8.15 Direct sums and limits

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (I, ⪯) be a nonempty pre-directed set, and let B be a nonempty set. Suppose that V_{j}^{β} is a module over A for each j ∈ I and β ∈ B. More precisely, these should be all left modules over A, or all right modules over A.

Suppose that for every β ∈ B and j, l ∈ I with j ⪯ l we have a homomorphism ν_{j,l}^{β} from V_{j}^{β} into V_{l}^{β}, as modules over A, with the usual two properties as in Section 3.2. Namely, ν_{j,j}^{β} should be the identity mapping on V_{j}^{β} for every j ∈ I and β ∈ B, and we should have that

ν_{l,r}^{β} ∘ ν_{j,l}^{β} = ν_{j,r}^{β} (8.15.1)

for every β ∈ B and j, l, r ∈ I with j ⪯ l ⪯ r. Let

V^{β} = lim_{⟶} V_{j}^{β} (8.15.2)

be the corresponding direct limit of the V_{j}^{β}’s, j ∈ I, for each β ∈ B, which is a module over A. If β ∈ B and l ∈ I, then we get a homomorphism ν_{l}^{β} from V_{l}^{β} into V^{β}, as modules over A, as before. Remember that this homomorphism satisfies ν_{l}^{β} = ν_{r}^{β} ∘ ν_{l,r}^{β} (8.15.3)

when r ∈ I and l ⪯ r.

If j ∈ I, then put

V_{j} = ⨁_{β∈B} V_{j}^{β}, (8.15.4)

which is a module over A. If j, l ∈ I and j ⪯ l, then let ν_{j,l} be the homomorphism from V_{j} into V_{l}, as modules over A, defined using ν_{j,l}^{β} on V_{j}^{β} for each β ∈ B. Thus ν_{j,j} is the identity mapping on V_{j} for each j ∈ I, and ν_{l,r} ∘ ν_{j,l} = ν_{j,r} for every j, l, r ∈ I with j ⪯ l ⪯ r, because of the analogous properties of the ν_{j,l}^{β}’.

Let

V = lim_{⟶} V_{j} (8.15.5)

be the direct limit of the V_{j}’s, j ∈ I, which is a module over A. If l ∈ I, then we get a homomorphism ν_{l} from V_{l} into V, as modules over A, with ν_{l} = ν_{r} ∘ ν_{l,r} when r ∈ I and l ⪯ r, as usual.

If α ∈ B and j ∈ I, then let i_{j}^{α} be the obvious mapping from V_{j}^{α} into V_{j}, which sends v_{j}^{α} ∈ V_{j}^{α} to the element of V_{j} whose component in V_{j}^{α} is equal to v_{j}^{α}, and whose component in V_{j}^{β} is equal to 0 when α ≠ β. If l ∈ I and j ⪯ l, then it is easy to see that i_{l}^{α} ∘ ν_{j,l}^{α} = ν_{j,l} ∘ i_{j}^{α}. (8.15.6)

Thus the family of i_{j}^{α}’s, j ∈ I, defines a homomorphism from the direct system of V_{j}^{α}’s into the direct system of V_{j}’s, j ∈ I, as in Section 3.4. This means that there is a unique homomorphism i^{α} = lim_{⟶} i_{j}^{α} from V^{α} into V, as modules over A, such that i^{α} ∘ ν_{l}^{α} = ν_{l} ∘ i_{l}^{α} (8.15.7)

for every l ∈ I, as before.

Similarly, if α ∈ B and j ∈ I, then let π_{j}^{α} be the usual coordinate projection from V_{j} onto V_{j}^{α}. If l ∈ I and j ⪯ l, then

π_{l}^{α} ∘ ν_{j,l} = ν_{j,l}^{α} ∘ π_{j}^{α}, (8.15.8)

by construction. This means that the family of π_{j}^{α}’s, j ∈ I, defines a homomorphism from the direct system of V_{j}’s into the direct system of V_{j}^{α}’s, j ∈ I, as in Section 3.4 again. It follows that there is a unique homomorphism π^{α} = lim_{⟶} π_{j}^{α} from V into V^{α}, as modules over A, such that

π^{α} ∘ ν_{l} = ν_{l}^{α} ∘ π_{l}^{α} (8.15.9)

for every l ∈ I.

Of course, (8.15.10) π_{j}^{α} ∘ i_{j}^{α} is the identity mapping on V_{j}^{α}

for every α ∈ B and j ∈ I. If α, γ ∈ B and α ≠ γ, then

π_{j}^{γ} ∘ i_{j}^{α} = 0 (8.15.11)

for every j ∈ I. This implies that

(8.15.12) π^{α} ∘ i^{α} is the identity mapping on V^{α}

for every α ∈ B, and that π^{γ} ∘ i^{α} = 0 (8.15.13)

for every α, γ ∈ B with α ≠ γ, as in Section 3.4.

Every v ∈ V can be expressed as v = ν_{l}(v_{l}) for some l ∈ I and v_{l} ∈ V_{l}, as in Section 3.2. In this case,

π^{α}(v) = π^{α}(ν_{l}(v_{l})) = ν_{l}^{α}(π_{l}^{α}(v_{l})) (8.15.14)

for every α ∈ B. In particular,

(8.15.15) π^{α}(v) = 0 for all but finitely many α ∈ B,

because π_{l}^{α}(v_{l}) = 0 for all but finitely many α ∈ B.

If ν_{l}^{α}(π_{l}^{α}(v_{l})) = 0 for some α ∈ B, then there is an r(α) ∈ I such that l ⪯ r(α) and ν_{l,r(α)}^{α}(π_{l}^{α}(v_{l})) = 0, (8.15.16)

as in Section 3.2. If ν_{l}^{α}(π_{l}^{α}(v_{l})) = 0 for every α ∈ B, then it follows that there is an r ∈ I such that l ⪯ r and

ν_{l,r}^{α}(π_{l}^{α}(v_{l})) = 0 (8.15.17) for every α ∈ B, because (I, ⪯) is a pre-directed system, and this holds automatically when π_{l}^{α}(v_{l}) = 0. Equivalently, this means that

π_{r}^{α}(ν_{l,r}(v_{l})) = 0 (8.15.18)

for every α ∈ B, so that ν_{l,r}(v_{l}) = 0. This implies that

v = ν_{l}(v_{l}) = ν_{r}(ν_{l,r}(v_{l})) = 0. (8.15.19)

Using the i^{α}’s, we get a homomorphism

(8.15.20) from ⨁_{β∈B} V^{β} into V,

as modules over A. Similarly, we get a homomorphism

(8.15.21) from V into ⨁_{β∈B} V^{β},

as modules over A, using the π^{α}’s and (8.15.15). The composiiton of these two homomorphisms is the identity mapping on ⨁_{β∈B} V^{β}, because of (8.15.12) and (8.15.13), as in Section 1.15. We also have that the homomorphism as in (8.15.21) is injective, by the remarks in the previous two paragraphs. One can use this to get that the homomorphism as in (8.15.20) is surjective, so that these two homomorphisms are inverses of each other.

Alternatively, one can check that ⨁_{β∈B} V^{β} satisfies conditions that characterize the direct limit of the V_{j}’s, j ∈ I, up to isomorphism, as in Section 3.2. More precisely, for each l ∈ I, we can get a homomorphism ν̃_{l} from V_{l} into ⨁_{β∈B} V^{β}, as modules over A, using the homomorphisms ν_{l}^{β} from V_{l}^{β} into V^{β} for each β ∈ B. It is easy to see that

ν̃_{l} = ν̃_{r} ∘ ν_{l,r} (8.15.22)

for every l, r ∈ I with l ⪯ r, using (8.15.3). The ν̃_{l}’s are the analogues of the ν_{l}’s for ⨁_{β∈B} V^{β}, as a version of the direct limit of the V_{l}’s. We would like to verify that ⨁_{β∈B} V^{β} has the same property as the direct limit, in terms of homomorphisms from the V_{l}’s into other modules over A that satisfy suitable compatibility conditions.

Let Z be another left or right module over A, as appropriate, and let ζ_{l} be a homomorphism from V_{l} into Z, as modules over A, for each l ∈ I. Suppose that ζ_{l} = ζ_{r} ∘ ν_{l,r} (8.15.23)

for every l, r ∈ I with l ⪯ r. If l ∈ I, then ζ_{l} determines a homomorphism ζ_{l}^{β} from V_{l}^{β} into Z, as modules over A, for each β ∈ B. We also have that

ζ_{l}^{β} = ζ_{r}^{β} ∘ ν_{l,r}^{β} (8.15.24) for every l, r ∈ I with l ⪯ r and every β ∈ B, because of (8.15.23), and the way that ν_{l,r} was defined. This implies that for each β ∈ B, there is a unique homomorphism ζ^{β} from V^{β} into Z, as modules over A, such that

ζ^{β} ∘ ν_{l}^{β} = ζ_{l}^{β} (8.15.25)

for every l ∈ I, as in Section 3.2.

Let ζ̃ be the homomorphism from ⨁_{β∈B} V^{β} into Z obtained from the ζ^{β}’s, β ∈ B, in the obvious way. Observe that

ζ̃∘ ν̃_{l} = ζ_{l} (8.15.26)

for every l ∈ I, by construction. It is easy to see that ζ̃ is uniquely determined by this property, because ζ^{β} is uniquely determined by (8.15.25) for each β ∈ B. This implies that ⨁_{β∈B} V^{β} is isomorphic to V, as modules over A, as in Section 3.2. Of course, ν̃_{l} corresponds to ν_{l} under this isomorphism, for each l ∈ I.

Let us now take B = Z, so that V_{j} may be considered as a graded module over A for each j ∈ I. More precisely, ν_{j,l} has degree 0 as a homomorphism from V_{j} into V_{l} for every j, l ∈ I with j ⪯ l, by construction. We may consider V as a graded module over A too, as in the preceding paragraph. If l ∈ I, then ν_{l} has degree 0 as a homomorphism from V_{l} into V, with respect to this grading on V. It is easy to see that this grading on V is uniquely determined by this property.

Suppose that V_{j} is in fact a graded module over A with differentiation that is a complex for each j ∈ I, and that the family of V_{j}’s is a direct system of modules with differentiation over (I, ⪯), as in the previous section. This means that ν_{j,l} is a homomorphism from V_{j} into V_{l} as modules over A with differentiation for each j, l ∈ I with j ⪯ l, as before. The direct limit V may be considered as a module over A with differentiation, as in the previous section. One can check that V is a complex, using the grading defined here. This means that the differentiation operator d on V has degree 1, which can be verified using the fact that the differentiation operator d_{j} on V_{j} has degree 1 for each j ∈ I, by hypothesis.

Part III

More rings, modules, and complexes Some rings and modules

9.1 A reformulation of projectivity

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that a left module V over A has the following property. Let W_{1} and Z_{1} be left modules over A, let ψ_{1} be a homomorphism from W_{1} onto Z_{1}, and let ϕ_{Z_{1}} be a homomorphism from V into Z_{1}, as left modules over A. If W_{1} is injective as a left module over A, then there is a homomorphism ϕ_{W_{1}} from V into W_{1}, as left modules over A, such that ϕ_{Z_{1}} = ψ_{1} ∘ ϕ_{W_{1}}. (9.1.1)

This is the same as the definition of projectivity of V, as in Section 2.7, restricted to the case where W_{1} is injective.

Proposition 5.1 on p12 of [50] states that this condition implies that V is projective. To see this, let W and Z be left modules over A, let ψ be a homomorphism from W onto Z, and let ϕ_{Z} be a homomorphism from V into Z, as left modules over A. Thus

W_{0} = ker ψ (9.1.2)

is a submodule of W, and we may as well take

Z = W/W_{0}, (9.1.3)

and ψ to be the natural quotient mapping from W onto W/W_{0}. We may suppose too that W is a submodule of an injective left module W_{1} over A, as mentioned in Section 2.8. Put Z_{1} = W_{1}/W_{0}, (9.1.4)

and let ψ_{1} be the natural quotient mapping from W_{1} onto Z_{1}.

Of course, Z may be considered as a submodule of Z_{1}, and so we may take

ϕ_{Z_{1}} = ϕ_{Z}, (9.1.5) considered as a homomorphism from V into Z_{1}. By hypothesis, there is a homomorphism ϕ_{W_{1}} from V into W_{1} that satisfies (9.1.1). Note that

ψ_{1}(ϕ_{W_{1}}(V)) = ϕ_{Z_{1}}(V) = ϕ_{Z}(V) ⊆ Z, (9.1.6)

which implies that ϕ_{W_{1}}(V) ⊆ W. (9.1.7)

Put ϕ_{W} = ϕ_{W_{1}}, (9.1.8)

considered as a mapping from V into W. Clearly ϕ_{Z} = ψ ∘ ϕ_{W}, because of (9.1.1), as desired.

There are analogous statements for right modules over A, as usual.

9.2 A reformulation of injectivity

Let k be a commutative ring with a multiplicative identity element again, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that a left module W over A has the following property. Let V_{1} be a left module over A, let V_{1,0} be a submodule of V_{1}, and let ϕ_{1,0} be a homomorphism from V_{1,0} into W, as modules over A. If V_{1} is projective as a left module over A, then we ask that there be a homomorphism ϕ_{1} from V_{1} into W, as left modules over A, such that ϕ_{1} = ϕ_{1,0} on V_{1,0}. (9.2.1)

This is the same as the injectivity of W, as in Section 2.8, with the additional hypotheses that V_{1} be projective.

Proposition 5.2 on p12 of [50] states that this condition implies that W is injective. To show this, let V be a left module over A, let V_{0}, be a submodule of V, and let ϕ_{0} be a homomorphism from V_{0} into W, as left modules over A. Remember that there is a projective left module V_{1} over A and a homomorphism ξ from V_{1} onto V, as left modules over A, as in Section 2.7. More precisely, one can take V_{1} to be a free left module over A. Of course,

V_{1,0} = ξ^{−1}(V_{0}) (9.2.2)

is a submodule of V_{1}.

Let ξ_{0} be the restriction of ξ to V_{1,0}, and put

ϕ_{1,0} = ϕ_{0} ∘ ξ_{0}. (9.2.3)

This is a homomorphism from V_{1,0} into W, as left modules over A, which can be extended to a homomorphism ϕ_{1} from V_{1} into W, as left modules over A, by hypothesis. Observe that the kernel of ξ is contained in V_{1,0}, by construction, so that (9.2.4) ker ξ = ker ξ_{0} ⊆ ker ϕ_{1,0} ⊆ ker ϕ_{1}.

This implies that there is a unique homomorphism ϕ from V into W, as left modules over A, such that ϕ_{1} = ϕ ∘ ξ. (9.2.5)

It is easy to see that ϕ = ϕ_{0} on V_{0}, because of the analogous property of ϕ_{1}. Of course, there are analogous statements for right modules over A.

9.3 Hereditary rings

Let A be a ring, with a nonzero multiplicative identity element e_{A}. We say that A is left hereditary if every left ideal in A is projective, as a left module over A, as on p13 of [50]. The definition of what it means for A to be right hereditary can be defined analogously, as mentioned on p15 of [50].

If x ∈ A, then A x = {a x : a ∈ A} (9.3.1)

is a left ideal in A. Of course, a ↦ a x (9.3.2)

defines a homomorphism from A onto A x, as left modules over A. Suppose for the moment that A has no nontrivial zero divisors, so that the product of any two nonzero elements of A is nonzero as well. If x ≠ 0, then (9.3.2) defines an isomorphism from A onto A x, as left modules over A. In particular, this means that A x is free as a left ideal over A, and thus projective.

If A has no nontrivial zero divisors, and every left ideal in A is of the form A x for some x ∈ A, then it follows that A is left hereditary. This corresponds to a remark at the bottom of p13 of [50].

Let V be a free left module over A, and let V_{0} be a submodule of V. If A is left hereditary, then a theorem of Kaplansky states that

(9.3.3) V_{0} is isomorphic to a direct sum of a family of left modules over A, each of which is isomorphic to a left ideal in A,

as a left module over A. This is Theorem 5.3 on p13 of [50], and we shall return to this in the next section.

Let V be a projective left module over A, so that V may be considered as a submodule of a free left module over A, as in Section 2.7. This means that every submodule of V may be considered as a submodule of a free left module over A. If A is left hereditary, then Kaplansky’s theorem implies that every submodule of V is isomorphic to a direct sum of projective left modules over A. It follows that

(9.3.4) every submodule of V is projective as a left module over A

under these conditions, as in Section 2.7.

Conversely, suppose that

(9.3.5) every submodule of a projective left module over A is projective as a left module over A.

Of course, A is free as a left module over itself, and thus projective. It follows that every left ideal in A is projective as a left module over A, so that A is left hereditary. This corresponds to the equivalence of parts (a) and (b) of Theorem 5.4 on p14 of [50].

Suppose that (9.3.5) holds, and let us show that

(9.3.6) every quotient of an injective left module over A is injective as a left module over A.

Let W be an injective left module over A, and let W_{0} be a submodule of W. To check that W/W_{0} is injective, we let V_{1} be a projective left module over A, V_{1,0} be a submodule of V_{1}, and ϕ_{1,0} be a homomorphism from V_{1,0} into W/W_{0}, as left modules over A. Using (9.3.5), we get that V_{1,0} is projective as a left module over A too. This implies that there is a homomorphism ϕ_{W} from V_{1,0} into W, as left modules over A, such that

(9.3.7) ϕ_{1,0} is the same as the composition of ϕ_{W} with the natural quotient mapping from W onto W/W_{0}.

Because W is injective, ϕ_{W} can be extended to a homomorphism ϕ from V_{1} into W, as left modules over A. The composition of ϕ with the natural quotient mapping from W onto W_{0} is a homomorphism from V into W/W_{0}, as left modules over A, that is equal to ϕ_{1,0} on V_{1,0}. It follows that W/W_{0} is injective as a left module over A, as in the previous section.

Conversely, suppose that (9.3.6) holds, and let us show that (9.3.5) holds. Let V be a projective left module over A, and let V_{0} be a submodule of V. To show that V_{0} is projective, let W_{1} and Z_{1} be left modules over A, with W_{1} injective, let ψ_{1} be a homomorphism from W_{1} onto Z_{1}, and let ϕ_{V_{0},Z_{1}} be a homomorphism from V_{0} into Z_{1}, as left modules over A. It follows that Z_{1} is injective as a left module over A, because of (9.3.6). Thus ϕ_{V_{0},Z_{1}} can be extended to a homomorphism ϕ_{V,Z_{1}} from V into Z_{1}, as left modules over A. Because V is projective, there is a homomorphism ϕ_{V,W_{1}} from V into W_{1}, as left modules over A, such that

ϕ_{V,Z_{1}} = ψ_{1} ∘ ϕ_{V,W_{1}}. (9.3.8)

The restriction of ϕ_{V,W_{1}} to V_{0} defines a homomorphism ϕ_{V_{0},W_{1}} from V_{0} into W_{1}, as left modules over A, such that

ϕ_{V_{0},Z_{1}} = ψ_{1} ∘ ϕ_{V_{0},W_{1}}. (9.3.9)

This implies that V_{0} is projective, as in Section 9.1. This corresponds to the equivalence of parts (b) and (c) in Theorem 5.4 on p14 of [50].

9.4 Kaplansky’s theorem

Let A be a ring with a nonzero multiplicative identity element e_{A} again, and let V be a free left module over A. More precisely, suppose that {x_{α}}_{α∈B} is a basis for V, as a free left module over A, and that B is well ordered by ⪯. If β ∈ B, then we let V_{β} be the submodule of V generated by x_{α}, with α ∈ B, α ⪯ β, and α ≠ β. Similarly, let \overline{V}_{β} be the submodule of V generated by x_{α} with α ∈ B and α ⪯ β.

Let U be a submodule of V. If β ∈ B and v ∈ U ∩ \overline{V}_{β}, then v can be expressed in a unique way as

v = w + a · x_{β}, (9.4.1)

where w ∈ V_{β} and a ∈ A. Note that v ↦ a defines a homomorphism from U ∩ \overline{V}_{β} into A, as left modules over A. Let I_{β} be the image of U ∩ \overline{V}_{β} under this homomorphism, which is a left ideal in A. The kernel of this homomorphism is equal to U ∩ V_{β}. (9.4.2)

Suppose that A is left hereditary, so that I_{β} is projective as a left module over A. This implies that U ∩ \overline{V}_{β} corresponds to the direct sum of U ∩ V_{β} and another submodule C_{β}, as left modules over A, as in Section 2.7. More precisely, C_{β} is isomorphic to I_{β}, as a left module over A. We would like to show that U corresponds to the direct sum of the submodules C_{β}, β ∈ B, as a left module over A.

Let β_{1}, …, β_{n} be finitely many distinct elements of B, with

β_{1} ⪯ β_{2} ⪯ ⋯ ⪯ β_{n}. (9.4.3)

Suppose that c_{j} ∈ C_{β_{j}}, 1 ≤ j ≤ n, satisfy

∑_{j=1}^{n} c_{j} = 0. (9.4.4)

Note that c_{j} ∈ U ∩ \overline{V}_{β_{j}} for each j = 1, …, n, so that c_{j} ∈ U ∩ V_{β_{n}} when j < n. Because U ∩ \overline{V}_{β_{n}} corresponds to the direct sum of U ∩ V_{β_{n}} and C_{β_{n}}, we get that

∑_{j=1}^{n−1} c_{j} = c_{n} = 0. (9.4.5)

One can repeat the process to get that c_{j} = 0 for each j = 1, …, n.

Let ∑_{β∈B} C_{β} be the submodule of U generated by the C_{β}’s, β ∈ B. We would like to verify that

U = ∑_{β∈B} C_{β}. (9.4.6)

Otherwise, there is an α ∈ B such that

U ∩ \overline{V}_{α} ⊈ ∑_{β∈B} C_{β}, (9.4.7)

and we may as well take α to be the minimal element of B with this property. One can check that

U ∩ V_{α} ⊆ ∑_{β∈B} C_{β}, (9.4.8)

using the minimality of α. This implies that

U ∩ \overline{V}_{α} = (U ∩ V_{α}) + C_{α} ⊆ ∑_{β∈B} C_{β}, (9.4.9)

which is a contradiction.

9.5 Semi-hereditary rings

Let A be a ring with a nonzero multiplicative identity element e_{A}. We say that A is left semi-hereditary if every left ideal in A that is finitely generated as a left module over A is projective, as a left module over A, as on p14 of [50]. The right-hereditary condition can be defined analogously, as mentioned on p15 of [50].

Let V be a free left module over A, and let V_{0} be a finitely-generated submodule of V. If A is left semi-hereditary, then

(9.5.1) V_{0} is isomorphic to a direct sum of finitely many left modules over A, each of which is isomorphic to a left ideal in A that is finitely generated as a left module over A.

More precisely, V_{0} is isomorphic to such a direct sum, as a left module over A. This is Proposition 6.1 on p14 of [50].

To see this, one can first reduce to the case where V is freely generated by finitely many elements x_{1}, …, x_{n}, as a left module over A. More precisely, it suffices to choose the x_{j}’s so that V_{0} is contained in the submodule of V generated by them.

The statement holds trivially if n = 0, with suitable interpretations. If n = 1, then it is easy to see that V_{0} is isomorphic to a left ideal in A, as a left module over A. Otherwise, we can use induction, as follows.

Let W be the submodule of V generated by x_{1}, …, x_{n−1}, which is interpreted as being {0} when n = 1. Thus every v ∈ V can be expressed in a unique way as v = w + a · x_{n}, (9.5.2)

with w ∈ W and a ∈ A. Of course, v ↦ a defines a homomorphism from V onto A, as left modules over A. This homomorphism maps V_{0} onto a left ideal I_{0} in A, and the kernel of the restriction of this homomorphism to V_{0} is equal to V_{0} ∩ W. (9.5.3)

Note that I_{0} is finitely generated as a left module over A, because V_{0} is finitely generated, by hypothesis.

If A is left semi-hereditary, then it follows that I_{0} is projective, as a left module over A. This implies that V_{0} corresponds to the direct sum of V_{0} ∩ W and another submodule C_{0}, as left modules over A, as in Section 2.7. We also have that C_{0} is isomorphic to I_{0}, as a left module over A.

Observe that V_{0} ∩ W is finitely generated as a left module over A, because there is a homomorphism from V_{0} onto V_{0}∩W. This permits us to use induction, to get that V_{0} ∩ W is isomorphic to a direct sum of finitely many left modules over A, each of which is isomorphic to a left ideal in A that is finitely generated as a left module over A. It follows that V_{0} has the same property, as desired.

Proposition 6.2 on p15 of [50] states that A is left semi-hereditary if and only if

(9.5.4) every finitely-generated submodule of a projective left module over A is projective as a left module over A.

Indeed, any projective left module over A may be considered as a submodule of a free left module over A, as in Section 2.7. Thus the “only if” part follows from (9.5.1), and the fact that direct sums of projective modules are projective. The “if” part follows from the fact that A is free as a left module over itself, and thus projective.

9.6 Hereditary rings and differentiation

Let A be a ring with a nonzero multiplicative identity element, and let (V, d_{V}) be a left or right module over A with differentiation. Suppose that A is left or right hereditary, as appropriate, and that V is projective as a module over A. This implies that

(9.6.1) B(V) = d_{V} (V) is projective as a module over A,

as in Section 9.3, because B(V) is a submodule of V. It follows that V corresponds to the direct sum of Z(V) = ker d_{V} and another submodule of V, as a module over A, as in Section 2.7.

Note that

(9.6.2) B′(V) = V/Z(V) = V/ ker d_{V} is projective as a module over A,

because it is isomorphic to B(V). Remember that there is a natural homomorphism from Z′(V) = V/d_{V} (V) onto B′(V), because D_{V} (V) ⊆ ker d_{V}, as in Section 5.1. One can use this to get that Z′(V) corresponds to the direct sum of H(V) = Z(V)/B(V) and another submodule of Z′(V), as a module over A, as in Section 2.7. Of course, this could also be obtained by expressing V as a direct sum of Z(V) and another submodule, as in the preceding paragraph.

Suppose that (V, d_{V}) is a graded module over A with differentiation that is a complex. In this case, V is projective as a module over A if and only if V^{j} is projective as a module over A for every j ∈ Z, as in Section 2.7. It is easy to see that V corresponds to the direct sum of Z(V) and another homogeneous submodule of V, as a module over A, under these conditions. We also have that Z′(V) corresponds to the direct sum of H(V) and another homogeneous submodule of Z′(V), as a module over A, as before. This is related to part of the proof of Theorem 3.2 on p113 of [50].

Let (V, d_{V}) be a left or right module over A with differentiation again, and suppose now that V is injective as a module over A. This implies that

(9.6.3) B(V) is injective as a module over A,

as in Section 9.3. It follows that V corresponds to the direct sum of B(V) and another submodule of V, as a module over A, as in Section 2.8. We also get that Z(V) corresponds to the direct sum of B(V) and another submodule of Z(V), as a module over A.

Suppose that (V, d_{V}) is a graded module over A with differentiation that is a complex, and that V^{j} is injective as a module over A for every j ∈ Z. This implies that

(9.6.4) B(V)^{j} is injective as a module over A for every j ∈ Z,

as before. It follows that V^{j} corresponds to the direct sum of B(V)^{j} and another submodule of V^{j}, as a module over A, for every j ∈ Z. We also get that Z(V)^{j} corresponds to the direct sum of B(V)^{j} and another submodule of Z(V)^{j}, as a module over A, for every j ∈ Z. This is related to Theorem 3.2a on p114 of [50].

9.7 Noetherian conditions

Let A be a ring with a multiplicative identity element e_{A}, and let V be a left or right module over A. We say that V is Noetherian if

(9.7.1) every submodule of V is finitely generated, as a module over A,

as on p15 of [50].

The Noetherian condition is often defined in terms of the ascending chain condition for the collection of submodules of V, as a partially-ordered set with respect to inclusion. This says that any monotonically-increasing sequence of submodules of V with respect to inclusion is eventually constant. Another formulation asks that any nonempty collection of submodules of V have a maximal element. The equivalence of these conditions can be shown using standard arguments, as in Propositions 6.1 and 6.2 on p74f of [8].

Let V_{0} be a submodule of V. If V_{0} and the corresponding quotient module V/V_{0} are finitely generated as modules over A, then one can check that V is finitely generated as a module over A, as in Exercise 9 on p32 of [8], and the first part of Exercise 2 on p16 of [50]. If V is finitely generated as a module over A, then it is easy to see that V/V_{0} is finitely generated too. Of course, if V is Noetherian, then V_{0} is finitely generated as well, as in the second part of Exercise 2 on p16 of [50].

One can check that V is Noetherian if and only if

(9.7.2) V_{0} and V/V_{0} are Noetherian,

as in part (i) of Proposition 6.3 on p75 of [8]. The argument given there uses the ascending chain condition, and one could also verify this using the definition in terms of finitely-generated submodules.

Let V_{1}, …, V_{n} be finitely many Noetherian modules over A, which are all left modules over A, or all right modules over A. Under these conditions, their direct sum

(9.7.3)  ⨁_{j=1}^{n} V_{j} is Noetherian as a module over A,

as in Corollary 6.4 on p76 of [8]. This can be seen using induction on n, and the statement in the preceding paragraph.

We say that A is left or right Noetherian as a ring if A is Noetherian as a left or right module over itself, as appropriate, as on p76 of [8], and p15 of [50]. This means that every left or right ideal in A is finitely generated as a module over A.

In this case, if V is a finitely-generated left or right module over A, as appropriate, then

(9.7.4) V is Noetherian as a module over A,

as in Proposition 6.5 on p76 of [8], and Proposition 7.1 on p15 of [50]. In the proof in [8], one expresses V as a quotient of A^{n} for some n ≥ 1, and one uses the results mentioned in the previous two paragraphs. The proof in [50] uses induction on the number of generators of V, and basically involves analogous arguments.

An example is given on p16 of [50] of a ring that is left Noetherian and not right Noetherian, and which is due to J. Dieudonné. Let k be a commutative ring with a nonzero multiplicative identity element, which one can take to be Z for the example. The example may be considered as an associative algebra A over k with a nonzero multiplicative identity element e_{A}, which is generated by e_{A} and two additional nonzero elements x, y, as an algebra over k. More precisely, x and y are supposed to satisfy

y x = y y = 0, (9.7.5)

and no other relations.

One might as well take x to be an indeterminate, so that the subalgebra of A generated by e_{A} and x may be identified with the usual algebra k[x] of formal polynomials in x with coefficients in k. Every element of A can be expressed in a unique way as an element of k[x] plus an element of k[x] times y, so that A corresponds to the direct sum of k[x] and (k[x]) y, as a module over k. If k is a Noetherian ring, then k[x] is a Noetherian ring, by Hilbert’s Basis Theorem, which is Theorem 7.5 on p81 of [8]. In particular, this holds when k = Z, or k is a field.

If we consider A as a module over k[x] with respect to multiplication by elements of k[x] on the left, then A is isomorphic to the direct sum of two copies of k[x]. If k is a Noetherian ring, so that k[x] is a Noetherian ring, then it follows that A is Noetherian as a module over k[x], as before. In this case, it is easy to see that A is Noetherian as a left module over itself, because every submodule of A as a left module over itself is also a submodule of A as a module over k[x] with respect to multiplication by elements of k[x] on the left. This means that A is left Noetherian as a ring under these conditions.

Observe that (k[x]) y is a right ideal in A, because of the relations (9.7.5). Using these relations, we also get that any set of generators for (k[x]) y as a right module over A generates (k[x]) y as a module over k. In particular, (k[x]) y is not finitely generated as a right module over A, because (k[x]) y is not finitely generated as a module over k. This means that A is not Noetherian as a right module over itself. Equivalently, A is not right Noetherian as a ring.

9.8 A criterion for finite generation

Let A be a ring with multiplicative identity element e_{A}, and let V be a left or right module over A. Also let V_{1} and V_{2} be submodules of V, so that V_{1} + V_{2} and V_{1} ∩ V_{2} are submodules of V as well. If V_{1} + V_{2} and V_{1} ∩ V_{2} are finitely generated as modules over A, then V_{1} and V_{2} are finitely generated as modules over A. This is Exercise 4 on p16 of [50].

This is a bit simpler when

V_{1} ∩ V_{2} = {0}. (9.8.1)

In this case, V_{1} + V_{2} corresponds to the direct sum of V_{1} and V_{2}, as modules over A. In particular, this leads to homomorphisms from V_{1} + V_{2} onto V_{1} and V_{2}, as modules over A. If w_{1}, …, w_{n} are finitely many elements of V_{1} + V_{2} that generate V_{1} + V_{2} as a module over A, then their images in V_{1} and V_{2} generate these modules.

Otherwise, consider the quotients

(V_{1} + V_{2})/V_{1} (9.8.2)

and (V_{1} + V_{2})/V_{2}, (9.8.3) which are modules over A. More precisely, these are finitely-generated modules over A, because V_{1} + V_{2} is finitely generated. Let q_{1}, q_{2} be the natural quotient mappings from V_{1} + V_{2} onto (9.8.2), (9.8.3), respectively. Observe that

q_{1}(V_{2}) = (V_{1} + V_{2})/V_{1}, (9.8.4)

and that the kernel of the restriction of q_{1} to V_{2} is V_{1} ∩ V_{2}. It follows that V_{2} is finitely generated as a module over A, because (9.8.2) and V_{1} ∩ V_{2} are finitely generated, as in the previous section. Similarly, one can get that V_{1} is finitely generated as a module over A, using the restriction of q_{2} to V_{1}. This is related to the diagram in Exercise 1 on p16 of [50].

Alternatively, consider the quotient

(V_{1} + V_{2})/(V_{1} ∩ V_{2}). (9.8.5)

As before, this is a finitely-generated module over A, because V_{1} + V_{2} is finitely generated. Let q_{1,2} be the natural quotient mapping from V_{1} + V_{2} onto (9.8.5). Of course,

q_{1,2}(V_{1}) + q_{1,2}(V_{2}) = q_{1,2}(V_{1} + V_{2}) = (V_{1} + V_{2})/(V_{1} ∩ V_{2}). (9.8.6)

Let us check that q_{1,2}(V_{1}) ∩ q_{1,2}(V_{2}) = {0}. (9.8.7)

If v_{1} ∈ V_{1} and v_{2} ∈ V_{2} satisfy q_{1,2}(v_{1}) = q_{1,2}(v_{2}), then

v_{1} − v_{2} ∈ V_{1} ∩ V_{2}. (9.8.8)

This implies that v_{1} ∈ V_{2} and v_{2} ∈ V_{1}, so that v_{1}, v_{2} ∈ V_{1} ∩ V_{2}, and thus q_{1,2}(v_{1}) = q_{1,2}(v_{2}) = 0, as desired.

This shows that (9.8.5) corresponds to the direct sum of q_{1,2}(V_{1}) and q_{1,2}(V_{2}), as a module over A. It follows that q_{1,2}(V_{1}) and q_{1,2}(V) are finitely generated as modules over A, because (9.8.5) is finitely generated as a module over A, as before. One can use this to get that V_{1} and V_{2} are finitely generated, because V_{1} ∩ V_{2} is finitely generated, as in the previous section.

9.9 Direct sums and injective modules

Let A be a ring with multiplicative identity element e_{A}, let B be a nonempty set, and let V_{β} be a left module over A for each β ∈ B. If V_{β} is an injective module over A for each β ∈ B, and A is left Noetherian, then

(9.9.1)  ⨁_{β∈B} V_{β} is injective as a left module over A.

Of course, there is an analogous statement for right modules. Remember that direct products of injective modules are injective, as in Section 2.8, without additional conditions on A. Of course, if B has only finitely many elements, then ⨁_{β∈B} V^{β} is the same as the direct product.

To show (9.9.1), let I be a left ideal in A, and let ϕ be a homomorphism from I into ⨁_{β∈B} V_{β}, as left modules over A. It suffices to find

v ∈ ⨁_{β∈B} V_{β} (9.9.2)

such that

ϕ(a) = a · v (9.9.3)

for every a ∈ I, as in Section 2.8. Let ϕ_{β}(a) be the component of ϕ(a) in V_{β} for each a ∈ I and β ∈ B. Thus ϕ_{β} defines a homomorphism from I into V_{β}, as left modules over A, for each β ∈ B. Note that (9.9.3) is the same as saying that

ϕ_{β}(a) = a · v_{β} (9.9.4)

for every a ∈ I and β ∈ B, where v_{β} is the component of v in V_{β}.

If A is left Noetherian, then there are finitely many elements x_{1}, …, x_{n} of I such that I is generated by x_{1}, …, x_{n}, as a left module over A. Put

B_{j} = {β ∈ B : ϕ_{β}(x_{j}) ≠ 0} (9.9.5)

for each j = 1, …, n, which is a finite subset of B, because ϕ(x_{j}) ∈ ⨁_{β∈B} V_{β}. Thus

⋃_{j=1}^{n} B_{j} (9.9.6)

is a finite subset of B too. If a ∈ I, then it is easy to see that

ϕ_{β}(a) = 0 when β ∈ B \ ( ⋃_{j=1}^{n} B_{j}), (9.9.7)

because I is generated by x_{1}, …, x_{n}, as a left module over A. Equivalently, ϕ_{β} = 0 when β ∈ B \ ( ⋃_{j=1}^{n} B_{j}).

If β ∈ ⋃_{j=1}^{n} B_{j}, then let v_{β} be an element of V_{β} such that (9.9.4) holds for every a ∈ I. This uses the injectivity of V_{β} as a left module over A, as in Section 2.8. If β ∈ B \ ( ⋃_{j=1}^{n} B_{j}), then we can take v_{β} = 0. This defines an element v of ⨁_{β∈B} V_{β}, whose component in V_{β} is equal to v_{β} for every β ∈ B. Under these conditions, (9.9.3) holds for every a ∈ I, as desired.

Note that if (9.9.1) holds, then V_{α} is injective as a left module over A for every α ∈ B. This can be seen using the same argument as for direct products in Section 2.8.

9.10 Some finitely-generated modules

Let k be a commutative ring with a multiplicative identity element, and let V, W be modules over k. If V and W are finitely generated as modules over k, then V ⨂_{k} W is finitely generated as a module over k. This is the first part of Exercise 6 on p32 of [50]. More precisely, suppose that V, W are generated by subsets E_{V}, E_{W}, respectively, as modules over k. Under these conditions, V ⨂_{k} W is generated by

{v ⊗ w : v ∈ E_{V} , w ∈ E_{W} }, (9.10.1)

as a module over k.

Let v_{1}, …, v_{n} be finitely many elements of V for some positive integer n, and let W^{n} be the usual space of n-tuples of elements of W. Thus W^{n} is a module over k, with respect to coordinatewise addition and scalar multiplication. If ϕ is a homomorphism from V into W, as modules over k, then put

L(ϕ) = (ϕ(v_{1}), …, ϕ(v_{n})), (9.10.2)

which is an element of W^{n}. This defines a homomorphism from Hom_{k}(V, W) into W^{n}, as modules over k. If V is freely generated by v_{1}, …, v_{n}, as a module over k, then L is an isomorphism from Hom_{k}(V, W) onto W^{n}, as modules over k.

Suppose that W is finitely generated, as a module over k. This implies that W^{n} is finitely generated too, as a module over k. Suppose that k is also Noetherian as a ring. It follows that W^{n} is Noetherian as a module over k, as in Section 9.7. In particular, this means that

(9.10.3) L(Hom_{k}(V, W)) is finitely generated, as a module over k.

If V is generated by v_{1}, …, v_{n}, as a module over k, then it is easy to see that L is injective, as a mapping from Hom_{k}(V, W) into W^{n}. In this case, (9.10.3) implies that

(9.10.4) Hom_{k}(V, W) is finitely generated, as a module over k.

Thus (9.10.4) holds when V and W are finitely generated as modules over k, and k is Noetherian. This is the second part of Exercise 6 on p32 of [50]. Of course, (9.10.4) also holds when V is freely generated by finitely many elements, and W is finitely generated, as modules over k.

If V is finitely generated as a module over k, then there is a homomorphism from a module X over k onto V, where X is freely generated by finitely many elements. If V is projective as a module over k, then X corresponds to the direct sum of two submodules, one of which is isomorphic to V, as in Section 2.7. This implies that Hom_{k}(X, W) is isomorphic to the direct sum of the corresponding spaces of homomorphisms from each of the two submodules just mentioned into W. In particular, Hom_{k}(X, W) is isomorphic to the direct sum of Hom_{k}(V, W) and another module over k. If W is finitely generated as a module over k, then Hom_{k}(X, W) is finitely generated as a module over k, as before. In this case, it follows that Hom_{k}(V, W) is finitely generated as a module over k. This means that (9.10.4) holds when V and W are finitely generated modules over k, and V is projective.

9.11 Some remarks about division rings

Let A be a ring, with a nonzero multiplicative identity element e_{A}. In this section, we suppose that A is a division ring, so that every nonzero element of A has a multiplicative inverse in A. This implies that A and {0} are the only left or right ideals in A. In particular, A is left and right hereditary, and left and right Noetherian.

Of course, if A is commutative, then A is a field. Sometimes the term “field” is used for division rings that may not be commutative, but here we mean a commutative division ring. We may also refer to a division algebra over a field k, which is a division ring that is an algebra over k as well.

A left or right module over A is sometimes called a left or right vector space over A, as appropriate. Many of the usual properties of vector spaces over fields work for left and right vector spaces over division rings too, suitably interpreted. Let us consider a left module V over A, the corresponding statements for right modules being analogous.

A collection v_{1}, …, v_{n} of finitely many elements of V is said to be linearly dependent if there are elements a_{1}, …, a_{n} of A, not all equal to 0, such that

∑_{j=1}^{n} a_{j} · v_{j} = 0. (9.11.1)

In this case, it is easy to see that a smaller collection of elements of V will generate the same submodule of V, as a left module over A.

A subset E of V is said to be linearly independent if there are no nonempty finite subsets of E that are linearly dependent. This means that the submodule of V generated by E is freely generated by E.

Let E_{0} be a linearly independent subset of V. It is well known that there is a linearly independent subset E of V such that

E_{0} ⊆ E, (9.11.2)

and E is maximal with respect to inclusion. This can be shown using Zorn’s lemma, or Hausdorff’s maximality principle. One can use maximality of E to get that V is generated by E, as a left module over A. This implies that V is freely generated by E, as a left module over A, as before.

If V is generated by E_{0} and finitely or countably many other elements of V, as a left module over A, then one can get a linearly independent set E ⊆ V such that (9.11.2) holds and V is generated by E more directly. More precisely, one can go through the list of new elements and drop those that are in the submodule of V generated by E_{0} and the previous elements in the list.

In particular, this implies that any left module over A is a free module over A. If V_{1} is any submodule of V, as a left module, then one can use this to get a linearly independent set E_{1} ⊆ V_{1} such that V_{1} is generated by E_{1}, as a left module over A. This can be extended to a linearly independent set E ⊆ V that generates V as a left module over A, as before. Let V_{2} be the submodule of V generated by E_{2} = E \ E_{1}. It is easy to see that

(9.11.3) V corresponds to the direct sum of V_{1} and V_{2},

as a left module over A, under these conditions.

Every left module over A is injective as a left module over A too. This follows easily from the characterization of injectivity in Section 2.8, because A and {0} are the only left ideals in A. Alternatively, this can be obtained from the remarks in the preceding paragraph.

9.12 Polynomials and division rings

Let A be a division ring, let T be an indeterminate, and let A[T] be the corresponding ring of formal polynomials in T with coefficients in A, as in Section 4.3. If f(T) = ∑_{j=0}^{n} f_{j} T^{j} ∈ A[T] and f_{n} ≠ 0, then the degree deg f(T) of f(T) is defined to be n, as usual. If f(T) = 0, then the degree of f(T) may be defined to be −∞. It is easy to see that the degree of the product of two elements of A[T] is equal to the sum of their degrees.

Let a(T) ∈ A[T] be given, with a(T) ≠ 0. If f(T) ∈ A[T], then there are b(T), r(T) ∈ A[T] such that

f(T) = b(T) a(T) + r(T) (9.12.1)

and deg r(T) < deg a(T). This can be seen in essentially the same way as when A is commutative. Of course, if deg f(T) < deg a(T), then one can simply take b(T) = 0.

Otherwise, one can find c ∈ A and a nonnegative integer j such that

deg(f(T) − (c T^{j}) a(T)) < deg f(T). (9.12.2)

One can repeat the process until the degree of the remainder is strictly less than deg a(T). The analogous statement with a(T) multiplied on the right by an element of A[T] can be shown in the same way. This corresponds to some remarks on p98 of [151], which also work for polynomials with coefficients in any ring with a multiplicative identity element when the leading coefficient of a(T) is invertible in the ring.

Let I be a left ideal in A[T], and let a(T) be a nonzero element of I of minimal degree. Under these conditions,

I = {b(T) a(T) : b(T) ∈ A[T]}. (9.12.3)

Clearly the right side of (9.12.3) is contained in I, because a(T) ∈ I. If f(T) is an element of I, then f(T) can be expressed as in (9.12.1). In this case, r(T) ∈ I, because I is a left ideal in A[T]. The minimality of the degree of a(T) implies that r(T) = 0, so that f(T) is in the right side of (9.12.3). Of course, there is an analogous statement for nonzero right ideals in A[T].

Note that b(T) ↦ b(T) a(T) (9.12.4)

defines a homomorphism from A[T] onto I, as left modules over A[T]. This homomorphism is injective, because products of nonzero elements of A[T] are nonzero, as before. Thus I is isomorphic to A[T] as a left module over A[T], and in particular I is projective as a left module over A[T]. This implies that A[T] is left hereditary as a ring. Similarly, A[T] is right hereditary as a ring.

Let A[[T]] be the ring of formal power series in T with coefficients in A, as in Section 4.3. If j_{0} is a nonnegative integer, then it is easy to see that

(A[[T]]) T^{j_{0}} (9.12.5)

is a left and right ideal in A[[T]]. Conversely, any nonzero left or right ideal I in A[[T]] is of this form. To see this, let f(T) = ∑_{j=j_{0}}^{∞} f_{j}T^{j} be an element of I with f_{j_{0}} ≠ 0 and j_{0} as small as possible. Thus I is contained in (9.12.5), by construction. Note that an element of A[[T]] is invertible when its coefficient of T^{0} is nonzero, because A is a division ring, as in Section 4.5. One can use this to check that I is equal to (9.12.5).

It is easy to see that the product of two nonzero elements of A[[T]] is nonzero as well, by considering the smallest powers of T for which the corresponding coefficients are not zero. It follows that any nonzero left or right ideal I in A[[T]] is isomorphic to A[[T]], as a left or right module over A[[T]], as appropriate. In particular, this means that I is projective as a left or right module over A[[T]], as appropriate. This shows that A[[T]] is left and right hereditary, as a ring.

9.13 An interesting family of rings

Let A_{1} be a commutative ring with a nonzero multiplicative identity element e. Also let ε be a ring homomorphism from A_{1} into itself, with

ε(e) = e. (9.13.1)

Put A_{2} = A_{1} × A_{1}, (9.13.2)

considered initially as a commutative group with respect to coordinatewise addition. Of course, this is the same as the direct sum of two copies of A_{1}, as a commutative group with respect to addition.

Let us define multiplication on A_{2} by putting

(a, b) (a′, b′) = (a a′, a b′ + ε(a′) b) (9.13.3)

for every a, a′, b, b′ ∈ A_{1}, as in Exercise 5 on p159 of [50]. This defines a mapping from A_{2} ×A_{2} into A_{2} that is additive in each of (a, b) and (a′, b′). One can check that this definition of multiplication on A_{2} is associative, so that A_{2} is a ring. If ε is the identity mapping on A_{1}, then multiplication on A_{2} is commutative.

It is easy to see that (e, 0) is the multiplicative identity element in A_{2}, because of (9.13.1). Observe that

a ↦ (a, 0) (9.13.4)

is an injective ring homomorphism from A_{1} into A_{2}. Thus A_{1} × {0} is a subring of A_{2} that is isomorphic to A_{1}.

Suppose for the moment that k is a commutative ring with a nonzero multiplicative identity element, and that A_{1} is the algebra k[x] of formal polynomials in an indeterminate x with coefficients in k. Let ε be the mapping from k[x] into itself that sends a formal polynomial in x to its constant term. In this case, A_{2} corresponds to the algebra over k in Dieudonné’s example, discussed in Section 9.7, as in Exercise 5 on p159 of [50].

Clearly (a, b) ↦ a (9.13.5)

defines a ring homomorphism from A_{2} onto A_{1}. If (a, b) ∈ A_{2} has a left or right inverse in A_{2}, then it follows that A is invertible in A_{1}. If b ∈ A_{1}, then

(e, b) (e, −b) = (e, −b) (e, b) = (e, 0). (9.13.6)

This means that (e, b) is invertible in A_{2}, with inverse (e, −b). If (a, b) ∈ A_{2} and a is invertible in A_{1}, then it follows that

(a, b) = (a, 0) (e, a^{−1} b) (9.13.7)

is invertible in A_{2}, as in Exercise 5 on p159 of [50].

Suppose for the moment that A_{1} is a local ring, and let I_{1} be the unique maximal ideal in A_{1}. Thus I_{1} consists exactly of the noninvertible elements of A_{1}, as in Section 4.13. Under these conditions,

I_{2} = {(a, b) ∈ A_{2} : a ∈ I_{1}} (9.13.8)

is a two-sided ideal in A_{2}, which is the same as the collection of elements of A_{2} without a left inverse, or without a right inverse. This means that A_{2} is a local ring as well, as in Section 4.13. Note that A_{1}/I_{1} is isomorphic to A_{2}/I_{2}, as in Exercise 5 on p159 of [50].

It is easy to see that {0}×A_{1} is a two-sided ideal in A_{2}, which is the same as the kernel of (9.13.5). We may consider A_{2} as a module over A_{1} with respect to both left and right multiplication, using (9.13.4). Clearly A_{1} ×{0} and {0}×A_{1} are both submodules of A_{2}, as a module over A_{1} with respect to each of left and right multiplication. As modules over A_{1} with respect to left multiplication, A_{1} × {0} and {0} × A_{1} are both isomorphic to A_{1}.

If A_{1} is Noetherian as a ring, then A_{1} × {0} and {0} × A_{1} are Noetherian as modules over A_{1} with respect to multiplication on the left. This implies that A_{2} is Noetherian as a module over A_{1} with respect to multiplication on the left, because A_{2} corresponds to the direct sum of A_{1} ×{0} and {0}×A_{1}. In this case, it follows that A_{2} is left Noetherian, because left ideals in A_{2} are submodules of A_{2} as a module over A_{1} with respect to multiplication on the left.

Let k be a commutative ring with a nonzero multiplicative identity element again, and suppose now that A_{1} is the algebra k[[x]] of formal power series in an indeterminate x with coefficients in k. Also let ε be the mapping from k[[x]] to itself that sends a formal power series in x to its constant term, so that A_{2} may be defined as before. If k is a field, then k[[x]] is a local ring, so that A_{2} is a local ring, as mentioned earlier.

If k is Noetherian, then it is well known that k[[x]] is a Noetherian ring too, as on p81 of [8]. This implies that A_{2} is left Noetherian, as before.

It is easy to see that any submodule W of

{0} × A_{1} = {0} × k[[x]], (9.13.9)

as a module over k, is a right ideal in A_{2}, because of (9.13.3) and the definition of ε. Similarly, any set of generators for W, as a right module over A_{2}, also generates W as a module over k. One can use this to get that A_{2} is not right Noetherian, as in Exercise 5 on p159 of [50].

9.14 Graded rings

Let A be a ring, with a multiplicative identity element e_{A}. Suppose that A is graded as a commutative group with respect to addition, or equivalently as a module over Z, as in Section 5.9. Suppose also that

(9.14.1) A^{j} = {0} for every j < 0,

and that A^{j} · A^{l} ⊆ A^{j+l} (9.14.2)

for every j, l ≥ 0. Under these conditions, A is said to be a graded ring, as on p146 of [50].

In this case, one can check that the component e_{A}^{0} of e_{A} in A^{0} is a multiplicative identity element of A. This means that e_{A}^{0} = e_{A}, so that e_{A} ∈ A^{0}. Note that A^{0} is a subring of A.

If a ∈ A, then let ε(a) = a^{0} (9.14.3)

be the component of a in A^{0}. This defines a ring homomorphism from A onto A^{0}. Of course, the kernel of ε corresponds to the direct sum of A^{j}, with j ≥ 1.

Let k be a commutative ring with a multiplicative identity element, and suppose that A is an associative algebra over k. If A is graded as a module over k, with respect to the same grading as before, then A is a graded as an associative algebra over k, as on p164 of [50].

Let B be a subring of A, and suppose that B is homogeneous as a subgroup of A, as a graded commutative group with respect to addition, as in Section 5.9. Thus B is a graded commutative group with respect to addition too, using the grading induced by the one on A. Clearly B is a graded ring with respect to this grading.

Let A be any ring with multiplicative identity element e_{A}, and let T_{1}, …, T_{n} be n distinct commuting indeterminates for some positive integer n. Thus the corresponding ring A[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in A may be defined as in Section 4.3. An element

f(T) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} T^{α} (9.14.4)

of A[T_{1}, …, T_{n}] is said to be homogeneous of degree j for some nonnegative integer j if f_{α} = 0 when α_{1} + ⋯ + α_{n} ≠ j. (9.14.5)

If we take (A[T_{1}, …, T_{n}])^{j} to be the set of elements of A[T_{1}, …, T_{n}] that are homogeneous of degree j for each j ≥ 0, and to be {0} when j < 0, then

(9.14.6) A[T_{1}, …, T_{n}] is a graded ring,

as on p146 of [50]. Of course, (A[T_{1}, …, T_{n}])^{0} can be identified with A in a natural way.

Let E be a nonempty set, and let Σ(E) be the free semigroup generated by E, as in Section 4.10. One can use Σ(E) to get the corresponding semigroup ring A(Σ(E)), as in Sections 4.9 and 4.10. This is the free ring over A with generating set E, as on p146, 148 of [50]. By construction, the elements of A(Σ(E)) correspond to finite sums of elements of A times formal products of finitely many elements of E. If j is a nonnegative integer, then let (A(Σ(E)))^{j} be the set of elements of A(Σ(E)) that can be expressed as a finite sum of elements of A times formal products of elements of E of length j. If we take (A(Σ(E)))^{j} = {0} when j < 0, then it is easy to see that

(9.14.7) A(Σ(E)) is a graded ring,

as on p146 of [50]. Note that (A(Σ(E)))^{0} can be identified with A in a natural way.

Let V be a bimodule over A, so that V is both a left and right module over A, and the actions of A on V on the left and right commute with each other. The corresponding tensor ring T (V) can be defined as in Section 4.11, and

(9.14.8) T (V) is a graded ring,

with (T (V))^{j} = T^{j}(V) for each j ≥ 0, and (T (V))^{j} = {0} when j < 0. In particular, (T (V))^{0} = T^{0}(V) = A, by construction.

Let n be a positive integer, and let x_{1}, …, x_{n} be n distinct indeterminates. Remember that the corresponding exterior ring E_{A}(x_{1}, …, x_{n}) in x_{1}, …, x_{n} with coefficients in A is defined as in Section 4.15. If 0 ≤ j ≤ n, then let (E_{V} (x_{1}, …, x_{n}))^{j} be the set of elements of E_{V} (x_{1}, …, x_{n}) of the form

∑_{{I⊆\{1,...,n\} \atop |I|=j}} a_{I} x_{I}, (9.14.9)

Here x_{I} is as in (4.15.2) for each I ⊆ {1, …, n}, |I| is the number of elements of I, and a_{I} ∈ A. If j < 0 or j > n, then we take (E_{A}(x_{1}, …, x_{n}))^{j} = {0}. It is easy to see that (9.14.10) E_{A}(x_{1}, …, x_{n}) is a graded ring,

as on p146 of [50]. Note that (E_{A}(x_{1}, …, x_{n}))^{0} can be identified with A, as before.

Remember that the ring A[d] of dual numbers associated to A as in Section 5.4 consists of expressions of the form a_{1} + a_{2} d, where a_{1}, a_{2} ∈ A, and d is a formal symbol that commutes with all elements of A and satisfies d^{2} = 0. This corresponds to taking n = 1 in the preceding paragraph, as on p146 of [50]. In particular, (9.14.11) A[d] is a graded ring,

with (A[d])^{0} = A, (A[d])^{1} = A d, and (A[d])^{j} = {0} otherwise.

9.15 Graded modules over graded rings

Let A be a ring, with a multiplicative identity element e_{A}, and suppose that A is graded, as in the previous section. Also let V be a right module over A. Suppose that V is graded as a commutative group with respect to addition, or equivalently as a module over Z, as in Section 5.9. Let us say that this grading on V is compatible with the grading on A if

V^{j} · A^{l} ⊆ V^{j+l} (9.15.1)

for every j, l ∈ Z, where of course we may as well take l ≥ 0. Equivalently, this means that v^{j} · a^{l} ∈ V^{j+l} (9.15.2)

for every v^{j} ∈ V^{j} and a^{l} ∈ A^{l}.

Remember that A^{0} is a subring of A, so that V may be considered as a right module over A^{0}. If the grading on V is compatible with the grading on A, then V is a graded module over A^{0} in the usual sense, as in Section 5.9.

Suppose that V is positive with respect to its grading, so that V^{j} = {0} when j < 0, as in Section 5.9. If the grading on V is compatible with the grading on A, then V is said to be graded as a module over A, as a graded ring, as on p154 of [50]. Of course, there are analogous notions for left modules. Note that A may be considered as a graded module over itself, as a graded ring.

Let U be a submodule of V, as a module over A. Suppose that U is homogeneous as a subgroup of V, as a graded commutative group with respect to addition, as in Section 5.9. If the grading on V is compatible with the grading on A, then the grading on U induced by the grading on V is compatible with the grading on A as well. In particular, if V is graded as a module over A, as a graded ring, then U is graded as a module over A, as a graded ring, with respect to the induced grading.

Let B be a subring of A that contains e_{A}, and suppose that B is homogeneous as a subgroup of A, as a commutative group with respect to addition, so that B may be considered as a graded ring with respect to the induced grading, as in the previous section. Of course, V may be considered as a module over B. If the grading on V is compatible with the grading on A, then the grading on V is compatible with the grading on B induced by the grading on A. If V is graded as a module over A, as a graded ring, then V is graded as a module over B, as a graded ring, with the induced grading.

Let us now take A to be any ring with a multiplicative identity element e_{A}, and let T_{1}, …, T_{n} be n distinct commuting indeterminates for some positive integer n, so that the corresponding ring A[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in A may be defined as in Section 4.3. This may be considered as a graded ring, as in the previous section. Let V be a left or right module over A, and let V [T_{1}, …, T_{n}] be the space of formal polynomials in T_{1}, …, T_{n} with coefficients in V, as in Section 4.3 again. This may be considered as a left or right module over A[T_{1}, …, T_{n}], as appropriate, as before.

Let us say that an element f(T) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} T^{α} of V [T_{1}, …, T_{n}] is homogeneous of degree j for some nonnegative integer j if f_{α} = 0 when

α_{1} + ⋯ + α_{n} ≠ j, (9.15.3)

as usual. Let (V [T_{1}, …, T_{n}])^{j} be the set of elements of V [T_{1}, …, T_{n}] that are homogeneous of degree j when j ≥ 0, and be equal to {0} when j < 0. Using this, it is easy to see that

(9.15.4) V [T_{1}, …, T_{n}] is a graded module over A[T_{1}, …, T_{n}], as a graded ring.

Let x_{1}, …, x_{n} be n distinct indeterminates, so that the exterior ring

E_{A}(x_{1}, …, x_{n}) (9.15.5)

in x_{1}, …, x_{n} with coefficients in A is defined as in Section 4.15. This may be considered as a graded ring, as in the previous section. Let V be a left or right module over A again, and let E_{V} (x_{1}, …, x_{n}) be as in Section 4.15. Remember that E_{V} (x_{1}, …, x_{n}) may be considered as a left or right module over E_{A}(x_{1}, …, x_{n}), as appropriate.

Let (E_{V} (x_{1}, …, x_{n}))^{j} be the set of elements of E_{V} (x_{1}, …, x_{n}) of the form

∑_{{I⊆\{1,...,n\} \atop |I|=j}} v_{I} x_{I} (9.15.6)

when 0 ≤ j ≤ n. As before, x_{I} is as in (4.15.2) for each I ⊆ {1, …, n}, |I| is the number of element of I, and v_{I} ∈ V. If we also take (E_{V} (x_{1}, …, n))^{j} = {0} when j < 0 or j > n, then it is easy to see that

(9.15.7) E_{V} (x_{1}, …, x_{n}) is a graded module over E_{A}(x_{1}, …, x_{n}), as a graded ring.

The ring A[d] of dual numbers associated to A as in Section 5.4 may be considered as a graded ring, as in Section 9.14. Remember that a left or right module (V, d_{V}) over A with differentiation corresponds exactly to a left or right module over A[d], as appropriate. Suppose that V is graded as a module over A, as in Section 5.9. Observe that this grading is compatible with the grading on A[d] exactly when d_{V} has degree 1, which means that V is a complex, as in Section 5.10. If V is positive as a graded module, then it follows that

(9.15.8) V is a graded module over A[d], as a graded ring.

Left and right complexes

10.1 Left complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a left or right module over A. As on p75 of [50], and mentioned in Section 7.5, we may consider V as a graded module over A with differentiation that this a complex, by taking V^{0} = V, V^{j} = {0} when j ≠ 0, and differentiation operator d_{V} = 0. In this case, Z(V), Z′(V), and H(V) are the same as V, and B(V), B′(V) = {0}.

Let (X, d_{X}) be a graded left or right module over A, as appropriate, with differentiation that is a complex. Suppose that X is negative as a graded module, so that X^{j} = {0} when j > 0, (10.1.1)

as in Section 5.9. Let ε be a map from X into V, as complexes. This means that ε is a homomorphism from X into V, as modules over A with differentiation, of degree 0, as in Section 5.11. The combination of X and ε is called a left complex over V, and ε is called the augmentation of the left complex, as on p75 of [50].

The condition that ε have degree 0 means that ε(X^{j}) ⊆ V^{j} for each j ∈ Z, as in Section 5.9. Remember that the restriction of ε to X^{j} is denoted ε^{j} for each j, as before. In this case,

ε^{j} = 0 when j ≠ 0, (10.1.2)

because V^{j} = {0} when j ≠ 0. Thus ε is determined by ε^{0}, which is a homomorphism from X^{0} into V^{0} = V, as modules over A.

The condition that ε be a homomorphism from X into V, as modules with differentiation, means that

ε ∘ d_{X} = d_{V} ∘ ε = 0, (10.1.3)

as in Section 5.2. This is the same as saying that

ε^{j} ∘ d_{X}^{j−1} = d_{V}^{j−1} ∘ ε^{j−1} = 0 (10.1.4) for each j, as in Section 5.11. This holds automatically when j ≠ 0 here, so that we are reduced to the condition that ε^{0} ∘ d_{X}^{−1} = 0, (10.1.5) as on p75 of [50].

The left complex X is said to be projective if X^{j} is projective as a module over A for each j, as on p75 of [50].

Remember that ε induces a homomorphism (10.1.6) from H(X) into H(V) of degree 0, as in Sections 5.2 and 5.11. Thus we get an induced homomorphism (10.1.7) from H(X)^{j} into H(V)^{j} for each j ∈ Z. The left complex X over V is said to be acyclic if the homomorphism as in (10.1.6) is an isomorphism, as on p75 of [50]. Equivalently, this means that the homomorphism as in (10.1.7) is an isomorphism for each j.

If j ≠ 0, then H(V)^{j} = {0}, and the condition that the homomorphism as in (10.1.7) be an isomorphism means that H(X)^{j} = {0}. (10.1.8) This holds automatically when j > 0, because of (10.1.1).

Note that d_{X}^{0} = 0, because X^{1} = {0}, as in (10.1.1). Thus Z(X)^{0} = X^{0}. (10.1.9) The condition (10.1.5) is the same as saying that B(X)^{0} = d_{X}^{−1}(X^{−1}) ⊆ ker ε^{0}. (10.1.10) The homomorphism as in (10.1.7) is injective when j = 0 if and only if d_{X}^{−1}(X^{−1}) = ker ε^{0}. (10.1.11) The homomorphism as in (10.1.7) is surjective when j = 0 if and only if ε^{0}(X^{0}) = V^{0} = V. (10.1.12)

It is often convenient to put X_{j} = X^{−j}, as in Section 5.9. Similarly, we may put d_{j} = d_{X,j} = d_{X}^{−j}, (10.1.13) which maps X^{−j} = X_{j} into X^{−j+1} = X_{j−1}. We may also put ε_{j} = ε^{−j}, (10.1.14) which maps X^{−j} = X_{j} into V^{−j} = V_{j}. Of course, this means that ε_{j} = 0 when j ≠ 0, and ε_{0} = ε^{0}. Using this notation, the acyclicity of X as a left complex over V is equivalent to the exactness of the sequence

⋯ ⟶ X_{n} ⟶^{d_{n}} X_{n−1} ⟶ ⋯ ⟶ X_{1} ⟶^{d_{1}} X_{0} ⟶^{ε_{0}} V ⟶ 0, (10.1.15)

as on p75 of [50].

10.2 Projective resolutions

Let us continue with the same notation and hypotheses as in the previous section. A left complex X over V is said to be a projective resolution of V if it is projective and acyclic, as on p75 of [50].

Every module V over A has a projective resolution, as in Proposition 1.2 on p77 of [50]. One can start with a projective module X_{0} over A and a homomorphism ε_{0} from X_{0} onto V, as in Section 2.7. Put

Z_{0} = ker ε_{0}. (10.2.1)

Similarly, let d_{1} be a homomorphism from a projective module X_{1} over A onto Z_{0}. More precisely, we shall consider d_{1} as a homomorphism from X_{1} into X_{0}, as modules over A.

Continuing in this way, suppose that d_{j} is a homomorphism from X_{j} into X_{j−1} for some j ≥ 1, and put

Z_{j} = ker d_{j}. (10.2.2)

We take d_{j+1} to be a homomorphism from a projective module X_{j+1} over A onto Z_{j}, considered as a homomorphism into X_{j}, as before.

To get a left complex X over V, we take X^{j} = {0} when j > 0, X^{−j} = X_{j} when j ≥ 0, ε^{0} = ε_{0}, and ε^{j} = 0 when j ≠ 0. We also take d_{X}^{j} = 0 when j ≥ 0, and d_{X}^{−j} = d_{j} when j ≥ 1. This is a projective resolution of V, by construction.

Note that we can take X_{j} to be a free module over A for each j ≥ 0, as in Section 2.7, and as mentioned on p77 of [50].

If V is projective as a module over A, then we can take X_{0} = V, and ε_{0} to be the identity mapping. In this case, Z_{0} = {0}, and we can take X_{j} = {0} and d_{j} = 0 when j ≥ 1.

If Z_{j} = {0} for any j ≥ 0, then we can take

X_{l} = {0}, d_{l} = 0 when l > j. (10.2.3)

If Z_{j} is a projective module over A for some j ≥ 0, then we can take X_{j+1} = Z_{j} and d_{j+1} to be the obvious inclusion mapping from Z_{j} into X_{j}. This means that Z_{j+1} = {0}, (10.2.4)

so that we can take

X_{l} = {0}, d_{l} = 0 when l > j + 1. (10.2.5)

Suppose for the moment that A is left or right hereditary, as appropriate, as in Section 9.3. We can start with a projective module X_{0} over A and a homomorphism ε_{0} from X_{0} onto V as before. In this case, Z_{0} is projective as a module over A, because Z_{0} is a submodule of X_{0}, as in Section 9.3. This permits us to take X_{1} = Z_{1}, and d_{1} to be the obvious inclusion mapping from X_{1} into X_{0}, as in the preceding paragraph. This means that

Z_{1} = ker d_{1} = {0}, (10.2.6) so that we can take X_{j} = {0} and d_{j} = 0 when j ≥ 2, as before.

If V is finitely generated as a module over A, then we can take X_{0} to be a finitely-generated free module over A. If A is left or right Noetherian as a ring, as appropriate, then Z_{0} is finitely generated as a module over A too. Under these conditions, we can repeat the process, and take X_{j} to be a finitely-generated free module over A for each j ≥ 0. This corresponds to Proposition 1.3 on p78 of [50].

Suppose now that A has no nontrivial zero divisors, so that if a, b ∈ A and a, b ≠ 0, then a b ≠ 0. Let x be a nonzero element of A, so that A x is a left ideal in A, and a ↦ a x is an isomorphism from A onto A x, as left modules over A. This implies that A x is free as a left module over A. Consider

V = A/A x, (10.2.7)

which is a left module over A. We can take X_{0} = A, considered as a free left module over A, and ε_{0} to be the natural quotient mapping from A onto V. The kernel of ε_{0} is A x, which is free as a left module over A, and thus projective. As before, we can take X_{1} = A x, d_{1} to be the obvious inclusion mapping from A x into A, and X_{j} = {0}, d_{j} = 0 when j ≥ 2. Of course, there are analogous statements for x A and A/x A, as right modules over A.

10.3 Maps over homomorphisms

Let us continue with the same notation and hypotheses as in Section 10.1. In particular, we let V be a left or right module over A, and we suppose that X is a left complex over V with augmentation ε. Let Ṽ be another left or right module over A, as appropriate, and suppose that X̃ is a left complex over Ṽ, with augmentation ε̃.

Let ϕ be a homomorphism from V into Ṽ, as modules over A. Suppose that Φ is a map from X into X̃, as complexes, as in Section 5.11. This means that Φ is a homomorphism from X into X̃, as modules over A with differentiation, and that Φ has degree 0. If ε̃∘ Φ = ϕ ∘ ε, (10.3.1)

then Φ is said to be a map over ϕ, as on p76 of [50].

Suppose that

(10.3.2) X is a projective left complex over V,

and that

(10.3.3)  X̃ is acyclic as a left complex over Ṽ.

Under these conditions,

(10.3.4) there is a map Φ from X into X̃ over ϕ,

as in Proposition 1.1 on p76 of [50].

Let

d = d_{X}, d̃= d_{X} (10.3.5)

be the differentiation operators on X, X̃, respectively. As in Section 10.1, we put X_{j} = X^{−j}, d_{j} = d^{−j}, and ε_{j} = ε^{−j}, and similarly for X̃. Thus X_{j} = X̃_{j} = {0} when j < 0, by hypothesis. The restriction of Φ to X_{j} is Φ_{j} = Φ^{−j}, which is 0 when j < 0. We want to define Φ_{j} recursively for j ≥ 0.

Note that ϕ ∘ ε_{0} is a homomorphism from X_{0} into Ṽ, as modules over A. We also have that ϵ̃(X̃_{0}) = Ṽ, (10.3.6)

because X̃ is acyclic as a left complex over Ṽ, as in (10.1.12). This implies that there is a homomorphism Φ_{0} from X_{0} into X̃_{0}, as modules over A, such that

ε̃_{0} ∘ Φ_{0} = ϕ ∘ ε_{0}, (10.3.7)

because X_{0} is projective as a module over A, by hypothesis.

Similarly, Φ_{0} ∘d_{1} defines a homomorphism from X_{1} into X̃_{0}, as modules over A. Observe that ε̃_{0} ∘ Φ_{0} ∘ d_{1} = ϕ ∘ ε_{0} ∘ d_{1} = 0, (10.3.8)

using (10.3.7) in the first step, and (10.1.5) in the second step. This means that

(Φ_{0} ∘ d_{1})(X_{1}) ⊆ ker ε̃_{0} = d̃_{1}(X̃_{1}), (10.3.9)

where the second step is as in (10.1.11), because X̃ is acyclic as a left complex over Ṽ. It follows that there is a homomorphism Φ_{1} from X_{1} into X̃_{1}, as modules over A, such that d̃_{1} ∘ Φ_{1} = Φ_{0} ∘ d_{1}, (10.3.10)

because X_{1} is projective as a module over A.

Suppose now that l ≥ 2 is an integer, and that Φ_{j} has been defined as a homomorphism from X_{j} into X̃_{j}, as modules over A, for j = 0, …, l − 1. Suppose also that d̃_{j} ∘ Φ_{j} = Φ_{j−1} ∘ d_{j} (10.3.11)

for j = 1, …, l − 1. Thus Φ_{l−1} ∘ d_{l} defines a homomorphism from X_{l} into X̃_{l−1}, as modules over A, and

d̃_{l−1} ∘ Φ_{l−1} ∘ d_{l} = Φ_{l−2} ∘ d_{l−1} ∘ d_{l} = 0. (10.3.12)

This implies that

(Φ_{l−1} ∘ d_{l})(X_{l}) ⊆ ker d̃_{l−1} = d̃_{l}(X̃_{l}), (10.3.13)

where the second step is as in (10.1.8), because X̃ is acyclic as a left complex over Ṽ. This means that there is a homomorphism Φ_{l} from X_{l} into X̃_{l}, as modules over A, such that

d̃_{l} ∘ Φ_{l} = Φ_{l−1} ∘ d_{l}, (10.3.14) because X_{l} is projective as a module over A.

Continuing in this way, we can define Φ_{j} for every j, to get a homomorphism Φ from X into X̃ of degree 0. More precisely, Φ is a homomorphism from X into X̃, as modules with differentiation, because (10.3.11) holds when j ≥ 1, by construction, and this also holds automatically when j ≤ 0. Thus Φ is a map from X into X̃, as complexes. Note that (10.3.1) follows from (10.3.7), so that Φ is a map over ϕ, as desired.

10.4 Some related homotopies

We continue with the same notation and hypotheses as in the previous section. Let ϕ be a homomorphism from V into Ṽ again, as modules over A, and suppose now that Φ, Ψ are maps from X into X̃ over ϕ. If X is a projective left complex over V, and X̃ is acyclic as a left complex over V, then

(10.4.1) there is a homotopy Σ between Φ and Ψ,

as maps between complexes, as in Section 5.11. This is another part of Proposition 1.1 on p76 of [50].

In particular, Σ is supposed to be a homomorphism from X into X̃ of degree −1, so that Σ^{j} maps X^{j} into X̃^{j−1} for every integer j. As before, we put

Σ_{j} = Σ^{−j}, (10.4.2)

which should map X_{j} into X̃_{j+1} for every j. This is automatically equal to 0 when j < 0, and we want to define Σ_{j} recursively when j ≥ 0.

Observe that

ε̃_{0} ∘ (Φ_{0} − Ψ_{0}) = ε̃∘ Φ_{0} − ε̃∘ Ψ_{0} = ϕ ∘ ϵ_{0} − ϕ ∘ ϵ_{0} = 0, (10.4.3)

because Φ and Ψ are both maps over ϕ. Thus

(Φ_{0} − Ψ_{0})(X_{0}) ⊆ ker ε̃_{0} = d̃_{1}(X̃_{1}), (10.4.4)

where the second step is as in (10.1.11), because X̃ is acyclic as a left complex over Ṽ. This implies that there is a homomorphism Σ_{0} from X_{0} into X̃_{1}, as modules over A, such that

d̃_{1} ∘ Σ_{0} = Φ_{0} − Ψ_{0}, (10.4.5)

because X_{0} is projective as a module over A, by hypothesis.

Let l be a positive integer, and suppose that Σ_{j} has been defined as a homomorphism from X_{j} into X̃_{j+1}, as modules over A, for j = 0, …, l − 1. Suppose also that d̃_{j+1} ∘ Σ_{j} + Σ_{j−1} ∘ d_{j} = Φ_{j} − Ψ_{j} (10.4.6)

when 0 < j ≤ l − 1. Note that Σ_{l−1} ∘ d_{l} defines a homomorphism from X_{l} into X̃_{l}, as modules over A.

Because Φ and Ψ are homomorphisms from X into X̃, as modules with differentiation, we have that

d̃_{l} ∘ (Φ_{l} − Ψ_{l} − Σ_{l−1} ∘ d_{l}) = d̃_{l} ∘ Φ_{l} − d̃_{l} ∘ Ψ_{l} − d̃_{l} ∘ Σ_{l−1} ∘ d_{l}
(10.4.7)  = Φ_{l−1} ∘ d_{l} − Ψ_{l−1} ∘ d_{l} − d̃_{l} ∘ Σ_{l−1} ∘ d_{l}.

If l = 1, then the right side is equal to 0, by (10.4.5). If l ≥ 2, then we can take j = l − 1 in (10.4.6), to get that

d̃_{l} ∘ Σ_{l−1} ∘ d_{l} = Φ_{l−1} ∘ d_{l} − Ψ_{l−1} ∘ d_{l}. (10.4.8)

This implies that the right side of (10.4.7) is equal to 0 when l ≥ 2 as well. It follows that

(Φ_{l} − Ψ_{l} − Σ_{l−1} ∘ d_{l})(X_{l}) ⊆ ker d̃_{l} = d̃_{l+1}(X_{l}), (10.4.9)

where the second step is as in (10.1.8), because X̃ is acyclic as a left complex over Ṽ. Thus there is a homomorphism Σ_{l} from X_{l} into X̃_{l+1}, as modules over A, such that d̃_{l+1} ∘ Σ_{l} = Φ_{l} − Ψ_{l} − Σ_{l−1} ∘ d_{l}, (10.4.10)

because X_{l} is projective as a module over A. This means that (10.4.6) holds, with j = l.

We can continue in this way to define Σ_{j} for every j, and get a homomorphism from X into X̃ that is homogeneous of degree −1. By construction, (10.4.6) holds when j > 0. If j = 0, then (10.4.6) reduces to (10.4.5), and (10.4.6) holds automatically when j < 0. This implies that Σ is a homotopy between Φ and Ψ, as desired.

10.5 Some remarks about compositions

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, Ṽ, and V̂ be all left or all right modules over A. Suppose that X, X̃, and X̂ are left complexes over V, Ṽ, and V̂, respectively, as in Section 10.1. Let ε, ε̃, and ε̂ be the corresponding augmentations on X, X̃, and X̂, respectively.

Let ϕ be a homomorphism from V into Ṽ, and let ψ be a homomorphism from Ṽ into V̂, as modules over A. Suppose that Φ is a map from X into X̃ over ϕ, and that Ψ is a map from X̃ into X̂ over ψ, as in Section 10.3. Note that Ψ∘Φ is a map from X into X̂, as complexes. Of course, ψ∘ϕ is a homomorphism from V into V̂, as modules over A. We also have that

ε̂∘ Ψ ∘ Φ = ψ ∘ ε̃∘ Φ = ψ ∘ ϕ ∘ ε, (10.5.1)

so that Ψ ∘ Φ is a map over ψ ∘ ϕ.

Now let X and Y be projective resolutions of V, as in Section 10.2. There is a map Φ from X into Y over the identity map on V, as in Section 10.3.

Similarly, there is a map Ψ from Y into X over the identity mapping on V. It follows that Ψ ∘ Φ is a map from X into itself over the identity mapping on V, as in the preceding paragraph. Similarly, Φ ∘ Ψ is a map from Y into itself over the identity mapping on V.

It is easy to see that the identity mapping on X is a map over the identity mapping on V, as a map from X into itself. It follows that Ψ ∘ Φ is homotopic to the identity on X, as maps from X into itself as a complex, as in the previous section. Similarly, Φ ∘ Ψ is homotopic to the identity map on Y, as maps from Y into itself, as a complex. This correspoonds to a remark on p77 of [50], after Proposition 1.2. One may say that X and Y have the same homotopy type under these conditions, as in [50].

10.6 Right complexes

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V be a left or right module over A again. As before, we may consider V to be a complex, with V^{0} = V, V^{j} = {0} when j ≠ 0, and d_{V} = 0.

Let (X, d_{X}) be a graded left or right module over A, as appropriate, with differentiation that is a complex, and suppose that X is positive as a graded module. This means that

X^{j} = {0} when j < 0, (10.6.1)

as in Section 5.9. Let ε be a map from V into X as complexes, so that ε is a homomorphism from V into X, as modules over A with differentiation, of degree 0, as in Section 5.11. The combination of X and ε is called a right complex over V, and ε is called the augmentation of the right complex, as on p78 of [50].

The condition that ε have degree 0 means that ε(V^{j}) ⊆ X^{j} for every j ∈ Z, as in Section 5.9. Of course, the restriction ε^{j} of ε to V^{j} is equal to 0 when j ≠ 0, so that ε is determined by ε^{0}.

The condition that ε be a homomorphism from V into X, as modules with differentiation, means that

d_{X} ∘ ε = ε ∘ d_{V} = 0, (10.6.2)

as in Section 5.2. This is the same as saying that

d_{X}^{j} ∘ ε^{j} = ε^{j+1} ∘ d_{V}^{j} = 0 (10.6.3)

for each j, as in Section 5.11. This reduces to the condition that

d_{X}^{0} ∘ ε^{0} = 0, (10.6.4)

because ε^{j} = 0 when j ≠ 0.

The right complex X is said to be injective if X^{j} is injective as a module over A for each j, as on p78 of [50].

Remember that ε induces a homomorphism

(10.6.5) from H(V) into H(X)

of degree 0, as in Sections 5.2 and 5.11. This leads to an induced homomorphism

(10.6.6) from H(V)^{j} into H(X)^{j}

for each j ∈ Z. The right complex X over V is said to be acyclic if the homomorphism as in (10.6.5) is an isomorphism, as on p78 of [50]. This is the same as saying that the homomorphism as in (10.6.6) is an isomorphism for every j.

If j ≠ 0, then this means that

H(X)^{j} = {0}, (10.6.7)

because H(V)^{j} = {0}. This holds automatically when j < 0, by (10.6.1).

Observe that B(X)^{0} = {0}, (10.6.8)

because X^{−1} = {0}, as in (10.6.1). The condition (10.6.4) is the same as saying that ε^{0}(V^{0}) ⊆ Z(X)^{0}. (10.6.9)

The homomorphism as in (10.6.6) is injective when j = 0 if and only if

ker ε^{0} = {0}, (10.6.10)

because of (10.6.8). The homomorphism as in (10.6.6) is surjective when j = 0 if and only if ε^{0}(V^{0}) = Z(X)^{0}. (10.6.11)

The acyclicity of X as a right complex is equivalent to the exactness of the sequence

0 ⟶ V ⟶^{ε} X^{0} ⟶^{d_{X}^{0}} X^{1} ⟶ ⋯ ⟶ X^{n} ⟶^{d_{X}^{n}} X^{n+1} ⟶ ⋯, (10.6.12)

as on p78 of [50].

10.7 Injective resolutions

We continue with the same notation and hypotheses as in the previous section. A right complex X over V is said to be an injective resolution of V if it is injective and acyclic, as on p78 of [50].

Every module V over A has an injective resolution, as in Proposition 1.2a on p78 of [50]. We can begin here with an injective homomorphism ε from V into an injective module X^{0} over A, as in Section 2.8. Put

Y^{0} = X^{0}/ε(V), (10.7.1) which is another module over A.

As before, there is an injective homomorphism

(10.7.2) from Y^{0} into an injective module X^{1} over A.

Let d^{0} be the composition of the natural quotient mapping from X^{0} onto Y^{0} with this injection, so that d^{0} is a homomorphism from X^{0} into X^{1}, as modules over A. Note that ker d^{0} = ε(V), (10.7.3)

by construction.

Suppose that modules X^{j}, X^{j+1} over A have already been chosen for some j ≥ 0, as well as a homomorphism d^{j} from X^{j} into X^{j+1}, as modules over A. Put Y^{j+1} = X^{j+1}/d^{j}(X^{j}), (10.7.4)

which is a module over A too. There is an injective homomorphism

(10.7.5) from Y^{j+1} into an injective module X^{j+2} over A,

as usual. Let d^{j+1} be the composition of the natural quotient mapping from X^{j+1} onto Y^{j+1} with this injection, so that d^{j+1} is a homomorphism from X^{j+1} into X^{j+2}, as modules over A. Thus

ker d^{j+1} = d^{j}(X^{j}), (10.7.6)

by construction.

Continuing in this way, we get injective modules X^{j} for each j ≥ 0. We take X^{j} = {0} when j < 0, to get a positive graded module X over A. More precisely, X is a complex, with d_{X}^{j} = d^{j} as before when j ≥ 0, and of course d_{X}^{j} = 0 when j < 0. We also take ε^{0} = ε, and ε^{j} = 0 when j ≠ 0. It is easy to see that this makes X a right complex over V, which is an injective resolution of V.

If V is an injective module over A, then we can take X^{0} = V, and ε to be the identity mapping. This means that Y^{0} = {0}, and we can take X^{j+1} = {0} and d^{j} = 0 for every j ≥ 0.

Similarly, if Y^{j} = {0} for any j ≥ 0, then we can take

X^{l+1} = {0}, d^{l} = 0 when l ≥ j. (10.7.7)

If Y^{j} is an injective module over A for some j ≥ 0, then we can take X^{j+1} = Y^{j}, and use the identity mapping on Y^{j} as the injective homomorphism from Y^{j} into X^{j+1}. In this case, d^{j} is the natural quotient mapping from X^{j} onto Y^{j}, so that d^{j}(X^{j}) = Y^{j} = X^{j+1}, and

Y^{j+1} = {0}. (10.7.8)

Thus we can take

X^{l+1} = {0}, d^{l} = 0 when l ≥ j + 1, (10.7.9) as before.

Suppose that A is left or right hereditary, as appropriate, as in Section 9.3. Let ε be an injective homomorphism from V into an injective module X^{0} over A, as before. If Y^{0} is as in (10.7.1), then Y^{0} is injective as a module over A, as in Section 9.3. This means that we can take X^{1} = Y^{0}, and use the identity mapping on Y^{0} as the injective homomorphism from Y^{0} into X^{1}, as in the preceding paragraph. It follows that d^{0} is the natural quotient mapping from X^{0} onto Y^{0}, and Y^{1} = {0}, (10.7.10)

so that we can take X^{l+1} = {0} and d^{l} = 0 when l ≥ 1, as before.

10.8 Maps and right complexes

Let us continue with the same notation and hypotheses as in Section 10.6. We let V be a left or right module over A again, and suppose that X is a right complex over V with augmentation ε. Let Ṽ be another left or right module over A, as appropriate, and suppose that X̃ is a right complex over Ṽ, with augmentation ε̃.

Let ϕ be a homomorphism from V into Ṽ, as modules over A. Suppose that Φ is a map from X into X̃, as complexes, as in Section 5.11, so that Φ is a homomorphism from X into X̃, as modules over A with differentiation, of degree 0. We say that Φ is a map over ϕ if

Φ ∘ ε = ε̃∘ ϕ, (10.8.1)

as on p78 of [50].

Suppose that

(10.8.2) X is acyclic as a right complex over V,

and that

(10.8.3)  X̃ is an injective right complex over Ṽ.

In this case, (10.8.4) there is a map Φ from X into X̃ over ϕ,

as in Proposition 1.1a on p78 of [50].

Of course, the restriction Φ^{j} of Φ to X^{j} is 0 when j < 0, because X^{j} = {0}. We would like to define Φ^{j} recursively when j ≥ 0.

We would like to begin by choosing Φ^{0} to be a homomorphism from X^{0} into X̃^{0}, as modules over A, such that

Φ^{0} ∘ ε^{0} = ε^{0} ∘ ϕ. (10.8.5)

Note that ε^{0} is injective, because X is acyclic as a right complex over V, as in (10.6.10). Thus (10.8.5) determines Φ^{0} on ε^{0}(V) ⊆ X^{0}, as a homomorphism into X̃^{1}, because ε^{0} ∘ ϕ is a homomorphism from V into X̃^{0}, as modules over A. It follows that there is a homomorphism Φ^{0} from X^{0} into X̃^{0} as in (10.8.5), because X̃^{0} is injective as a module over A, by hypothesis.

Similarly, we would like to choose Φ^{1} to be a homomorphism from X^{1} into X̃^{1}, as modules over A, such that

Φ^{1} ∘ d_{X}^{0} = d0 ; _{X} ∘ Φ^{0}. (10.8.6)

Observe that

d0 ; _{X} ∘ Φ^{0} ∘ ε^{0} = d0 ; _{X} ∘ ε^{0} ∘ ϕ = 0, (10.8.7)

using (10.8.5) in the first step, and the analogue of (10.6.4) for ε^{0} in the second step. This implies that the right side of (10.8.6) is equal to 0 on the kernel of d_{X}^{0}, because of (10.6.11). This means that Φ^{1} is well-defined on d_{X}^{0}(X^{0}) by (10.8.6). Thus there is a homomorphism Φ^{1} from X^{1} into X̃^{1} as in (10.8.6), because X̃^{1} is injective as a module over A.

Suppose that l ≥ 2 is an integer, and that Φ^{j} has been defined as a homomorphism from X^{j} into X̃^{j}, as modules over A, for j = 0, …, l − 1. Suppose also that

Φ^{j} ∘ d_{X}^{j−1} = dj−1 ; _{X} ∘ Φ^{j−1} (10.8.8)

for j = 1, …, l − 1. We would like to choose Φ^{l} to be a homomorphism from X^{l} into X̃^{l}, as modules over A, such that

Φ^{l} ∘ d_{X}^{l−1} = dl−1 ; _{X} ∘ Φ^{l−1}, (10.8.9)

which is the same as (10.8.8) with j = l. Observe that

dl−1 ; _{X} ∘ Φ^{l−1} ∘ d_{X}^{l−2} = dl−1 ; _{X} ∘ dl−2 ; _{X} ∘ Φ^{l−2} = 0, (10.8.10)

using (10.8.8) with j = l − 1 in the first step. This implies that

dl−1 ; _{X} ∘ Φ^{l−1} = 0 (10.8.11)

on d_{X}^{l−2}(X^{l−2}), which is the same as ker d_{X}^{l−1}, because H(X)^{l−1} = {0}, as in (10.6.7). Thus Φ^{l} is well-defined on d_{X}^{l−1}(X^{l−1}) by (10.8.9). It follows that there is a homomorphism Φ^{l} from X^{l} into X̃^{l} that satisfies (10.8.9), because X̃^{l} is injective as a module over A.

We can continue in this way to define Φ^{j} for every j, to get a homomorphism from X into X̃ of degree 0. We also have that Φ is a homomorphism from X into X̃ as modules with differentiation, because (10.8.8) holds when j ≥ 1 by construction, and automatically when j ≤ 0. This means that Φ is a map from X into X̃, as complexes. Of course, (10.8.1) follows from (10.8.5), so that Φ is a map over ϕ.

10.9 Homotopies and right complexes

Let us continue with the same notation and hypotheses as in the previous section, with ϕ a homomorphism from V into Ṽ, as modules over A, in particular. Suppose that Φ and Ψ are maps from X into X̃ over ϕ, in the sense defined in the previous section. If X is acyclic as a right complex over V, and X̃ is an injective right complex over Ṽ, then

(10.9.1) there is a homotopy Σ between Φ and Ψ,

as maps between complexes, as in Section 5.11. This is the second part of Proposition 1.1a on p78 of [50].

By definition of a homotopy, Σ is supposed to be a homomorphism from X into X̃ of degree −1, so that Σ^{j} maps X^{j} into X̃^{j−1} for every j. Of course, Σ^{j} is automatically 0 when j ≤ 0, and we would like to define it recursively when j > 0.

Note that

(Φ^{0} − Ψ^{0}) ∘ ε^{0} = Φ^{0} ∘ ε^{0} − Ψ^{0} ∘ ε^{0} = ε^{0} ∘ ϕ − ε^{0} ∘ ϕ = 0, (10.9.2)

because Φ and Ψ are maps over ϕ. This means that

ker(Φ^{0} − Ψ^{0}) ⊇ ε^{0}(X^{0}) = Z(X)^{0}, (10.9.3)

using the acyclicity of X as a right complex over V in the second step, as in (10.6.11). We would like to choose Σ^{1} to be a homomorphism from X^{1} into X̃^{0}, as modules over A, such that

Σ^{1} ∘ d_{X}^{0} = Φ^{0} − Ψ^{0}. (10.9.4)

It follows from (10.9.3) that Σ^{1} is well defined on d_{X}^{0}(X^{0}) by (10.9.4). Thus there is a homomorphism Σ^{1} from X^{1} into X̃^{0} that satisfies (10.9.4), because X̃^{0} is injective as a module over A.

Let l ≥ 2 be an integer, and suppose now that Σ^{j} has been defined as a homomorphism from X^{j} into X̃^{j−1}, as modules over A, for j = 1, …, l − 1. Suppose also that

Σ^{j} ∘ d_{X}^{j−1} + dj−2 ; _{X} ∘ Σ^{j−1} = Φ^{j−1} − Ψ^{j−1} (10.9.5)

when 2 ≤ j ≤ l − 1. We would like to choose Σ^{l} to be a homomorphism from X^{l} into X̃^{l−1}, as modules over A, such that

Σ^{l} ∘ d_{X}^{l−1} = −dl−2 ; _{X} ∘ Σ^{l−1} + Φ^{l−1} − Ψ^{l−1}, (10.9.6)

which is equivalent to (10.9.5), with j = l.

Observe that

(−dl−2 ; _{X} ∘ Σ^{l−1} + Φ^{l−1} − Ψ^{l−1}) ∘ d_{X}^{l−2} (10.9.7)

= −dl−2 ; _{X} ∘ Σ^{l−1} ∘ d_{X}^{l−2} + Φ^{l−1} ∘ d_{X}^{l−2} − Ψ^{l−1} ∘ d_{X}^{l−2}
= −dl−2 ; _{X} ∘ Σ^{l−1} ∘ d_{X}^{l−2} + dl−2 ; _{X} ∘ Φ^{l−2} − dl−2 ; _{X} ∘ Ψ^{l−2},

because Φ, Ψ are homomorphisms from X into X̃, as modules over A with differentiation. If l = 2, then the right side of (10.9.7) is equal to 0, by (10.9.4). If l ≥ 3, then we can take j = l − 1 in (10.9.5), to get that

dl−2 ; _{X} ∘ Σ^{l−1} ∘ d_{X}^{l−2} = dl−2 ; _{X} ∘ Φ^{l−2} − dl−2 ; _{X} ∘ Ψ^{l−2}. (10.9.8)

This implies that the right side of (10.9.7) is equal to 0 in this case too.

It follows that the right side of (10.9.6) is equal to 0 on d_{X}^{l−2}(X^{l−2}). This is the same as saying that the right side of (10.9.6) is equal to 0 on ker d_{X}^{l−1}, because H(X)^{l−1} = {0}, as in (10.6.7). This means that Σ^{l} is well-defined on d_{X}^{l−1}(X^{l−1}) by (10.9.6). Thus we can get a homomorphism Σ^{l} from X^{l} into X̃^{l−1}, as modules over A, that satisfies (10.9.6), because X̃^{l−1} is injective as a module over A.

Continuing in this way, we can define Σ^{j} for every j, to get a homomorphism Σ from X into X̃ that is homogeneous of degree −1. We also have that (10.9.5) holds when j ≥ 2, by construction. If j = 1, then (10.9.5) follows from (10.9.4). If j ≤ 0, then (10.9.5) holds automatically, so that Σ is a homotopy between Φ and Ψ, as desired.

10.10 Compositions and right complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, Ṽ, and V̂ be all left or all right modules over A, and suppose that X, X̃, and X̂ are right complexes over V, Ṽ, and V̂, respectively, as in Section 10.6. These include the augmentations ε, ε̃, and ε̂, as before.

Let ϕ, ψ be homomorphisms from V, Ṽ into Ṽ, V̂, respectively, as modules over A, and suppose that Φ, Ψ are maps fom X, X̃ into X̃, X̂, over ϕ, ψ, respectively, as in Section 10.8. Thus ψ ∘ ϕ is a homomorphism from V into V̂, as modules over A, and Ψ ∘ Φ is a map from X into X̂, as complexes. Observe that

Ψ ∘ Φ ∘ ε = Ψ ∘ ε̃∘ ϕ = ε̂∘ ψ ∘ ϕ, (10.10.1)

so that Ψ ∘ Φ is a map over ψ ∘ ϕ.

Suppose now that X, Y are injective resolutions of V, as in Section 10.7. There are maps Φ, Ψ from X, Y into Y, X, respectively, over the identity map on V, as in Section 10.8. It follows that Ψ ∘ Φ and Φ ∘ Ψ are maps from X and Y into themselves, respectively, over the identity mapping on V, as in the preceding paragraph.

Note that the identity mappings on X, Y are maps over the identity mapping on V, as maps from X, Y into themselves. Using the remarks in the previous section, we get that Ψ ∘ Φ and Φ ∘ Ψ are homotopic to the identity mappings on X and Y, respectively, as maps from X, Y into themselves, as complexes.

10.11 Tensor products and left complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a right module over A, let W be a left module over A, and let V ⨂_{A} W be a tensor product of V and W, as modules over A. Suppose that X, Y are left complexes over V, W, as in Section 10.1, with augmentations ε_{X}, ε_{Y}, respectively.

Under these conditions, X ⨂_{A} Y can be defined as a doubly-graded module over k, with (X ⨂_{A}Y )^{j,l} = X^{j} ⨂_{A}Y^{l} (10.11.1)

for every j, l ∈ Z, as in Section 6.1. Of course, this is equal to {0} when j > 0 or l > 0, because X and Y are negative as graded modules, by hypothesis. One can get a single grading on X ⨂_{A} Y from the double grading as in Section 5.13, as usual. It is easy to see that X ⨂_{A} Y is negative with respect to this single grading, because of the analogous property of the double grading just mentioned.

More precisely, X ⨂_{A} Y can be defined as a double complex over k, as in Section 6.2. The two differentiation operators on X ⨂_{A} Y, as a double complex, can be combined to get a single differentiation operator, so that X ⨂_{A} Y becomes a single complex with respect to the single grading mentioned in the preceding paragraph, as in Section 5.14. Thus

H(X ⨂_{A}Y )_{n} = H(X ⨂_{A}Y )^{−n} (10.11.2)

can be defined as a module over k in the usual way for every integer n. This is equal to {0} when n < 0, because X ⨂_{A} Y is negative with respect to its single grading.

Remember that V, W may be considered as complexes, as in Section 10.1. Using this, one can consider V ⨂_{A} W as a double complex, which leads to a single complex, as before. This is the same as the single complex obtained from V ⨂_{A} W as a module over k, as in Section 10.1 again. One can use the augmentations ε_{X}, ε_{Y} to get a map

(10.11.3) ε from X ⨂_{A}Y into V ⨂_{A}W,

as complexes, as in Sections 5.15, 6.1, and 6.2. This means that

(10.11.4) X ⨂_{A}Y is a left complex over V ⨂_{A}W with respect to ε,

as in Section 10.1.

Let Ṽ be another right module over A, let W̃ be another left module over A, and let Ṽ ⨂_{A} W̃ be a tensor product of Ṽ and W̃, as modules over A. Also let X̃, Ỹ be left complexes over Ṽ, Ỹ, respectively, so that X̃ ⨂_{A} Ỹ is a left complex over Ṽ ⨂_{A} W̃, as before. Suppose that ϕ, ψ are homomorphisms from V, W into Ṽ, W̃, respectively, as modules over A, and that Φ, Ψ are maps from X, Y into X̃, Ỹ over ϕ, ψ, respectively, as in Section 10.3.

Using ϕ and ψ, we get a homomorphism

(10.11.5) from V ⨂_{A}W into Ṽ ⨂_{A}W̃,

as modules over k, as in Section 1.9. Similarly, we get a map

(10.11.6) Φ ⊗ Ψ from X ⨂_{A}Y into X̃ ⨂_{A}Ỹ,

as double complexes, as in Section 6.2. One can check that

(10.11.7) Φ ⊗ Ψ is a map over the homomorphism as in (10.11.5),

as in Section 10.3. Note that Φ ⊗ Ψ is a map from X ⨂_{A} Y into X̃ ⨂_{A} Ỹ, considered as single complexes, as in Section 5.15. Thus Φ ⊗ Ψ induces a homomorphism from (10.11.2) into its analogue for X̃ ⨂_{A} Ỹ for each n, as in Section 5.11.

Let Φ′, Ψ′ be another pair of maps from X, Y into X̃, Ỹ over ϕ, ψ, respectively, which leads to a map

(10.11.8) Φ′ ⊗ Ψ′ from X ⨂_{A}Y into X̃ ⨂_{A}Ỹ,

as double complexes, as before. Suppose that Φ, Ψ are homotopic to Φ′, Ψ′, respectively, as maps between complexes, as in Section 5.11. This implies that

(10.11.9) Φ ⊗ Ψ is homotopic to Φ′ ⊗ Ψ′,

as maps between double complexes, as in Section 6.2. It follows that Φ ⊗ Ψ is homotopic to Φ′ ⊗ Ψ′ as maps between single complexes, as in Section 5.15. This means that Φ ⊗ Ψ and Φ′ ⊗ Ψ′ induce the same mappings on (10.11.2) for each n, as in Section 5.11.

We may be particularly interested in (10.11.2) when X and Y are projective resolutions of V and W, respectively. In this case,

(10.11.10) (10.11.2) is uniquely determined by V and W

for each n, up to isomorphism. This follows from the fact that projective resolutions are unique up to homotopy equivalence, as in Section 10.5.

These remarks correspond to some of those on p82-4 of [50] in the left case, and for tensor products. This is also related to some remarks on p107 of [50].

10.12 Left complexes in one factor

Let us return to the same notation and hypotheses as at the beginning of the previous section. Remember that X ⨂_{A} W and V ⨂_{A} Y can be defined as graded modules over k, with

(X ⨂_{A}W )^{j} = X^{j} ⨂_{A}W, (V ⨂_{A}Y )^{j} = V ⨂_{A}Y^{j} (10.12.1) for every integer j, as in Section 7.5. More precisely, X ⨂ W and V ⨂_{A} Y are negative as graded modules, because X and Y are negative, by hypothesis. In fact, X ⨂_{A} W and V ⨂_{A} Y can be defined as complexes over k, because X and Y are complexes, as in Section 7.5 again.

One can define suitable augmentations on X ⨂_{A} W and V ⨂_{A} Y, so that

(10.12.2) X ⨂_{A}W and V ⨂_{A}Y are left complexes over V ⨂_{A}W,

as in Section 10.1. If one considers V as a complex, as in Section 10.1, then one can consider V ⨂_{A} W as a complex too, as before. In this case, one can get an augmentation map from X ⨂_{A} W into V ⨂_{A} W using the augmentation map ε_{X} on X and the identity mapping on W. This is a map from X ⨂_{A} W into V ⨂_{A} W, as complexes, as in Section 7.6. The augmentation on V ⨂_{A} Y is defined analogously, by considering W as a complex, and using the augmentation ε_{Y} on Y and the identity mapping on V.

Let Ṽ be another right module over A again, and let Ṽ ⨂_{A} W be a tensor product of Ṽ and W, as modules over A. Also let X̃ be a left complex over Ṽ, so that X̃ ⨂_{A} W is a left complex over Ṽ ⨂_{A} W, as before. Suppose that ϕ is a homomorphism from V into Ṽ, as modules over A, and that Φ is a map from X into X̃ over ϕ, as in Section 10.3. Using ϕ and the identity mapping on W, we get a homomorphism

(10.12.3) from V ⨂_{A}W into Ṽ ⨂_{A}W,

as modules over k. Similarly, we can use Φ and the identity mapping on W to get a map (10.12.4) from X ⨂_{A}W into X̃ ⨂_{A}W,

as complexes, as in Section 7.6. One can check that this is a map over the homomorphism as in (10.12.3), as in Section 10.3. Of course, this uses augmentations on X ⨂_{A} W and X̃ ⨂_{A} W as in the preceding paragraph.

Let Φ′ be another map from X into X̃ over ϕ, which leads to a map from X ⨂_{A} W into X̃ ⨂_{A} W, as complexes, as before. Suppose that Φ is homotopic to Φ′, as maps between complexes, as in Section 5.11. This implies that the corresponding maps from X ⨂_{A} W into X̃ ⨂_{A} W are homotopic, as maps between complexes, as in Section 7.6. Of course, there are analogous statements when W̃ is another left module over A, Ỹ is a left complex over W̃, ψ is a homomorphism from W into W̃, as modules over A, and Ψ is a map from Y into Ỹ over ψ. In this case, we consider the corresponding mappings from V ⨂_{A} W into V ⨂_{A} W̃, and from V ⨂_{A} Y into V ⨂_{A} Ỹ.

If n is an integer, then

H(X ⨂_{A}W )_{n} = H(X ⨂_{A}W )^{−n} (10.12.5)

and H(V ⨂_{A}Y )_{n} = H(V ⨂_{A}Y )^{−n} (10.12.6) can be defined as modules over k in the usual way. These are both equal to {0} when n < 0, because X ⨂_{A} W and V ⨂_{A} Y are negative as graded modules. We may be particularly interested in (10.12.5) and (10.12.6) when X and Y are projective resolutions of V and W, respectively. In this case,

(10.12.5) and (10.12.6) are uniquely determined, up to

(10.12.7)

isomorphism, by V and W instead of X and Y, respectively,

for each n, because projective resolutions are unique up to homotopy equivalence, as in Section 10.5. These remarks correspond to some of those on p82-4 of [50] in the left case again, where modules are transformed by taking the tensor product with a fixed module.

If X and Y are projective resolutions of V and W, respectively, then (10.12.5) and (10.12.6) can also be described in terms of “satellites”, which are discussed in Chapter III of [50]. This uses Theorem 6.1 on p90 of [50].

Remember V, W can be used to define complexes in a simple way, as in Section 10.1. Using these complexes, X ⨂_{A} W and V ⨂_{A} Y may be defined as double complexes, as in Sections 6.1 and 6.2. These double complexes lead to single complexes in the usual way, as in Sections 5.13 and 5.14. The single complexes that we get this way are the same as before, using the gradings in (10.12.1), as in Section 7.5.

The complexes associated to V, W as in the preceding paragraph may be considered as left complexes over V, W, respectively, where the augmentation mappings correspond to the identity mappings on these complexes. The augmentation maps ε_{X}, ε_{Y} may be considered as maps from X, Y into the complexes associated to V, W over the identity mappings on V, W, respectively, as in Section 10.3. We can use these maps and the identity mappings on X and Y to get maps (10.12.8) from X ⨂_{A}Y into X ⨂_{A}W

and (10.12.9) from X ⨂_{A}Y into V ⨂_{A}Y,

as double complexes, as in the previous section. These maps induce homomorphisms

from (10.11.2) into (10.12.5)

(10.12.10)

and

from (10.11.2) into (10.12.6)

(10.12.11)

for each n, as in Sections 5.11 and 5.15. This corresponds to the homomorphisms as in (1a) on p97 of [50], in the case of tensor products.

If X and Y are projective resolutions of V and W, respectively, then it is well known that

the homomorphisms as in (10.12.10) and (10.12.11)

(10.12.12)

are isomorphisms for each n. This uses the analogue of Theorem 8.1 on p95 of [50] in the left case. This also uses Proposition 1.1a on p106 of [50] to get the “left balanced” condition condition defined on p97 of [50], which was discussed near the end of Section 2.7. This fact is mentioned on p107 of [50] too, as well as the description in terms of satellites. Of course, (10.12.12) implies that

(10.12.5) and (10.12.6) are isomorphic to each other

(10.12.13)

for each n.

10.13 Homomorphisms and resolutions

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V and W be both left or both right modules over A. Suppose that X is a left complex over V with augmentation ε_{X}, and that Y is a right complex over W with an augmentation ε_{Y}, as in Sections 10.1 and 10.6, respectively.

We can define Hom_{A}^{gr}(X, Y) as a doubly-graded module over k, with

(Hom_{A}^{gr}(X, Y))^{j,l} = Hom_{A}(X^{−j}, Y^{l}) (10.13.1)

for every j, l ∈ Z, as in Section 6.3. This is equal to {0} when j < 0 or l < 0, because X is negative and Y is positive as graded modules, by hypothesis. Remember that there is a single grading on Hom_{A}^{gr}(X, Y) obtained from the double grading as in Section 5.13. Note that Hom_{A}^{gr}(X, Y) is positive with respect to this single grading, because of the analogous property of the double grading just mentioned.

In fact, Hom_{A}^{gr}(X, Y) can be defined as a double complex over k, as in Section 6.5. As usual, the two differentiation operators on Hom_{A}^{gr}(X, Y), as a double complex, can be combined to get a single differentiation operator, which makes Hom_{A}^{gr}(X, Y) a single complex with respect to the single grading mentioned in the previous paragraph, as in Section 5.14. This means that

H(Hom_{A}^{gr}(X, Y))^{n} (10.13.2)

can be defined as a module over k in the usual way for every n ∈ Z. This is equal to {0} when n < 0, because Hom_{A}^{gr}(X, Y) is positive with respect to its single grading.

We may consider V, W as complexes, as in Sections 10.1 and 10.6. We can use this to consider Hom_{A}^{gr}(V, W) as a double complex, which leads to a single complex, as before. This is the same as the single complex obtained from Hom_{A}(V, W), as a module over k, in the usual way. We can use the augmentations ε_{X} and ε_{Y} to get a map

(10.13.3) ε from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(X, Y), as complexes, as in Sections 5.15, 6.4, and 6.5. It follows that

(10.13.4) Hom_{A}^{gr}(X, Y) is a right complex over Hom_{A}(V, W)

with respect to ε,

as in Section 10.6.

Let Ṽ, W̃ be additional left or right modules over A, as appropriate. Also let X̃ be a left complex over Ṽ, and let Ỹ be a right complex over W̃. Thus Hom_{A}^{gr}(X̃, Ỹ) is a right complex over Hom_{A}(V, W), as before. Suppose that ϕ is a homomorphism from Ṽ into V, and that ψ is a homomorphism from W into W̃, as modules over A. Let Φ be a map from X̃ into X over ϕ, and let Ψ be a map from Y into Ỹ over ψ, as in Sections 10.3 and 10.8, respectively.

Using ϕ and ψ, we get a homomorphism

(10.13.5) Hom(ϕ, ψ) from Hom_{A}(V, W) into Hom_{A}(Ṽ, W̃),

as modules over k, as in Section 6.3. Similarly, we can use Φ and Ψ to get a map (10.13.6) Hom^{gr}(Φ, Ψ) from Hom_{A}^{gr}(X, Y) into Hom_{A}^{gr}(X̃, Ỹ),

as double complexes, as in Sections 6.4 and 6.5. One can verify that

(10.13.7) Hom^{gr}(Φ, Ψ) is a map over Hom(ϕ, ψ),

as in Section 10.8. Remember that Hom^{gr}(Φ, Ψ) may also be considered as a map from Hom_{A}^{gr}(X, Y) into Hom_{A}^{gr}(X̃, Ỹ) as single complexes, as in Section 5.15. This means that Hom^{gr}(Φ, Ψ) induces a homomorphism from (10.13.2) into its analogue for Hom_{A}^{gr}(X̃, Ỹ) for each n, as in Section 5.11.

Let Φ′ be another map from X̃ into X over ϕ, and let Ψ′ be another map from Y into Ỹ ove ψ. This leads to a map

(10.13.8) Hom^{gr}(Φ′, Ψ′) from Hom_{A}^{gr}(X, Y) into Hom_{A}^{gr}(X̃, Ỹ),

as double complexes, as before. Suppose that Φ, Ψ are homotopic to Φ′, Ψ′, respectively, as maps between complexes, as in Section 5.11. This implies that

(10.13.9) Hom^{gr}(Φ, Ψ) is homotopic to Hom^{gr}(Φ′, Ψ′),

as maps between double complexes, as in Section 6.5, and thus as maps between single complexes, as in Section 5.15. It follows that Hom^{gr}(Φ, Ψ) and Hom^{gr}(Φ′, Ψ′) induce the same mappings on (10.13.2) for each n, as in Section 5.11.

We may be particularly interested in (10.13.2) when X is a projective resolution of V, and Y is an injective resolution of W. Under these conditions,

(10.13.10) (10.13.2) is uniquely determined by V and W

for each n, up to isomorphism. This uses the uniqueness of projective and injective resolutions up to homotopy equivalence, as in Sections 10.5 and 10.10.

These remarks correspond to some of those on p82f of [50], for spaces of homomorphisms. This is related to some remarks on p107 of [50] as well.

10.14 Left complexes and Hom(·, W)

We return to the same notation and hypotheses as at the beginning of the previous section. Remember that Hom_{A}^{gr}(X, W) can be defined as a graded module over k, with

(Hom_{A}^{gr}(X, W))^{j} = Hom_{A}(X^{−j}, W) (10.14.1)

for every integer j, as in Section 8.4. Note that Hom_{A}^{gr}(X, W) is positive as a graded module, because X is negative, by hypothesis. More precisely, Hom_{A}^{gr}(X, W) can be defined as a complex over k, because X is a complex, as in Section 8.4.

One can define an augmentation on Hom_{A}^{gr}(X, W), so that

(10.14.2) Hom_{A}^{gr}(X, W) is a right complex over Hom_{A}(V, W),

as in Section 10.6. If V is considered to be a complex, as in Section 10.6, then Hom_{A}^{gr}(V, W) may be defined as a complex, as in the preceding paragraph. This is the same as the complex associated to Hom_{A}(V, W) in the usual way. One can get an augmentation map from Hom_{A}(V, W) into Hom_{A}^{gr}(X, W) using the augmentation map ε_{X} on X and the identity mapping on W. This corresponds to a map from Hom_{A}^{gr}(V, W) into Hom_{A}^{gr}(X, W), as complexes, as in Section 8.5.

Let Ṽ be another left or right module over A, as appropriate, and let X̃ be a left complex over Ṽ. Thus Hom_{A}^{gr}(X̃, W) can be defined as a right complex over Hom_{A}(Ṽ, W), as before. Suppose that ϕ is a homomorphism from Ṽ into V, as modules over A, and let Φ be a map from X̃ into X over ϕ, as in Section 10.3. Using ϕ and the identity mapping on W, we get a homomorphism

(10.14.3) from Hom_{A}(V, W) into Hom_{A}(Ṽ, W),

as modules over k, as in Section 6.3. Similarly, we can use Φ and the identity mapping on W to get a map

(10.14.4) from Hom_{A}^{gr}(X, W) into Hom_{A}^{gr}(X̃, W),

as complexes, as in Section 8.5. One can check that this is a map over the homomorphism as in (10.14.3), as in Section 10.8. This uses augmentations on Hom_{A}^{gr}(X, W) and Hom_{A}^{gr}(X̃, W) as in the previous paragraph.

Let Φ′ be another map from X̃ into X over ϕ, which leads to a map from Hom_{A}^{gr}(X, W) into Hom_{A}^{gr}(X̃, W), as complexes, as before. If Φ is homotopic to Φ′, as maps between complexes, as in Section 5.11, then the corresponding maps from Hom_{A}^{gr}(X, W) into Hom_{A}^{gr}(X̃, W) are homotopic, as maps between complexes, as in Section 8.5.

Because Hom_{A}^{gr}(X, W) is a complex over k,

10.15. RIGHT COMPLEXES AND HOM(V, ·) 293

H(Hom_{A}^{gr}(X, W))^{n} (10.14.5) can be defined as a module over k for each integer n in the usual way. This is equal to {0} when n < 0, because Hom_{A}^{gr}(X, W) is positive as a graded module. We may be particularly interested in (10.14.5) when X is a projective resolution of V, in which case

(10.14.6) (10.14.5) is uniquely determined by V

for each n, up to isomorphism, because projective resolutions are unique up to homotopy equivalence, as in Section 10.5. These remarks correspond to some of those on p82f of [50] again.

Note that (10.14.5) can also be described in terms of “satellites” under these conditions, as in Theorem 6.1 on p90 of [50].

If we consider W as a complex in the usual way, as in Section 10.6, then Hom_{A}^{gr}(X, W) can be defined as a double complex, as in Sections 6.3 and 6.5. This double complex leads to a single complex, as in Sections 5.13 and 5.14. This single complex is the same as before, with the grading as in (10.14.1), as in Section 8.4.

The complex associated to W as in the previous paragraph may be considered as a right complex over W, where the augmentation mapping corresponds to the identity mapping on this complex. The augmentation map ε_{Y} may be considered as a map from the complex associated to W into Y over the identity mapping on W, as in Section 10.8. We can use this map and the identity mapping on X to get a map

(10.14.7) from Hom_{A}^{gr}(X, W) into Hom_{A}^{gr}(X, Y),

as double complexes, as in the previous section. This map induces a homomorphism

from (10.14.5) into (10.13.2)

(10.14.8)

for each n, as in Sections 5.11 and 5.15. This corresponds to a homomorphism as in (1) on p94 of [50], in the case of spaces of homomorphisms.

If X is a projective resolution of V, and Y is an injective resolution of W, then it is well known that

the homomorphism as in (10.14.8) is an isomorphism

(10.14.9)

for each n, as in Theorem 8.1 on p95 of [50]. This is mentioned on p107 of [50] as well.

10.15 Right complexes and Hom(V, ·)

Let us return to the same notation and hypotheses as at the beginning of Section 10.13. Note that Hom_{A}^{gr}(V, Y) can be defined as a graded module over k, with

(Hom_{A}^{gr}(V, Y))^{l} = Hom_{A}(V, Y^{l}) (10.15.1) for every integer l, as in Section 7.12. This is positive as a graded module, because Y is positive, by hypothesis. In fact, Hom_{A}^{gr}(V, Y) can be defined as a complex over k, because Y is a complex, as in Section 7.12.

One can define an augmentation on Hom_{A}^{gr}(V, Y), so that

(10.15.2) Hom_{A}^{gr}(V, Y) is a right complex over Hom_{A}(V, W),

as in Section 10.6. If W is considered as a complex, as in Section 10.6, then Hom_{A}^{gr}(V, W) may be defined as a complex, as in the previous paragraph. This is the same as the complex associated to Hom_{A}(V, W) in the analogous way. An augmentation map from Hom_{A}(V, W) into Hom_{A}^{gr}(V, Y) can be obtained from the identity mapping on V and the augmentation map ε_{Y} from W into Y. This corresponds to a map from Hom_{A}^{gr}(V, W) into Hom_{A}(V, Y), as complexes, as in Section 7.13.

Let W̃ be another left or right module over A, as appropriate, and let Ỹ be a right complex over W̃. This means that Hom_{A}^{gr}(V, Ỹ) can be defined as a right complex over Hom_{A}(V, W̃), as in the previous two paragraphs. Let ψ be a homomorphism from W into W̃, as modules over A, and let Ψ be a map from Y into Ỹ over ψ, as in Section 10.8. We can use ψ and the identity mapping on V to get a homomorphism

(10.15.3) from Hom_{A}(V, W) into Hom_{A}(V, W̃),

as modules over k, as in Section 6.3. We can also use Ψ and the identity mapping on V to get a map

(10.15.4) from Hom_{A}^{gr}(V, Y) into Hom_{A}^{gr}(V, Ỹ),

as complexes, as in Section 7.13. One can verify that this is a map over the homomorphism as in (10.15.3), as in Section 10.8. This uses the augmentations on Hom_{A}^{gr}(V, Y) and Hom_{A}^{gr}(V, Ỹ) that are defined as in the preceding paragraph.

Let Ψ′ be another map from Y into Ỹ over ψ, which leads to a map from Hom_{A}^{gr}(V, Y) into Hom_{A}^{gr}(V, Ỹ), as complexes, as before. If Ψ is homotopic to Ψ′, as maps between complexes, as in Section 5.11, then the corresponding maps from Hom_{A}^{gr}(V, Y) into Hom_{A}^{gr}(V, Ỹ) are homotopic, as maps between complexes, as in Section 7.13.

We can define H(Hom_{A}^{gr}(V, Y))^{n} (10.15.5)

as a module over k for every integer n in the usual way, because Hom_{A}^{gr}(V, Y) is a complex over k. This module is {0} when n < 0, because Hom_{A}^{gr}(V, Y) is positive, as a graded module. We may be particularly interested in this module when Y is an injective resolution of W, in which case

(10.15.6) (10.15.5) is uniquely determined by W

for each n, up to isomorphism, because injective resolutions are unique up to homotopy equivalence, as in Section 10.10. These remarks correspond to some of those on p82f of [50].

Under these conditions, (10.15.5) can be described in terms of “satellites” too, as in Theorem 6.1 on p90 of [50].

We may consider V as a complex in the usual way, as in Section 10.6, so that Hom_{A}^{gr}(V, Y) may be defined as a double complex, as in Sections 6.3 and 6.5. Using this double complex, we get a single complex, as in Sections 5.13 and 5.14. This single complex is the same as at the beginning of the section, with the grading as in (10.15.1), as in Section 7.12.

The complex associated to V as in the preceding paragraph may be considered as a left complex over V, where the augmentation mapping corresponds to the identity mapping on this complex. The augmentation map ε_{X} may be considered as a map from X into the complex associated to V over the identity mapping on V, as in Section 10.3. We can use this map and the identity mapping on Y to get a map

(10.15.7) from Hom_{A}^{gr}(V, Y) into Hom_{A}^{gr}(X, Y),

as double complexes, as in Section 10.13. This map induces a homomorphism

from (10.15.5) into (10.13.2)

(10.15.8)

for each n, as in Sections 5.11 and 5.15. This corresponds to a homomorphism as in (1) on p94 of [50] again, in the case of spaces of homomorphisms.

If X is a projective resolution of V, and Y is an injective resolution of W, then

the homomorphism as in (10.15.8) is an isomorphism

(10.15.9)

for each n, as in Theorem 8.1 on p95 of [50]. This is also mentioned on p107 of

Integral domains

11.1 Torsion and divisible elements

Let k be an integral domain, which is to say a commutative ring with a nonzero multiplicative identity element such that the product of any two nonzero elements of k is not zero too. Sometimes an integral domain is simply called a domain, as on p12 of [92], or an entire ring, as on p61 of [177]. Also let V be a module over k.

An element v of V is said to be a torsion element of V if there is a t ∈ k such that t ≠ 0 and t · v = 0. (11.1.1)

The collection τ(V) of torsion elements of V is a submodule of V, as in Exercise 12 on p45 of [8], and as mentioned on p127 of [50].

If τ(V) = V, (11.1.2)

then V is said to be a torsion module, as a module over k. It is easy to see that τ(V) is a torsion module, as on p127 of [50].

If τ(V) = {0}, (11.1.3)

then V is said to be torsion-free, as a module over k. One can check that

V/τ(V) (11.1.4)

is torsion-free as a module over k, as in the first part of Exercise 12 on p45 of [8], and as mentioned on p127 of [50]. Note that submodules of torsion-free modules are torsion-free as well.

Let I be a nonempty set, and let V_{j} be a module over k for each j ∈ I. One can verify that

τ( ⨁_{j∈I} V_{j}) = ⨁_{j∈I} τ(V_{j}), (11.1.5)

as mentioned on p127 of [50].

Of course, k is torsion-free as a module over itself, by hypothesis. Thus free modules over k are torsion-free.

If V is projective as a module over k, then V is isomorphic to a submodule of a free module over k, as in Section 2.7. This implies that V is torsion-free, as in Proposition 1.1 on p127 of [50].

An element v of V is said to be divisible if for every t ∈ k with t ≠ 0 there is a w ∈ V such that v = t · w. (11.1.6)

The collection δ(V) of divisible elements of V is a submodule of V, as on p127f of [50].

If δ(V) = V, (11.1.7)

then V is said to be divisible, as a module over k. In this case, it is easy to see that every quotient of V is divisible too, as on p128 of [50].

One can check that δ(V/δ(V)) = {0}, (11.1.8)

as mentioned on p128 of [50]. More precisely, let q be the natural quotient mapping from V onto V/δ(V). If v ∈ V and q(v) is divisible in V/δ(V), then one can verify that v is divisible in V.

Let I be a nonempty set again, and let V_{j} be a module over k for each j ∈ I. One can check that

δ( ∏_{j∈I} V_{j}) = ∏_{j∈I} δ(V_{j}), (11.1.9)

as on p128 of [50].

11.2 More on torsion and divisibility

Let k be an integral domain again, and let V be a module over k. If V is injective as a module over k, then V is divisible as a module over k, as in Proposition 1.2 on p128 of [50]. To see this, let v ∈ V and t_{0} ∈ k with t_{0} ≠ 0 be given, and note that t_{0} k = {t_{0} x : x ∈ k} is an ideal in k. If t ∈ t_{0} k, then we would like to put

ϕ(t t_{0}) = t · v. (11.2.1)

It is easy to see that this is well defined, because k has no nontrivial zero divisors.

More precisely, ϕ defines a homomorphism from t_{0} k into V, as modules over k. Because V is injective, there is a w ∈ V such that

ϕ(t t_{0}) = (t t_{0}) · w (11.2.2)

for every t ∈ k, as in Section 2.8. It follows that

v = t_{0} · w, (11.2.3) as desired.

If V is torsion-free and divisible, then V is injective as a module over k, as in Proposition 1.3 on p128 of [50]. Indeed, let I be an ideal in k, and let ϕ be a homomorphism from I into V, as modules over k. It suffices to show that there is a w ∈ V such that ϕ(t) = t · w (11.2.4)

for every t ∈ I, as in Section 2.8. Of course, if I = {0}, then we can take w = 0, and so we may suppose that I ≠ {0}.

If t ∈ I and t ≠ 0, then there is a unique w_{t} ∈ V such that

ϕ(t) = t · w_{t}, (11.2.5)

because V is torsion-free and divisible. If t_{1}, t_{2} ∈ I and t_{1}, t_{2} ≠ 0, then

(11.2.6) (t_{1} t_{2}) · w_{t_{1}} = t_{2} · (t_{1} · w_{t_{1}}) = t_{2} · ϕ(t_{1}) = ϕ(t_{1} t_{2})
= t_{1} · ϕ(t_{2}) = t_{1} · (t_{2} · w_{t_{2}}) = (t_{1} t_{2}) · w_{t_{2}}.

This implies that w_{t_{1}} = w_{t_{2}}, because t_{1} t_{2} ≠ 0 in k, and V is torsion-free. If we let w be the common value of w_{t}, t ∈ I, t ≠ 0, then (11.2.4) holds for every t ∈ I, as desired.

If t ∈ k, then v ↦ t · v (11.2.7)

defines a homomorphism from V into itself, as a module over k. The condition that V be torsion-free says exactly that this mapping is injective when t ≠ 0. Similarly, V is divisible exactly when (11.2.7) maps V onto itself when t ≠ 0.

Let W be another module over k, and let V ⨂_{k} W be a tensor product of V and W, as modules over k. If t ∈ k, then the action of t on V ⨂_{k} W is the unique homomorphism from V ⨂_{k} W into itself, as a module over k, such that

v ⊗ w ↦ t · (v ⊗ w) (11.2.8)

for every v ∈ V and w ∈ W. This is the same as the homomorphism from V ⨂_{k} W into itself, as a module over k, corresponding to (11.2.7) and the identity mapping on W.

If V is divisible, then (11.2.7) maps V onto itself when t ≠ 0, as before. This implies that the action of t on V ⨂_{k} W is surjective when t ≠ 0, as in Section 1.3. It follows that (11.2.9) V ⨂_{k}W is divisible

as a module over k in this case, as in Corollary 1.5 on p128 of [50]. Similarly, (11.2.9) holds when W is divisible, as a module over k.

If V is torsion-free and divisible, then (11.2.7) defines an isomorphism from V onto itself, as a module over k, when t ≠ 0. This implies that the action of t on V ⨂_{k} W is an isomorphism from V ⨂_{k} W onto itself, as a module over k, when t ≠ 0, as in Section 1.3. This means that

(11.2.10) V ⨂_{k}W is torsion-free and divisible as a module over k, as in the last part of Proposition 1.4 on p128 of [50]. This also works when W is torsion-free and divisible.

If W is torsion-free as a module over k, then it is easy to see that

(11.2.11) Hom_{k}(V, W) is torsion-free,

as a module over k. One can check that this also holds when V is divisible, using the surjectivity of (11.2.7) when t ≠ 0. This is the second part of Corollary 1.5 on p128 of [50].

If V is torsion-free and divisible, then the action of t on Hom_{k}(V, W) is an isomorphism when t ≠ 0, because (11.2.7) is an isomorphism from V onto itself. This implies that

(11.2.12) Hom_{k}(V, W) is torsion-free and divisible,

as a module over k. Similarly, if W is torsion-free and divisible, then the action of t on W is an isomorphism from W onto itself, as a module over k, when t ≠ 0. This implies that the action of t on Hom_{k}(V, W) is an isomorphism from Hom_{k}(V, W) onto itself, as a module over k, when t ≠ 0, so that (11.2.12) holds. This corresponds to the last part of Proposition 1.4 on p128 of [50] again.

If V is a torsion module and W is divisible, then

V ⨂_{k}W = {0}, (11.2.13)

as in Proposition 1.8 on p129 of [50]. To see this, let v ∈ V and w ∈ W be given, and let us check that v ⊗ w = 0 (11.2.14)

in V ⨂_{k} W. Because V is a torsion module, there is a t ∈ k such that t ≠ 0 and t · v = 0. There is also a u ∈ W such that w = t · u, because W is divisible. It follows that v ⊗ w = v ⊗ (t · u) = (t · v) ⊗ u = 0, (11.2.15)

as desired.

If ϕ is a homomorphism from V into W, as modules over k, then it is easy to see that ϕ(τ(V)) ⊆ τ(W), (11.2.16)

as in the second part of Exercise 12 on p45 of [8]. Similarly,

ϕ(δ(V)) ⊆ δ(W). (11.2.17)

If V is a torsion module and W is torsion-free, then (11.2.16) implies that

ϕ = 0. (11.2.18)

This means that Hom_{k}(V, W) = {0} (11.2.19)

under these conditions, as in Proposition 1.8 on p129 of [50].

11.3 Some easy homomorphisms

Let k be an integral domain, and let V be a module over k. If v_{1}, …, v_{n} are finitely many torsion elements of V, then it is easy to see that there is a t ∈ k such that t ≠ 0 and t · v_{j} = 0 (11.3.1)

for every j = 1, …, n. Of course, this implies that t · v = 0 for every v in the submodule of V generated by v_{1}, …, v_{n}. If V is finitely generated and torsion, as a module over k, then there is a t ∈ k such that t ≠ 0 and

(11.3.2) t · v = 0 for every v ∈ V.

This corresponds to part of the proof of Proposition 1.6 on p128 of [50].

If t ∈ k, then t · V = {t · v : v ∈ V } (11.3.3)

is a submodule of V. Similarly,

tV = {v ∈ V : t · v = 0} (11.3.4)

is a submodule of V. Put V_{t} = V/(t · V), (11.3.5)

which is another module over k.

Note that x ↦ t x (11.3.6)

defines a homomorphism from k into itself, as a module over itself. This homomorphism is injective when t ≠ 0, because k is an integral domain.

Remember that V satisfies the requirements of a tensor product of k with V, as modules over k, as in Section 1.4. Using (11.3.6) and the identity mapping on V, we get a homomorphism

(11.3.7) from k ⨂_{k}V into itself,

as a module over k, as in Section 1.3. This corresponds to v ↦ t · v, as a homomorphism from V into itself. Of course, (11.3.4) is the kernel of this homomorphism on V.

If v ∈ V, then y ↦ y · v (11.3.8)

defines a homomorphism from k into V, as modules over k. Every homomorphism from k into V, as modules over k, corresponds to a unique v ∈ V in this way. The restriction of (11.3.8) to y ∈ t k defines a homomorphism from t k into V, as modules over k. This homomorphism is equal to 0 on t k exactly when v ∈ _{t}V.

Suppose that t ≠ 0, and let ϕ be a homomorphism from t k into V, as modules over k. Thus ϕ(t x) = x · ϕ(t) (11.3.9) for every x ∈ k. Note that ϕ(t) can be any element of V. It is easy to see that (11.3.8) is equal to ϕ on t k if and only if

ϕ(t) = t · v. (11.3.10)

In particular, ϕ has an extension to a homomorphism from k into V, as modules over k, exactly when ϕ(t) ∈ t · V.

These remarks correspond to some of those on p129 of [50].

11.4 Fields of fractions

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions, or quotients. We may identify k with a subring of Q_{k}, as usual. Note that Q_{k} is torsion-free and divisible, and thus injective, as a module over k. It follows that the quotient Q_{k}/k is divisible, as a module over k.

If t ∈ k and t ≠ 0, then

(1/t) k = {x/t : x ∈ k} (11.4.1)

is a submodule of Q_{k}, as a module over k. We also have that

Q_{k} = ⋃{(1/t) k : t ∈ k, t ≠ 0}, (11.4.2)

by the definition of Q_{k}.

Note that (1/r) k, (1/t) k ⊆ (1/(r t)) k (11.4.3)

when r, t ∈ k and r, t ≠ 0. More precisely, if t_{1}, t_{2} ∈ k and t_{1}, t_{2} ≠ 0, then

(1/t_{1}) k ⊆ (1/t_{2}) k (11.4.4)

if and only if 1/t_{1} ∈ (1/t_{2}) k, which means that

t_{2} = r t_{1} (11.4.5)

for some r ∈ k.

Put t_{1} ⪯ t_{2} (11.4.6)

when t_{1}, t_{2} ∈ k \ {0} satisfy (11.4.5) for some r ∈ k. This defines a pre-order on k \ {0}, which makes k \ {0} a directed system, because

t_{1}, t_{2} ⪯ t_{1} t_{2} (11.4.7)

for every t_{1}, t_{2} ∈ k \ {0}.

One may identify Q_{k}, as a module over k, with the direct limit of the submodules (11.4.1), as in Exercise 12 on p45 of [8], and mentioned on p130 of [50]. More precisely, one may consider these submodules to be indexed by k \ {0}, or simply ordered by inclusion, as in Section 3.3.

Let V be a module over k, and let Q_{k} ⨂_{k} V be a tensor product of Q_{k} and V, as modules over k. There is a natural homomorphism

(11.4.8) ϕ from V into Q_{k} ⨂_{k}V,

as modules over k, with ϕ(v) = 1 ⊗ v (11.4.9)

for every v ∈ V. If we identify V with k ⨂_{k} V, as in Section 1.4, then ϕ corresponds to the homomorphism from k ⨂_{k} V into Q_{k} ⨂_{k} V associated to the natural inclusion mapping from k into Q_{k} and the identity mapping on V, as in Section 1.3. We would like to show that

ker ϕ = τ(V), (11.4.10)

as in the fourth part of Exercise 12 on p45 of [8], and Proposition 2.1 on p130 of [50].

If t ∈ k and t ≠ 0, then let ((1/t) k) ⨂_{k} V be a tensor product of (1/t) k and V, as modules over k. There is a natural homomorphism

(11.4.11) from ((1/t) k) ⨂_{k}V into Q_{k} ⨂_{k}V,

as modules over k, associated to the obvious inclusion mapping from (1/t) k into Q_{k} and the identity mapping on V, as in Section 1.3. There is also a natural homomorphism (11.4.12) ϕ_{t} from V into ((1/t) k) ⨂_{k}V,

as modules over k, with ϕ_{t}(v) = 1 ⊗ v (11.4.13)

for every v ∈ V. As before, ϕ_{t} corresponds to the homomorphism from k ⨂_{k} V into ((1/t) k) ⨂_{k} V associated to the natural inclusion mapping from k into (1/t) k and the identity mapping on V. Observe that

(11.4.14) ϕ is the same as the composition of ϕ_{t} with the homomorphism as in (11.4.11).

In particular, ker ϕ_{t} ⊆ ker ϕ. (11.4.15)

In fact, ker ϕ = ⋃{ker ϕ_{t} : t ∈ k, t ≠ 0}. (11.4.16)

This can be seen by identifying Q_{k} ⨂_{k} V with the direct limit of ((1/t) k) ⨂_{k} V, as in Section 3.6.

Let t ∈ k \ {0} be given, and note that

(1/t) x ↦ x (11.4.17) defines an isomorphism from (1/t) k onto k, as modules over k. This leads to an isomorphism (11.4.18) ψ_{t} from ((1/t) k) ⨂_{k}V onto k ⨂_{k}V,

as modules over k, using the identity mapping on V, as in Section 1.3. This may be considered as an isomorphism from ((1/t) k) ⨂_{k} V onto V, by identifying k ⨂_{k} V with V, as before. It follows that

ker ϕ_{t} = ker(ψ_{t} ∘ ϕ_{t}). (11.4.19)

One can check that ψ_{t} ∘ ϕ_{t} corresponds to

v ↦ t · v, (11.4.20)

as a homomorphism from V into itself. Thus

ker(ψ_{t} ∘ ϕ_{t}) = {v ∈ V : t · v = 0}. (11.4.21)

This implies (11.4.10), as desired.

11.5 Some injective mappings

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions. If V is a finitely-generated torsion-free module over k, then there is an injective homomorphism from V into a free module over k with a finite basis, as in Proposition 2.4 on p131 of [50]. To see this, let Q_{k} ⨂_{k} V be a tensor product of Q_{k} and V, as modules over k, as in the previous section. Thus (11.4.9) defines an injective homomorphism ϕ from V into Q_{k} ⨂_{k} V, as modules over k, as before.

Note that Q_{k} ⨂_{k} V may be considered as a module over Q_{k}, as in Section 1.10. This means that Q_{k} ⨂_{k} V may be considered as a vector space over Q_{k}, because Q_{k} is a field. Let v_{1}, …, v_{n} be generators for V, as a module over k. It is easy to see that Q_{k} ⨂_{k} V is spanned by ϕ(v_{1}), …, ϕ(v_{n}), as a vector space over Q_{k}. In particular, Q_{k} ⨂_{k} V has finite dimension, as a vector space over Q_{k}.

Let e_{1}, …, e_{m} be a basis for Q_{k} ⨂_{k} V, as a vector space over Q_{k}. Thus, for each j = 1, …, n, we can express ϕ(v_{j}) as

ϕ(v_{j}) = ∑_{l=1}^{m} q_{j,l} e_{l}, (11.5.1)

where q_{j,l} ∈ Q_{k} for every l = 1, …, m. It is easy to see that there is a t ∈ k such that t ≠ 0 and t q_{j,l} ∈ k (11.5.2)

for every j = 1, …, n and l = 1, …, m. Of course, (11.5.1) is the same as saying that

ϕ(v_{j}) = ∑_{l=1}^{n} t q_{j,l} (t^{−1} e_{l}) (11.5.3)

for every j = 1, …, n.

Let W be the submodule of Q_{k} ⨂_{k} V, as a module over k, generated by

t^{−1} e_{1}, …, t^{−1} e_{m}. (11.5.4)

Observe that W is freely generated by (11.5.4), because e_{1}, …, e_{m} is a basis for Q_{k} ⨂_{k} V, as a vector space over Q_{k}. We also have that

ϕ(V) ⊆ W, (11.5.5)

because ϕ(v_{j}) ∈ W for each j = 1, …, n. This means that ϕ may be considered as a homomorphism from V into W, as modules over k, as desired.

Let U be a projective module over k, and let U_{0} be a submodule of U that is also projective, as a module over k. Let V be another module over k, and let U_{0} ⨂_{k} V, U ⨂_{k} V be tensor products of U_{0}, U with V, respectively, as modules over k. This leads to a homomorphism

(11.5.6) from U_{0} ⨂_{k}V into U ⨂_{k}V,

as modules over k, using the obvious inclusion mapping from U_{0} into U, and the identity mapping on V. If V is torsion-free, then Proposition2.5 on p131 of [50] states that

(11.5.7) the homomorphism as in (11.5.6) is injective.

To show this, we start with the case where V is finitely generated as a module over k, so that there is an injective homomorphism from V into a free module W over k, as before. Let U_{0} ⨂_{k} W, U ⨂_{k} W be tensor products of U_{0}, U with W, respectively, as modules over k. We get homomorphisms

(11.5.8) from U_{0} ⨂_{k}V into U_{0} ⨂_{k}W

and (11.5.9) from U ⨂_{k}V into U ⨂_{k}W,

as modules over k, using the identity mappings on U_{0}, U, respectively, and the homomorphism from V into W mentioned before. We also get a homomorphism

(11.5.10) from U_{0} ⨂_{k}W into U ⨂_{k}W,

as modules over k, using the obvious inclusion mapping from U_{0} into U, and the identity mapping on W.

Under these conditions, each of the three homomorphisms mentioned in the preceding paragraph is injective, by the remarks near the end of Section 2.7. This uses the projectivity of U_{0}, U, and W, and the injectivity of the inclusion mapping from U_{0} into U, and the homomorphism from V into W being considered.

It is easy to see that the composition of the homomorphism as in (11.5.6) with the homomorphism as in (11.5.9) is the same as the composition of the homomorphism as in (11.5.8) with the homomorphism as in (11.5.10). More precisely, both compositions are the same as the homomorphism

(11.5.11) from U_{0} ⨂_{k}V into U ⨂_{k}W,

as modules over k, obtained from the obvious inclusion mapping from U_{0} into U, and the homomorphism from V into W mentioned earlier. One can use this and the injectivity conditions mentioned in the previous paragraph to get (11.5.7) in this case.

If V is not necessarily finitely generated as a module over k, then one may identify it with a direct limit of finitely-generated submodules, as in Section 3.12. We may also identify U_{0} ⨂_{k} V, U ⨂_{k} V with direct limits of tensor products of U_{0}, U with these finitely-generated submodules of V, as in Section 3.6. Of course, these submodules of V are torsion-free too. One can use this to obtain (11.5.7) from the analogous statement for finitely-generated torsion-free modules.

11.6 Another injective mapping

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions again. If Y is any module over k, then there is an injective homomorphism from Y into a divisible module over k, as in Proposition 2.6 on p132 of [50]. To show this, one can start with a homomorphism η from a torsion-free module V over k onto Y. More precisely, one can take V to be a free module over k, as in Section 2.7, for instance.

Let Q_{k} ⨂_{k} V be a tensor product of Q_{k} and V, as modules over k, and let ϕ be the homomorphism from V into Q_{k} ⨂_{k} V with ϕ(v) = 1 ⊗ v for every v ∈ V, as in Section 11.4. Thus ϕ is injective, because V is torsion-free, as before. If V is a free module over k, then Q_{k} ⨂_{k} V is a direct sum of a family of copies of Q_{k}, and the injectivity of ϕ can be seen more directly.

Using ϕ, we get a homomorphism

(11.6.1) from V/ ker η into (Q_{k} ⨂_{k}V )/ϕ(ker η),

as modules over k. It is easy to see that this homomorphism is injective, because ϕ is injective. Of course,

ϕ(ker η) = ker(ϕ ∘ η), (11.6.2)

because ϕ is injective. Note that Q_{k} ⨂_{k} V is divisible as a module over k, because Q_{k} is divisible as a module over k, as in Section 11.2. This implies that

(11.6.3) (Q_{k} ⨂_{k}V )/ϕ(ker η) is divisible

as a module over k, as in Section 11.1.

Using η, we get an isomorphism

(11.6.4) from V/ ker η onto Y,

as modules over k. Thus the homomorphism as in (11.6.1) leads to an injective homomorphism (11.6.5) from Y into (Q_{k} ⨂_{k}V )/ϕ(ker η),

as modules over k, as desired.

Now let k be any commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a left module over A. It is well known that V is projective as a module over A if and only if there is a family {v_{j}}_{j∈I} of elements of V and a family {ϕ_{j}}_{j∈I} of homomorphisms from V into A, as left modules over A, with the following properties: for every v ∈ V,

(11.6.6) ϕ_{j}(v) = 0 for all but finitely many j ∈ I,

and

v = ∑_{j∈I} ϕ_{j}(v) · v_{j}. (11.6.7)

This is Proposition 3.1 on p132 of [50]. Of course, there is an analogous statement for right modules over A.

If V is projective, then we can start with a free left module U over A and a homomorphism ψ from U onto V, as modules over A, as in Section 2.7. Projectivity of V implies that there is a homomorphism ϕ from V into U, as modules over A, such that ψ ∘ ϕ is the identity mapping on V. Let {e_{j}}_{j∈I} be a basis for U as a free left module over A, and put

v_{j} = ψ(e_{j}) (11.6.8)

for each j ∈ I. Every u ∈ U can be expressed in a unique way as

u = ∑_{j∈I} u_{j} · e_{j}, (11.6.9)

where u_{j} ∈ A for every j ∈ I, and u_{j} = 0 for all but finitely many j ∈ I. If v ∈ V, then we can take u = ψ(v), to get that

v = ϕ(ψ(v)) = ∑_{j∈I}(ψ(v))_{j} v_{j}, (11.6.10)

with (ψ(v))_{j} = 0 for all but finitely many j ∈ I.

More precisely, in this argument, we can start with any family {v_{j}}_{j∈I} of elements of V such that V is generated by the v_{j}’s, j ∈ I, as a left module over A. We can take U to be a free left module over A with a basis {e_{j}}_{j∈I}, using the same set I. In this case, there is a unique homomorphism ψ from U into V, as left modules over A, that satisfies (11.6.8), and which maps U onto V.

Conversely, suppose that there are families {v_{j}}_{j∈I} of elements of V and {ϕ_{j}}_{j∈I} of homomorphisms from V into A that satisfy (11.6.6) and (11.6.7). Let U be a free left module over A with basis {e_{j}}_{j∈I}, for some family of elements indexed by the same set I. Thus there is a unique homomorphism ψ from U onto V, as left modules over A, as in (11.6.8). Note that

ϕ(v) = ∑_{j∈I} ϕ_{j}(v) · e_{j} (11.6.11)

defines a homomorphism from V into U, as left modules over A. If v ∈ V, then

ψ(ϕ(v)) = ∑_{j∈I} ϕ_{j}(v) v_{j} = v, (11.6.12)

by (11.6.7). This implies that ϕ is injective, and that U corresponds to the direct sum of ϕ(V) and ker ψ, as a left module over A. It follows that V is projective, as in Section 2.7.

11.7 Fractional and invertible ideals

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions. A submodule M of Q_{k}, as a module over k, is called a fractional ideal of k if there is an x ∈ k such that x ≠ 0 and

x M ⊆ k, (11.7.1)

as on p96 of [8]. The ordinary ideals in k satisfy this condition with x = 1, and may be called integral ideals of k.

If (11.7.1) holds for some x ∈ Q_{k} with x ≠ 0, then it is easy to see that M is a fractional ideal of k. More precisely, one can multiply x by a suitable nonzero element of k, to get another nonzero element of k.

If M is a fractional ideal of k, then we put

(k : M) = {x ∈ Q_{k} : x M ⊆ k}, (11.7.2)

as on p96 of [8]. It is easy to see that this is a submodule of Q_{k}, as a module over k. Note that (k : M) ≠ {0}, by the definition of a fractional ideal.

If y ∈ M, then y (k : M) ⊆ k, (11.7.3)

by the definition of (k : M). This implies that (k : M) is a fractional ideal of k when M ≠ {0}.

Let M be a submodule of Q_{k}, as a module over k. If M is finitely generated as a module over k, then M is a fractional ideal of k, as on p96 of [8]. Indeed, suppose that M is generated by y_{1}, …, y_{n} ∈ Q_{k} as a module over k. It is easy to see that there are x, z_{1}, …, z_{n} ∈ k such that

y_{j} = z_{j}/x (11.7.4) for each j = 1, …, n. This implies that (11.7.1) holds, as desired.

Of course, if k is Noetherian as a ring, then every integral ideal of k is finitely generated as a module over k. This implies that every fractional ideal of k is finitely generated as a module over k, as on p96 of [8].

Let M, N be submodules of Q_{k}, as a module over k. The product M N is defined to be the subset of Q_{k} consisting of finite sums of products of elements of M and N. This is a submodule of Q_{k}, as a module over k, as well. Note that multiplication in this sense is commutative and associative, with k as the identity element.

We say that M is an invertible ideal of k if there is a submodule N of Q_{k}, as a module over k, such that M N = k, (11.7.5)

as on p96 of [8]. In particular, this implies that M, N ≠ {0}, so that M, N are fractional ideals of k. We also get that

N ⊆ (k : M), (11.7.6)

and that k = M N ⊆ M (k : M) ⊆ k. In fact, we have that

(k : M) ⊆ (k : M) M N ⊆ k N = N, (11.7.7)

so that N = (k : M), (11.7.8)

as in [8].

If M is an invertible ideal, then there are finitely many elements x_{1}, …, x_{n} of M and y_{1}, …, y_{n} of (k : M) such that

∑_{j=1}^{n} x_{j} y_{j} = 1. (11.7.9)

This implies that

x = ∑_{j=1}^{n}(x y_{j}) x_{j} (11.7.10)

for every x ∈ M, with x y_{j} ∈ k for each j = 1, …, n, because y_{j} ∈ (k : M). It follows that M is generated by x_{1}, …, x_{n} as a module over k, and in particular that M is finitely generated as a module over k, as on p96 of [8].

Conversely, if M is a fractional ideal of k, and there are x_{1}, …, x_{n} ∈ M and y_{1}, …, y_{n} ∈ (k : M) such that (11.7.9) holds, then it is easy to see that M (k : M) = k, so that M is invertible.

11.8 Inversible ideals

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions. An ideal I of k is said to be inversible if there are finitely many elements a_{1}, …, a_{n} of I and q_{1}, …, q_{n} of Q_{k} such that

q_{j} I ⊆ k (11.8.1) for every j = 1, …, n, and

∑_{j=1}^{n} q_{j} a_{j} = 1, (11.8.2)

as on p132 of [50]. This is the same as the characterization of the invertibility of I as a fractional ideal of k mentioned in the previous section. Although these conditions are practically the same, it is convenient to use the slightly different terms to reflect the slightly different situations in which they are used. Note that I ≠ {0} in this case.

Proposition 3.2 on p132 of [50] states that a nonzero ideal I in k is inversible if and only if it is projective as a module over k. Suppose that I is inversible, and let a_{1}, …, a_{n} and q_{1}, …, q_{n} be as in the preceding paragraph. Put

ϕ_{j}(x) = q_{j} x (11.8.3)

for each j = 1, …, n and x ∈ I, which defines a homomorphism from I into k, as modules over k. If x ∈ I, then

x = ∑_{j=1}^{n} q_{j} x a_{j} = ∑_{j=1}^{n} ϕ_{j}(x) a_{j}, (11.8.4)

using (11.8.2) in the first step. This implies that I is projective as a module over k, as in Section 11.6.

Before showing the converse, let us mention a helpful fact. Let I be a nonzero ideal in k, and let ϕ be a homomorphism from I into k. If x, y ∈ I, then

x ϕ(y) = ϕ(x y) = y ϕ(x). (11.8.5)

It follows that there is a q ∈ Q_{k} such that

q = ϕ(x)/x (11.8.6)

for every x ∈ I with x ≠ 0. This means that

ϕ(x) = q x (11.8.7)

for every x ∈ I.

Suppose now that I is a nonzero ideal in k that is projective as a module over k. This implies that there is a nonempty family {a_{j}}_{j∈L} of elements of I and a family {ϕ_{j}}_{j∈L} of homomorphisms from I into k, as modules over k, such that for every x ∈ I, ϕ_{j}(x) = 0 for all but finitely many j ∈ L, and

x = ∑_{j∈L} ϕ_{j}(x) a_{j}, (11.8.8)

as in Section 11.6. Using the remarks in the preceding paragraph, we get that for each j ∈ L, there is a unique q_{j} ∈ Q_{k} such that (11.8.3) holds for every x ∈ I. Note that q_{j} = 0 for all but finitely many j ∈ L, because (11.8.3) is supposed to be equal to 0 for all but finitely many j ∈ L, and we can take x ≠ 0. We may as well replace L with the finite subset of j ∈ L such that q_{j} ≠ 0.

It follows from (11.8.8) that

x = ∑_{j∈L} x q_{j} a_{j} = x ∑_{j∈L} q_{j} a_{j} (11.8.9)

for every x ∈ I. Taking x ≠ 0, we obtain that

∑_{j∈L} q_{j} a_{j} = 1. (11.8.10)

Note that (11.8.1) holds for every j ∈ L, because ϕ_{j}(I) ⊆ k.

Proposition 3.3 on p132 of [50] states that inversible ideals in k are finitely generated as modules over k. This was mentioned already in the previous section, for invertible fractional ideals of k. One could also have used this in the previous argument, to get L to be finite at the beginning.

Let I be an ideal in k, and let V be a module over k. There is a natural homomorphism (11.8.11) from Hom_{k}(k, V) into Hom_{k}(I, V),

as modules over k, which sends a homomorphism from k into V, as modules over k, to its restriction to I. If I is inversible and V is divisible, then Proposition 3.4 on p133 of [50] states that

(11.8.12) the homomorphism as in (11.8.11) is surjective.

Equivalently, let ϕ be a homomorphism from I into V, as modules over k. We would like to show that there is a v ∈ V such that

ϕ(x) = x · v (11.8.13)

for every x ∈ I.

Because I is inversible, there are a_{1}, …, a_{n} ∈ I and q_{1}, …, q_{n} ∈ Q_{k} satisfying (11.8.1) and (11.8.2). Let us choose, for each j = 1, …, n, an element v_{j} of V such that ϕ(a_{j}) = a_{j} · v_{j}. (11.8.14)

More precisely, we can take v_{j} = 0 when a_{j} = 0, and otherwise use the divisibility of V. We may also require that a_{j} ≠ 0 for each j, by dropping and a_{j}’s equal to 0.

If x ∈ I, then

ϕ(x) = ϕ( ∑_{j=1}^{n} q_{j} x a_{j}) = ∑_{j=1}^{n}(q_{j} x) · ϕ(a_{j}), (11.8.15)

using (11.8.2) in the first step, and the fact that q_{j} x ∈ k for each j = 1, …, n in the second step. This implies that

ϕ(x) = ∑_{j=1}^{n}(q_{j} x) · (a_{j} · v_{j}) = ∑_{j=1}^{n}(q_{j} x a_{j}) · v_{j}, (11.8.16)

using (11.8.14) in the first step. It follows that

ϕ(x) = x · ( ∑_{j=1}^{n}(q_{j} a_{j}) · v_{j}), (11.8.17)

because q_{j} a_{j} ∈ k for every j = 1, …, n. Thus (11.8.13) holds, with v = ∑_{j=1}^{n}(q_{j} a_{j}) · v_{j}.

11.9 Prüfer rings

Let k be a commutative ring with a multiplicative identity element. As in Section 9.5, we say that k is semi-hereditary if every ideal in k that is finitely generated as a module over k is projective as a module over k.

Suppose now that k is an integral domain. In this case, k is semi-hereditary if and only if every nonzero ideal in k that is finitely generated as a module over k is inversible, because inversibility is equivalent to projectivity, as in the previous section. Under these conditions, k is said to be a Prüfer ring, as on p133 of [50].

Proposition 4.1 on p133 of [50] states that k is a Prüfer ring if and only if

(11.9.1) every finitely-generated torsion-free module over k is projective.

Note that ideals in k are torsion-free as modules over k, because k is an integral domain. Thus (11.9.1) implies that k is semi-hereditary.

Conversely, let V be a finitely-generated torsion-free module over k. This implies that there is an injective homomorphism from V into a free module over k with a finite basis, as in Section 11.5. Thus we may identify V with a submodule of a free module over k. If k is semi-hereditary, then every finitelygenerated submodule of a free module over k is projective, as in Section 9.5. This means that V is projective as a module over k, as desired.

Suppose that k is a Prüfer ring, and let V be a torsion-free module over k. Also let U_{1}, U_{2}, U_{3} be modules over k, and let θ θ′ be homomorphisms from U_{1}, U_{2} into U_{2}, U_{3}, respectively, as modules over k. If V_{0} is a submodule of V, then we let U_{1} ⨂_{k} V_{0}, U_{2} ⨂_{k} V_{0}, U_{3} ⨂_{k} V_{0} be tensor products of U_{1}, U_{2}, U_{3} with V_{0}, as modules over k, respectively. This leads to homomorphisms

(11.9.2) Θ_{V_{0}} from U_{1} ⨂_{k}V_{0} into U_{2} ⨂_{k}V_{0}

and (11.9.3) Θ′_{V_{0}} from U_{2} ⨂_{A}V_{0} into U_{3} ⨂_{k}V_{0},

as modules over k, using θ, θ′, respectively, and the identity mapping on V. Suppose that

U_{1} ⟶^{θ} U_{2} ⟶^{θ′} U_{3} (11.9.4)

is exact, so that θ(U_{1}) = ker θ′.

If V_{0} is a finitely-generated submodule of V, then V_{0} is projective as a module over k, because V_{0} is torsion-free. In this case,

U_{1} ⨂_{A}V_{0} ⟶^{Θ_{V_{0}}} U_{2} ⨂_{A}V_{0} ⟶^{Θ′_{V_{0}}} U_{3} ⨂_{A}V_{0} (11.9.5)

is exact, as in Section 2.7. We can use this to get that

U_{1} ⨂_{k}V ⟶^{Θ_{V}} U_{2} ⨂_{k}V ⟶^{Θ′_{V}} U_{3} ⨂_{k}V (11.9.6)

is exact, as in Corollary 4.3 on p134 of [50].

More precisely, V corresponds to the direct limit of its finitely-generated submodules, as in Section 3.12. Similarly, for each j = 1, 2, 3, U_{j} ⨂_{k} V may be considered as a direct limit of U_{j} ⨂_{k} V_{0}, where V_{0} is a finitely-generated submodule of V, as in Section 3.6. We may consider Θ_{V}, Θ′_{V} as direct limits of Θ_{V_{0}}, Θ′_{V_{0}}, respectively, where V_{0} is a finitely-generated submodule of V, as in Section 3.4. This permits us to obtain the exactness of (11.9.6) from the exactness of (11.9.5), as before. Of course, there are analogous statements for V ⨂_{k} U_{j} instead of U_{j} ⨂_{k} V.

Suppose that k is a Prüfer ring again, let V, W be torsion-free modules over k, and let V ⨂_{k} W be a tensor product of V and W over k. Under these conditions, (11.9.7) V ⨂_{k}W is torsion-free

as a module over k, as in Proposition 4.5 on p134 of [50]. To see this, let Q_{k} be the field of fractions corresponding to k, and let Q_{k} ⨂_{k} V be a tensor product of Q_{k} and V, as modules over k. The natural homomorphism from V into Q_{k} ⨂_{k} V discussed in Section 11.4 is injective, because V is torsion-free.

Let (Q_{k} ⨂_{k} V ) ⨂_{k} W be a tensor product of Q_{k} ⨂ V and W, as modules over k. Consider the homomorphism

(11.9.8) from V ⨂_{k}W into (Q_{k} ⨂_{k}V ) ⨂_{k}W,

as modules over k, obtained from the mapping from V into Q_{k} ⨂_{k} V mentioned in the preceding paragraph, and the identity mapping on W. This homomorphism is injective, because the mapping from V into Q_{k} ⨂_{k} V is injective, and W is torsion-free, by the previous statement about exactness.

Of course, we may identify (Q_{k} ⨂_{k} V ) ⨂_{k} W with Q_{k} ⨂_{k} (V ⨂_{k} W ). Using this identification, one can check that the homomorphism as in (11.9.8) corresponds to the natural mapping from V ⨂_{k} W into Q_{k} ⨂_{k} (V ⨂_{k} W ), as in Section 11.4. The injectivity of this mapping implies (11.9.7), as before.

11.10 Dedekind rings

Let k be a commutative ring with a multiplicative identity element. We say that k is hereditary if every ideal in k is projective as a module over k, as in Section 9.3. Similarly, k is Noetherian as a ring if k is Noetherian as a module over itself, as in Section 9.7. This means that every ideal in k is finitely generated as a module over k, as before.

Suppose that k is an integral domain. Observe that

(11.10.1) k is hereditary if and only if every nonzero ideal in k is inversible,

because inversibility is equivalent to projectivity, as in Section 11.8. In this case, k is said to be a Dedekind ring, as on p134 of [50].

Equivalently, k is a Dedekind ring if and only if

(11.10.2) every nonzero fractional ideal of k is invertible.

Remember that an ideal in k is inversible if and only if it is invertible as a fractional ideal, as in Section 11.8. If M is a fractional ideal of k, x ∈ k, x ≠ 0, and x M ⊆ k, then it is easy to see that M is invertible if and only if x M is invertible.

Dedekind domains are defined another way on p95 of [8]. This formulation includes the condition that k have dimension one in a standard sense involving prime ideals that will be discussed in Section 14.8, and which implies that k is not a field. The definition of a Dedekind domain on p10 of [263] is equivalent to the one in [8], except that k is only asked to have dimension less than or equal to one in this sense, so that k may be a field.

If k is not a field, then the equivalence of these definitions of Dedekind rings and domains is given in Theorem 9.8 on p97 of [8]. The fact that Dedekind domain satisfy (11.10.2) is also mentioned in Proposition 5 on p11 of [263].

Note that (11.10.3) Dedekind rings are Noetherian, because inversible ideals in k are finitely generated as modules over k, as in Section 11.8. More precisely, Dedekind rings are the same as Prüfer rings that are Noetherian, as on p134 of [50].

If k is a Dedekind ring and V is a divisible module over k, then Proposition 5.1 on p134 of [50] states that

(11.10.4) V is injective as a module over k.

To see this, it suffices to show that if I is an ideal in k and ϕ is a homomorphism from I into V, as modules over k, then there is a v ∈ V such that ϕ(x) = x · v for every x ∈ I, as in Section 2.8. Of course, this holds trivially when I = {0}. Otherwise, if I ≠ {0}, then I is inversible as an ideal in k, by hypothesis. In this case, the condition was shown in Section 11.8.

Conversely, if k is an integral domain, and every divisible module over k is injective, then k is a Dedekind ring. This is another part of Proposition 5.1 on p134 of [50]. Remember that injective modules over k are divisible, as in Section 11.2. We also have that quotients of divisible modules are divisible, as in Section 11.1. Thus our hypothesis implies that quotients of injective modules over k are injective too. This implies that k is hereditary, as in Section 9.3. This means that k is a Dedekind ring, as desired.

11.11 Some injective modules

If k is an integral domain and Y is a module over k, then we have seen that there is an injective homomorphism from Y into a divisible module W over k, as in Section 11.6. If k is a Dedekind ring, then W is injective as a module over k, as in the previous section. This gives another approach to the analogous result for modules over arbitrary rings, as in Section 2.8. This corresponds to the remark on p135 of [50].

Let A be a ring with a nonzero multiplicative identity element e_{A}. There is a natural ring homomorphism ϕ from Z into A with ϕ(1) = e_{A}. Let V be a left module over A, which may be considered as a module over Z. It is easy to see that Z is a Dedekind ring. Thus one can use the argument described in the preceding paragraph to get an injective module W over Z and an injective homomorphism from V into W, as modules over Z.

Remember that the contravariant ϕ-extension of W is the space

(ϕ)W = Hom_{Z}(A, W) (11.11.1)

of homomorphisms from A into W, as modules over Z, as in Section 2.11, which is a left module over A. Let V (Z) be V considered as a module over Z, so that (ϕ)Z(V) may be defined as a left module over A in the same way. There is a natural injective homomorphism

(11.11.2) from V into ^{(ϕ)}V (Z),

as left modules over A, because V is a left module over A, as in Section 2.13. The injective homomorphism from V into W, as modules over Z, leads to an injective homomorphism

(11.11.3) from ^{(ϕ)}V (Z) into ^{(ϕ)}W,

as left modules over A, using composition of homomorphisms. The composition of these two homomorphisms defines an injective homomorphism

(11.11.4) from V into ^{(ϕ)}W,

as left modules over A.

We also have that ^{(ϕ)}W is injective as a left module over A, because W is injective as a module over Z, as in Section 2.11. This is another approach to getting an injective homomorphism from a left module over A into an injective left module over A, as in Section 2.8. Of course, there is an analogous argument for right modules. This corresponds to the remarks after Proposition 6.3 on p31 of [50]. As mentioned in [50], this proof was communicated to the authors by B. Echmann, and a similar proof was also found by H. A. Forrester.

11.12 Some divisible modules

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions. We may consider k and Q_{k} as modules over k, so that the quotient Q_{k}/k is a module over k too. In fact, Q_{k}/k is divisible, as mentioned in Section 11.4. It is easy to see that Q_{k}/k is a torsion module, as a module over k. If k is a Dedekind ring, then Q_{k}/k is injective as a module over k, as in Section 11.10.

Let Y be a module over k, and let η be a homomorphism from a torsion-free module V over k onto Y, as in Section 11.6. Also let Q_{k} ⨂_{k} V be a tensor product of Q_{k} and V, as modules over k, and let ϕ be the homomorphism from V into Q_{k} ⨂_{k} V with ϕ(v) = 1 ⊗ v for every v ∈ V, as before. Remember that ϕ is injective, so that ϕ leads to an injective homomorphism from V/ ker η into

(Q_{k} ⨂_{k}V )/ϕ(ker η), (11.12.1)

as modules over k. We have seen that Q_{k} ⨂_{k} V is divisible as a module over k, so that (11.12.1) is divisible too. Of course, V/ ker η is isomorphic to Y, as a module over k, so that we get an injective homomorphism from Y into (11.12.1), as modules over k, as before.

Suppose now that Y is a torsion module, as a module over k. If v ∈ V, then it follows that there is a t ∈ k such that t ≠ 0 and

η(t · v) = t · η(v) = 0, (11.12.2)

so that t · v ∈ ker η. In this case,

t · (1 ⊗ v) = 1 ⊗ (t · v) = ϕ(t · v) ∈ ϕ(ker η). (11.12.3)

Using this, one can check that

(11.12.4) (11.12.1) is a torsion module,

as a module over k.

Suppose that k is a Dedekind ring as well. Thus (11.12.1) is injective as a module over k, because it is divisible, as in Section 11.10. This shows that if Y is a torsion module over k, then there is an injective homomorphism from Y into an injective torsion module over k.

One can use this to get an injective resolution of Y consisting of torsion modules, as in Section 10.7. This corresponds to the first part of Exercise 2 on p139 of [50]. Of course, injective resolutions are simpler for hereditary rings, as before.

Commutative rings and fractions

12.1 Rings of fractions

Let k be a commutative ring with a multiplicative identity element. A subset S of k is said to be multiplicatively closed in k if 1 ∈ S (12.1.1) and S is closed under multiplication in the usual sense that (12.1.2) x y ∈ S for every x, y ∈ S. Thus S is a commutative semigroup with multiplicative identity element 1, which is a sub-semigroup of k, as a semigroup with respect to multiplication.

Consider the binary relation ≃ defined on k × S by putting (x, r) ≃ (y, t) (12.1.3) when x t v − y r v = 0 (12.1.4) for some v ∈ S. This relation is clearly reflexive and symmetric on k × S. To show that ≃ is transitive on k × S, suppose that (12.1.3) and (12.1.4) hold, and that (y, t) ≃ (z, u) for some (z, u) ∈ k × S, so that y u w − z t w = 0 (12.1.5) for some w ∈ S. It follows that (x t v − y r v) u w = (y u w − z t w) r v = 0, (12.1.6) so that x u t v w − z r t v w = 0. (12.1.7)

This means that (x, r) ≃ (z, u), because t v w ∈ S, as desired.

Thus ≃ defines an equivalence relation on k × S. The corresponding set of equivalence classes may be denoted S^{−1} k, as on p36 of [8]. This construction is also mentioned in Exercise 9 on p141 of [50], with the additional condition that

0 ∉ S. (12.1.8)

Note that the multiplicative identity element in a ring is normally supposed to be nonzero in [50], as on p3 of [50], whereas 1 = 0 in k is permitted in [8], as on p1 of [8]. The set of equivalence classes is denoted k_{S} in [50], but we shall normally use the notation S^{−1} k here, to avoid confusion with some other notation in [8].

If x ∈ k and r ∈ S, then we let x/r denote the equivalence class of (x, r) with respect to ≃, as on p36 of [8]. If y ∈ k and t ∈ S as well, then we would like to define the sum and product of x/r and y/t in S^{−1} k by

(x/r) + (y/t) = (x r + y t)/(r t) (12.1.9)

and (x/r) (y/t) = (x y)/(r t). (12.1.10)

One can check that these determine well-defined operations of addition and multiplication on S^{−1} k, which make S^{−1} k a commutative ring, as in [8, 50]. This is called the ring of fractions of k with respect to S, as on p37 of [8]. Clearly 1/1 is the multiplicative identity element in S^{−1} k.

We also have that x ↦ x/1 (12.1.11)

is a ring homomorphism from k into S^{−1} k, as on p37 of [8]. If r ∈ S, then r/1 has multplicative inverse 1/r in S^{−1} k. If x ∈ k, then it is easy to see that

x/1 = 0 (12.1.12)

in S^{−1} k if and only if x t = 0 (12.1.13)

in k for some t ∈ S.

If 0 ∈ S, then (12.1.4) holds automatically with v = 0, so that S^{−1} k = {0}. Conversely, if S^{−1} k = {0}, then one can check that 0 ∈ S, as in Example 2 on p38 of [8].

Observe that k is an integral domain exactly when

S = k \ {0} (12.1.14)

is multiplicatively closed in k. In this case, S^{−1} k is the same as the usual field Q_{k} of fractions of k, as in the remark on p37 of [8].

12.2 Modules of fractions

Let k be a commutative ring with a multiplicative identity element, let S be a multiplicatively closed subset of k, and let V be a module over k. Consider the binary relation ≅ defined on V × S by putting

(v, r) ≅ (v′, r′) (12.2.1)

when t r′ v − t r v′ = 0 (12.2.2)

for some t ∈ S. One can check that this defines an equivalence relation on V ×S, as in the previous section. The corresponding set of equivalence classes may be denoted S^{−1} V, as on p38 of [8]. If v ∈ V and r ∈ S, then we let v/r denote the equivalence class of (v, r) with respect to ≅, as in [8].

One can define addition and scalar multiplication on S^{−1} V in natural ways, so that S^{−1}V becomes a module over S^{−1} k, as in [8]. In Exercise 9 on p141 of [50], S^{−1} V is defined initially as a module over k, before S^{−1} k is defined as a ring. This module is denoted V_{S} in [50], but we shall not use this notation here, to avoid confusion with some other notation from [8], as before. In particular, one can use this to define S^{−1} k initially as a module over k, by considering k as a module over itself. One can define multiplication on S^{−1} k as in (12.1.10) again to get a commutative ring with a multiplicative identity element, and define scalar multiplication on S^{−1} V so that it becomes a module over S^{−1} k, as before.

If S^{−1} V is defined as a module over S^{−1} k in this way, then S^{−1} V may also be considered as a module over k, using the ring homomorphism (12.1.11) from k into S^{−1} k. This is the same as defining S^{−1} V initially as a module over k, as in [50]. It is easy to see that

v ↦ v/1 (12.2.3)

defines a homomorphism from V into S^{−1} V, as modules over k, as on p142 of [50]. Note that v/1 = 0 (12.2.4)

in S^{−1} V if and only if t v = 0 (12.2.5)

for some t ∈ S, as in [50].

Let W be another module over k, and let f be a homomorphism from V into W, as modules over k. This leads to a homomorphism S^{−1} f from S^{−1} V into S^{−1} W, as modules over S^{−1} k, with

(S^{−1} f)(v/r) = f(v)/r (12.2.6)

for every v ∈ V and r ∈ S, as on p38 of [8].

12.2. MODULES OF FRACTIONS 319

Let Z be a third module over k, and let g be a homomorphism from W into Z, as modules over k. Thus g ∘f is a homomorphism from V into Z, as modules over k, and S^{−1} (g ∘ f) = (S^{−1} g) ∘ (S^{−1} f), (12.2.7) as on p38 of [8]. If g ∘ f = 0, then it follows that (S^{−1} g) ∘ (S^{−1} f) = 0. (12.2.8)

Suppose that f(V) = ker g, (12.2.9) and let us check that (S^{−1} f)(S^{−1} V) = ker(S^{−1} g). (12.2.10) This corresponds to Proposition 3.3 on p39 of [50], and the first part of Exercise 10 on p142 of [50]. Of course, (12.2.9) implies that g ∘ f = 0, so that (S^{−1} f)(S^{−1} V) ⊆ ker(S^{−1}g), (12.2.11) by (12.2.8). To get the opposite inclusion, let w ∈ W and r ∈ S be given, with g(w)/r = (S^{−1} g)(w/r) = 0 (12.2.12) in S^{−1} Z. This implies that there is a t ∈ S such that g(t w) = t g(w) = 0 (12.2.13) in Z. It follows that there is a v ∈ V such that f(v) = t w. (12.2.14) This means that (S^{−1} f)(v/(r t)) = f(v)/(r t) = (t w)/(r t) = w/r (12.2.15) in S^{−1} W, as desired.

In particular, if g is injective, then S^{−1} g is injective, which can also be seen a bit more directly. If W is a submodule of Z, then we can apply this to the obvious inclusion mapping from W into Z. This permits us to consider S^{−1} W as a submodule of S^{−1} Z in this case, as on p39 of [8].

Let V_{1}, V_{2} be submodules of V, so that S^{−1} V_{1} and S^{−1} V_{2} may be considered as submodules of S^{−1} V, as in the preceding paragraph. It is easy to see that S^{−1} (V_{1} + V_{2}) = (S^{−1} V_{1}) + (S^{−1} V_{2}), (12.2.16) as submodules of S^{−1} V, as in part (i) of Corollary 3.4 on p39 of [8].

Similarly, let us check that S^{−1} (V_{1} ∩ V_{2}) = (S^{−1} V_{1}) ∩ (S^{−1} V_{2}), (12.2.17) as in part (ii) of Corollary 3.4 on p39 of [8]. The left side is clearly contained in the right side, and so it suffices to verify the opposite inclusion. Let v_{1} ∈ V_{1}, v_{2} ∈ V_{2}, and r_{1}, r_{2} ∈ S be given, with

v_{1}/r_{1} = v_{2}/r_{2}. (12.2.18)

This means that t r_{2} v_{1} = t r_{1} v_{2} (12.2.19)

for some t ∈ S. If u denotes the common value in (12.2.19), then u ∈ V_{1} ∩ V_{2}. The common value in (12.2.18) is equal to

u/(t r_{1} r_{2}). (12.2.20)

This is an element of S^{−1} (V_{1} ∩ V_{2}), as desired.

Let q_{1} be the natural quotient mapping from V onto the quotient module V/V_{1}. This leads to a homomorphism

(12.2.21) S^{−1} q_{1} from S^{−1} V onto S^{−1} (V/V_{1}),

as modules over S^{−1} k, as before. We also have that

ker(S^{−1} q_{1}) = S^{−1} V_{1}, (12.2.22)

as in (12.2.10). This means that S^{−1} q_{1} induces an isomorphism

(12.2.23) from (S^{−1} V)/(S^{−1} V_{1}) onto S^{−1} (V/V_{1}),

as modules over S^{−1} k, as in part (iii) of Corollary 3.4 on p39 of [8].

12.3 Fractions and tensor products

Let k be a commutative ring with a multiplicative identity element, let S be a multiplicatively closed subset of k, and let V be a module over k. Also let (S^{−1} k) ⨂_{k} V be a tensor product of S^{−1} k and V, as modules over k. An arbitrary element of this tensor product may be expressed as a finite sum

∑_{j=1}^{n}(x_{j}/r_{j}) ⊗ v_{j}, (12.3.1)

where x_{j} ∈ k, r_{j} ∈ S, and v_{j} ∈ V for each j = 1, …, n. Put

r = ∏_{j=1}^{n} r_{j} (12.3.2)

and

t_{l} = ∏_{j≠l} r_{j} (12.3.3)

for each l = 1, …, n. These are elements of S, with r = r_{l} t_{l} for each l. Thus (12.3.1) is equal to

(12.3.4)  ∑_{j=1}^{n}(t_{j} x_{j}/r) ⊗ v_{j} = ∑_{j=1}^{n}(1/r) ⊗ ((t_{j} x_{j}) · v_{j})

= (1/r) ⊗ ( ∑_{j=1}^{n}(t_{j} x_{j}) · v_{j}).

This shows that every element of (S^{−1} k) ⨂_{k} V may be expressed as (1/r) ⊗ v for some r ∈ S and v ∈ V. This corresponds to part of the proof of Proposition 3.5 on p39 of [8].

One can check that (x/r, v) ↦ (x · v)/r (12.3.5)

defines a mapping from (S^{−1} k) × V into S^{−1} V that is bilinear over k. This leads to a unique homomorphism

(12.3.6) from (S^{−1} k) ⨂_{k}V into S^{−1} V,

as modules over k, with (x/r) ⊗ v ↦ (x · v)/r (12.3.7)

for every x ∈ k, r ∈ S, and v ∈ V. It is easy to see that this homomorphism is surjective, and we would like to check that it is injective as well.

Let r ∈ S and v ∈ V be given, and suppose that

v/r = 0 (12.3.8)

in S^{−1} V. This implies that t · v = 0 for some t ∈ S. It follows that

(1/r) ⊗ v = (t/(r t)) ⊗ v = (1/(r t)) ⊗ (t · v) = 0 (12.3.9)

in (S^{−1} k) ⨂_{k} V. This shows that the kernel of the homomorphism mentioned in the preceding paragraph is trivial.

Note that (S^{−1} k) ⨂_{k} V may be considered as a module over S^{−1} k, because S^{−1} k may be considered as a module over itself, as in Section 1.10. This may also be considered as an example of extension of scalars, as in Section 2.9. It is easy to see that the homomorphism as in (12.3.6) is linear over S^{−1} k. Thus we get an isomorphism as in (12.3.6), as modules over S^{−1} k. This corresponds to Proposition 3.5 on p39 of [8], and to part of Exercise 9 on p141 of [50].

Let W be another module over k, and let V ⨂_{k} W be a tensor product of V and W. This is a module over k, so that S^{−1} (V ⨂_{k} W) may be defined as a module over S^{−1} k, as before. Let (S^{−1} V) ⨂_{S^{−1} k}(S^{−1} W) be a tensor product of S^{−1} V and S^{−1} W, as modules over S^{−1} k. One can check that

((v/r), (w/t)) ↦ (v ⊗ w)/(r t) (12.3.10) defines a mapping from (S^{−1} V) × (S^{−1} W) into S^{−1} (V ⨂_{k} W) that is bilinear over S^{−1} k. This leads to a unique homomorphism

(12.3.11) from (S^{−1} V) ⨂_{S^{−1} k}(S^{−1} W) into S^{−1} (V ⨂_{k}W),

as modules over S^{−1} k, with

(v/r) ⊗ (w/t) ↦ (v ⊗ w)/(r t) (12.3.12)

for every v ∈ V, w ∈ W, and r, t ∈ k.

It is easy to see that this homomorphism is surjective. Proposition 3.7 on p40 of [8] states that this homomorphism is an isomorphism. This can be obtained from the previous result and isomorphisms between double tensor products, as in Section 1.12.

If Z is a module over S^{−1} k, then Z may be considered as a module over k, using the natural ring homomorphism from k into S^{−1} k. If Z is considered as a module over k, then S^{−1} Z may be defined as a module over S^{−1} k, as before. Under these conditions, one can check that the natural homomorphism from Z into S^{−1} Z, as modules over k, is an isomorphism between modules over S^{−1} k. This is another part of Exercise 9 on p141 of [50].

12.4 Fractions and prime ideals

Let k be a commutative ring with a multiplicative identity element, and let I be a proper ideal in k. Note that I is a prime ideal in k if and only if

S = k \ I (12.4.1)

is multiplicatively closed in k, as in Example 1 on p38 of [8]. Supposing that this is the case, we may put k_{I} = S^{−1} k, (12.4.2)

as in [8].

It is easy to see that

I_{1} = {x/r : x ∈ I, r ∈ k \ I} (12.4.3)

is an ideal in k_{I}. One can check that 1/1 ∉ I_{1}, so that I_{1} ≠ k_{I}. If y/t ∈ k_{I} \I_{1}, then y ∈ k \ I, and y/t is invertible in k_{I}. This implies that I_{1} is a maximal ideal in k_{I}, and in fact k_{I} is a local ring, as in Section 4.13. Equivalently, I_{1} is the unique maximal proper ideal in k_{I}, as on p38 of [8].

If V is a module over k, then we may put

V_{I} = S^{−1} V, (12.4.4)

with S as in (12.4.1), as on p38 of [8]. Of course, if V = {0}, then V_{I} = {0}. Suppose that V ≠ {0}, and let v be a nonzero element of V. Thus

{t ∈ k : t · v = 0} (12.4.5) is a proper ideal in k. Let I_{0} be a proper maximal ideal in k that contains (12.4.5), which can be obtained using Zorn’s lemma or Hausdorff’s maximality principle.

Of course, maximal ideals are prime ideals, by a standard argument, so that V_{I_{0}} may be defined as before. If

v/1 = 0 in V_{I_{0}}, (12.4.6)

then r · v = 0 (12.4.7)

for some r ∈ k \ I_{0}. However, this would contradict the fact that (12.4.5) is contained in I_{0}. This shows that

V = {0} (12.4.8)

when V_{I} = {0} (12.4.9)

for all maximal proper ideals I in k. This corresponds to Proposition 3.8 on p40 of [8], and the first part of Exercise 11 on p142 of [50].

Let W be another module over k, and let ϕ be a homomorphism from V into W, as modules over k. If I is a proper prime ideal in k and S is as in (12.4.1), then we get an induced homomorphism ϕ_{I} = S^{−1} ϕ from V_{I} into W_{I}, as in Section 12.2. If ϕ is surjective, then it is easy to see that ϕ_{I} is surjective. If ϕ is injective, then ϕ_{I} is injective too, as before. This corresponds to part of Proposition 3.9 on p40 of [8].

More precisely, ker ϕ_{I} = (ker ϕ)_{I}. (12.4.10)

This follows from (12.2.10), with f equal to the obvious inclusion mapping from ker g into V, and g = ϕ. If (12.4.10) is equal to {0} for all maximal proper ideals I in k, then it follows that ker g = {0}, as in (12.4.8). Equivalently, if ϕ_{I} is injective for all maximal proper ideals I in k, then ϕ is injective. This is another part of Proposition 3.9 on p40 of [8].

Similarly, if ϕ_{I} is surjective for all maximal proper ideals I in k, then ϕ is surjective, as in Proposition 3.9 on p40 of [8].

12.5 The nilradical

Let k be a commutative ring with a multiplicative identity element. An element x if k is said to be nilpotent if x^{n} = 0 (12.5.11)

in k for some positive integer n, as on p2 of [8]. The set N of all nilpotent elements of k is called the nilradical of k, as on p5 of [8].

Suppose that x, y ∈ N, so that x^{m} = y^{n} = 0 for some positive integers m, n. Under these conditions, m + n − 1 is a positive integer, and one can check that (x + y)^{m+n−1} = 0. (12.5.12)

Indeed, the left side can be expanded into a sum of terms of the form

x^{j} y^{l}, (12.5.13)

where j and l are nonnegative integers with j + l = m + n − 1. It is easy to see that j ≥ m or l ≥ n, so that (12.5.13) is equal to 0. This implies (12.5.12).

One can use this to verify that N is an ideal in k. One can also check that every nilpotent element of the quotient ring k/N is equal to 0. This is Proposition 1.7 on p5 of [8].

Propostion 1.8 on p5 of [8] states that N is equal to the intersection of all the prime ideals in k. It is easy to see that every nilpotent element of k is contained in every prime ideal of k.

Suppose that x ∈ k is not nilpotent, so that for each positive integer n, x^{n} ≠ 0. One can use Zorn’s lemma or Hausdorff’s maximality principle to get an ideal I in k such that for each positive integer n,

x^{n} ∉ I, (12.5.14)

and I is maximal with respect to inclusion. We would like to check that I is a prime ideal in k.

Let y, z ∈ k \ I be given, and let I_{y}, I_{z} be the ideals in k generated by I and y, z, respectively. There are positive integers m, n such that

x^{m} ∈ I_{y}, x^{n} ∈ I_{z}, (12.5.15)

by the maximality of I.

Let I_{yz} be the ideal in k generated by I and y z. One can verify that

x^{m+n} ∈ I_{yz}, (12.5.16)

using (12.5.15).

This implies that y z ∉ I, so that I is prime. Another version of this using a suitable ring of fractions of k is mentioned in Remark 2 on p42 of [8], and this will be discussed in the next section.

12.6 Using powers to get fractions

Let k be a commutative ring with a multiplicative identity element, and let x ∈ k be given. Observe that

S = {x^{n} : n ∈ (Z_{+} ∪ {0})} (12.6.1)

is a multiplicatively closed subset of k. This is Example 3 on p38 of [8]. We also have that 0 ∈ S if and only if x is nilpotent in k.

Suppose now that x is not nilpotent in k, so that 0 ∉ S. This implies that S^{−1} k ≠ {0}, as in Section 12.1. It follows that there is a maximal proper ideal I_{0} in S^{−1} k. In particular, I_{0} is a prime ideal in S^{−1} k.

Put I_{1} = {w ∈ k : w/1 ∈ I_{0}}. (12.6.2)

It is easy to see that this is an ideal in k. Note that 1 ∉ I_{1}, because 1/1 ∉ I_{0}. One can check that I_{1} is a prime ideal in k, because I_{0} is a prime ideal in S^{−1} k.

If n is a positive integer, then x^{n}/1 is invertible in S^{−1} k, because x^{n} ∈ S, as in Section 12.1. This means that

x^{n}/1 ∉ I_{0}, (12.6.3)

because I_{0} is a proper ideal in S^{−1} k. It follows that

x^{n} ∉ I_{1}. (12.6.4)

This is the argument from Remark 2 on p42 of [8] that was mentioned in the previous section.

12.7 Extensions and contractions of ideals

Let k_{1} and k_{2} be commutative rings with multiplicative identity elements 1_{k_{1}}, 1_{k_{2}}, respectively, and let f be a ring homomorphism from k_{1} into k_{2}, with f(1_{k_{1}}) = 1_{k_{2}}. Also let I_{1} be an ideal in k_{1}, and note that f(I_{1}) is not necessarily an ideal in k_{2}. Under these conditions, the extension of I_{1} with respect to f is the ideal I_{1}^{e} in k_{2} generated by f(I_{1}). Equivalently, I_{1}^{e} consists of finite sums of products of elements of f(I_{1}) and k_{2}, as on p9 of [8].

If I_{2} is an ideal in k_{2}, then f^{−1}(I_{2}) is an ideal in k_{1}. This is the contraction of I_{2} with respect to f, which may be denoted I_{2}^{c}, as on p9 of [8]. If I_{2} ≠ k_{2}, then 1_{k_{2}} ∉ I_{2}, which implies that 1_{k_{1}} ∉ I_{2}^{c}. If I_{2} is a prime ideal in k_{2}, then it is easy to see that I_{2}^{c} is a prime ideal in k_{1}.

It is easy to see that I_{1} ⊆ I_{1}^{ec}, (12.7.1)

where I_{1}^{ec} is the contraction of I_{1}^{e}. Similarly,

I_{2}^{ce} ⊆ I_{2}, (12.7.2)

where I_{2}^{ce} is the extension of I_{2}^{c}. This is part (i) of Proposition 1.17 on p10 of [8]. Part (ii) of this proposition states that

I_{2}^{cec} = I_{2}^{c} (12.7.3)

and I_{1}^{ece} = I_{1}^{e}, (12.7.4)

where the contraction and extension operations are used again, as indicated. This follows from the previous part, as in [8].

326 CHAPTER 12. COMMUTATIVE RINGS AND FRACTIONS

Let C be the collection of all ideals in k_{1} that are contractions of ideals in k_{2}, and let E be the collection of all ideals in k_{2} that are extensions of ideals in k_{1}. One can check that C is the same as the collection of all ideals I_{1} in k_{1} such that I_{1}^{ec} = I_{1}. (12.7.5) Similarly, one can verify that E is the same as the collection of all ideals I_{2} in k_{2} such that I_{2}^{ce} = I_{2}. (12.7.6) This is part of part (iii) of Proposition 1.17 in [8].

In fact, I_{1} ↦ I_{1}^{e} (12.7.7) defines a one-to-one mapping from C onto E. Similarly, I_{2} ↦ I_{2}^{c} (12.7.8) is a one-to-one mapping from E onto C, which is the inverse of the previous mapping. This is the other part of part (iii) of Proposition 1.17 in [8].

12.8 Products and ideal quotients

Let k be a commutative ring with a multiplicative identity element, and let I_{1}, I_{2} be ideals in k. The product I_{1} I_{2} is the ideal in k generated by products of elements of I_{1} and I_{2}, as on p6 of [8]. Equivalently, I_{1} I_{2} consists of finite sums of products of elements of I_{1} and I_{2}. Of course, I_{1} I_{2} ⊆ I_{1} ∩ I_{2}. (12.8.1)

Let V be a module over k, and let V_{1}, V_{2} be submodules of V. Put (V_{1} : V_{2}) = {t ∈ k : t · V_{2} ⊆ V_{1}}, (12.8.2) where t · V_{2} = {t · v_{2} : v_{2} ∈ V_{2}}. It is easy to see that (12.8.2) is an ideal in k, as on p19 of [8]. Note that this is not the same as in Section 11.7, although the notation is similar.

If we take V_{1} = {0} in (12.8.2), then we get the annihilator Ann(V_{2}) = {t ∈ k : t · V_{2} = {0}} (12.8.3) of V_{2} in k. Observe that (12.8.4) Ann(V_{1} + V_{2}) = Ann(V_{1}) ∩ Ann(V_{2}). One can also verify that (V_{1} : V_{2}) = Ann((V_{1} + V_{2})/V_{1}). (12.8.5)

These two statements correspond to parts (i) and (ii) of Exercise 2.2 on p20 of [8], respectively.

The product I_{1} ·V_{2} of I_{1} and V_{2} may be defined as the subset of V_{2} consisting of finite sums of elements of the form t_{1} · v_{2}, where t_{1} ∈ I_{1} and v_{2} ∈ V_{2}. This is submodule of V_{2}, as on p19 of [8]. Note that

(V_{1} : V_{2}) · V_{2} ⊆ V_{1}, (12.8.6)

as in part (ii) of Exercise 1.12 on p8 of [8].

Of course, I_{1}, I_{2} may be considered as submodules of k, as a module over itself. Their ideal quotient is defined to be (I_{1} : I_{2}), as in (12.8.2), with V = k, as on p8 of [8]. Clearly I_{1} ⊆ (I_{1} : I_{2}), (12.8.7)

as in part (i) of Exercise 1.12 on p8 of [8]. If x ∈ k, then we put

(I_{1} : x) = {t ∈ k : t x ∈ I_{1}}. (12.8.8)

This is the same as (I_{1} : I_{2}), with I_{2} taken to be the ideal in k generated by x, as in [8].

12.9 More on extensions and contractions

Let k_{1}, k_{2} be commutative rings with multiplicative identity elements 1_{k_{1}}, 1_{k_{2}}, respectively, and let f be a ring homomorphism from k_{1} into k_{2} with f(1_{k_{1}}) = 1_{k_{2}}, as in Section 12.7. Also let I_{1}, Ĩ_{1} be ideals in k_{1}, and let I_{2}, Ĩ_{2} be ideals in k_{2}. We would like to mention some properties of extensions and contractions with respect to f, as in Exercise 1.18 on p10 of [8].

Note that I_{1} + Ĩ_{1} and I_{2} + Ĩ_{2} are ideals in k_{1} and k_{2}, respectively. It is easy to see that (I_{1} + Ĩ_{1})^{e} = I_{1}^{e} + Ĩ_{1}^{e}. (12.9.1)

We also have that I_{2}^{c} + Ĩ_{2}^{c} ⊆ (I_{2} + Ĩ_{2})^{c}. (12.9.2)

Of course, I_{1} ∩ Ĩ_{1} and I_{2} ∩ Ĩ_{2} are ideals in k_{1} and k_{2}, respectively. One can check that (I_{1} ∩ Ĩ_{1})^{e} ⊆ I_{1}^{e} ∩ Ĩ_{1}^{e}. (12.9.3)

It is easy to see that (I_{2} ∩ Ĩ_{2})^{c} = I_{2}^{c} ∩ Ĩ_{2}^{c}. (12.9.4)

The products I_{1} Ĩ_{1} and I_{2} Ĩ_{2} are defined as ideals in k_{1} and k_{2}, respectively, as in the previous section. One can verify that

(I_{1} Ĩ_{1})^{e} = I_{1}^{e} Ĩ_{1}^{e}. (12.9.5)

Clearly I_{2}^{c} Ĩ_{2}^{c} ⊆ (I_{2} Ĩ_{2})^{c}. (12.9.6)

The quotient ideals (I_{1} : Ĩ_{1}) and (I_{2} : Ĩ_{2}) in k_{1} and k_{2}, respectively, may be defined as in the previous section. Similarly, (I_{1}^{e} : Ĩ_{1}^{e}) is an ideal in k_{2}, and (I_{2}^{c} : Ĩ_{2}^{c}) is an ideal in k_{1}. One can check that

(I_{1} : Ĩ_{1})^{e} ⊆ (I_{1}^{e} : Ĩ_{1}^{e}). (12.9.7)

Similarly, (I_{2} : Ĩ_{2})^{c} ⊆ (I_{2}^{c} : Ĩ_{2}^{c}). (12.9.8)

12.10 Radicals of ideals

Let k be a commutative ring with a multiplicative identity element, and let I be an ideal in k. The radical r(I) of I in k is defined by

(12.10.1) r(I) = {x ∈ k : x^{n} ∈ I for some n ∈ Z_{+}},

as on p8 of [8].

Let q be the natural quotient homomorphism from k onto k/I, and let N (k/I) be the nilradical of k/I, as in Section 12.5. It is easy to see that

r(I) = q^{−1}(N (k/I)), (12.10.2)

as on p8 of [8]. This implies that r(I) is an ideal in k as well, because N (k/I) is an ideal in k/I, as before. In particular, r({0}) is the same as the nilradical of k.

Let us mention some basic properties of the radical, as in Exercise 1.13 on p9 of [8]. Clearly I ⊆ r(I). (12.10.3)

It is easy to see that r(r(I)) = r(I). (12.10.4)

Let Ĩ be another ideal in k. One can check that

r(I Ĩ) = r(I ∩ Ĩ). (12.10.5)

We also have that r(I ∩ Ĩ) = r(I) ∩ r(Ĩ). (12.10.6)

It is easy to see that r(I) = k if and only if I = k. Indeed, if 1 ∈ r(I), then 1 ∈ I. One can verify that

r(I + Ĩ) = r(r(I) + r(Ĩ)). (12.10.7)

If n is a positive integer, then let I^{n} be the nth power of I with respect to multiplication of ideals as in Section 12.8. Equivalently, I^{n} consists of all finite sums of products of n elements of I, as on p6 of [8]. This is interpreted as being equal to k when n = 0, as in [8].

It is easy to see that r(I^{n}) = r(I) (12.10.8)

for every n ≥ 1. If I is a prime ideal in k, then one can check that

r(I) = I. (12.10.9)

This implies that r(I^{n}) = I (12.10.10)

for each n ≥ 1.

Proposition 1.14 on p9 of [8] states that r(I) is the same as the intersection of all of the prime ideals in k that contain I. This can be obtained from the analogous statement for the nilradical in Section 12.5 and (12.10.2).

Let k_{1}, k_{2} be commutative rings with multiplicative identity elements 1_{k_{1}}, 1_{k_{2}}, respectively, and let f be a ring homomorphism from k_{1} into k_{2} with f(1_{k_{1}}) = 1_{k_{2}}. If I_{1} is an ideal in k_{1}, then one can check that

r(I_{1})^{e} ⊆ r(I_{1}^{e}), (12.10.11)

where the radicals are taken in k_{1} and k_{2}, respectively. If I_{2} is an ideal in k_{2}, then r(I_{2})^{c} = r(I_{2}^{c}), (12.10.12)

where the radicals are taken in k_{2} and k_{1}, respectively. This is another part of Exercise 1.18 on p10 of [8].

12.11 Ring homomorphisms and fractions

Let k_{1}, k_{2} be commutative rings with multiplicative identity elements 1_{k_{1}}, 1_{k_{2}}, respectively, and let ϕ be a ring homomorphism from k_{1} into k_{2} with ϕ(1_{k_{1}}) = 1_{k_{2}}. Also let S_{1} be a multiplicatively closed subset of k_{1}, and observe that

S_{2} = ϕ(S_{1}) (12.11.1)

is a multiplicatively closed subset of k_{2}. Thus S_{1}^{−1} k_{1} and S_{2}^{−1} k_{2} may be defined as commutative rings with multiplicative identity elements as in Section 12.1. We also get the corresponding ring homomorphisms

x_{1} ↦ x_{1}/1_{k_{1}} (12.11.2)

and x_{2} ↦ x_{2}/1_{k_{2}} (12.11.3)

from k_{1}, k_{2} into S_{1}^{−1} k_{1}, S_{2}^{−1} k_{2}, respectively, as before.

We would like to define a mapping Φ from S_{1}^{−1} k_{1} into S_{2}^{−1} k_{2} by putting

Φ(x_{1}/r_{1}) = ϕ(x_{1})/ϕ(r_{1}) (12.11.4) for every x_{1} ∈ k_{1} and r_{1} ∈ S_{1}. It is easy to see that this is a well-defined mapping from S_{1}^{−1} k_{1} into S_{2}^{−1} k_{2}, and a ring homomorphism. Note that the composition of (12.11.2) with Φ is the same as the composition of ϕ with (12.11.3). If

ϕ(k_{1}) = k_{2}, (12.11.5)

then Φ(S_{1}^{−1} k_{1}) = S_{2}^{−1} k_{2}. (12.11.6)

Of course, the kernel of ϕ is an ideal in k_{1}, which may be considered as a submodule of k_{1}, as a module over itself. Thus S_{1}^{−1}(ker ϕ) may be considered as a submodule of S_{1}^{−1} k_{1}, as a module over itself, as in Section 12.2. This means that S_{1}^{−1}(ker ϕ) may be considered as an ideal in S_{1}^{−1} k_{1}.

It is easy to see that (12.11.4) is equal to 0 in S_{2}^{−1} k_{2} if and only if there is a t_{1} ∈ S_{1} such that ϕ(t_{1} x_{1}) = ϕ(t_{1}) ϕ(x_{1}) = 0 (12.11.7)

in k_{2}. One can use this to get that

ker Φ = S_{1}^{−1}(ker ϕ). (12.11.8)

Let V be k_{2}, considered as a module over k_{1}, using ϕ. Thus we can define S_{1}^{−1} V as a module over S_{1}^{−1} k_{1}, as in Section 12.2. This corresponds to considering S_{2}^{−1} k_{2} as a module over S_{1}^{−1} k_{1} using Φ, as in Exercise 4 on p44 of [8].

12.12 More on rings of fractions

Let k_{1}, k be commutative rings with multiplicative identity elements 1_{k_{1}}, 1_{k}, respectively, and let g be a ring homomorphism from k_{1} into k with g(1_{k_{1}}) = 1_{k}. Also let S_{1} be a multiplicatively closed subset of k_{1}, and suppose that for each r_{1} ∈ S_{1}, (12.12.1) g(r_{1}) has a multiplicative inverse in k.

Under these conditions, Proposition 3.1 on p37 of [8] states that there is a unique ring homomorphism h from S_{1}^{−1} k_{1} into k such that

h(x_{1}/1_{k_{1}}) = g(x_{1}) (12.12.2)

for every x_{1} ∈ k_{1}.

To get uniqueness, observe that for each r_{1} ∈ S_{1}, we have that

h(1_{k_{1}}/r_{1}) = h(r_{1}/1_{k_{1}})^{−1} = g(r_{1})^{−1}. (12.12.3)

This implies that

h(x_{1}/r_{1}) = h(x_{1}/1_{k_{1}}) h(1_{k_{1}}/r_{1}) = g(x_{1}) g(r_{1})^{−1} (12.12.4)

for every x_{1} ∈ k_{1} and r_{1} ∈ S_{1}.

To get existence, we would like to define h as in (12.12.4), and we need to check that h is well-defined on S_{1}^{−1} k_{1}. Let x_{1}, x′_{1} ∈ k_{1} and r_{1}, r′_{1} ∈ S_{1} be given, with x_{1}/r_{1} = x′_{1}/r′_{1}. (12.12.5)

This means that there is a t_{1} ∈ S_{1} such that

x_{1} r′_{1} t_{1} = x′_{1} r_{1} t_{1}, (12.12.6)

as in Section 12.1. Thus

g(x_{1}) g(r′_{1}) g(t_{1}) = g(x′_{1}) g(r_{1}) g(t_{1}), (12.12.7)

so that g(x_{1}) g(r_{1})^{−1} = g(x′_{1}) g(r′_{1})^{−1}, (12.12.8)

because g(t_{1}) is invertible in k. This implies that h is well-defined on S_{1}^{−1} k_{1}, and it is easy to see that h is a ring homomorphism.

Suppose now that g satisfies the following two additional conditions. First, if x_{1} ∈ k_{1} and g(x_{1}) = 0, then x_{1} t_{1} = 0 (12.12.9)

in k_{1} for some t_{1} ∈ S_{1}. Second, every element of k is of the form

g(x_{1}) g(r_{1})^{−1} (12.12.10)

for some x_{1} ∈ k_{1} and r_{1} ∈ S_{1}. In this case, Corollary 3.2 on p37 of [8] states that h is an isomorphism from S_{1}^{−1} k_{1} onto k. Indeed, the first condition implies that the kernel of h is trivial, and the second condition implies that h is surjective.

Let k_{2} be another commutative ring with multiplicative identity element 1_{k_{2}}, and let ϕ be a ring homomorphism from k_{1} into k_{2} with ϕ(1_{k_{1}}) = 1_{k_{2}}. Remember that S_{2} = ϕ(S_{1}) is a multiplicatively closed subset of k_{2}, as in Section 12.11. Let us take k = S_{2}^{−1} k_{2}, (12.12.11)

and let g be the ring homomorphism from k_{1} into k defined by

g(x_{1}) = ϕ(x_{1})/1_{k_{2}} (12.12.12)

for every x_{1} ∈ k_{1}. This is the same as the composition of ϕ with the usual homomorphism x_{2} ↦ x_{2}/1_{k_{2}} from k_{2} into S_{2}^{−1} k_{2}. This homomorphism satisfies (12.12.1) by construction, and the corresponding homomorphism h is the same as the homomorphism Φ discussed in the previous section.

12.13 Extensions, contractions, and fractions

Let k be a commutative ring with a multiplicative identity element, and let S be a multiplicatively closed subset of k. Put

f(x) = x/1 (12.13.1) for every x ∈ k, which defines a ring homomorphism from k into S^{−1} k, as in Section 12.1. If I_{1} is an ideal in k, then we let I_{1}^{e} be its extension in S^{−1} k with respect to f, as in Section 12.7. Similarly, if I_{2} is an ideal in S^{−1} k, then we let I_{2}^{c} be its contraction in k with respect to f. Let C be the collection of ideals in k that are contractions of ideals in S^{−1} k, and let E be the collection of ideals in S^{−1} k that are extensions of ideals in k, as before.

If I_{1} is any ideal in k, then we would like to check that

I_{1}^{e} = S^{−1} I_{1}, (12.13.2)

as on p41 of [8]. More precisely, I_{1} may be considered as a submodule of k, as a module over itself, so that S^{−1} I_{1} is defined as a module over S^{−1} k. This may be identified with a submodule of S^{−1} k, as a module over itself, as in Section 12.2. Thus S^{−1} I_{1} is an ideal in S^{−1} k, which contains f(I_{1}), and thus I_{1}^{e}. It is clear that S^{−1} I_{1} is contained in I_{1}^{e}, so that they are equal to each other.

Let I_{2} be an ideal in S^{−1} k, and let us check that I_{2}^{ce} = I_{2}. This means that I_{2} is the extension of an ideal in k, as in part (i) of Proposition 3.11 on p41 of [8]. It suffices to show that I_{2} ⊆ I_{2}^{ce}, (12.13.3)

because of (12.7.2). Let x/r be an element of I_{2}, where x ∈ k and r ∈ S, as usual. Observe that x/1 ∈ I_{2}, so that x ∈ I_{2}^{c}, and x/r ∈ I_{2}^{ce}, as desired.

Let I_{1} be an ideal in k again, and let us verify that

I_{1}^{ec} = ⋃_{r∈S}(I_{1} : r), (12.13.4)

where (I_{1} : r) is as in (12.8.8). Of course,

I_{1}^{ec} = (S^{−1} I_{1})^{c}, (12.13.5)

by (12.13.2). Thus x ∈ k is an element of I_{1}^{ec} if and only if

x/1 = y/t (12.13.6)

in S^{−1} k for some y ∈ I_{1} and t ∈ S. Remember that (12.13.6) means that x t v − y v = 0 for some v ∈ S, as in Section 12.1. One can use this to get (12.13.4).

This is the first part of part (ii) of Proposition 3.11 on p41 of [8], and the second part says that

(12.13.7) I_{1}^{e} = S^{−1} k if and only if I_{1} ∩ S ≠ ∅.

More precisely, if I_{1} ∩ S ≠ ∅, then 1/1 ∈ I_{1}^{e}. Conversely, if I_{1}^{e} = S^{−1} k, then I_{1}^{ec} = k. This means that 1 ∈ I_{1}^{ec}, which implies that I_{1} ∩ S ≠ ∅, by (12.13.4).

Note that (12.13.8) I_{1} ∈ C if and only if I_{1}^{ec} ⊆ I_{1}, as in Section 12.7. Let q_{1} be the natural quotient mapping from k onto k/I_{1}. Part (iii) of Proposition 3.11 on p41 of [8] says that

(12.13.9) I_{1} ∈ C if and only if q_{1}(S) has no zero-divisors in k/I_{1}.

Indeed, I_{1} ∈ C if and only if

⋃_{r∈S}(I_{1} : r) ⊆ I_{1}, (12.13.10)

by (12.13.4) and (12.13.8). This condition says exactly that for every r ∈ S, if x ∈ k and r x ∈ I_{1}, then x ∈ I_{1}, as desired.

Consider the mapping I_{1} ↦ I_{1}^{e} = S^{−1} I_{1} (12.13.11)

from ideals in k to ideals in S^{−1} k. Part (iv) of Proposition 3.11 on p41 of [8] states that this defines a one-to-one mapping from the set of prime ideals in k that are disjoint from S onto the set of proper prime ideals in S^{−1} k. Of course, if 0 ∈ S, then there are no ideals in k that are disjoint from S, and there are no proper ideals in S^{−1} k = {0}. Thus we may as well suppose that 0 ∉ S.

Remember that (12.13.11) is one-to-one on C, as in Section 12.7. If I_{1} is a prime ideal in k, then k/I_{1} has no nonzero zero-divisors. If I_{1} is also disjoint from S, then it follows that I_{1} ∈ C, by (12.13.9). Thus (12.13.11) is one-to-one on the set of prime ideals in k that are disjoint from S.

If I_{2} is any ideal in S^{−1} k, then we have already seen that I_{2}^{ce} = I_{2}. If I_{2} ≠ S^{−1} k, then it follows that

I_{2}^{c} ∩ S = ∅. (12.13.12)

If I_{2} is a prime ideal in S^{−1} k, then I_{2}^{c} is a prime ideal in k, as in Section 12.7. This shows that every proper prime ideal in S^{−1} k corresponds to a prime ideal in k that is disjoint from S as in (12.13.11).

Let I_{1} be a proper prime ideal in k, and let q_{1} be the natural quotient mapping from k onto k/I_{1}, which is an integral domain. Note that q_{1}(S) is a multiplicatively closed subset of k/I_{1}, as in Section 12.11. Suppose that

I_{1} ∩ S = ∅, (12.13.13)

so that 0 ∉ q_{1}(S). (12.13.14)

This means that q_{1}(S)^{−1}(k/I_{1}) ≠ {0}, (12.13.15)

as in Section 12.1. In fact, one can check that

(12.13.16) q_{1}(S)^{−1}(k/I_{1}) is an integral domain,

because k/I_{1} is an integral domain.

Using q_{1}, we get a ring homomorphism

(12.13.17) Q_{1} from S^{−1} k onto q_{1}(S)^{−1}(k/I_{1}),

with kernel S^{−1} I_{1}, as in Section 12.11. This leads to a ring isomorphism

(12.13.18) from (S^{−1} k)/(S^{−1} I_{1}) onto q_{1}(S)^{−1}(k/I_{1}).

It follows that (S^{−1} k)/(S^{−1} I_{1}) is an integral domain, which could also be verified more directly from the definitions. This implies that

(12.13.19) S^{−1} I_{1} is a proper prime ideal in S^{−1} k,

so that (12.13.11) maps the set of prime ideals in k that are disjoint from S into the set of proper prime ideals in S^{−1} k.

Part (v) of Proposition 3.11 on p42 of [8] states that (12.13.11) commutes with taking finite sums, products, intersections, and radicals of ideals in k. This was already mentioned in Section 12.9 for finite sums and products, and it follows from analogous statements in Section 12.2 for finite sums and intersections. If I_{1} is an ideal in k, then

S^{−1}(r(I_{1})) = r(I_{1})^{e} ⊆ r(I_{1}^{e}) = r(S^{−1} I_{1}), (12.13.20)

as in Section 12.10. The opposite inclusion can be verified directly in this case. If N (k) is the nilradical of k, then Corollary 3.12 on p42 of [8] states that

(12.13.21) the nilradical N (S^{−1} k) of S^{−1} k is the same as S^{−1} N (k).

This follows from the previous statement about radicals, because the nilradical is the same as the radical of {0}.

Let I_{0} be a proper prime ideal in k, and let us take S = k \ I_{0}, which is a multiplicatively closed subset of k, as in Section 12.4. In this case, (12.13.11) defines a one-to-one mapping from the set of prime ideals in k that are contained in I_{0} onto the set of proper prime ideals in S^{−1} k = k_{I_{0}}, as before. This is Corollary 3.13 on p42 of [8].

12.14 More on modules of fractions

Let k be a commutative ring with a multiplicative identity element, and let S be a multiplicatively closed subset of k. If V is a finitely-generated module over k, then Proposition 3.14 on p 43 of [8] states that

S^{−1} Ann(V) = Ann(S^{−1} V). (12.14.1)

Let W and Z be submodules of V, and suppose that they satisfy the corresponding analogues of (12.14.1). Let us first check that W + Z has the same property, as in [8]. Observe that

(12.14.2) S^{−1} Ann(V + W) = S^{−1}(Ann(W) ∩ Ann(Z))
= (S^{−1} Ann(W)) ∩ (S^{−1} Ann(Z))
= (Ann(S^{−1} W)) ∩ (Ann(S^{−1} Z)),

where the first step is as in Section 12.8, the second step is as in Section 12.2, and the third step is by hypothesis. The right side is equal to

Ann((S^{−1} W) + (S^{−1} Z)) = Ann(S^{−1}(W + Z)), (12.14.3)

as in Sections 12.8 and 12.2 again.

This permits one to reduce to the case where V is generated by a single element, as a module over k. If I = Ann(V), then it follows that V is isomorphic to k/I, as a module over k. In this case, S^{−1} V is isomorphic to (S^{−1} k)/(S^{−1} I), as a module over S^{−1} k, as in Section 12.2. This implies that

Ann(S^{−1} V) = S^{−1} I = S^{−1} Ann(V), (12.14.4)

as desired.

Now let V be any module over k, and let W, Z be submodules of V. If Z is finitely generated as a module over k, then

S^{−1} (W : Z) = (S^{−1} W : S^{−1} Z), (12.14.5)

as in Corollary 3.15 on p43 of [8]. Remember that (W : Z) is an ideal in k, as in Section 12.8, so that S^{−1} (W : Z) is an ideal in S^{−1} k. Similarly, S^{−1} W and S^{−1} Z are submodules of S^{−1} V, as a module over S^{−1} k, so that (S^{−1} W : S^{−1} Z) is an ideal in S^{−1}k. To show (12.14.5), we use the fact that

(W : Z) = Ann((W + Z)/W), (12.14.6)

as in Section 12.8.

It is easy to see that (W + Z)/W is finitely generated as a module over k, because Z is finitely generated, by hypothesis. It follows that

S^{−1} (W : Z) = Ann(S^{−1} ((W + Z)/W)), (12.14.7)

by (12.14.1). We also have that S^{−1} ((W + Z)/W) is isomorphic to

((S^{−1} W) + (S^{−1} Z))/(S^{−1} W), (12.14.8)

as modules over S^{−1} k, as in Section 12.2. This means that

S^{−1} (W : Z) = Ann(((S^{−1} W) + (S^{−1} Z))/(S^{−1} W)), (12.14.9)

by (12.14.7). The right side of (12.14.9) is the same as the right side of (12.14.5), as in Section 12.8 again.

12.15 Contractions of prime ideals

Let k_{1} and k_{2} be commutative rings with multiplicative identity elements 1_{k_{1}} and 1_{k_{2}}, respectively, and let f be a ring homomorphism from k_{1} into k_{2}, with f(1_{k_{1}}) = 1_{k_{2}}. Also let I_{1} be an ideal in k_{1}, and remember that extensions of ideals in k_{1} and contractions of ideals in k_{2} with respect to f are defined as in Section 12.7. In particular, if I_{1} is the contraction of an ideal in k_{2} with respect to f, then I_{1} = I_{1}^{ec}, (12.15.1)

as before.

Suppose now that I_{1} is a proper prime ideal in k_{1} that satisfies (12.15.1). We would like to show that I_{1} is the contraction of a prime ideal in k_{2} with respect to f, as in Proposition 3.16 on p43 of [8]. Remember that

S_{1} = k_{1} \ I_{1} (12.15.2)

is multiplicatively closed in k_{1}, because I_{1} is a prime ideal, as in Section 12.4. This implies that S_{2} = f(S_{1}) = f(k_{1} \ I_{1}) (12.15.3)

is multiplicatively closed in k_{2}, as in Section 12.11.

Clearly I_{1}^{ec} ∩ S_{1} = ∅, (12.15.4)

by (12.15.1). Using this, one can check that

I_{1}^{e} ∩ S_{2} = ∅. (12.15.5)

Of course, x_{2} ↦ x_{2}/1_{k_{2}} (12.15.6)

defines a ring homomorphism from k_{2} into S_{2}^{−1} k_{2}, as in Section 12.1. Let (I_{1}^{e})^{e} be the ideal in S_{2}^{−1} k_{2} which is the extension of I_{1}^{e} with respect to (12.15.6). Observe that (I_{1}^{e})^{e} ≠ S_{2}^{−1} k_{2}, (12.15.7)

by (12.13.7) and (12.15.5).

Let I_{3} be a maximal proper ideal in S_{2}^{−1} k_{2} with

(I_{1}^{e})^{e} ⊆ I_{3}. (12.15.8)

In particular, I_{3} is a prime ideal in S_{2}^{−1} k_{2}, because it is maximal. Consider

I_{2} = (I_{3})^{c} = {x_{2} ∈ k_{2} : x_{2}/1_{k_{2}} ∈ I_{3}}, (12.15.9)

which is the contraction of I_{3} in k_{2} with respect to (12.15.6). Thus I_{2} is a prime ideal in k_{2}, because I_{3} is a prime ideal in S_{2}^{−1} k_{2}, as in Section 12.7. Note that

I_{1}^{e} ⊆ I_{2}, (12.15.10)

because of (12.15.8). We also have that

I_{2} ∩ S_{2} = ∅, (12.15.11)

because I_{3} is a proper ideal in S_{2}^{−1} k_{2}. This means that I_{2}^{c} ⊆ I_{1}, by (12.15.3). It follows that I_{2}^{c} = I_{1}, (12.15.12)

by (12.15.10).

Part IV

More on rings and modules Nakayama’s lemma and other matters

13.1 The Jacobson radical

Let k be a commutative ring with a multiplicative identity element. The Jacobson radical R in k is defined to be the intersection of all of the maximal proper ideals in k, as on p5 of [8]. Of course, this is an ideal in k. Note that k has a proper maximal ideal when k ≠ {0}. If k = {0}, then we interpret R to be {0}.

If x ∈ R and y ∈ k, then

(13.1.1) 1 − x y has a multiplicative inverse in k,

as in Proposition 1.9 on p6 of [8]. Otherwise, the ideal generated by 1 − x y would be a proper subset of k, which is contained in a maximal proper ideal I in k. We also have that x ∈ I, because R ⊆ I, by construction, so that x y ∈ I. This means that 1 ∈ I, which is a contradiction, because I is supposed to be a proper subset of k.

Conversely, suppose that x ∈ k has the property that (13.1.1) holds for every y ∈ k. We would like to check that x ∈ R, which is the other part of Proposition 1.9 on p6 of [8]. Let I be a maximal proper ideal in k, and suppose that x ∈ k \ I. This means that the ideal in k generated by I and x is equal to k. It follows that there is a w ∈ I and k ∈ k such that

w + x y = 1. (13.1.2)

This implies that w = 1 − x y is not invertible in k, because I ≠ k. Of course, x ∈ k \ R exactly when x ∈ k \ I for some maximal proper ideal I in k.

Remember that N denotes the nilradical of k, consisting of all nilpotent elements of k, as in Section 12.5. Equivalently, N is equal to the intersection of all prime ideals in k, as before. This implies that

N ⊆ R, (13.1.3) because maximal ideals in k are prime ideals.

If z ∈ N, then

(13.1.4) 1 − z has a multiplicative inverse in k,

by (13.1.1) and (13.1.3). This corresponds to the first part of Exercise 1 on p10 of [8]. Alternatively,

∑_{j=0}^{∞} z^{j} (13.1.5)

reduces to a finite sum in k, because z is nilpotent. This sum is the multiplicative inverse of 1 − z in k, by a standard argument. If x ∈ N and y ∈ k, then x y ∈ N, so that one can take z = x y. It follows that (13.1.1) holds under these conditions. This is another way to get (13.1.3), using the earlier characterization of R.

Let I be an ideal in k. Of course, I ⊆ R exactly when I is contained in every maximal proper ideal in k.

If I ⊆ R and x ∈ I, then 1 − x has a multiplicative inverse in k, as in (13.1.1). Conversely, if for every x ∈ I, 1 − x has a multiplicative inverse in k, then one can check that I ⊆ R, using the characterization of R in terms of (13.1.1).

13.2 Ideals, modules, and homomorphisms

Let k be a commutative ring with a multiplicative identity element, and let I be an ideal in k. Also let V be a module over k, and remember that I · V is the submodule of V consisting of finite sums of elements of the form t · v, where t ∈ I and v ∈ V, as in Section 12.8.

Suppose that ϕ is a homomorphism from V into itself, as a module over k, such that ϕ(V) ⊆ I · V. (13.2.1)

Suppose that V is finitely generated as a module over k as well. Under these condition, there are a positive integer n and elements a_{1}, …, a_{n} of I such that

ϕ^{n} + a_{1} ϕ^{n−1} + ⋯ + a_{n−1} ϕ + a_{n} I_{V} = 0, (13.2.2)

where I_{V} is the identity mapping on V. This is Proposition 2.4 on p21 of [8].

Remember that the space Hom_{k}(V, V) of homomorphisms from V into itself, as a module over k, is an associative algebra over k with respect to composition of mappings. Let A_{ϕ} be the subalgebra of Hom_{k}(V, V) generated by ϕ and I_{V}. This is a commutative subalgebra of Hom_{k}(V, V), consisting of finite linear combinations of I_{V} and powers of ϕ, with coefficients in k.

Let v_{1}, …, v_{n} be generators of V, as a module over k. Thus every element of I · V can be expressed as

∑_{l=1}^{n} t_{l} · v_{l}, (13.2.3)

where t_{1}, …, t_{n} ∈ I. In particular, for each j = 1, …, n, ϕ(v_{j}), ϕ(v_{j}) can be expressed as

ϕ(v_{j}) = ∑_{l=1}^{n} a_{j,l} · v_{l}, (13.2.4)

where a_{j,1}, …, a_{j,n} ∈ I.

Let δ_{j,l} be the usual n × n Kronecker delta matrix with entries in k, which is equal to 1 when j = l and to 0 when j ≠ l. Thus

δ_{j,l} ϕ − a_{j,l} I_{V} (13.2.5)

defines an n × n matrix with entries in A_{ϕ}. This means that the determinant

ψ = det(δ_{j,l} ϕ − a_{j,l} I_{V}) (13.2.6)

of this matrix can be defined as an element of A_{ϕ} in the usual way. In fact, ψ can be expressed as in the left side of (13.2.2). We would like to show that

ψ = 0. (13.2.7)

Note that δ_{j,l} I_{V} (13.2.8)

is the identity matrix, as an n×n matrix with entries in A_{ϕ}. It is well known that the product of the cofactor transpose of (13.2.5) with (13.2.5), as n×n matrices with entries in A_{ϕ}, is equal to the determinant times the identity matrix, which is to say δ_{j,l} ψ, (13.2.9)

as in Cramer’s rule. The cofactor transpose of a matrix is sometimes called the adjoint or classical adjoint of the matrix, which is different from other uses of the term adjoint for linear mappings.

Let us reexpress (13.2.4) as saying that

∑_{l=1}^{n}(δ_{j,l} ϕ − a_{j,l} I_{V})(v_{l}) = 0 (13.2.10)

for each j = 1, …, n. This implies that

∑_{l=1}^{n} δ_{r,l} · ψ(v_{l}) = 0 (13.2.11)

for every r = 1, …, n, by the remarks in the preceding paragraph. This means that ψ(v_{r}) = 0 for every r = 1, …, n. It follows that (13.2.7) holds, because V is generated by v_{1}, …, v_{n} as a module over k, by hypothesis.

13.3 Nakayama’s lemma

Let k be a commutative ring with a multiplicative identity element, and let V be a finitely-generated module over k. Also let I be an ideal in k, and suppose that I · V = V. (13.3.1)

Under these conditions, there is an x ∈ k such that

x − 1 ∈ I (13.3.2)

and x · V = 0. (13.3.3)

This is Corollary 2.5 on p21 of [8]. To see this, one can take ϕ to be the identity mapping on V in (13.2.1). Let a_{1}, …, a_{n} ∈ I be as in (13.2.2), and put

x = 1 + a_{1} + ⋯ + a_{n}. (13.3.4)

Thus (13.3.2) holds by construction, and (13.3.3) follows from (13.2.2). Suppose now that we also have that

I ⊆ R, (13.3.5)

where R is the Jacobson radical of k, as in Section 13.1. In this case, Nakayama’s lemma states that V = {0}, (13.3.6)

as in Proposition 2.6 on p21 of [8]. Indeed, (13.3.2) and (13.3.5) imply that x has a multiplicative inverse in k, as in Section 13.1. This means that (13.3.6) follows from (13.3.3). This is the first proof on p21 of [8].

Alternatively, suppose that V is generated by v_{1}, …, v_{n}, as a module over k, for some positive integer n. One can check that every element of V can be expressed as a linear combination of v_{1}, …, v_{n}, with coefficients in I, because of (13.3.1). In particular,

v_{n} = a_{1} · v_{1} + ⋯ + a_{n} · v_{n} (13.3.7)

for some a_{1}, …, a_{n} ∈ I. Equivalently,

(1 − a_{n}) · v_{n} = a_{1} · v_{1} + ⋯ a_{n−1} · v_{n−1}. (13.3.8)

Note that a_{n} ∈ R, by (13.3.5), so that 1 − a_{n} has a multiplicative inverse in k, as in Section 13.1. Using this and (13.3.8), we get that V is generated by v_{1}, …, v_{n−1}, as a module over k. We can repeat the process as needed to get that (13.3.6) holds. This corresponds to the second proof on p22 of [8].

Let W be a finitely-generated module over k, and let Z be a submodule of W. Also let I be an ideal in k again, and suppose that

I · W + Z = W. (13.3.9)

If (13.3.5) holds, then Z = W, (13.3.10)

as in Corollary 2.7 on p22 of [8]. This can be obtained from Nakayama’s lemma, with V = W/Z. (13.3.11)

More precisely, this uses the fact that

I · (W/Z) = (I · W + Z)/Z. (13.3.12)

13.4 A noncommutative version

Let A be a ring with a nonzero multiplicative identity element e, and let I be a left ideal in A. Suppose that for every x ∈ I,

(13.4.1) e + x has a right inverse in A.

If A is a commutative ring, then this is the same as saying that I is contained in the Jacobson radical of A, as in Section 13.1.

Suppose for the moment that A is a local ring, as in Section 4.13. Thus the collection I_{0} of a ∈ A such that a does not have a left inverse in A is a left ideal in A. This implies that I_{0} is a two-sided ideal in A, and that every element of A\I_{0} has a two-sided inverse in A, as before. If a ∈ I_{0}, then e+a ∉ I_{0}, because e ∉ I_{0}, so that e + a is invertible in A. In particular, I_{0} satisfies (13.4.1).

Let V be a right module over A, and let V · I be the subset of V consisting of finite sums of elements of the form v · a, with v ∈ V and a ∈ I. This is a subgroup of V, as a commutative group with respect to addition. Suppose that

V · I = V. (13.4.2)

If V is finitely generated as a right module over A, then it follows that V = {0}. This is the same as Nakayama’s lemma when A is commutative. If A is a local ring, and I = I_{0} is as before, then this is the first part of Proposition 5.1’ on p155 of [50]. The formulation using (13.4.1) is part of Exercise 9 on p160 of [50].

The proof in [50] is analogous to the second proof of Nakayama’s lemma in the previous section. Suppose that V is generated by v_{1}, …, v_{n}, as a right module over A, for some positive integer n. Using this and (13.4.2), one can check that every element of V can be expressed as

∑_{j=1}^{n} v_{j} · a_{j} (13.4.3)

for some a_{1}, …, a_{n} ∈ I. This also uses the hypothesis that I be a left ideal in A.

In particular, v_{n} can be expressed as (13.4.3) for some a_{1}, …, a_{n} ∈ I. This means that

v_{n} · (e − a_{n}) = ∑_{j=1}^{n−1} v_{j} · a_{j}. (13.4.4)

Let b be a right inverse of e − a_{n} in A, as in (13.4.1). Observe that

v_{n} = (v_{n} · (e − a_{n})) · b = ∑_{j=1}^{n−1}(v_{j} · a_{j}) · b = ∑_{j=1}^{n−1} v_{j} · (a_{j} b). (13.4.5)

This implies that V is generated by v_{1}, …, v_{n−1}, as a right module over A, and one can repeat the process to get that V = {0}. Of course, there are analogous statements for left modules.

13.5 Ideals and fractions

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. Also let S be a multiplicatively closed subset of k, so that S^{−1} k and S^{−1} V may be defined as in Sections 12.1 and 12.2. If

r · V = {0} (13.5.1)

for some r ∈ S, then it is easy to see that

S^{−1} V = {0}. (13.5.2)

If (13.5.2) holds, and if V is finitely generated as a module over k, then one can check that there is an r ∈ S such that (13.5.1) holds. This corresponds to Exercise 1 on p43 of [8].

Let I be an ideal in k, so that I · V may be defined as a submodule of V as in Section 12.8. If I is considered as a submodule of k, as a module over itself, then S^{−1} I may be considered as a submodule of S^{−1} k, as a module over itself, as in Section 12.2. Equivalently, S^{−1} I may be considered as an ideal in S^{−1} k. Thus (S^{−1} I) · (S^{−1} V) may be defined as a submodule of S^{−1} V, as a module over S^{−1} k, in the usual way. One can check that

S^{−1} (I · V) = (S^{−1} I) · (S^{−1} V). (13.5.3)

Let us now take S = 1 + I = {1 + a : a ∈ I}. (13.5.4)

It is easy to see that this is a multiplicatively closed subset of k, as in Example 4 on p38 of [8]. In this case, every element of S^{−1} I can be expressed as

a/(1 − b), (13.5.5) where a, b ∈ I. Note that

(13.5.6) (1/1) − (a/(1 − b)) = ((1 − b)/(1 − b)) − (a/(1 − b))
= (1 − a − b)/(1 − b).

This is invertible as an element of S^{−1} k, because 1 − b, 1 − a − b ∈ S. This implies that S^{−1} I is contained in the Jacobson radical of S^{−1} k, as in Section 13.1. This is the first part of Execise 2 on p43 of [8].

Let V be a module over k again, and suppose that

V = I · V. (13.5.7)

This implies that S^{−1} V = (S^{−1} I) · (S^{−1} V), (13.5.8)

by (13.5.3). Suppose that V is finitely generated as a module over k, which implies that S^{−1} V is finitely generated as a module over S^{−1} k. Under these conditions, we get that (13.5.2) holds, by Nakayama’s lemma, because S^{−1} I is contained in the Jacobson radical of S^{−1} k, as in the preceding paragraph. This implies that (13.5.1) holds for some r ∈ S, as before.

This is another way to get (13.3.3), as in the second part of Exercise 2 on p43 of [8]. Of course, one should use the second proof of Nakayama’s lemma in Section 13.3 here, since the first proof uses (13.3.3).

13.6 Modules over quotient rings

Let k be a commutative ring with a multiplicative identity element, let I be an ideal in k, and let V be a module over k. Suppose for the moment that

I ⊆ Ann(V), (13.6.1)

where the annihilator Ann(V) of V in k is as in Section 12.8. This means that the action of each element of I on V is equal to 0. It follows that the action of any t ∈ k on V only depends on the image of t in the quotient ring k/I. This permits one to consider V as a module over k/I, as on p19 of [8].

Let W be a module over k, and remember that I · W is a submodule of W, as in Section 12.8. Thus the quotient W/(I · W) is defined as a module over k. Of course, I ⊆ Ann(W/(I · W)), (13.6.2)

by construction. This means that W/(I · W) may be considered as a module over k/I, as in the preceding paragraph.

Let q_{W} be the natural quotient mapping from W onto W/(I ·W), as modules over k. Suppose that w_{1}, …, w_{n} are finitely many elements of W such that

(13.6.3) W/(I · W) is generated by q_{W} (w_{1}), …, q_{W} (w_{n}), as a module over k.

Equivalently, this means that

(13.6.4) W/(I · W) is generated by q_{W} (w_{1}), …, q_{W} (w_{n}), as a module over k/I.

Let Z be the submodule of W, as a module over k, generated by w_{1}, …, w_{n}. Thus q_{W} (Z) = W/(I · W). (13.6.5)

This is the same as saying that

Z + (I · W) = W. (13.6.6)

Suppose now that W is also finitely generated, as a module over k. If I is contained in the Jacobson radical of k, then it follows that W = Z, as in Section 13.3. This means that

(13.6.7) W is generated by w_{1}, …, w_{n}, as a module over k.

In particular, if k is a local ring, and I is the unique maximal proper ideal in k, then I is the same as the Jacobson radical of k. This corresponds to Proposition 2.8 on p22 of [8].

13.7 Another noncommutative version

Let A be a ring with a nonzero multiplicative identity element e_{A}, and let I be a proper two-sided ideal in A. Thus the quotient A/I is a ring with a nonzero multiplicative identity element e_{A/I}.

Let W be a right module over A, and let W · I be as in Section 13.4. It is easy to see that

(13.7.1) W · I is a submodule of W, as a right module over A,

because I is a right ideal in A.

This means that the quotient W/(W · I) is defined as a right module over A. The action of any element of I on W/(W · I) on the right is equal to 0, by construction. Equivalently, the action of a ∈ A on W/(W · I) on the right only depends on the image of a in A/I, under the natural quotient homomorphism from A onto A/I. It follows that W/(W ·I) may be considered as a right module over A/I as well, with respect to this action.

Let q_{W} be the natural quotient homomorphism from W onto W/(W · I), as right modules over A. Suppose that w_{1}, …, w_{n} are finitely many elements of W such that

(13.7.2) W/(W · I) is generated by q_{W} (w_{1}), …, q_{W} (w_{n}), as a right module over A.

This is the same as saying that

(13.7.3) W/(W · I) is generated by q_{W} (w_{1}), …, q_{W} (w_{n}), as a right module over A/I.

Let Z be the submodule of W, as a right module over A, generated by w_{1}, …, w_{n}. Note that q_{W} (Z) = W/(W · I), (13.7.4)

so that Z + (W · I) = W. (13.7.5)

Consider the quotient V = W/Z, which is another right module over A. Thus V · I may be defined in the usual way, and is a submodule of V, as a right module over A. It is easy to see that

V · I = (W/Z) · I = (W · I + Z)/Z = W/Z = V, (13.7.6)

because of (13.7.5).

Suppose that for every x ∈ I,

(13.7.7) e_{A} + x has a right inverse in A.

In particular, this holds when A is a local ring, and I is the corresponding two-sided ideal in A, as in Sections 4.13 and 13.4.

Suppose also that W is finitely generated as a right module over A. This implies that V is finitely generated as a right module over A too. Under these conditions, we get that V = {0}, because of (13.7.6), as in Section 13.4. This means that W = Z, so that

(13.7.8) W is generated by w_{1}, …, w_{n}, as a right module over A.

This basically corresponds to a simplification of the first part of Proposition 5.2 on p155 of [50]. This also uses the first part of Proposition 5.1’ on p155 of [50], and the corresponding part of Exercise 9 on p160 of [50], as in Section 13.4. There are analogous statements for left modules, as usual.

13.8 Some quotients by ideals

Let A be a ring with a nonzero multiplicative identity element e_{A} again, and let I_{R} be a right ideal in A, so that

W = A/I_{R} (13.8.1)

is a right module over A. Also let I be a left ideal in A, and let I_{R} I be the subset of A consisting of finite sums of products of elements of I_{R} and I. Note that I_{R} I ⊆ I_{R} ∩ I. (13.8.2)

If I_{R} is considered as a right module over A, then I_{R} I is the same as the subset I_{R} ·I of I_{R} defined in Section 13.4. Similarly, if I is considered as a left module over A, then I_{R} I is the same as the subset I_{R} · I of I defined analogously.

Let

W · I = (A/I_{R}) · I (13.8.3)

be as in Section 13.4 again. This is a subgroup of W, as a commutative group with respect to addition. Of course, I_{R}+I is a subgroup of A, as a commutative group with respect to addition. It is easy to see that

W · I = (I_{R} + I)/I_{R}. (13.8.4)

The quotients

W/(W · I) (13.8.5)

and

A/(I_{R} + I) (13.8.6)

are defined as commutative groups with respect to addition in the usual way. These quotients are isomorphic to each other in a natural way, because of (13.8.4), by standard arguments.

Suppose now that I is a proper two-sided ideal in A, so that the quotient A/I is a ring with a nonzero multiplicative identity element. Observe that I_{R} + I, I_{R} ∩ I, and I_{R} I are right ideals in A, because I is a right ideal. We also have that W · I is a submodule of W, as a right module over A, as in the previous section. Thus the quotients (13.8.5) and (13.8.6) are defined as right modules over A in this case. These quotients are isomorphic as right modules over A in a natural way, as before.

In fact, (13.8.5) and (13.8.6) may be considered as right modules over A/I, as in the previous section. They are isomorphic as right modules over A/I in a natural way, as usual. Let q_{W} be the natural quotient mapping from W onto (13.8.5), and let q_{0} be the natural quotient mapping from A onto W. Put

q_{1} = q_{W} ∘ q_{0}, (13.8.7)

which corresponds to the natural quotient mapping from A onto (13.8.6). If x ∈ I, then

q_{1}(e_{A} + x) = q_{1}(e_{A}). (13.8.8)

This implies that (13.8.5) is generated by q_{1}(e_{A} + x), as a right module over A.

If e_{A} + x has a right inverse in A, then A is generated by e_{A} + x, as a right module over itself. This implies that W is generated by q_{0}(e_{A} + x), as a right module over A. Of course, one could consider left modules over A obtained by taking quotients of A by left ideals as well.

13.9 Some freely-generated modules

Let A be a ring, with a nonzero multiplicative identity element e_{A}. Suppose that A is a local ring, as in Sections 4.13 and 13.4, with corresponding twosided ideal I_{0}. Let W ≠ {0} be a finitely-generated right module over A, and let W · I_{0} be as in Section 13.4. This is a submodule of W, as a right module over A, as in Section 13.7. Let q_{W} be the natural quotient homomorphism from W onto W/(W · I_{0}), as right modules over A.

Remember that W/(W · I_{0}) may be considered as a right module over A/I_{0} in a natural way, as in Section 13.7. Let w_{1}, …, w_{n} be finitely many elements of W such that

(13.9.1) W/(W · I_{0}) is generated by q_{W} (w_{1}), …, q_{W} (w_{n}), as a right module over A/I_{0}.

This implies that W is generated by w_{1}, …, w_{n}, as a right module over A, as in Section 13.7.

We may as well ask that q_{W} (w_{1}), …, q_{W} (w_{n}) be a minimal set of generators of W/(W · I_{0}), as a right module over A/I_{0}. This implies that

(13.9.2) W/(W · I_{0}) is freely generated by q_{W} (w_{1}), …, q_{W} (w_{n}), as a right module over A/I_{0},

because A/I_{0} is a division ring. This can be verified in basically the same way as for vector spaces over (commutative) fields.

Let A^{n} be the usual space of n-tuples of elements of A, considered as a right module over A. If a ∈ A^{n}, then put

ϕ(a) = ∑_{j=1}^{n} w_{j} · a_{j}. (13.9.3)

This defines a homomorphism from A^{n} onto W, as right modules over A.

By considering A^{n} as a right module over A, A^{n}·I_{0} is defined as a submodule of A^{n} as in Sections 13.4 and 13.7. This is the same as the set of n-tuples of elements of I_{0}, but it is better not to use the notation I_{0}^{n} for this set, because that may also be used for the nth power of I_{0} in A. Observe that

ker ϕ ⊆ A^{n} · I_{0}. (13.9.4)

More precisely, ker ϕ ⊆ ker(q_{W} ∘ ϕ) = A^{n} · I_{0}, (13.9.5)

using (13.9.2) in the second step.

Suppose now in addition that

(13.9.6) W is projective as a right module over A.

This implies that A^{n} corresponds to the direct sum of ker ϕ and another submodule of A^{n}, as a right module over A, as in Section 2.7. This means that there is a homomorphism ψ from A^{n} onto ker ϕ, as right modules over A, that is equal to the identity mapping on ker ϕ. In particular, it follows that

(13.9.7) ker ϕ is finitely generated, as a right module over A.

As usual, (ker ϕ) · I_{0} is defined as a submodule of ker ϕ, as a right module over A, as in Sections 13.7 and 13.7. One can check that

ker ϕ = (ker ϕ) · I_{0}, (13.9.8)

using (13.9.4) and the remarks in the preceding paragraph. More precisely,

(ker ϕ) · I_{0} = ψ(A^{n}) · I_{0} = ψ(A^{n} · I_{0}) = ker ϕ, (13.9.9)

using (13.9.4) and the fact that ψ is the identity mapping on ker ϕ in the last step.

Under these conditions, we get that

ker ϕ = {0}, (13.9.10)

as in Section 13.4. This means that

(13.9.11) ϕ is an isomorphism from A^{n} onto W,

as right modules over A. This is the same as saying that

(13.9.12) W is freely generated by w_{1}, …, w_{n}, as a right module over A.

One could just as well consider left modules over A, as usual.

The remarks in this section basically correspond to parts of Propositions 5.1’ and 5.2 and Theorem 5.3 on p155f of [50], and their proofs. This may be considered as a simplification of Theorem 6.1’ on p157 of [50]. More precisely, (13.9.6) was not part of the hypotheses in [50], and was used here to get (13.9.7) and (13.9.8). In [50], (13.9.7) was obtained by asking A to be right Noetherian as a ring. Another condition was used to get (13.9.8) in [50], which is implied by (13.9.6).

The condition used in [50] to get (13.9.8) may be described in terms of “augmented rings”, which are discussed in Chapter 33. Here A may be considered as an augmented ring, with augmentation ideal I_{0}. The condition used in [50] may be described as saying that the first homology group of A, with respect to this augmentation, and with coefficients in W, is equal to {0}. This uses the definition of the homology groups on p143 of [50].

13.10 Complementary ideals

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a nonzero multiplicative identity element e_{A}. Let us say that left ideals I_{1}, I_{2} in A are complementary in A if A corresponds to the direct sum of I_{1} and I_{2}, as a left module over itself. It is easy to see that this happens if and only if A corresponds to the direct sum of I_{1} and I_{2}, as a module over k. Of course, there are analogous notions and statements for right ideals and two-sided ideals in A. If I_{1} and I_{2} are complementary two-sided ideals in A, then A corresponds to the direct sum of I_{1} and I_{2}, as an algebra over k.

If I_{1}, I_{2} are left ideals in A, then the quotients A/I_{1}, A/I_{2} may be considered as left modules over A. If I_{1} and I_{2} are complementary left ideals in A, then A/I_{1}, A/I_{2} are isomorphic to I_{2}, I_{1}, respectively, as left ideals over A. In this case, I_{1} and I_{2} are projective as left modules over A, so that A/I_{1} and A/I_{2} are projective as left modules over A.

Let I_{1} be a left ideal in A, and suppose that A/I_{1} is projective as a left module over A. This implies that there is a left ideal I_{2} in A that is complementary to I_{1}, as in Section 2.7.

Let θ_{1} be an idempotent element of A, so that

θ_{1}^{2} = θ_{1}. (13.10.1)

Note that θ_{2} = e_{A} − θ_{1} (13.10.2)

is idempotent in A as well. We also have that

θ_{1} θ_{2} = θ_{2} θ_{1} = 0. (13.10.3)

Put I_{j} = {a θ_{j} : a ∈ A} (13.10.4)

for j = 1, 2, which are left ideals in A. One can check that I_{1} and I_{2} are complementary left ideals in A under these conditions.

Conversely, suppose that I_{1} and I_{2} are complementary left ideals in A. This implies that there are unique elements θ_{1}, θ_{2} of I_{1}, I_{2} such that

θ_{1} + θ_{2} = e_{A}. (13.10.5)

If a ∈ A, then a θ_{1} ∈ I_{1}, a θ_{2} ∈ I_{2}, and

a = a θ_{1} + a θ_{2}. (13.10.6)

In fact, a θ_{1} and a θ_{2} are uniquely determined by these properties, by hypothesis. In particular, if a ∈ I_{1}, then

a θ_{1} = a and a θ_{2} = 0, (13.10.7)

and similarly for elements of I_{2}. It follows that θ_{1}, θ_{2} are idempotent elements of A that satisfy (13.10.3), and that I_{1}, I_{2} are as in (13.10.4). This corresponds to a remark on p164 of [262].

13.11 A class of modules

Let A be a ring with a nonzero multiplicative identity element e_{A}, and let I be a proper two-sided ideal in A. If W is a right module over A, then W · I may be defined as in Section 13.4, and is a submodule of W, as in Section 13.7. Let us say that W is proper with respect to I if either W = {0} or

W ≠ W · I, (13.11.1)

as on p154 of [50]. Thus Nakayama’s lemma and its analogue for noncommutative rings give criteria for modules to be proper in this sense. Note that free modules over A are proper with respect to I, as in [50].

Suppose now that A is a graded ring, as in Section 9.14. Remember that A^{0} is a subring of A that contains e_{A}, and that ε(a) = a^{0} defines a ring homomorphism from A onto A^{0}. In this case, we take

I = ker ε. (13.11.2)

This is a proper two-sided ideal in A, which corresponds to the direct sum of A^{j}, j ≥ 1, as a subgroup of A, as a commutative group with respect to addition.

Let W be a graded right module over A, as a graded ring, as in Section 9.15. Under these conditions,

(13.11.3) W is proper with respect to I,

as in Proposition 5.1 on p154 of [50]. To see this, suppose that W ≠ {0}, and let m be the smallest nonnegative integer such that

W^{m} ≠ {0}. (13.11.4)

It is easy to see that W · I is contained in the subgroup of W generated by W^{j}, j ≥ m = 1, as a commutative group with respect to addition. This implies (13.11.1), because of (13.11.4).

Of course, W^{j} ∩ (W · I) (13.11.5)

is a subgroup of W, as a commutative group with respect to addition, for every j ≥ 0. This subgroup consists of finite sums of elements of the form w · a, where w ∈ W^{l} and a ∈ A^{r} for some l ≥ 0 and r ≥ 1 with j = l + r. In particular, W^{0} = {0}. Observe that W · I corresponds to the direct sum of these subgroups, with j ≥ 1. Thus W · I is homogeneous as a subgroup of W, as a graded commutative group with respect to addition, as in Section 5.9.

This defines a grading on W · I, as a commutative group with respect to addition. This grading on W · I is compatible with the grading on A, as in Section 9.15. Thus W · I may be considered as a graded module over A, as a graded ring, as before.

13.12 Quotients and gradings

Let us continue with the same notation and hypotheses as in the previous section, so that A is a graded ring, W is a graded right module over A, as a graded ring, and so on.

Let U be a submodule of W, as a right module over A, that is homogeneous as a subgroup of W, as a graded commutative group with respect to addition, as in Section 5.9. The grading on U induced by the grading on W is compatible with the grading on A, so that

(13.12.1) U is graded as a module over A, as a graded ring,

as in Section 9.15. The quotient W/U is graded as a commutative group with respect to addition in a natural way, as in Section 5.9 again. Of course, W/U is also a right module over A, and the grading on W/U is compatible with the grading on A. This means that

(13.12.2) W/U is graded as a module over A, as a graded ring.

If we take U = W · I, then we get that W/(W · I) is a graded module over A, as a graded ring. If a ∈ I, then the action of a on the right on W/(W · I) is equal to 0, by construction. The action of A on W/(W · I) on the right leads to an action of A/I on W/(W ·I) on the right, as in Section 13.7. This corresponds to the action of A^{0} on W/(W · I) on the right, using ε. Note that W/(W · I) is a graded right module over A^{0} in the usual sense, as in Sections 5.9 and 9.15.

If W is any right module over A^{0}, then W may be considered as a right module over A, using ϵ. In this case, if W is considered as a right module over A, then W · I = {0}, (13.12.3)

by construction. If W is a graded right module over A^{0} in the usual sense, then W may be considered as a graded module over A, as a graded ring, in this way.

Let W be any graded right module over A, as a graded ring, again, and let U be a submodule of W, as a right module over A, that is homogeneous as a subgroup of W, as a graded commutative group with respect to addition. Thus W/U is a graded right module over A, as a graded ring, as before, so that (W/U) · I is a submodule of W/U, as a graded right module over U, that is homogeneous as a subgoup of W/U, as a graded commutative group with respect to addition, as in the previous section. If W/U ≠ {0}, then

W/U ≠ (W/U) · I, (13.12.4)

as in (13.11.1). Equivalently, this means that if U ≠ W, then

U + (W · I) ≠ W. (13.12.5)

Let E be a nonempty subset of W consisting of homogeneous elements of W, as in Section 5.9. Thus each x ∈ E is an element of W^{j} for some nonnegative integer j. Let U_{E} be the submodule of W, as a right module over A, generated by E. It is easy to see that U_{E} is homogeneous as a subgroup of W, as a graded commutative group with respect to addition, under these conditions.

Let q_{W} be the natural quotient mapping from W onto W/(W · I). Note that W/(W · I) is generated by q_{W} (E), as a right module over A, if and only if W/(W · I) is generated by q_{W} (E), as a right module over A/I. This is also the same as saying that U_{E} + (W · I) = W. (13.12.6)

This implies that U_{E} = W, (13.12.7)

as before. This corresponds to the first part of Proposition 5.2 on p155 of [50], in the case of graded modules over graded rings.

13.13 Some graded free modules

Let A be a ring with a nonzero multiplicative identity element e_{A}. Also let E be a nonempty set, and let F_{E} be the free right module over A generated by E. This may be defined precisely as the space c_{00}(E, A) of A-valued functions on E with finite support, on which A acts by multiplication on the right. Let us identify each element of E with the A-valued function on E equal to e_{A} at that element, and to 0 otherwise. Thus F_{E} is freely generated by the elements of E, as a right module over A.

Suppose now that A is a graded ring, as in Section 9.14. Let λ be a function on E with values in the set Z_{+}∪{0} of nonnegative integers. If j is a nonnegative integer, then let (F_{E})^{j} be the subset of F_{E} consisting of finite sums of elements of the form x · a, where x ∈ E, a ∈ A^{r} for some nonnegative integer r, and

j = λ(x) + r. (13.13.1)

It is easy to see that this defines a grading on F_{E}, as a commutative group with respect to addition, as in Section 5.9. This grading is compatible with the grading on A, by construction. Thus

(13.13.2) F_{E} may be considered as a graded right module over A, as a graded ring,

as in Section 9.15. Of course, one can always take λ(x) = 0 for every x ∈ E, in which case F_{E} corresponds to a direct sum of copies of A, indexed by E, as graded right modules over itself.

Let W be a right module over A. Any mapping ϕ from E into W has a unique extension to a homomorphism Φ from F_{E} into W, as right modules over A. Suppose now that W is a graded right module over A, as a graded ring, and that ϕ(x) ∈ W^{λ(x)} (13.13.3) for every x ∈ E. This implies that

(13.13.4) Φ has degree 0 as a homomorphism from F_{E} into W,

as graded commutative groups with respect to addition, as in Section 5.9. This corresponds to a remark on p155 of [50], in the proof of Proposition 5.1.

It follows that

(13.13.5) ker Φ and Φ(F_{E}) are homogeneous as subgroups of F_{E} and W, respectively,

as graded commutative groups with respect to addition. This means that

(13.13.6) ker Φ and Φ(F_{E}) are graded modules over A, as a graded ring,

with respect to the gradings induced by those on F_{E} and W, respectively, as in Section 9.15. This corresponds to another remark on p155 of [50].

Suppose that E is a nonempty subset of W consisting of homogeneous elements, with x ∈ W^{λ(x)} (13.13.7)

for every x ∈ E. This is the same as (13.13.3), with ϕ equal to the obvious inclusion mapping from E into W. In this case, Φ(F_{E}) is the same as the submodule U_{E} of W, as a right module over A, generated by E.

13.14 Some more freely-generated modules

Let A be a ring with a nonzero multiplicative identity element e_{A}, and suppose that A is a graded ring, as in Section 9.14. Thus A^{0} is a subring of A that contains e_{A}, ε(a) = a^{0} defines a ring homomorphism from A onto A^{0}, and I = ker ε is a proper two-sided ideal in A. In this section, we also ask that

(13.14.1) A^{0} be a division ring.

Let W ≠ {0} be a graded right module over A, as a graded ring, as in Section 9.15. Remember that W · I is as in Section 13.4, and is a submodule of W, as a right module over A, as in Section 13.7. Let q_{W} be the natural quotient homomorphism from W onto W/(W · I), as right modules over A. We have seen that W/(W · I) may be considered as a right module over A/I, which corresponds to considering W/(W · I) as a right module over A^{0}, using ε, as in Section 13.12.

Let E be a nonempty subset of W consisting of homogeneous elements. Suppose that W/(W · I) is generated by q_{W} (E), as a right module over A/I. This is the same as saying that W/(W · I) is generated by q_{W} (E), as a right module over A, as before.

One can show that W/(W · I) is freely generated by a subset of q_{W} (E), as a right module over A/I, because A/I is a division ring. This is analogous to standard arguments for vector spaces over (commutative) fields.

13.14. SOME MORE FREELY-GENERATED MODULES 355

This implies that there is a nonempty subset Ẽ of E such that (13.14.2) W/(W · I) is freely generated by q_{W} (Ẽ),

as a right module over A/I,

and (13.14.3) q_{W} is one-to-one on Ẽ. It follows that (13.14.4) W is generated by Ẽ, as a right module over A, as in Section 13.12.

If x ∈ Ẽ, then let λ(x) be a nonnegative integer such that x ∈ W^{λ(x)}. Let F_{E} be the free right module over A generated by Ẽ, with the grading defined using λ, as in the previous section. The obvious inclusion mapping ϕ̃ from Ẽ into W has a unique extension to a homomorphism Φ̃ from F_{E} into W, as right modules over A. We also have that (13.14.5) Φ̃ has degree 0 as a homomorphism from F_{E} into W, as graded commutative groups with respect to addition, as in the previous section.

Note that

Φ̃(F_{E}) = W, (13.14.6)

by (13.14.4). As in the previous section, ker Φ̃ is homogeneous as a subgroup of F_{E}, as a graded commutative group with respect to addition. This implies that (13.14.7) ker Φ̃ is a graded module over A, as a graded ring, with respect to the grading induced by the one on F_{E}, as before.

Of course, F_{E} · I can be defined in the usual way, because F_{E} is a right module over A. In this case, F_{E} · I consists of finite sums of elements of the form x · a, where x ∈ Ẽ and a ∈ I. Observe that

ker(q_{W} ∘ Φ̃) = F_{E} · I, (13.14.8)

because of (13.14.2) and (13.14.3). This means that

ker Φ̃ ⊆ F_{E} · I. (13.14.9)

Suppose that (13.14.10) W is projective as a right module over A. It follows that F_{E} corresponds to the direct sum of ker Φ̃ and another submodule of F_{E}, as a right module over A, as in Section 2.7. This implies that there is a homomorphism Ψ from F_{E} onto ker Φ̃, as right modules over A, that is equal to the identity mapping on ker Φ̃.

Note that (ker Φ̃) · I may be defined in the usual way, and is a submodule of ker Φ̃, as a right module over A. Under these conditions, we have that

(13.14.11) (ker Φ̃) · I = Ψ(F_{E}) · I = Ψ(F_{E} · I) = ker Φ̃,

using (13.14.9) and the fact that Ψ is the identity mapping on ker Φ̃ in the last step.

This implies that ker Φ̃ = {0}, (13.14.12)

because of (13.14.7), as in Section 13.11. It follows that Φ̃ is an isomorphism from F_{E} onto W, as right modules over A.

This basically corresponds to parts of Propositions 5.1 and 5.2 and Theorem 5.3 on p154ff of [50], and their proofs. This may also be considered as a simplification of Theorem 6.1 on p156 of [50]. More precisely, in [50], the first homology group of A, as an augmented ring with augmentation ideal I, and with coefficients in W, was asked to be {0}, instead of (13.14.10). This condition is implied by (13.14.10), and was used to get the same conclusion as in (13.14.11). Of course, there are analogous statements for left moduules.

One might note that some of the arguments used here and in Section 13.12 are similar to arguments in Sections 13.7 and 13.9. In [50], some conditions are formulated on p154 that include both cases, and Proposition 5.2 and Theorem 5.3 on p155f are stated in terms of these conditions.

13.15 Some quotients as tensor products

Let A be a ring with a multiplicative identity element e_{A}, and let I be a left ideal in A. If W is a right module over A, then W · I is defined as in Section 13.4, and is a subgroup of W, as a commutative group with respect to addition. This means that the quotient W/(W · I) (13.15.1)

is defined as a commutative group as well. If I is a two-sided ideal in A, then W · I is a submodule of W, as a right module over A, as in Section 13.7. In this case, (13.15.1) may be considered as a right module over A too.

Put Q = A/I, (13.15.2)

which may be considered as a left module over A. If I is a two-sided ideal in A, then Q may be considered as a ring, and both a left and right module over A.

Let W ⨂_{A} I and W ⨂_{A} Q be tensor products of W with I and Q, respectively, which are commutative groups with respect to addition. If I is a two-sided ideal in A, then W ⨂_{A} I and W ⨂_{A} Q may be considered as right modules over A, as in Section 1.10.

Using the identity mapping on W and the obvious inclusion mapping from I into A, we get a homomorphism

(13.15.3) from W ⨂_{A}I into W ⨂_{A}A,

as commutative groups with respect to addition. This may be identified with a homomorphism (13.15.4) from W ⨂_{A}I into W,

as commutative groups with respect to addition, because of the usual identification of W ⨂_{A} A with W. The image of this homomorphism is equal to W · I. If I is a two-sided ideal in A, then we get module homomorphisms in (13.15.3) and (13.15.4).

Similarly, we can use the identity mapping on W and the natural quotient mapping from A onto Q to get a homomorphism

(13.15.5) from W ⨂_{A}A onto W ⨂_{A}Q,

as commutative groups with respect to addition. This may be identified with a homomorphism (13.15.6) from W onto W ⨂_{A}Q,

as commutative groups with respect to addition, by identifying W ⨂_{A} A with W again. If I is a two-sided ideal in A, then we get module homomorphisms in (13.15.5) and (13.15.6), as before.

The image of the homomorphism as in (13.15.3) is equal to the kernel of the homomorphism as in (13.15.5), as in Section 2.5. Equivalently, the image of the homomorphism as in (13.15.4) is equal to the kernel of the homomorphism as in (13.15.6). This leads to an isomorphism

(13.15.7) from W/(W · I) onto W ⨂_{A}Q,

as commutative groups with respect to addition. If I is a two-sided ideal in A, then this is an isomorphism between right modules over A.

This corresponds to Exercise 2 on p31 of [8] when A is commutative. This also corresponds to (2) on p144 of [50], and to (1) on p154 of [50]. Of course, there are analogous statements for left modules over A and right ideals in A.

Fractions, dimension, and valuations

14.1 Pairs of multiplicatively closed sets

Let k_{1}, k_{2} be commutative rings with multiplicative identity elements 1_{k_{1}}, 1_{k_{2}}, respectively, and let ϕ be a ring homomorphism from k_{1} into k_{2} with ϕ(1_{k_{1}}) = 1_{k_{2}}. Also let S_{1}, S_{2} be multiplicatively closed subsets of k_{1}, k_{2}, respectively, and suppose that ϕ(S_{1}) ⊆ S_{2}. (14.1.1)

Thus S_{1}^{−1} k_{1}, S_{2}^{−1} k_{2} may be defined as in Section 12.1, with the corresponding ring homomorphisms x_{1} ↦ x_{1}/1_{k_{1}} (14.1.2)

and x_{2} ↦ x_{2}/1_{k_{2}} (14.1.3)

from k_{1}, k_{2} into S_{1}^{−1} k_{1}, S_{2}^{−1} k_{2}, respectively. As in Section 12.11,

Φ(x_{1}/r_{1}) = ϕ(x_{1})/ϕ(r_{1}) (14.1.4)

defines a ring homomorphism from S_{1}^{−1} k_{1} into S_{2}^{−1} k_{2}. Note that

S_{1}^{−1}(ker ϕ) ⊆ ker Φ. (14.1.5)

Alternatively, x_{1} ↦ ϕ(x_{1})/1_{k_{2}} (14.1.6)

defines a ring homomorphism from k_{1} into S_{2}^{−1} k_{2}, which is the same as the composition of ϕ with (14.1.3). This homomorphism sends elements of S_{1} to invertible elements of S_{2}^{−1} k_{2}, because of (14.1.1). This leads to a unique ring homomorphism Φ from S_{1}^{−1} k_{1} into S_{2}^{−1} k_{2} such that

Φ(x_{1}/1_{k_{1}}) = ϕ(x_{1})/1_{k_{2}} (14.1.7) for every x_{1} ∈ k_{1}, as in Section 12.12. Of course, this is the same as the homomorphism Φ mentioned in the preceding paragraph.

Let k be a commutative ring with a multiplicative identity element 1_{k}, and let S, T be multiplicatively closed subsets of k. This implies that

S T = {r t : r ∈ S, t ∈ T } (14.1.8)

is a multiplicatively closed set in k. Let U be the image of T under the natural homomorphism from k into S^{−1} k, which is a multiplicatively closed set in S^{−1} k. Exercise 3 on p43 of [8] asks one to show that

(14.1.9) (S T)^{−1} k and U^{−1} (S^{−1} k) are isomorphic as rings.

There is a natural ring homomorphism

(14.1.10) from k into U^{−1} (S^{−1} k),

obtained by composing the natural homomorphism from k into S^{−1} k with the natural homomorphism from S^{−1} k into U^{−1} (S^{−1} k). It is easy to see that this homomorphism sends elements of S T to invertible elements of U^{−1} (S^{−1} k). This leads to a natural ring homomorphism

(14.1.11) from (S T)^{−1} k into U^{−1} (S^{−1} k),

as in Section 12.12. More precisely, the natural homomorphism as in (14.1.10) is equal to the composition of the natural homomorphism from k into (S T)^{−1} k with the homomorphism as in (14.1.11).

Note that S ⊆ S T. This leads to a natural ring homomorphism

(14.1.12) from S^{−1} k into (S T)^{−1} k,

as discussed at the beginning of the section. More precisely, the composition of the natural homomorphism from k into S^{−1} k with the homomorphism as in (14.1.12) is the same as the natural homomorphism from k into (S T)^{−1} k. In particular, the image of U under the homomorphism as in (14.1.12) is the same as the image of T under the natural homomorphism from k into (S T)^{−1} k.

Of course, the natural homomorphism from k into (S T)^{−1} k sends elements of T to invertible elements of (S T)^{−1} k. This means that the homomorphism as in (14.1.12) sends elements of U to invertible elements of (S T)^{−1} k. Thus the homomorphism as in (14.1.12) leads to a natural ring homomorphism

(14.1.13) from U^{−1} (S^{−1} k) into (S T)^{−1} k,

as in Section 12.12. One can check that this is the inverse of the homomorphism as in (14.1.11).

If S ⊆ T, then S T = T. (14.1.14)

This means that (S T)^{−1} k = T^{−1} k, so that we get an isomorphism between T^{−1} k and U^{−1} (S^{−1} k), as before. If T ⊆ S, then S T = S, and we get an isomorphism between S^{−1} k and U^{−1} (S^{−1} k). This is basically trivial, because the elements of U are invertible in S^{−1} k in this case.

14.2 Fractions and integral domains

Let k be an integral domain, so that k is a commutative ring with a nonzero multiplicative identity element and no nonzero zero divisors. Thus

S_{0} = k \ {0} (14.2.1)

is a multiplicatively closed set in k, so that the corresponding ring of fractions S_{0}^{−1} k may be defined as in Section 12.1. In this case, the natural homomorphism from k into S_{0}^{−1} k is injective, so that k may be identified with a subring of S_{0}^{−1} k. In fact, S_{0}^{−1} k is the same as the usual field Q_{k} of fractions associated to k.

Let S be another multiplicatively closed set in k, with 0 ∉ S. Note that the natural homomorphism from k into S^{−1} k is injective, so that k may be identified with a subring of S^{−1} k, as before. There is also a natural ring homomorphism from S^{−1} k into Q_{k}, as in the previous section. One can check that this homomorphism is injective too, so that S^{−1} k may be identified with a subring of Q_{k} as well. More precisely, S^{−1} k corresponds to the set of elements of Q_{k} of the form x/r, with x ∈ k and r ∈ S.

Of course, S^{−1} k is an integral domain, and one can define its field of fractions in the usual way. In fact, the field of fractions of S^{−1} k is isomorphic to Q_{k} in a natural way.

We may consider Q_{k} as a module over k, and as a module over S^{−1} k. If we consider Q_{k} as a module over k, then S^{−1} Q_{k} may be defined as a module over S^{−1} k, as in Section 12.2. One can check that S^{−1} Q_{k} is isomorphic to Q_{k}, considered as a module over S^{−1} k, in a natural way.

Let M be a submodule of Q_{k}, as a module over k. Remember that S^{−1} M may be considered as a submodule of S^{−1} Q_{k}, as in Section 12.2. In this case, S^{−1} M may be identified with a submodule of Q_{k}, as a module over S^{−1} k. Using this identification, S^{−1} M consists of elements of Q_{k} of the form y/r, with y ∈ M and r ∈ S.

Remember that M is called a fractional ideal of k if there is an x ∈ k such that x ≠ 0 and x M ⊆ k, as in Section 11.7, and on p96 of [8]. Similarly, a submodule N of Q_{k}, as a module over S^{−1} k, is a fractional ideal of S^{−1} k if there are x ∈ k and r ∈ S such that x ≠ 0 and

(x/r) N ⊆ S^{−1} k. (14.2.2)

This implies that x N ⊆ S^{−1} k, (14.2.3)

so that we might as well take r = 1.

If (14.2.3) holds for some x ∈ Q_{k} with x ≠ 0, then N is a fractional ideal of S^{−1} k, as in Section 11.7. More precisely, we can multiply x by a nonzero element of k to get another nonzero element of k in this case, as before.

If M is a fractional ideal of k, then it is easy to see that S^{−1} M is a fractional ideal of S^{−1} k.

14.3 Fractions and invertible ideals

Let us continue with the same notation and hypotheses as in the previous section. If M, N are submodules of Q_{k}, as a module over k, then M N is defined to be the set of finite sums of products of elements of M and N, as in Section 11.7. This is a submodule of Q_{k}, as a module over k, as before. Similarly, if M and N are submodules of Q_{k}, as a module over S^{−1} k, then M N can be defined in the same way, and is a submodule of Q_{k}, as a module over S^{−1} k.

Remember that a submodule M of Q_{k}, as a module over k, is said to be an invertible ideal of k if there is a submodule N of Q_{k}, as a module over k, such that M N = k, as in Section 11.7, and on p96 of [8]. This implies that M, N are fractional ideals of k, as before. Similarly, a submodule M of Q_{k}, as a module over S^{−1} k, is said to be an invertible ideal of S^{−1} k if there is a submodule N of Q_{k}, as a module over S^{−1} k, such that

M N = S^{−1} k. (14.3.1)

This implies that M and N are fractional ideals of S^{−1} k, for the same reasons as before.

If M, N are submodules of Q_{k}, as a module over k, then S^{−1} M, S^{−1} N are submodules of Q_{k}, as a module over S^{−1} k, as in the previous section. It is easy to see that S^{−1}(M N) = (S^{−1} M) (S^{−1} N). (14.3.2)

If M is an invertible ideal of k, then there is a submodule N of Q_{k}, as a module over k, such that M N = k, and we get that

(S^{−1} M) (S^{−1} N) = S^{−1} k. (14.3.3)

This implies that S^{−1} M is an invertible ideal of S^{−1} k, as in Proposition 9.6 on p97 of [8].

If M is a submodule of Q_{k}, as a module over k, then put

(k : M)_{Q_{k}} = {x ∈ Q_{k} : x M ⊆ k}. (14.3.4)

This was denoted (k : M) in Section 11.7, and we include the subscript Q_{k} on the left here, because similar notation was used in Section 12.8 for something else. Note that (k : M)_{Q_{k}} is a submodule of Q_{k}, as a module over k, that is nonzero exactly when M is a fractional ideal of k, as before. If M ≠ {0}, then (k : M)_{Q_{k}} is a fractional ideal of k, as before.

Similarly, if M is a submodule of Q_{k}, as a module over S^{−1} k, then put

(S^{−1} k : M)_{Q_{k}} = {x ∈ Q_{k} : x M ⊆ S^{−1} k}. (14.3.5)

This is a submodule of Q_{k}, as a module over S^{−1} k, that is nonzero exactly when M is a fractional ideal of S^{−1} k. If M ≠ {0}, then (S^{−1} k : M)_{Q_{k}} is a fractional ideal of S^{−1} k.

Suppose that M is a submodule of Q_{k}, as a module over k, that is an invertible ideal of k, so that there is a submodule N of Q_{k}, as a module over k, such that M N = k. Under these conditions,

N = (k : M)_{Q_{k}}, (14.3.6)

as in Section 11.7. Similarly, let M be a submodule of Q_{k}, as a module over S^{−1} k, that is an invertible ideal of S^{−1} k, so that there is a submodule N of Q_{k}, as a module over S^{−1} k, that satisfies (14.3.1). In this case, we have that

N = (S^{−1} k : M)_{Q_{k}}, (14.3.7)

for the same reasons as before.

If M_{1}, M_{2} are submodules of Q_{k}, as a module over k, then it is easy to see that (k : M_{1} + M_{2})_{Q_{k}} = (k : M_{1})_{Q_{k}} ∩ (k : M_{2})_{Q_{k}}. (14.3.8)

Similarly, if M_{1}, M_{2} are submodules of Q_{k}, as a module over ^{−1} k, then

(S^{−1} k : M_{1} + M_{2})_{Q_{k}} = (S^{−1} k : M_{1})_{Q_{k}} ∩ (S^{−1} k : M_{2})_{Q_{k}}. (14.3.9)

14.4 Fractions and (k : M)_{Q_{k}}

We continue with the same notation and hypotheses as in the previous two sections. Let M be a submodule of Q_{k}, as a module over k, and suppose that M is finitely generated, as a module over k. We would like to check that

(S^{−1} k : S^{−1} M)_{Q_{k}} = S^{−1}(k : M)_{Q_{k}}. (14.4.1)

This is analogous to Proposition 3.15 on p43 of [8], which was discussed in Section 12.14.

Let M_{1}, M_{2} be submodules of Q_{k}, as a module over k, and suppose that they satisfy the corresponding analogues of (14.4.1). We would like to verify that M_{1} + M_{2} has the same property. Using (14.3.8), we get that

S^{−1}(k : M_{1} + M_{2})_{Q_{k}} = S^{−1}((k : M_{1})_{Q_{k}} ∩ (k : M_{2})_{Q_{k}}). (14.4.2)

This is equal to (S^{−1}(k : M_{1})_{Q_{k}}) ∩ (S^{−1}(k : M_{2})_{Q_{k}}), (14.4.3)

as in Section 12.2. This reduces to

(S^{−1} k : S^{−1} M_{1})_{Q_{k}} ∩ (S^{−1} k : S^{−1} M_{2})_{Q_{k}}, (14.4.4)

by hypothesis. This is the same as

(S^{−1} k : (S^{−1} M_{1}) + (S^{−1} M_{2}))_{Q_{k}}, (14.4.5)

because of (14.3.9). This is equal to

(S^{−1} k : S^{−1}(M_{1} + M_{2}))_{Q_{k}}, (14.4.6)

14.5. PRIMARY IDEALS 363 as in Section 12.2 again.

The remarks in the preceding paragraph permit one to reduce to the case where M is generated by a single element, as a module over k. In this case, (14.4.1) can be verified directly.

Alternatively, it is easy to see that S^{−1}(k : M)_{Q_{k}} ⊆ (S^{−1} k : S^{−1} M)_{Q_{k}} (14.4.7) for any submodule of Q_{k}, as a module over k. Thus it suffices to show that (S^{−1} k : S^{−1} M)_{Q_{k}} ⊆ S^{−1}(k : M)_{Q_{k}} (14.4.8) when M is finitely generated as a module over k.

Suppose that M is generated by y_{1}, …, y_{n} ∈ M, as a module over k, for some positive integer n. If x ∈ Q_{k} is in the left side of (14.4.8), then x y_{j} ∈ S^{−1} k (14.4.9) for each j = 1, …, n. This means that for each j = 1, …, n there is an r_{j} ∈ S such that r_{j} x y_{j} ∈ k. (14.4.10) Put r = ∏_{j=1}^{n} r_{j}, so that r ∈ S, and r x y_{j} ∈ k (14.4.11) for each j = 1, …, n. This implies that r x ∈ (k : M)_{Q_{k}}, (14.4.12) so that x ∈ S^{−1}(k : M)_{Q_{k}}.

14.5 Primary ideals

Let k be a commutative ring with a multiplicative identity element 1_{k} = 1. An ideal I ≠ k in k is said to be primary if for every x, y ∈ k with x y ∈ I, we have that (14.5.1) x ∈ I or y^{n} ∈ I for some n ∈ Z_{+}, as on p50 of [8]. This is the same as saying that k/I ≠ {0}, and that (14.5.2) every zero-divisor in k/I is nilpotent, as in [8]. Note that proper prime ideals are primary.

Let k_{2} be another commutative ring with multiplicative identity element 1_{k_{2}}, and let f be a ring homomorphism from k into k_{2} with f(1_{k}) = 1_{k_{2}}. Let I_{2} be an ideal in k_{2}, and remember that f^{−1}(I_{2}) is an ideal in k, which is the contraction I_{2}^{c} of I_{2} with respect to f, as in Section 12.7. If I_{2} ≠ k_{2}, then 1_{k_{2}} ∉ I_{2}, which implies that 1_{k} ∉ f^{−1}(I_{2}), so that f^{−1}(I_{2}) ≠ k. If

(14.5.3) I_{2} is a primary ideal in k_{2},

then (14.5.4) f^{−1}(I_{2}) is a primary ideal in k,

as mentioned on p50 of [8]. Although this can be verified directly, it can also be obtained by observing that f induces a ring isomorphism from k/f^{−1}(I_{2}) onto a subring of k_{2}/I_{2}, as in [8].

Let I be a primary ideal in k, and remember that r(I) is the radical of I in k, as in Section 12.10. Let us check that

(14.5.5) r(I) is a prime ideal in k,

as in Proposition 4.1 on p50 of [8]. If x, y ∈ k and x y ∈ r(I), then

x^{m} y^{m} = (x y)^{m} ∈ I (14.5.6)

for some positive integer m. This implies that either x^{m} ∈ I, or that y^{m n} = (y^{m})^{n} ∈ I for some positive integer n, because I is primary. This means that either x or y is an element of r(I), as desired.

Remember that r(I) is the same as the intersection of all of the prime ideals in k that contain I, as in Section 12.10. It follows that r(I) is the smallest prime ideal in k that contains I in this case, as in Proposition 4.1 on p50 of [8]. If I_{0} = r(I), (14.5.7)

then I is said to be I_{0}-primary, as on p51 of [8].

Let I be a proper ideal in k, and suppose that

(14.5.8) r(I) is a maximal proper ideal in k.

Remember that r(I) ≠ k when I ≠ k, as in Section 12.10. Under these conditions, (14.5.9) I is a primary ideal in k,

as in Proposition 4.2 on p51 of [8]. To see this, note that the image of r(I) in k/I under the natural quotient mapping is the nilradical N (k/I) of k/I, as in Section 12.10. Of course, the image of r(I) in k/I is also a maximal proper ideal in k/I, so that

(14.5.10) N (k/I) is a maximal proper ideal in k/I.

Remember that N (k/I) is the same as the intersection of all prime ideals in k/I, as in Section 12.5. In particular, every prime ideal in k/I contains N (k/I). It follows that

(14.5.11) N (k/I) is the only proper prime ideal in k/I, 14.6. MORE ON S^{−1} K 365

because it is maximal.

This means that N (k/I) is the only proper maximal ideal in k/I, because maximal ideals are prime. Thus every element of k/I that is not in N (k/I) has a multiplicative inverse in k/I. This implies that every zero-divisor in k/I is an element of N (k/I). It follows that I is a primary ideal in k, as before.

If I_{0} is a maximal proper ideal in k, then I_{0} is a prime ideal in k, and

r(I_{0}^{n}) = I_{0} (14.5.12)

for every positive integer n, as in Section 12.10. This implies that I_{0}^{n} is a primary ideal in k, which is another part of Proposition 4.2 on p51 of [8].

14.6 More on S^{−1} k

Let k be a commutative ring with a multiplicative identity element. Also let S be a multiplicatively closed subset of k, so that S^{−1} k is the corresponding ring of fractions, as in Section 12.1.

Suppose that I_{0} is a prime ideal in k, and that I is an I_{0}-primary ideal in k. If S ∩ I_{0} ≠ ∅, (14.6.1)

then S^{−1} I = S^{−1} k, (14.6.2)

as in part (i) of Proposition 4.8 on p53 of [8]. Indeed, if r ∈ S ∩ I_{0}, then r^{n} ∈ S ∩ I for some positive integer n. This means that r^{n}/1 is an element of S^{−1} I that is invertible in S^{−1} k, so that (14.6.2) holds.

Suppose now that S ∩ I_{0} = ∅, (14.6.3)

so that S^{−1} I_{0} is a proper prime ideal in S^{−1} k, as in Section 12.13. Part (ii) of Proposition 4.8 on p53 of [8] states that

(14.6.4) S^{−1} I is (S^{−1} I_{0})-primary in S^{−1} k,

and that (14.6.5) the contraction of S^{−1} I is equal to I.

More precisely, this means the contraction of S^{−1} I with respect to the natural homomorphism from k into S^{−1} k, as in Section 12.7. Remember that S^{−1} I is the same as the extension I^{e} of I with respect to the natural homomorphism from k into S^{−1} k, as in Section 12.13.

Thus (14.6.5) is the same as saying that the contraction I^{ec} of I^{e} is the same as I. Of course, I ⊆ I^{ec}, and so we only need to check the opposite inclusion. To do this, we use the characterization of I^{ec} in Section 12.13. Let t ∈ S and x ∈ k be given, with t x ∈ I. It suffices to show that

x ∈ I. (14.6.6)

If x ∉ I, then t^{n} ∈ I for some positive integer n, because I is primary. This would imply that t ∈ r(I) = I_{0}, which would contradict (14.6.3).

Observe that r(S^{−1} I) = S^{−1} r(I) = S^{−1} I_{0}, (14.6.7)

where the first step is as in Section 12.13, and the second step follows from the fact that r(I) = I_{0}, by hypothesis. One can check directly that S^{−1} I is primary in S^{−1} k, to get (14.6.4).

If I_{2} is a primary ideal in S^{−1} k, then its contraction I_{2}^{c} with respect to the natural homomorphism from k into S^{−1} k is a primary ideal in k, as in the previous section. Remember that there is a natural one-to-one correspondence between contracted ideals in k and extended ideals in S^{−1} k, as in Section 12.7. In this case, all ideals in S^{−1} k are extended, as in Section 12.13. Using the previous statements, we get a one-to-one correspondence between the primary ideals in S^{−1} k and the primary ideals in k that are disjoint from S. This is the last part of Proposition 4.8 on p53 of [8].

Remember that k is Noetherian if every ideal in k is finitely generated as a module over k, as in Section 9.7. This is equivalent to the ascending chain condition for the collection of ideals in k, as before. This is also equivalent to asking that any nonempty collection of ideals in k have a maximal element.

If k is Noetherian, then

(14.6.8) S^{−1} k is Noetherian,

as in Proposition 7.3 on p80 of [8]. Remember that every ideal in S^{−1} k is the extension of its contraction with respect to the natural homomorphism x ↦ x/1 from k into S^{−1} k, as in Section 12.13. There is also a one-to-one correspondence between contractions of ideals in S^{−1} k and extensions of ideals in k, as in Section 12.7. One can use this to get the ascending chain condition for ideals in S^{−1} k, or the existence of maximal elements in nonempty collections of ideals in S^{−1} k, from the corresponding property of k.

Alternatively, every ideal in S^{−1} k is of the form S^{−1} I_{1} for some ideal I_{1} in k, as in Section 12.13. More precisely, any ideal in S^{−1} k is the extension of an ideal I_{1} in k with respect to the natural homomorphism from k into S^{−1} k, and the extension of I_{1} is equal to S^{−1} I_{1}, as before. If k is Noetherian, then I_{1} is generated by finitely many elements x_{1}, …, x_{n}, as an ideal in k. In this case, it is easy to see that S^{−1} I_{1} is generated by x_{1}/1, …, x_{n}/1, as an ideal in S^{−1} k, as in [8].

14.7 Powers of radicals

Let k be a commutative ring with a multiplicative identity element, let I be an ideal in k, and let r(I) be the radical of I in k, as in Section 12.10. Suppose that I_{1} is an ideal in k such that

(14.7.1) I_{1} is finitely generated, as an ideal in k and I_{1} ⊆ r(I). (14.7.2)

Under these conditions, there is a positive integer m such that

I_{1}^{m} ⊆ I, (14.7.3)

as in Proposition 7.14 on p83 of [8]. Remember that powers of ideals in k are defined using products of ideals as in Section 12.8. Of course, one might as well take I_{1} = r(I) when r(I) is finitely generated as an ideal in k, which holds automatically when k is Noetherian.

Let x_{1}, …, x_{l} be finitely many elements of I_{1} that generate I_{1}, as an ideal in k. Thus, for each j = 1, …, l, there is a positive integer n_{j} such that

x_{j}^{n_{j}} ∈ I, (14.7.4)

because x_{j} ∈ r(I). If m is any positive integer, then I_{1}^{m} is generated as an ideal in k by elements of the form

x_{1}^{r_{1}} x_{2}^{r_{2}} ⋯ x_{l}^{r_{l}}, (14.7.5)

where r_{1}, …, r_{l} are nonnegative integers such that

∑_{j=1}^{l} r_{j} = m. (14.7.6)

If

m = ∑_{j=1}^{l}(n_{j} − 1) + 1, (14.7.7)

then (14.7.6) implies that r_{j} ≥ n_{j} for some j. This means that (14.7.5) is an element of I, as desired.

If I = {0}, then r(I) is the same as the nilradical N of k, as in Section 12.5. If (14.7.8) N is finitely generated, as an ideal in k,

then it follows that N^{m} = {0} (14.7.9)

for some positive integer m, as before. In particular, this holds when k is Noetherian, as in Corollary 7.15 on p83 of [8].

Let I_{0} be an ideal in k. If r(I) = I_{0}, (14.7.10)

and if I_{0} is finitely generated as an ideal in k, then

I_{0}^{m} ⊆ I ⊆ I_{0} (14.7.11)

for some positive integer m, as in Proposition 7.16 on p83 of [8]. In particular, (14.7.10) implies that (14.7.11) holds for some m ≥ 1 when k is Noetherian.

Of course, the second inclusion in (14.7.11) follows directly from (14.7.10). The first inclusion follows from (14.7.3) under these conditions.

If I_{0} is any ideal in k, and (14.7.11) holds for some m ≥ 1, then we get that

r(I) = r(I_{0}), (14.7.12)

because r(I_{0}^{m}) = r(I_{0}), as in Section 12.10. If I_{0} is a prime ideal in k, then r(I_{0}) = I_{0}, as before, so that (14.7.10) holds. Of course, if I_{0} is a maximal proper ideal in k, then I_{0} is prime. This corresponds to another part of Proposition 7.16 on p83 of [8].

14.8 Dimension via prime ideals

Let k be a commutative ring with a multiplicative identity element. Suppose that I_{0} ⊆ I_{1} ⊆ ⋯ ⊆ I_{n} (14.8.1)

is a strictly increasing sequence of proper prime ideals in k for some nonnegative integer n, so that I_{j} ≠ I_{j+1} for each j = 0, 1, …, n − 1. Such a sequence is said to be a chain of prime ideals in k of length n. Note that if k ≠ {0}, then k has a proper maximal ideal, which is prime.

If k ≠ {0}, then the dimension of k is defined to be the supremum of the lengths of chains of prime ideals in k, as on p90 of [8]. This is either a nonnegative integer, or +∞ if the lengths of the chains of prime ideals in k are unbounded. The dimension of a field in this sense is equal to 0.

If k ≠ {0}, then (14.8.2) k has dimension 0

if and only if (14.8.3) every proper prime ideal in k is maximal.

This uses the facts that every proper ideal in k is contained in a proper maximal ideal, and that proper maximal ideals are prime.

Suppose for the moment that k is an integral domain, so that {0} is a proper prime ideal in k. If the dimension of k is 0, then {0} is the only proper prime ideal in k. Remember that every non-invertible element of k is contained in a proper maximal ideal in k, which is prime. It follows that k is a field in this case.

Suppose that k ≠ {0}, and remember that N denotes the nilradical of k, as in Section 12.5. If (14.8.4) N is a maximal proper ideal in k,

then (14.8.5) N is the only proper prime ideal in k.

This follows from the earlier remarks about (14.5.10), with I = {0}, because N is the same as the radical of {0} in k. This also corresponds to part of Exercise 10 on p11 of [8]. Of course, this implies that the dimension of k is 0.

Remember that N is equal to the intersection of all of the prime ideals in k, as in Section 12.5. More precisely, N is the intersection of all of the proper prime ideals in k, because k ≠ {0}. If

(14.8.6) k has only one proper prime ideal,

then that ideal is equal to N. Of course, k has dimension 0 in this case, and (14.8.4) holds. This corresponds to another part of Exercise 10 on p11 of [8].

Suppose now that k is a local ring, and let I be the unique maximal proper ideal in k. If k has dimension 0, then I is the unique proper prime ideal in k. This implies that I = N , (14.8.7)

as in the preceding paragraph. This is related to some remarks on p90 of [8].

14.9 Noetherian local rings

Let k be a commutative ring with a multiplicative identity element, and let I be an ideal in k. Remember that the nth power I^{n} of I is defined for each positive integer n using multiplication of ideals, as in Section 12.10. Note that

I^{n+1} ⊆ I^{n} (14.9.1)

for each n ≥ 1. Of course, we either have that

I^{n} ≠ I^{n+1} for each n ≥ 1, (14.9.2)

or I^{n} = I^{n+1} for some n ≥ 1, (14.9.3)

as on p90 of [8].

If I is finitely generated as an ideal in k, then it is easy to see that I^{m} is finitely generated for every m ≥ 1. Of course, this holds when k is Noetherian, as in Section 9.7.

Suppose that k is a local ring, with unique maximal proper ideal I. Suppose also that (14.9.3) holds, which may be reexpressed as

I I^{n} = I^{n}. (14.9.4)

Note that I is the same as the Jacobson radical of k in this case. If I^{n} is finitely generated as an ideal in k, then we get that

I^{n} = {0}, (14.9.5)

by Nakayama’s lemma, as in Section 13.3. This corresponds to part of Proposition 8.6 on p90 of [8].

In particular, (14.9.5) implies that every element of I is nilpotent, so that I is contained in the nilradical N of k. This implies that I = N, because the opposite inclusion follows from the fact that I is a prime ideal. It follows that I is the only prime ideal in k, so that k has dimension 0, as in the previous section. This corresponds to another part of Proposition 8.6 on p90 of [8]. Alternatively, one can use the fact that any prime ideal in k contains I^{n} to get that it contains I, and thus is equal to I, as in [8].

Suppose for the moment that I is generated, as an ideal in k, by a single element x of k. This means that I^{n} is generated by x^{n}, as an ideal in k, for every positive integer n. Thus (14.9.5) is the same as saying that

x^{n} = 0. (14.9.6)

If (14.9.4) holds, then x^{n} = a x^{n+1} (14.9.7)

for some a ∈ k. This means that

(1 − a x) x^{n} = 0, (14.9.8)

which implies (14.9.6), because 1 − a x is invertible in k under these conditions.

Suppose now for the moment that k is an integral domain. In this case, (14.9.5) implies that I = {0}. This means that k is a field, which shows more directly that k has dimension 0.

If x is any element of k, and the ideal in k generated by x^{n} is equal to the ideal in k generated by x^{n+1} for some positive integer n, then (14.9.7) holds for some a ∈ k, as before. If k is an integral domain, then (14.9.8) implies that either x = 0, or x is invertible in k.

14.10 Local rings of dimension 0

Let k be a commutative ring with a multiplicative identity element, and suppose that k is a local ring, with unique maximal proper ideal I_{0}. Thus

k_{0} = k/I_{0} (14.10.1)

is a field. Note that I_{0}/I_{0}^{2} may be considered as a vector space over k_{0}, as in Section 13.6. Let us suppose throughout this section that

(14.10.2) I_{0} is finitely generated as an ideal in k,

which happens in particular when k is Noetherian. This implies that I_{0}/I_{0}^{2} has finite dimension, as a vector space over k_{0}, as on p91 of [8].

Suppose for the moment that

I_{0}/I_{0}^{2} = {0}, (14.10.3)

which is to say that I_{0}^{2} = I_{0}. (14.10.4)

Nakayama’s lemma implies that

I_{0} = {0}, (14.10.5)

as in Section 13.3, because I is the same as the Jacobson radical of k. This means that k is a field in this case. This corresponds to part of Proposition 8.8 on p91 of [8], and its proof.

Suppose now that dim_{k_{0}}(I_{0}/I_{0}^{2}) = 1, (14.10.6)

which is to say that the dimension of I_{0}/I_{0}^{2}, as a vector space over k_{0}, is equal to one. Let q_{0} be the natural quotient mapping from I_{0} onto I_{0}/I_{0}^{2}, and let x be an element of I_{0} \ I_{0}^{2}, so that q_{0}(x) ≠ 0. This means that I_{0}/I_{0}^{2} is spanned by q_{0}(x), as a vector space over k_{0}, by (14.10.6). It follows that

(14.10.7) I_{0} is generated by x, as an ideal in k,

as in Section 13.6. This corresponds to another part of Proposition 8.8 on p91 of [8].

Let us suppose in addition that

(14.10.8) k has dimension 0,

in the sense of Section 14.8. This implies that I_{0} is the unique proper prime ideal in k, and that I_{0} is the same as the nilradical N in k, as before. It follows that I_{0}^{m} = {0} (14.10.9)

for some positive integer m, as in Section 14.7. Note that for each positive integer n, (14.10.10) I_{0}^{n} is generated by x^{n}, as an ideal in k,

by (14.10.7).

Let I be a proper ideal in k, and observe that

I ⊆ I_{0}, (14.10.11)

because I is contained in a maximal proper ideal in k. If I ≠ {0}, then there is a positive integer r such that

I ⊆ I_{0}^{r} and I ⊈ I_{0}^{r+1}. (14.10.12)

Let y be an element of I \ I_{0}^{r+1}. Thus y can be expressed as

y = a x^{r} (14.10.13)

for some a ∈ k, by (14.10.10). We also have that a ∉ I_{0}, because y ∉ I_{0}^{r+1}.

This means that a is invertible in k, because I_{0} is the unique maximal proper ideal in k. It follows that x^{r} ∈ I, which implies that I_{0}^{r} ⊆ I, by (14.10.10). This shows that I = I_{0}^{r}, (14.10.14)

as in Proposition 8.8 on p91 of [8], and its proof.

14.11 One-dimensional integral domains

Let k be a commutative ring with a multiplicative identity element that is an integral domain, so that 1 ≠ 0 in k, and k has no nonzero zero-divisors. Equivalently, this means that {0} is a proper prime ideal in k. Remember that k has dimension 0, in the sense of Section 14.8, if and only if k is a field. In this section, we ask that (14.11.1) k have dimension 1,

in the sense of Section 14.8. This is the same as saying that k is not a field, and that (14.11.2) every proper prime ideal ≠ {0} in k is maximal,

as on p93 of [8].

Let I_{1} be a proper ideal in k with I_{1} ≠ {0}. It is easy to see that

(14.11.3) k/I_{1} has dimension 0,

in the sense of Section 14.8. More precisely, proper prime ideals in k/I_{1} correspond to proper prime ideals in k that contain I_{1}, and are not equal to {0} in particular. Thus proper prime ideals in k/I_{1} are maximal, because of (14.11.2).

Suppose now that k is also a local ring, with unique maximal proper ideal I_{0}. The condition that k not be a field means that

I_{0} ≠ {0}. (14.11.4)

Note that (14.11.5) I_{0} is the only proper prime ideal ≠ {0} in k,

because of (14.11.2). We ask that I_{0} be finitely generated as an ideal in k, which happens in particular when k is Noetherian.

Put k_{0} = k/I_{0} again, which is a field. Remember that I_{0}/I_{0}^{2} may be considered as a vector space over k_{0}, as in Section 13.6. This vector space has finite dimension, because I_{0} is finitely generated.

Let I ≠ {0} be a proper ideal in k. Of course, I ⊆ I_{0}, because I is contained in a maximal proper ideal in k. Remember that the radical of I in k is equal to the intersection of all of the prime ideals in k that contain I, as in Section 12.10. It follows that r(I) = I_{0} (14.11.6)

in this case, because of (14.11.5). This implies that I is a primary ideal in k, as in Section 14.5. We also get that

I_{0}^{m} ⊆ I ⊆ I_{0} (14.11.7)

for some positive integer m, as in Section 14.7. This corresponds to Remark (A) on p95 of [8].

Remark (B) on p95 of [8] states that

I_{0}^{n} ≠ I_{0}^{n+1} (14.11.8) for each positive integer n. Otherwise, k would have dimension 0, as in Section 14.9. More precisely, because k is an integral domain, one could simply say that k would have to be a field otherwise.

Suppose for the moment that

(14.11.9) I_{0} is generated by a single element of k, as an ideal in k.

This implies that the dimension of I_{0}/I_{0}^{2}, as a vector space over k_{0}, is less than or equal to one. In fact, we get that

dim_{k_{0}}(I_{0}/I_{0}^{2}) = 1, (14.11.10)

because of (14.11.8). This is part of Proposition 9.2 on p94 of [8].

14.12 One-dimensional local domains

Let us continue with the same notation and hypotheses as in the previous section. We would like to show that if (14.11.10) holds, then

(14.12.1) every ideal I ≠ {0} in k is of the form I_{0}^{r}

for some nonnegative integer r. This is another part of Proposition 9.2 on p94 of [8]. Remember that I_{0}^{r} is interpreted as being equal to k when r = 0.

Let I ≠ {0} be a proper ideal in k, so that (14.11.7) holds for some positive integer m. Observe that k/I_{0}^{m} is a commutative ring with a multiplicative identity element that is a local ring, with unique maximal proper ideal I_{0}/I_{6}^{m}. Of course, (14.12.2) I_{0}/I_{0}^{m} is finitely generated as an ideal in k/I_{0}^{m},

because I_{0} is finitely generated as an ideal in k, by hypothesis. Clearly

(k/I_{0}^{m})/(I_{0}/I_{0}^{m}) (14.12.3)

is isomorphic to k_{0} as a field.

We may as well suppose that m ≥ 2, because otherwise I = I_{0}. It is easy to see that (I_{0}/I_{0}^{m})^{2} = I_{0}^{2}/I_{0}^{m}, (14.12.4)

where the left side is defined as an ideal in k/I_{0}^{m} in the usual way. Note that

(I_{0}/I_{0}^{m})/(I_{0}/I_{0}^{m})^{2} (14.12.5)

may be considered as a vector space over (14.12.3), as before. This is the same as (I_{0}/I_{0}^{m})/(I_{0}^{2}/I_{0}^{m}), (14.12.6)

because of (14.12.4). This may be identified with I/I_{0}^{2}, as a vector space over k_{0}.

Thus (14.11.10) implies the analogous condition for (14.12.5). Note that I_{0}^{m} ≠ {0}, because I_{0} ≠ {0}, and k is an integral domain. This implies that k/I_{0}^{m} has dimension 0, as in (14.11.3). We may as well suppose that I ≠ I_{0}^{m}, which means that I/I_{0}^{m} ≠ {0}. It follows that

I/I_{0}^{m} = (I_{0}/I_{0}^{m})^{r} (14.12.7)

in k/I_{0}^{m} for some positive integer r, as in Section 14.10.

We also have r < m in this case, because otherwise the right side of (14.12.7) would be {0}. Thus the right side of (14.12.7) is equal to I_{0}^{r}/I_{0}^{m}. This implies that I = I_{0}^{r}, as desired.

Suppose now that (14.12.1) holds, and let us show that

(14.12.8) there is an x ∈ k such that every nonzero ideal in k is generated by x^{l} for some nonnegative integer l.

This is another part of Proposition 9.2 on p94 of [8]. Note that (14.12.8) implies (14.11.9).

If we take n = 1 in (14.11.8), then we get that I_{0} ≠ I_{0}^{2}. Thus there is an x ∈ I_{0} such that x ∉ I_{0}^{2}. In particular, x ≠ 0, and (14.12.1) implies that there is a nonnegative integer r such that I_{0}^{r} is generated by x. It is clear in this case that r = 1. This shows directly that (14.12.1) implies (14.11.9).

If l is a nonnegative integer, then it follows that I_{0}^{l} is generated by x^{l}. This and (14.12.1) imply (14.12.8), as desired.

14.13 Valuations

Let k_{1} be a field. A real-valued function v on k_{1} \ {0} is said to be a valuation if it satisfies the following two conditions. First,

v(x y) = v(x) + v(y) (14.13.1)

for every x, y ∈ k_{1} \ {0}. Of course, this is the same as saying that v is a homomorphism from k_{1} \{0}, as a commutative group with respect to multiplication, into R, as a commutative group with respect to addition. In particular, this implies that v(1) = 0. The second condition is that

v(x + y) ≥ min(v(x), v(y)) (14.13.2)

for every x, y ∈ k_{1} \ {0} such that x + y ≠ 0. It is customary to put

v(0) = +∞, (14.13.3)

so that v is a function on k_{1} with values in R ∪ {+∞}. Note that (14.13.1) and (14.13.2) hold, with suitable interpretations, when x, y, or x + y is 0.

This definition is mentioned on p24 of [114]. This corresponds to a valuation of k_{1} with values in R, as in Exercise 31 on p72 of [8]. This may also be called a real valuation of k_{1}, as on p26 of [263].

This is closely related to ultrametric or non-archimedean absolute value functions on fields, as in Definition 2.1.1 on p21f of [114], and Problems 28 and 63 on p24, 39 of [114], respectively. This is mentioned on p26 of [263] as well. The term “valuation” is used a bit differently in Definition 1.1 on p12 of [52], which basically corresponds to a quasimetric version of an absolute value function. A non-archimedean valuation in the sense of Definition 1.3 on p15 of [52] is the same as an ultrametric or non-archimedean absolute value function, which corresponds to a valuation in the sense used here as before.

Note that we can get a valuation on k_{1} by putting

v(x) = 0 (14.13.4)

for every x ∈ k_{1} \ {0}, and using (14.13.3), as before. This may be described as the trivial valuation on k_{1}. This corresponds to the trivial absolute value function in Example 2 on p22 of [114], or the trivial valuation on p12 of [52].

If v is a valuation on k_{1} and a is a positive real number, then it is easy to see that a v(x) (14.13.5)

defines a valuation on k_{1} as well. This valuation is said to be equivalent to v on k_{1}. This corresponds to equivalence of absolute value functions, as in Lemma 3.1.2 on p42 of [114]. This also corresponds to equivalence of valuations in the sense of [52], as in Definition 1.2 on p13 of [52].

Let v be a valuation on k_{1}, and note that

(14.13.6) v(k_{1} \ {0}) is a subgroup of R,

as a commutative group with respect to addition. This is called the value group of v, as in Exercise 31 on p72 of [8], and Problem 29 on p24 of [114]. This corresponds to the valuation group of a valuation in the sense of [52], as on p42 of [52]. More precisely, the valuation group in this sense is a subgroup of the multiplicative group of positive real numbers.

If A is any subgroup of R, as a commutative group with respect to addition, then it is well known and not difficult to show that

(14.13.7) 0 is not a limit point of A,

with respect to the standard topology on R, if and only if

A = {0} (14.13.8)

or (14.13.9) A = a Z for some positive real number a.

Otherwise, if 0 is a limit point of A, then one can check that

(14.13.10) A is dense in R,

with respect to the standard topology.

If A = v(k_{1} \ {0}) satisfies (14.13.7), then v corresponds to a “discrete valuation” in the sense defined on p42 of [52]. However, we shall use this term a bit differently here, as on p94 of [8], and p5, 26 of [263].

14.14 Some related subgroups and ideals

Let us continue with the same notation and hypotheses as in the previous section. Note that v(1) = 0. (14.14.1)

If x ∈ k_{1} and x^{n} = 1 for some positive integer n, then

v(x) = 0, (14.14.2)

because n v(x) = v(x^{n}) = 0. In particular, v(−1) = 0. This implies that

v(−y) = v(y) (14.14.3)

for every y ∈ k_{1}.

If t ∈ R, then it is easy to see that

{x ∈ k_{1} : v(x) > t} (14.14.4)

and {x ∈ k_{1} : v(x) ≥ t} (14.14.5)

are subgroups of k_{1}, as a commutative group with respect to addition. In fact,

{x ∈ k_{1} : v(x) ≥ 0} (14.14.6)

is a subring of k_{1}. This is called the valuation ring of v, as in Exercise 31 on p72 of [8], and on p94 of [8]. This corresponds to the valuation ring of an ultrametric absolute value function, as in Definition 2.4.2 on p38 of [114]. This is the same as the ring of (valuation-)integers with respect to a non-archimedean valuation on p41 of [52].

If x ∈ k_{1} \ {0}, then v(x^{−1}) = −v(x). (14.14.7)

In particular, if v(x) = 0, then

v(x^{−1}) = 0. (14.14.8)

This implies that x is invertible as an element of the valuation ring of v. Conversely, if x is an invertible element of the valuation ring of v, then it is easy to see that v(x) = 0, using (14.14.7).

If t ≥ 0, then (14.14.4) and (14.14.5) are ideals in the valuation ring of v. One can check that {x ∈ k_{1} : v(x) > 0} (14.14.9)

is a maximal proper ideal in the valuation ring (14.14.6). More precisely, if x is an element of the valuation ring of v that is not in (14.14.9), then v(x) = 0, so that x is invertible in the valuation ring. This means that the valuation ring of v is a local ring, with (14.14.9) as its unique maximal proper ideal. This corresponds to the valuation ideal of an ultrametric absolute value function, as in Definition 2.4.2 on p38 of [114].

If v is not the trivial valuation on k_{1}, then there is an x ∈ k_{1} such that x ≠ 0 and v(x) ≠ 0. This implies that there are y, z ∈ k_{1} \ {0} such that

v(y) > 0 and v(z) < 0, (14.14.10)

using x and 1/x. This means that (14.14.9) is not equal to {0}, and in particular that the valuation ring (14.14.6) is not a field. Of course, if v is the trivial valuation on k_{1}, then the valuation ring (14.14.6) is equal to k_{1}, and (14.14.9) is equal to {0}.

Suppose for the moment that A = v(k_{1} \ {0}) satisfies (14.13.7), and thus either (14.13.8) or (14.13.9). Of course, (14.13.8) says exactly that v is the trivial valuation on k_{1}. If (14.13.9) holds, then it is easy to see that every nonzero ideal in the valuation ring (14.14.6) is of the form (14.14.5), where t is a nonnegative integer multiple of a. In particular, the valuation ring (14.14.6) is Noetherian in these two cases. Otherwise, if (14.13.10) holds, then one can check that the valuation ring (14.14.6) is not Noetherian.

If v(k_{1} \ {0}) = Z, (14.14.11)

then v is called a discrete valuation, as on p94 of [8]. This implies that v is not the trivial valuation on k_{1}, and that A = v(k_{1} \ {0}) satisfies (14.13.7). If v is a nontrivial valuation on k_{1} such that A = v(k_{1} \ {0}) satisfies (14.13.7), then v is equivalent to a discrete valuation in this sense.

14.15 Valuations on integral domains

Let k be an integral domain. A real-valued function v on k \ {0} is said to be a valuation if it satisfies the same two conditions as before, as in Problem 28 on p24 of [114]. Namely, if x, y ∈ k \ {0}, then we ask that

v(x y) = v(x) + v(y), (14.15.1)

and that v(x + y) ≥ min(v(x), v(y)) (14.15.2)

when x + y ≠ 0. Observe that v(1) = 0, by taking x = y = 1 in (14.15.1). It is customary to put v(0) = +∞, as before.

If x ∈ k and x^{n} = 1 for some positive integer n, then v(x) = 0, as in the previous section. It follows that v(−1) = 0, so that v(−y) = v(y) for every y ∈ k, as before. We also have that

{x ∈ k : v(x) > t} (14.15.3)

and {x ∈ k : v(x) ≥ t} (14.15.4)

are subgroups of k, as a commutative group with respect to addition, for every t ∈ R. Similarly, {x ∈ k : v(x) ≥ 0} (14.15.5)

378 CHAPTER 14. FRACTIONS, DIMENSION, AND VALUATIONS is a subring of k. If t ≥ 0, then (14.15.3) and (14.15.4) are ideals in (14.15.5), as before.

If x is an invertible element of k, then v(x^{−1}) = −v(x). (14.15.6) If we also have that v(x) = 0, then v(x^{−1}) = 0, so that x is an invertible element of (14.15.5). Conversely, if x is an invertible element of (14.15.5), then x is an invertible element of k, and v(x) = 0. Suppose for the moment that (14.15.7) v(y) ≥ 0 for every y ∈ k, so that (14.15.5) is the same as k. If x is an invertible element of k, then we get that v(x) = 0.

Let Q_{k} be the field of fractions corresponding to k. One can extend v to Q_{k} \ {0} in such a way that v(y/z) = v(y) − v(z) (14.15.8) for every y, z ∈ k with y, z ≠ 0. One can check that this defines a valuation on Q_{k}, as in Problem 28 on p24 of [114]. More precisely, any pair of nonzero elements of Q_{k} may be expressed as x/z, y/z for some x, y, z ∈ k \ {0}. If x + y ≠ 0, then it is easy to see that v((x/z) + (y/z)) ≥ min(v(x/z), v(y/z)), (14.15.9) using (14.15.2) and (14.15.8).

Suppose for the moment that if y, z ∈ k \ {0} satisfy v(y) ≥ v(z), (14.15.10) then there is a w ∈ k such that y = w z. (14.15.11) This is the same as saying that if v(y/z) ≥ 0, (14.15.12) then y/z ∈ k. (14.15.13) This means that the valuation ring of v in Q_{k} is contained in k.

Discrete valuations and Dedekind domains

15.1 Discrete valuation rings

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions. Suppose that v is a discrete valuation on Q_{k}, as in Section 14.14. Suppose also that k is the valuation ring of v, so that

k = {x ∈ Q_{k} : v(x) ≥ 0}. (15.1.1)

Under these conditions, k is said to be a discrete valuation ring, as on p94 of [8]. An equivalent formulation is given on p5 of [263], and we shall say more about that in Section 15.4.

Of course, v(Q_{k} \ {0}) = Z, (15.1.2)

by the definition of a discrete valuation. This implies that

v(k \ {0}) = Z_{+} ∪ {0}. (15.1.3)

As in Section 14.14, k is a local ring, with

{x ∈ k : v(x) > 0} (15.1.4)

as its unique maximal proper ideal. If l is a nonnegative integer, then

{x ∈ k : v(x) ≥ l} (15.1.5)

is an ideal in k, as before. Note that these ideals are all nonzero in this case, and that (15.1.5) is equal to (15.1.4) when l = 1. We also have that l is uniquely determined by (15.1.5), because of (15.1.3).

380CHAPTER 15. DISCRETE VALUATIONS AND DEDEKIND DOMAINS

If x, y ∈ k, x, y ≠ 0, and v(x) = v(y), (15.1.6) then v(x y^{−1}) = 0, so that x y^{−1} is an invertible element of k. This implies that x and y generate the same ideal in k. The ideal that they generate is the same as (15.1.5), with l = v(x) = v(y).

Let I be a nonzero ideal in k, and let l be the smallest nonnegative integer for which there is a y ∈ I such that v(y) = l. One can check that I is equal to (15.1.5), as on p94 of [8]. It follows that k is Noetherian, as in [8]. More precisely, I is equal to (15.1.5) for only one nonnegative integer l.

Suppose that x ∈ k satisfies

v(x) = 1. (15.1.7)

This implies that v(x^{l}) = l (15.1.8)

for every integer l. It follows that (15.1.5) is the same as the ideal in k generated by x^{l} when l ≥ 0, as before. This corresponds to a remark on p94 of [8].

It follows from the remarks in the previous paragraphs that (15.1.4) is the only nonzero proper prime ideal in k. In particular, k has dimension one in the sense of Section 14.8, as on p94 of [8].

If k is a discrete valuation ring, then it is easy to see that the corresponding valuation v is unique. More precisely, one can check that v is uniquely determined on k, and thus on Q_{k}.

15.2 Getting discrete valuation rings

Let k_{1} be a field, and let v_{1} be a valuation on k_{1}. Remember that

{w ∈ k_{1} : v_{1}(w) ≥ 0} (15.2.1)

is a subring of k_{1}, which is the valuation ring of v_{1}. It is easy to see that k_{1} may be identified with the field of fractions of this ring. If v_{1} is a discrete valuation on k_{1}, then it follows that

(15.2.2) (15.2.1) is a discrete valuation ring.

This is related to Proposition 1 on p6 of [263], although simpler, with the formulation of the definition of a discrete valuation ring in the previous section.

Let k be an integral domain with Q_{k} as its field of fractions again. Suppose that

(15.2.3) there is an x ∈ k such that every nonzero ideal in k is generated by x^{l} for some nonnegative integer l,

as in (14.12.8). Suppose also that k is not a field, so that x ≠ 0, and x is not invertible in k. This implies that the ideal in k generated by x^{l+1} is a proper subset of the ideal generated by x^{l} for each l ≥ 0, because k is an integral domain, as mentioned at the end of Section 14.9.

If y ∈ k and y ≠ 0, then there is a unique nonnegative integer l such that the ideal in k generated by y is the same as the ideal generated by x^{l}, and we put v(y) = l. (15.2.4)

This happens exactly when y is equal to the product of x^{l} and an invertible element of k, as in the next section. If z is another nonzero element of k, then one can check that v(y z) = v(y) + v(z). (15.2.5)

One can also verify that

v(y + z) ≥ min(v(y), v(z)) (15.2.6)

when y + z ≠ 0. Thus v defines a valuation on k, as in Section 14.15.

We can extend v to a valuation on Q_{k}, as before. Note that every nonzero element of Q_{k} can be expressed as u x^{l} for some invertible element u of k and integer l, in which case we have that

v(u x^{l}) = l. (15.2.7)

It follows that v is a discrete valuation on Q_{k}, and that (15.1.1) holds. This means that k is a discrete valuation ring, as in Proposition 9.2 on p94 of [8].

Remember that some conditions under which (15.2.3) holds were discussed in Sections 14.11 and 14.12. Thus these are conditions under which k is a discrete valuation ring, as in [8].

15.3 Some remarks about integral domains

Let k be a commutative ring with a multiplicative identity element. One may say that x, y ∈ k are associates in k if

x = a y, y = b x (15.3.1)

for some a, b ∈ k, as on p111 of [195]. Of course, if

y = x u (15.3.2)

for some invertible element u of k, then

x = y u^{−1}, (15.3.3)

and x, y are associates in k in the sense just mentioned. Sometimes x and y are said to be associates in k when this stronger condition holds. This corresponds to the notion of equivalence mentioned on p72 of [177].

Equivalently, x, y ∈ k are associates in the sense of (15.3.1) when x is an element of the ideal in k generated by y, and y is an element of the ideal generated by x. This is the same as saying that the ideals in k generated by x and y are the same.

Suppose for the moment that k is an integral domain. If x, y ∈ k are associates in k in the sense of (15.3.1), then it is well known that (15.3.1) holds for some invertible element u of k, as on p111 of [195]. More precisely, if a, b ∈ k are as in (15.3.1), then x = a b x, so that

(a b − 1) x = 0. (15.3.4)

If x ≠ 0, then it follows that a b = 1, so that a and b are inverses of each other. Otherwise, if x = 0, then y = 0, and the statement is trivial.

An element x ≠ 0 of k is said to be reducible if

(15.3.5) x can be expressed as the product of two non-invertible elements of k.

Otherwise, one may say that x is irreducible. Irreducible elements of k may also be asked to not be invertible, as on p13 of [92], p70 of [177], and p111 of [195]. Somewhat different terminology is used in [195], and we shall say more about that in a moment. Strictly speaking, 0 is the product of two non-invertible elements of k, except when k = {0}, so that 0 is invertible in k.

An element x of k is said to be prime if the ideal in k generated by x is a proper prime ideal, as on p12 of [92]. This means that x is not invertible in k, and that if y, z ∈ k and y z is a multiple of x in k, then y or z is a multiple of x in k. Thus 0 is a prime in k exactly when k is an integral domain, and one may often be concerned with nonzero prime elements in integral domains.

If k is an integral domain, x ∈ k is prime, and x ≠ 0, then

(15.3.6) x is irreducible in k,

as on p70 of [177]. Indeed, if x = y z (15.3.7)

for some y, z ∈ k, then y = b x or z = c x (15.3.8)

for some b or c in k. This means that at least one of y and z is an associate of x in k in the first sense discussed earlier, because of (15.3.7). This implies that y or z is an associate of x in the stronger sense mentioned earlier, and that the other one is invertible in k, as before.

The term “prime” is used for irreducible elements of an integral domain on p111 of [195], but this terminology does not seem to be completely standard, and we shall not use it here. In fact, it is mentioned on p117 of [195] that the term “prime” is often defined as it is here. It is well known that irreducible elements are prime in some situations, as in Corollary 1 on p117 of [195], and as mentioned on p70 of [177]. An instance of this will be mentioned here in a moment.

Suppose for the rest of the section that k is an integral domain. Let x, y ∈ k be given, and suppose that z ∈ k has the property that x and y are each multiples of z in k. Suppose also that if w ∈ k and x, y are each multiples of w in k, then z is a multiple of w in k. Under these conditions, z ∈ k is said to be a greatest common divisor of x, y ∈ k, as on p117 of [195]. In the analogous definition on p71 of [177], one asks that x, y, z, w ≠ 0.

Note that any other greatest common divisor of x, y in k is an associate of z, as mentioned in [195]. Suppose now that

(15.3.9) k is a principal ideal domain,

so that k is an integral domain, and every ideal in k is generated by a single element. Let x, y ∈ k be given, with x, y ≠ 0, and consider the ideal in k generated by x, y. By hypothesis, there is a z ∈ k such that the ideal in k generated by z is the same as the ideal generated by x, y. It is well known and not difficult to show that z is a greatest common divisor of x and y, as in Proposition 2 on p71 of [177], and Theorem 23 on p117 of [195].

If x ≠ 0 is an irreducible element of k, then a greatest common divisor of x and any other nonzero element of k is an associate of x, or an invertible element of k. If x is also not invertible in k, then one can use this to show that

(15.3.10) x is prime in k,

as in Corollary 1 on p117 of [195]. This is also mentioned at the top of p72 of [177]. Another argument is described on p14 of [92].

It is well known that every nonzero element of k is either invertible or a product of finitely many noninvertible irreducible elements of k, as mentioned on p14 of [92], and as in Theorem 1 on p71 of [177], Theorem 24 on p118 of [195]. These arguments use the fact that k is Noetherian, so that an increasing sequence of ideals in k is eventually constant. This is a bit simpler for principal ideal domains, because the union of the ideals in such a sequence is an ideal in k, and thus generated by a single element.

This factorization is unique, up to permutations and using other associates of the irreducible factors, as in [92, 177, 195]. This uses the fact that noninvertible irreducible nonzero elements of k are prime, as before.

15.4 More on principal ideal domains

Let k be a principal ideal domain. It is well known that

(15.4.1) k has dimension less than or equal to one,

in the sense of Section 14.8. This is the same as saying that every nonzero proper prime ideal in k is maximal. This corresponds to Example (3) on p5 of [8], as mentioned on p96 of [8]. This also follows from a remark on p125 of [195].

To see this, let I_{1} be a nonzero proper prime ideal in k, and let I_{2} be an ideal in k that contains I_{1}. By hypothesis, I_{1} and I_{2} are generated by nonzero elements x and y of k, respectively. Because I_{1} ⊆ I_{2}, we have that x ∈ I_{2}, so that x = y z for some z ∈ k. This implies that y or z is invertible in k, because x is irreducible, as in the previous section. This means that I_{2} is equal to I_{1} or k, as desired.

Of course, k is a field if and only if {0} is the only proper ideal in k. Suppose that k is a local ring that is not a field, so that k has a unique nonzero maximal proper ideal I_{0}. Of course, I_{0} is a prime ideal in k, and in fact

(15.4.2) I_{0} is the unique nonzero proper prime ideal in k,

by the previous remarks. Conversely, if k has a unique proper prime ideal, then that is the unique maximal proper ideal in k.

This can be used as another way to define discrete valuation rings, as on p5 of [263]. Note that discrete valuation rings have these properties, as in Section 15.1. This corresponds to Proposition 1 on p6 of [263].

Conversely, suppose that k satisfies the conditions mentioned before, and that (15.4.3) I_{0} is generated by x ∈ k, as an ideal in k.

Note that x ≠ 0, because I_{0} ≠ {0}, and that x is irreducible, because I_{0} is prime. The condition that I_{0} be the unique nonzero prime ideal in k implies that every irreducible element of k is an associate of x.

If y ∈ k and y ≠ 0, then y is invertible, or y can be expressed as a product of irreducible elements of k, as in the previous section. This implies that y is an associate of x^{l} for some nonnegative integer l in this case. It follows that every nonzero ideal in k is generated by x^{l} for some nonnegative integer l, because k is a principal ideal domain. This implies that k is a discrete valuation ring, as in Section 15.2.

Alternatively, if k satisfies the conditions mentioned before, then there is an x ∈ k such that every nonzero ideal in k is generated by x^{l} for some nonnegative integer l, as in Sections 14.11 and 14.12. More precisely, this uses the fact that I_{0} is generated by a single element of k, as an ideal in k, because k is a principal ideal domain. This means that k is a discrete valuation ring, as in Section 15.2 again.

15.5 Local rings and valuations

Let k be a commutative ring with a nonzero multiplicative identity element, and suppose that k is a local ring that is not a field. Thus k has a unique maximal proper ideal I_{0}, with I_{0} ≠ {0}. Let us suppose for the rest of the section that

(15.5.1) I_{0} is generated by w_{0} ∈ k, as an ideal in k.

Note that w_{0} ≠ 0, because I_{0} ≠ {0}.

Suppose in addition that

⋂_{j=1}^{∞} I_{0}^{j} = {0}. (15.5.2)

This holds automatically when k is Noetherian, as in the proof of Proposition 2 on p7 of [263]. This will be discussed further in the next section. More precisely, (15.5.2) holds when k is Noetherian, without asking that I_{0} be generated by a single element, as mentioned in [263].

If k is an integral domain, then

(15.5.3) k is a discrete valuation ring,

as in Exercise 4 on p99 of [8]. This is related to part of the proof of Proposition 2 on p7 of [263]. In [263], one asks that w_{0} not be nilpotent, instead of asking that k be an integral domain. Of course, if k is an integral domain, then w_{0} is not nilpotent, because w_{0} ≠ 0. Note that discrete valuation rings satisfy all of these conditions.

Let y ∈ k \ {0} be given. Observe that there is a largest nonnegative integer l such that y ∈ I_{0}^{l}, (15.5.4)

because of (15.5.2). Thus y = a w_{0}^{l} (15.5.5)

for some a ∈ k with a ∉ I_{0}. This means that a is invertible in k.

If k is an integral domain, then we get (15.5.3), as in Section 15.2. If we only ask that w_{0} not be nilpotent, then one can use (15.5.5) to get that k is an integral domain, as in [263]. Note that there is no condition here on the dimension of k, in the sense of Section 14.8, which is a bit different from the discussion in Sections 14.11 and 14.12.

15.6 More on the additional condition

Let k be as at the beginning of the prevous section again, and suppose now that k is Noetherian as well. Consider the set I of x ∈ k such that

x w_{0}^{m} = 0 (15.6.1)

for some nonnegative integer m, and thus all sufficiently large m. It is easy to see that this is an ideal in k. It follows that I is finitely generated as an ideal in k, because k is Noetherian. This implies that there is a nonnegative integer m_{0} such that x w_{0}^{m_{0}} = 0 (15.6.2)

for every x ∈ I, as in the proof of Proposition 2 on p7 of [263].

To show (15.5.2), let y ∈ ⋂_{j=1}^{∞} I_{0}^{j} be given. Thus, for each positive integer j, there is an x_{j} ∈ k such that

y = x_{j} w_{0}^{j}. (15.6.3)

Observe that (x_{j} − x_{j+1} w_{0}) w_{0}^{j} = 0 (15.6.4) for each j, which implies that

x_{j} − x_{j+1} w_{0} ∈ I. (15.6.5)

Let I(j) be the ideal in k generated by I and x_{j} for each j. It follows that

I(j) ⊆ I(j + 1) (15.6.6)

for each j, because of (15.6.5). The sequence of I(j)’s is eventually constant, because k is Noetherian. This means that

x_{j+1} ∈ I(j) (15.6.7)

when j is sufficiently large.

If (15.6.7) holds for some j, then

x_{j+1} − t_{j} x_{j} ∈ I (15.6.8)

for some t_{j} ∈ k, by the definition of I(j). Note that t_{j} x_{j} − t_{j} x_{j+1} w_{0} ∈ I, by (15.6.5). It follows that x_{j+1} (1 − t_{j} w_{0}) ∈ I. (15.6.9)

We also have that 1 − t_{j} w_{0} is invertible in k, because t_{j} w_{0} ∈ I_{0}. This implies that x_{j+1} ∈ I (15.6.10)

for all sufficiently large j. Thus

x_{j+1} w_{0}^{m_{0}} = 0 (15.6.11)

for all sufficiently large j, because of (15.6.2). This means that y = 0, as desired, because of (15.6.3).

If k is an integral domain, then the previous argument can be simplified, as mentioned on p7 of [263]. In this case, I = {0}, and x_{j} = x_{j+1} w_{0} for each j. Similarly, I(j) reduces to the ideal in k generated by x_{j} for each j.

15.7 Discrete valuation rings and invertibility

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions. Remember that a submodule M of Q_{k}, as a module over k, is called a fractional ideal of k if there is a y ∈ k such that y ≠ 0 and y M ⊆ k, as in Section 11.7. On p8 of [263], a submodule M of Q_{k}, as a module over k, is called a fractional ideal of Q_{k} with respect to k if M is finitely generated as a module over k. This implies that M is a fractional ideal of k in the sense used here, and the converse holds when k is Noetherian, as in Section 11.7.

If there is a submodule N of Q_{k}, as a module over k, such that M N = k, then M is said to be an invertible ideal of k, as before. In this case, N is equal to the set (k : M)_{Q_{k}} of x ∈ Q_{k} such that x M ⊆ k, as in Section 11.7.

Remember that this set was initially denoted (k : M), and that we used the notation (k : M)_{Q_{k}} in Section 14.3, because similar notation was used in Section 12.8 for something else.

If M ≠ {0}, then (k : M)_{Q_{k}} is a fractional ideal of k, as in Section 11.7. If k is Noetherian, then (k : M)_{Q_{k}} is a fractional ideal of Q_{k} with respect to k, as before. A nonzero ideal I of k is said to be invertible in the remark on p9 of [263] when I (k : I)_{Q_{k}} = k. (15.7.1)

This is equivalent to the invertibility of I as a fractional ideal of k.

Suppose for the moment that k is a discrete valuation ring, and let us show that (15.7.2) every nonzero fractional ideal of k is invertible.

This corresponds to part of Proposition 9.7 on p97 of [8], and it is also mentioned in the proof of Proposition 5 on p11 of [263]. Remember that there is an x ∈ k such that every nonzero ideal in k is generated by x^{l} for some nonnegative integer l, as in Section 15.1. Let M be a nonzero fractional ideal in k, so that y M ⊆ k for some y ∈ k with y ≠ 0. This means that y M is a nonzero ordinary or integral ideal in k, which is generated by x^{l} for some l ≥ 0, as before.

Similarly, the ideal in k generated by y is the same as the ideal generated by x^{n} for some nonnegative integer n. It follows that M is the submodule of Q_{k}, as a module over k, generated by x^{l−n}. If N is the submodule of Q_{k}, as a module over k, generated by x^{n−l}, then we get that M N = k, as desired.

15.8 Local domains and fractional ideals

Let k be an integral domain that is not a field, with the corresponding field Q_{k} of fractions. Suppose that k is also a local ring, with unique maximal proper ideal I_{0}, that satisfies (15.7.2). We would like to show that k is a discrete valuation ring, which is the other part of Proposition 9.7 on p97 of [8]. Another approach, based on some arguments on p8f of [263], will be discussed in the next section.

Remember that invertible ideals of k are finitely generated as modules over k, as in Section 11.7. It follows that any ideal in k is finitely generated as a module over k, so that k is Noetherian.

We would like to show that

(15.8.1) every nonzero ideal in k is af the form I_{0}^{r}

for some nonnegative integer r, as in (14.12.1). Let Σ be the collection of nonzero ideals in k that cannot be expressed as a nonnegative power of I_{0}, and suppose for the sake of a contradiction that Σ ≠ ∅. This implies that Σ has a maximal element I_{1}, because k is Noetherian, as in Section 9.7. Of course, I_{1} ≠ k, so that I_{1} is contained in a maximal proper ideal in k. This means that I_{1} ⊆ I_{0}, and we also have that I_{1} ≠ I_{0}.

Note that I_{0} ≠ {0}, because k is not a field, so that I_{0} is invertible as a fractional ideal in k, by hypothesis. This means that there is a submodule N_{0} of Q_{k}, as a module over k, such that

I_{0} N_{0} = k. (15.8.2)

In particular, I_{1} N_{0} ⊆ k, (15.8.3)

so that I_{1} N_{0} is an ideal in k. We also have that

I_{1} N_{0} ≠ k, (15.8.4)

because I_{0} N_{0} I_{1} = I_{1}.

Remember that N_{0} = (k : I_{0})_{Q_{k}}, so that 1 ∈ N_{0}. This implies that

I_{1} ⊆ I_{1} N_{0}, (15.8.5)

which could also be obtained from the fact that I_{1} I_{0} ⊆ I_{1}. If I_{1} = I_{1} N_{0}, then I_{1} I_{0} = I_{1}. This would imply that I_{1} = {0}, by Nakayama’s lemma, as in Section 13.3. This also uses the fact that k is Noetherian, to get that I_{1} is finitely generated as an ideal in k. However, I_{1} ≠ {0}, by construction. Thus we get that I_{1} ≠ I_{1} N_{0}.

It follows that I_{1} N_{0} is not an element of Σ, by the maximality of I_{1}. This means that I_{1} N_{0} is a nonnegative power of I_{0}. In fact, I_{1} N_{0} is a positive power of I_{0}, because of (15.8.4). This implies that I_{1} is a nonnegative power of I_{0}, which is a contradiction.

Thus (15.8.1) holds under these conditions. One can use this to get that there is an x ∈ k such that every nonzero ideal in k is generated by x^{l} for some nonnegative integer l, as in (14.12.8). More precisely, k was asked to have dimension one, in the sense of Section 14.8, in Sections 14.11 and 14.12. This was not needed for the argument just mentioned, and one could also obtain it from (15.8.1) here. Indeed, if I is a nonzero proper prime ideal in k and I = I_{0}^{j} for some positive integer j, then one can get I = I_{0}, by taking radicals, as in Section 12.10.

This implies that k is a discrete valuation ring, as in Section 15.2.

15.9 Local domains and invertible ideals

Let k be an integral domain, with the corresponding field of fractions Q_{k}, and suppose that k is a local ring, with unique maximal proper ideal I_{0}. Also let M be a nonzero fractional ideal of k that is invertible, so that there is a submodule N of Q_{k}, as a module over k, such that M N = k. We would like to show that

(15.9.1) M is generated by a single element, as a module over k.

This is mentioned in the remark on p9 of [263].

There are finitely many elements x_{1}, …, x_{l} of M and y_{1}, …, y_{l} of N such that

∑_{j=1}^{l} x_{j} y_{j} = 1, (15.9.2)

because 1 ∈ M N. Note that x_{j} y_{j} ∈ k for each j = 1, …, l, because M N ⊆ k. It follows that x_{j_{0}} y_{j_{0}} is not in I_{0} for some j_{0}. This means that x_{j_{0}} y_{j_{0}} is invertible in k in this case.

Using this, we get x ∈ M and y ∈ N such that x y = 1. If z ∈ M, then

z = x (y z), (15.9.3)

and y z ∈ M N = k. This implies that M is generated by x, as a module over k. This corresponds to the proof of I on p8 of [263], as mentioned in the remark on p9 of [263].

Suppose now that k is not a field, and that every nonzero fractional ideal of k is invertible. In particular, this means that I_{0} is invertible, because I_{0} ≠ {0}. The previous argument implies that I_{0} is generated by a single element, as an ideal in k.

Remember that k is Noetherian under these conditions, as in the previous section. It follows that k is a discrete valuation ring, as in Sections 15.5 and 15.6. This corresponds to a statement on p8 of [263], shortly before the proof of the statement I.

Alternatively, if every nonzero ideal in k is invertible, then the previous argument shows that k is a principal ideal domain. If k is not a field, then it follows that k is a discrete valuation ring, as in Section 15.4.

If k is a local domain of dimension one, in the sense of Section 14.8, then one could get that k is a discrete valuation ring from the condition that I_{0} be generated by a single element, as an ideal in k, as in Sections 14.11 and 14.12. More precisely, in Sections 14.11 and 14.11, one only ever seems to use a Noetherian condition on k to get that I_{0} is finitely generated, as an ideal in k.

15.10 A criterion for invertibility

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions. Remember that a submodule M of Q_{k}, as a module over k, is said to be a fractional ideal of k if there is an x ∈ k such that x ≠ 0 and x M ⊆ k, as in Section 11.7. If there is another submodule N of Q_{k}, as a module over k, such that M N = k, then M is said to be invertible, as before. In this case, we have seen that N is equal to the set (k : M)_{Q_{k}} of x ∈ Q_{k} such that x M ⊆ k.

Of course, M (k : M)_{Q_{k}} ⊆ k (15.10.1)

holds automatically, by definition of (k : M)_{Q_{k}}. This means that M (k : M)_{Q_{k}} is an ideal in k, because it is a submodule of Q_{k}, as a module over k. Invertibility of M is equivalent to M (k : M)_{Q_{k}} = k. (15.10.2)

390CHAPTER 15. DISCRETE VALUATIONS AND DEDEKIND DOMAINS

Let I be a maximal proper ideal in k, which is prime in particular. Thus S_{I} = k \ I (15.10.3) is multiplicatively closed in k, as in Section 12.4. The corresponding ring of fractions S_{I}^{−1} k, as in Section 12.1, is an integral domain whose field of fractions may be identified with Q_{k}, as in Section 14.2. The module of fractions S_{I}^{−1} M of M with respect to S_{I}, as in Section 12.2, may be identified with a submodule of Q_{k}, as a module over S_{I}^{−1} k, as before. This is a fractional ideal of S_{I}^{−1} k.

Suppose that (15.10.4) S_{I}^{−1} M is invertible as a fractional ideal of S_{I}^{−1} k. Remember that (S_{I}^{−1} k : S_{I}^{−1} M)_{Q_{k}} is the set of x ∈ Q_{k} such that x (S_{I}^{−1} M) ⊆ S_{I}^{−1} k, (15.10.5) as in Section 14.3. This is a submodule of Q_{k}, as a module over S_{I}^{−1} k, and (15.10.4) is the same as saying that (S_{I}^{−1} M) (S_{I}^{−1} k : S_{I}^{−1} M)_{Q_{k}} = S_{I}^{−1} k. (15.10.6)

Suppose that (15.10.7) M is finitely generated, as a module over k. This implies that (S_{I}^{−1} k : S_{I}^{−1} M)_{Q_{k}} = S_{I}^{−1} (k : M)_{Q_{k}}, as in Section 14.4. It follows that (S_{I}^{−1} M) (S_{I}^{−1} (k : M)_{Q_{k}}) = S_{I}^{−1} k, (15.10.8) by (15.10.6). This means that S_{I}^{−1}(M (k : M)_{Q_{k}}) = S_{I}^{−1} k, (15.10.9) as in Section 14.3.

In particular, 1/1 is an element of S_{I}^{−1}(M (k : M)_{Q_{k}}). This implies that there are v ∈ M (k : M)_{Q_{k}} and r, t ∈ S_{I} such that t v = t r ∈ S_{I}, as in Section 12.2. Note that v ∈ k, by (15.10.1). It follows that v ∉ I, by the definition of S_{I}, and because I is an ideal in k. Thus M (k : M)_{Q_{k}} ⊈ I. (15.10.10)

If this holds for every maximal proper ideal I in k, then it follows that (15.10.2) holds, because M (k : M)_{Q_{k}} is an ideal in k. This corresponds to part of Proposition 9.6 on p97 of [8].

15.11 Dedekind domains

Let k be a Noetherian integral domain. Remember that if I is a proper prime ideal in k, then S_{I} = k \ I is multiplicatively closed in k, as in Section 12.4. In this case, k_{I} = S_{I}^{−1} k (15.11.1)

may be defined as in Section 12.1.

Suppose that for every proper prime ideal I ≠ {0} in k,

(15.11.2) k_{I} is a discrete valuation ring,

as in Section 15.1. Under these conditions, k is said to be a Dedekind domain, as in p95 of [8], and p10 of [263]. More precisely, the formulation in [8] also asks that k not be a field. If we do not include this condition, then this definition is equivalent to the definition of a Dedekind ring in Section 11.10, as before.

The formulation in [8] also includes the condition that k have dimension one, in the sense of Section 14.8. Of course, this implies that k not be a field, which would have dimension 0. It is easy to see that the definition in the preceding paragraph implies that

(15.11.3) k have dimension less than or equal to one,

which is also mentioned in [263].

Indeed, let I be a nonzero maximal proper ideal in k. Note that k_{I} has dimension one, by hypothesis, as in Section 15.1. There is also a one-to-one correspondence between the prime ideals in k that are contained in I and the proper prime ideals in k_{I}, as mentioned at the end of Section 12.13. It follows that I is the only nonzero prime ideal in k contained in I, as desired.

Remember that an integral domain has dimension less than or equal to one if and only if every proper nonzero prime ideal is maximal, as in Section 14.11. If (15.11.2) holds for every nonzero maximal proper ideal I in k, then every nonzero proper prime ideal in k is maximal, as in the preceding paragraph. This means that (15.11.2) holds for every nonzero proper prime ideal in k.

Suppose now that k is a principal ideal domain. It is easy to see that k is a hereditary ring, as in Section 9.3, and thus a Dedekind ring, as in Section 11.10.

Let us verify that k is a Dedekind domain, without using the equivalence with Dedekind rings mentioned earlier. This is Example (1) on p96 of [8], which is also mentioned on p10 of [263]. Note that k is Noetherian, because every ideal in k is finitely generated.

Let S be a multiplicatively closed set in k, with 0 ∉ k, and remember that S^{−1} k is an integral domain, as in Section 14.2. We have also seen that every ideal in S^{−1} k is the extension of an ideal in k, as in Section 12.13. Using this, one can check that every ideal in S^{−1} k is generated by a single element, because of the analogous property of k, by hypothesis. This means that S^{−1} k is a principal ideal domain, and in particular that S^{−1} k has dimension less than or equal to one, as before.

Let I be a nonzero proper prime ideal in k, and let S_{I} and k_{I} be as before. Remember that k_{I} is a local ring, as in Section 12.4. Note that k_{I} is not a field under these conditions. More precisely, S_{I}^{−1} I is a nonzero proper ideal in k_{I}. We also have that k_{I} is a principal ideal domain, as in the preceding paragraph. It follows that k_{I} is a discrete valuation ring, as in Sections 14.11, 14.12, and 15.2, or Section 15.4. Of course, if k is a field, then there are no nonzero proper ideals in k, and the previous statement holds vacuously.

15.12 Dedekind domains and invertibility

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions. Suppose that k is a Dedekind domain, and let us show that

(15.12.1) every nonzero fractional ideal of k is invertible.

This corresponds to part of Theorem 9.8 on p97 of [8], and to Proposition 5 on p11 of [263]. Remember that “fractional ideals” are defined a bit differently in [263], as in Section 15.7. However, the difference does not matter here, because k is Noetherian, by hypothesis.

Let M be a nonzero fractional ideal of k, so that M is a submodule of Q_{k}, as a module over k, such that x M ⊆ k for some x ∈ k with x ≠ 0, as in Section 11.7. Remember that k is supposed to be Noetherian, as in Section 15.11. This implies that M is finitely generated as a module over k, as in Section 11.7.

Let I be a maximal proper ideal in k. Remember that S_{I} = k \ I is multiplicatively closed in k, because I is a prime ideal, as in Section 12.4. The ring of fractions k_{I} = S_{I}^{−1} k, as in Section 12.1, is an integral domain whose field of fractions may be identified with Q_{k}, as in Section 14.2.

The module of fractions M_{I} = S_{I}^{−1} M, (15.12.2)

as in Section 12.2, may be identified with a submodule of Q_{k}, as a module over k_{I}, as before. This is a fractional ideal of k_{I}, as in Section 15.10.

Let us check that

(15.12.3) M_{I} is invertible as a fractional ideal of k_{I}.

Of course, if k is a field, then I = {0}, and there is nothing to do.

If k is not a field, then I ≠ {0}, and k_{I} is a discrete valuation ring, as in Section 15.11. It is easy to see that

M_{I} ≠ {0}, (15.12.4)

because M ≠ {0}, by hypothesis, and k is an integral domain. It follows that (15.12.3) holds under these conditions, as in Section 15.7.

This implies that M is invertible as a fractional ideal of k, as in Section 15.10. This uses the fact that M is finitely generated as a module over k, as before.

15.13 Dedekind domains from invertibility

Let k be an integral domain, and let Q_{k} be the corresponding field of fractions, as usual. In this section, we suppose that every nonzero fractional ideal of k is invertible, and we would like to show that k is a Dedekind domain. This is the other part of Theorem 9.8 on p97 of [8].

Remember that invertible ideals of k are finitely generated as modules over k, as in Section 11.7. This implies that k is Noetherian, as in Section 15.8.

Let I be a nonzero proper prime ideal in k, and remember that S_{I} = k \ I is multiplicatively closed in k, as in Section 12.4. Thus k_{I} = S_{I}^{−1} k may be defined as in Section 12.1, and we would like to show that k_{I} is a discrete valuation ring.

Remember that k_{I} is a local ring, as in Section 12.4. More precisely, the unique maximal proper ideal in k_{I} is S_{I}^{−1} I, using the notation mentioned near the beginning of Section 12.13. We also have that k_{I} is an integral domain, as mentioned in Section 14.2, and that k_{I} is Noetherian, as in Section 14.6. Note that k_{I} is not a field, because S_{I}^{−1} I is a nonzero proper ideal in k_{I}, as in Section 15.11.

It suffices to show that every nonzero fractional ideal of k_{I} is invertible, as in Sections 15.8 and 15.9. It is easy to reduce to the case of a nonzero ordinary (integral) ideal in k_{I}.

Let I_{2} be a nonzero ideal in k_{I}, and remember that I_{2}^{ce} = I_{2}, as in Section 12.13. Here I_{2}^{c} is the contraction of I_{2} with respect to the natural homomorphism from k into k_{I}, and I_{2}^{ce} is the extension of I_{2}^{c} in k_{I}. In this case, k_{I} may be identified with a subring of Q_{k}, as in Section 14.2, so that

I_{2}^{c} = I_{2} ∩ k. (15.13.1)

It is easy to see that I_{2}^{c} ≠ {0}, because I_{2} ≠ {0}.

It follows that I_{2}^{c} is an invertible ideal of k, by hypothesis. This implies that S_{I}^{−1} I_{2}^{c} is an invertible ideal of k_{I}, as in Section 14.3. Note that

S_{I}^{−1} I_{2}^{c} = I_{2}^{ce} = I_{2}, (15.13.2)

as in Section 12.13. This means that I_{2} is an invertible ideal of k_{I}, as desired.

15.14 Primary ideals in Dedekind domains

Let k be a commutative ring with a multiplicative identity element, and let I_{1} ≠ {0} be a primary ideal in k, as in Section 14.5. Remember that r(I_{1}) is the radical of I_{1}, as in Section 12.10, and note that r(I_{1}) ≠ {0}, because I_{1} ≠ {0}. We also have that r(I_{1}) ≠ k, because I_{1} ≠ k, by the definition of a primary ideal. In fact, r(I_{1}) is a prime ideal in k, as in Section 14.5. Thus

S = k \ r(I_{1}) (15.14.1)

is a multiplicatively closed set in k, as in Section 12.4.

Let S^{−1} k be as in Section 12.1, and remember that x ↦ x/1 defines a ring homomorphism from k into S^{−1} k. The extensions I_{1}^{e} and r(I_{1})^{e} of I_{1} and r(I_{1}), respectively, with respect to this homomorphism are ideals in S^{−1} k, as in Section 12.7. These are the same as the ideals S^{−1} I_{1} and S^{−1} r(I_{1}) in S^{−1} k, as in Section 12.13. In this case, S^{−1} k is a local ring, with S^{−1} r(I_{1}) as its unique maximal proper ideal, as in Section 12.4.

The contractions of S^{−1} I_{1} and S^{−1} r(I_{1}) in k with respect to the natural homomorphism from k into S^{−1} k are equal to I_{1} and r(I_{1}), respectively, as in Section 14.6. This uses the fact that I_{1} and r(I_{1}) are disjoint from S, by construction.

If k is a Dedekind domain, then S^{−1} k is a discrete valuation ring, as in Section 15.11. This implies that

S^{−1} I_{1} = (S^{−1} r(I_{1}))^{l} (15.14.2)

for a unique positive integer l, as in Section 15.1.

Note that (S^{−1} r(I_{1}))^{l} = S^{−1} r(I_{1})^{l}, (15.14.3)

as in Sections 12.9 and 12.13, and which can be verified directly anyway. Thus (15.14.2) is the same as saying that

S^{−1} I_{1} = S^{−1} r(I_{1})^{l}. (15.14.4)

This holds for a unique positive integer l, as before.

If k is a Dedekind domain, then r(I_{1}) is a maximal proper ideal in k, because r(I_{1}) is a nonzero proper prime ideal in k. This implies that r(I_{1})^{l} is a primary ideal in k, as in Section 14.5. It follows that the contraction of S^{−1} r(I_{1})^{l} in k with respect to the natural homomorphism from k into S^{−1} k is equal to r(I_{1})^{l}, as in Section 14.6 again. Using this, we get that

I_{1} = r(I_{1})^{l}. (15.14.5)

This corresponds to part of Theorem 9.3 on p95 of [8].

Of course, (15.14.5) implies (15.14.4). This means that l is uniquely determined by (15.14.5).

15.15 One-dimensional Noetherian domains

Let k be an integral domain of dimension one in the sense of Section 14.8. Thus k is not a field, and every nonzero proper prime ideal in k is maximal. If k is Noetherian, and if every primary ideal in k is a power of a prime ideal, then k is a Dedekind domain, as in Theorem 9.3 on p95 of [8].

More precisely, if I_{1} is a primary ideal in k, then the radical r(I_{1}) is a proper prime ideal in k, as in Section 14.5. The hypothesis mentioned in the preceding paragraph means that I_{1} = r(I_{1})^{l} (15.15.1) for some positive integer l.

Let I ≠ {0} be a proper prime ideal in k, so that S_{I} = k \ I is multiplicatively closed in k, as in Section 12.4. We would like to show that k_{I} = S_{I}^{−1} k is a discrete valuation ring, as in Section 15.11. The hypothesis that k have dimension one implies that I is a maximal proper ideal in k, and it is sufficient to consider only that case anyway, as before.

Remember that k_{I} is a local ring, with unique maximal proper ideal S_{I}^{−1} I, as in Section 12.4. We have seen that k_{I} is an integral domain, because k is an integral domain, as in Section 14.2. We also have that k_{I} is Noetherian, because k is Noetherian, as in Section 14.6. Note that S_{I}^{−1} I ≠ {0}, because I ≠ 0, and k is an integral domain.

There is a one-to-one correspondence between the set of prime ideals in k that are contained in I and the set of proper prime ideals in k_{I}, as mentioned at the end of Section 12.13. This implies that

(15.15.2) k_{I} has dimension one,

because k has dimension one. In fact, S_{I}^{−1} I is the only nonzero proper prime ideal in k_{I}.

We would like to show that if I_{2} ≠ {0} is an ideal in k_{I}, then

I_{2} = (S_{I}^{−1} I)^{l} (15.15.3)

for some nonnegative integer l. This will imply that k_{I} is a discrete valuation ring, as in Sections 14.12 and 15.2. We may as well suppose that I_{2} ≠ k_{I}, since otherwise we could take l = 0 in (15.15.3). This means that

I_{2} ⊆ S_{I}^{−1} I, (15.15.4)

because S_{I}^{−1} I is the unique maximal proper ideal in k_{I}.

Remember that the radical r(I_{2}) of I_{2} is the same as the intersection of all of the prime ideals in k_{I} that contain I_{2}, as in Section 12.10. This implies that

r(I_{2}) = S_{I}^{−1} I, (15.15.5)

because I_{2} ≠ {0}, and S_{I}^{−1} I is the only nonzero proper prime ideal in k_{I}. It follows that I_{2} is a primary ideal in k_{I}, because S_{I}^{−1} I is a maximal proper ideal in k_{I}, as in Section 14.5.

Let I_{2}^{c} be the contraction of I_{2} in k, with respect to the natural homomorphism x ↦ x/1 from k into k_{I}, as in Section 12.7. Note that I_{2}^{c} is a primary ideal in k, because I_{2} is a primary ideal in k_{I}, as in Section 14.5. It is easy to see that I_{2}^{c} ≠ {0}, because I_{2} ≠ {0}. We also have that that I_{2}^{c} ≠ k, because I_{2} ≠ k_{I}.

Remember that a prime ideal in k that is disjoint from S_{I} is the contraction of an ideal in k_{I}, as in Section 12.13. This implies that a prime ideal in k that is disjoint from S_{I} is equal to the contraction of its extension, as in Section 12.7. In particular, I is a prime ideal in k that is disjoint from S_{I}, so that

I = I^{ec} = (S_{I}^{−1} I)^{c}, (15.15.6)

396CHAPTER 15. DISCRETE VALUATIONS AND DEDEKIND DOMAINS where the second step is as in Section 12.13. It follows that I_{2}^{c} ⊆ (S_{I}^{−1} I)^{c} = I, (15.15.7) using (15.15.4) in the first step.

The radical r(I_{2}^{c}) of I_{2}^{c} is the smallest prime ideal in k that contains I_{2}^{c}, because I_{2}^{c} is a primary ideal in k, as in Section 14.5. Note that r(I_{2}^{c}) ⊆ I, (15.15.8) by (15.15.7), and that r(I_{2}^{c}) ≠ {0}, because I_{2}^{c} ≠ {0}. This means that r(I_{2}^{c}) = I, (15.15.9) because k has dimension one, by hypothesis.

Thus I_{2}^{c} = I^{l} (15.15.10) for some positive integer l, as in (15.15.1). Remember that I_{2} = I_{2}^{ce}, as in Section 12.13. Using this, we get that I_{2} = (I^{l})^{e} = (I^{e})^{l} = (S_{I}^{−1} I)^{l}, (15.15.11) where the second step is as in Section 12.9, and the third step is as in Section 12.13. This shows that (15.15.3) holds, as desired.

Valuation rings and integral elements

16.1 Valuation rings

An integral domain k is said to be a valuation ring of its field Q_{k} of fractions if for every x ∈ Q_{k} with x ≠ 0, we have that x ∈ k, or x^{−1} ∈ k, or both, as on p65 of [8].

Let k_{1} be a field, and let k be a subring of k_{1} that contains the multiplicative identity element 1 = 1_{k_{1}} of k_{1}. This implies that k is an integral domain, for which the corresponding field of fractions may be identified with the subfield of k_{1} generated by k. One might say that k is a valuation ring of k_{1} if for every x ∈ k_{1} with x ≠ 0, we have that x ∈ k, x^{−1} ∈ k, or both. This would imply in particular that k_{1} is generated as a field by k, so that k_{1} may be identified with the field Q_{k} of fractions of k. This means that k would be a valuation ring of Q_{k}, as in the preceding paragraph.

Let v be a valuation on k_{1}, as in Section 14.13. Remember that

{x ∈ k_{1} : v(x) ≥ 0} (16.1.1)

is a subring of k_{1} that contains 1_{k_{1}}, called the valuation ring of v, as in Section 14.14. If x ∈ k_{1} and x ≠ 0, then x or x^{−1} is an element of (16.1.1), depending on whether v(x) ≥ 0 or v(x) ≤ 0. This implies that k_{1} may be identified with the field of fractions of (16.1.1), and that (16.1.1) is a valuation ring in its field of fractions, as in the previous paragraph. This corresponds to a remark on p94 of [8].

Let k be a commutative ring with a multiplicative identity element. If x, y are elements of k and x y is invertible in k, then it is easy to see that

(16.1.2) x and y are invertible in k.

Put (16.1.3) I_{0} = {x ∈ k : x is not invertible in k}.

398 CHAPTER 16. VALUATION RINGS AND INTEGRAL ELEMENTS

If x ∈ I_{0} and y ∈ k, then x y ∈ I_{0}, (16.1.4)

by (16.1.2).

Now let k be an integral domain, with field of fractions Q_{k}. In this case,

I_{0} = {x ∈ k : x = 0 or x^{−1} ∈ Q_{k} \ k}. (16.1.5)

Suppose that k is a valuation ring of Q_{k}. If x, y ∈ k and x, y ≠ 0, then

x y^{−1} ∈ k or x^{−1} y ∈ k. (16.1.6)

Let x, y ∈ I_{0} be given, and let us check that

x + y ∈ I_{0}. (16.1.7)

This follows automatically when x = 0 or y = 0, and so we may suppose that x, y ≠ 0. If x y^{−1} ∈ k, then

x + y = (x y^{−1} + 1) y ∈ I_{0}, (16.1.8)

as in (16.1.4). Similarly, (16.1.7) holds when x^{−1} y ∈ k.

This shows that I_{0} is an ideal in k, as in the proof of the first part of Proposition 5.18 on p65 of [8]. This means that I_{0} is the unique maximal proper ideal in k, so that k is a local ring, as in [8].

16.2 Noetherian valuation rings

Let k be an integral domain that is a valuation ring of its field Q_{k} of fractions. If x, y ∈ k and x, y ≠ 0, then either x = t y for some t ∈ k, or y = t′ x for some t′ ∈ k, by (16.1.6). This implies that the ideals in k that are generated by a single element are linearly ordered by inclusion.

Let I be an ideal in k generated by finitely many elements x_{1}, …, x_{n}. It is easy to see that (16.2.1) I is generated by x_{j}

for some j, using the remark in the preceding paragraph.

Suppose now that k is also Noetherian, so that every ideal in k is finitely generated. This implies that every ideal in k is generated by a single element, as in the previous paragraph, so that k is a principal ideal domain.

If k is not a field, then k is a discrete valuation ring under these conditions, as in Exercise 3 on p99 of [8]. Remember that principal ideal domains have dimension less than or equal to one, in the sense of Section 14.8, as in Section 15.4. This means that k has dimension one, because k is not a field.

Let I_{0} be the unique maximal proper ideal in k, as in the previous section. Note that I_{0} is generated as an ideal in k by a single element, as before. One can get that k is a discrete valuation ring using the arguments in Sections 14.11, 14.12, and 15.2.

Alternatively, one could use the characterization of discrete valuation rings in Section 15.4. One could also use the criterion for a commutative ring to be a discrete valuation ring discussed in Sections 15.5 and 15.6.

Remember that discrete valuation rings are Noetherian, as in Section 15.1.

16.3 Integral elements of larger rings

Let k_{1} be a commutative ring with a multiplicative identity element 1 = 1_{k_{1}}, and let k be a subring of k_{1} that contains 1_{k_{1}}. An element x of k_{1} is said to be integral over k if it satisfies a monic polynomial equation with coefficients in k. This means that there are a positive integer n and n elements a_{1}, …, a_{n} of k such that x^{n} + a_{1} x^{n−1} + ⋯ + a_{n−1} x + a_{n} = 0, (16.3.1)

as on p59 of [8], and p8 of [263]. Note that every element of k is integral over k in this sense.

If every element of k_{1} that is integral over k is an element of k, then k is said to be integrally closed in k_{1}, as on p60 of [8], and p8 of [263].

Suppose now that k is an integral domain, and let Q_{k} be its field of fractions. If k is integrally closed in Q_{k}, then k is simply said to be integrally closed, as on p62 of [8], and p8 of [263].

It is well known that the ring Z of integers is integrally closed in the field Q of rational numbers, as in Example 5.0 on p59 of [8]. The same argument can be used to show that any unique factorization domain is integrally closed, as mentioned on p63 of [8].

Suppose for the moment that k is a valuation ring of Q_{k}, as in Section 16.1. In this case, k is integrally closed in Q_{k}, as in Proposition 5.18 on p65 of [8]. Indeed, suppose that x ∈ Q_{k} is integral over k, so that x satisfies a monic polynomial equation with coefficients in k as in (16.3.1). If x ∉ k, then x ≠ 0 and x^{−1} ∈ k, and we can multiply both sides of (16.3.1) by x^{1−n} to get that

x = −a_{1} − ⋯ − a_{n−1} x^{2−n} − a_{n} x^{1−n}. (16.3.2)

This implies that x ∈ k.

Let k_{1} be a field, and let v be a valuation on k_{1}, as in Section 14.13. If x, y ∈ k_{1} and v(x) ≤ v(y), then

v(x + y) ≥ v(x), (16.3.3)

by definition of a valuation. We also have that

v(x) ≥ min(v(x + y), v(−y)) = min(v(x + y), v(y)). (16.3.4)

If v(x) > v(y), then it follows that

v(x) ≥ v(x + y). (16.3.5)

This means that v(x + y) = v(x) (16.3.6)

when v(x) > v(y).

Remember that the valuation ring of v,

{x ∈ k_{1} : v(x) ≥ 0}, (16.3.7)

is a valuation ring of k_{1}, as in Section 16.1. Thus (16.3.7) is integrally closed in k_{1}, as before. Another way to see this is mentioned on p8 of [263]. We can do this using the remarks in the preceding paragraph, as follows.

Suppose that x ∈ k_{1} is integral over (16.3.7), so that x satisfies a monic polynomial equation with coefficients in (16.3.7), as in (16.3.1). If v(x) < 0 and 1 ≤ l ≤ n, then

v(x^{n}) = n v(x) > (n − l) v(x) ≥ v(a_{l} x^{n−l}). (16.3.8)

This implies that

v(x^{n} + a_{1} x^{n−1} + ⋯ + a_{n−1} x + a_{n}) = v(x^{n}), (16.3.9)

as in the preceding paragraph. This contradicts (16.3.1), so that v(x) ≥ 0, as desired.

16.4 Some properties of integral elements

Let k_{0} be a commutative ring with a multiplicative identity element, and let V be a module over k_{0}. Remember that the annihilator Ann(V) of V in k_{0} consists of t ∈ k_{0} such that t · V = {0}, as in Section 12.8. If

Ann(V) = {0}, (16.4.1)

then V is said to be faithful as a module over k_{0}, as on p20 of [8].

Let k_{1} be a commutative ring with a multiplicative identity element 1 = 1_{k_{1}}, and let k be a subring of k_{1} that contains 1_{k_{1}}. If x ∈ k_{1}, then let k[x] be the subring of k_{1} generated by k and x. This is the same as the submodule of k_{1}, as a module over k, generated by the nonnegative integer powers of x in k_{1}. Proposition 5.1 on p59f of [8] gives some properties of k[x] that are equivalent to saying that (16.4.2) x is integral over k.

If (16.4.2) holds, then it is easy to see that

(16.4.3) k[x] is finitely generated as a module over k.

More precisely, suppose that x satisfies a monic polynomial equation of degree n with coefficients in k, as in (16.3.1). If r is a nonnegative integer, then we get that x^{n+r} = −a_{1} x^{n+r−1} − ⋯ − a_{n−1} x^{r+1} − a_{n} x^{r}. (16.4.4)

One can use this to check that k[x] is generated by 1, x, …, x^{n−1}, as a module over k, as on p60 of [8].

Clearly (16.4.3) implies that

(16.4.5) k[x] is contained in a subring C of k_{1} such that C is finitely generated as a module over k,

by taking C = k[x]. Let us check that (16.4.5) implies that

(16.4.6) there is a faithful module V over k[x] that is finitely generated as a module over k.

Of course, if (16.4.5) holds, then C is a module over k[x] that is finitely generated as a module over k. It is easy to see that C is also faithful as a module over k[x], because 1_{k_{1}} ∈ C, as on p60 of [8].

We would like to show that (16.4.6) implies (16.4.2), to get that these four conditions are equivalent, as in [8]. To do this, we use the remarks in Section 13.2, with V considered as a module over k. The ideal I of k mentioned in Section 13.2 is taken to be k here, so that I · V = V.

If y ∈ k[x] and v ∈ V, then put

ϕ_{y}(v) = y · v, (16.4.7)

where the right side is defined as an element of V, because V is a module over k[x]. Note that ϕ_{y} is a homomorphism from V into itself, as a module over k. Thus y ↦ ϕ_{y} (16.4.8)

defines a mapping from k[x] into the space Hom_{k}(V, V) of homomorphisms from V into itself, as a module over k. More precisely, (16.4.8) is a homomorphism from k[x] into Hom_{k}(V, V), as associative algebras over k. This homomorphism is injective, because V is faithful as a module over k[x], by hypothesis.

If we take ϕ = ϕ_{x}, then the remarks in Section 13.2 imply that ϕ satisfies a monic polynomial equation with coefficients in k. This uses the hypothesis that V be finitely generated as a module over k. It follows that x satisfies the analogous polynomial equation, with the same coefficients, because of the remarks in the preceding paragraph, as on p60 of [8].

16.5 Some corollaries

Let us continue with the same notation and hypotheses as in the previous section. Let x_{1}, …, x_{n} be finitely many elements of k_{1}, and let k[x_{1}, …, x_{n}] be the subring of k_{1} generated by k and x_{1}, …, x_{n}. If

(16.5.1) x_{j} is integral over k

for each j = 1, …, n, then Corollary 5.2 on p60 of [8] states that

(16.5.2) k[x_{1}, …, x_{n}] is finitely generated as a module over k.

Note that this reduces to (16.4.3) when n = 1.

Put A_{r} = k[x_{1}, …, x_{r}] for each r = 1, …, n. Suppose that n > 1, and observe that A_{n} is the same as the subring A_{n−1}[x_{n}] of k_{1} generated by A_{n−1} and x_{n}. Of course, x_{n} is integral over A_{n−1}, because x_{n} is integral over k, by hypothesis. This implies that A_{n−1}[x_{n}] is finitely generated as a module over A_{n−1}, as in (16.4.3).

If we use induction, then we may suppose that A_{n−1} is finitely generated as a module over k. One can use this to get that A_{n} = A_{n−1}[x_{n}] is finitely generated as a module over k, as on p60 of [8].

If x, y ∈ k_{1} are integral over k, then k[x, y] is finitely generated as a module over k, as in (16.5.2). This implies that x±y and x y are integral over k, because (16.4.5) implies (16.4.2). It follows that

(16.5.3) C = {x ∈ k_{1} : x is integral over k}

is a subring of k_{1}, as in Corollary 5.3 on p60 of [8].

The subring C of k_{1} is called the integral closure of k in k_{1}, as on p60 of [8]. Note that k is integrally closed in k_{1} exactly when C = k. If C = k_{1}, then k_{1} is said to be integral over k, as in [8]. Of course, C is integral over k.

Let k_{2} be a commutative ring with a multiplicative identity element 1 = 1_{k_{2}}, and suppose that k_{1} is a subring of k_{2} that contains 1_{k_{2}}. If k_{1} is integral over k and k_{2} is integral over k_{1}, then

(16.5.4) k_{2} is integral over k,

as in Corollary 5.4 on p60 of [8]. To see this, let x ∈ k_{2} be given, so that x satisfies a monic polynomial equation

x^{n} + b_{1} x^{n−1} + ⋯ + b_{n−1} x + b_{n} = 0 (16.5.5)

with coefficients in k_{1}. These coefficients are integral over k, by hypothesis, so that (16.5.6) k[b_{1}, …, b_{n}] is finitely generated as a module over k,

as in (16.5.2). Note that

(16.5.7) x is integral over k[b_{1}, …, b_{n}].

It follows that

(16.5.8) (k[b_{1}, …, b_{n}])[x] is finitely generated as a module over k[b_{1}, …, b_{n}],

as in the previous section. This implies that

(16.5.9) (k[b_{1}, …, b_{n}])[x] is finitely generated as a module over k,

because of (16.5.6). Using this, we get that x is integral over k, as in the previous section again.

Let C be as in (16.5.3), and let C̃ be the integral closure of C in k_{1}. We would like to check that C̃ = C, (16.5.10)

as in Corollary 5.5 on p61 of [8]. Of course, C ⊆ C̃ automatically. We also have that C is integral over k, as before, and similarly that C̃ is integral over C. This implies that C̃ is integral over k, as in (16.5.4), so that C̃ ⊆ C.

16.6 Getting a discrete valuation

Let k be a Noetherian integral domain. Suppose that k is a local ring, and let I_{0} be the unique maximal proper ideal in k. Suppose also that k has dimension one in the sense of Section 14.8. Equivalently, this means that I_{0} ≠ {0}, and that I_{0} is the only nonzero proper prime ideal in k, as in Section 14.11.

Suppose in addition that k is integrally closed. We would like to show that

(16.6.1) I_{0} is a principal ideal in k.

This will imply that k is a discrete valuation ring, using the results discussed in Sections 14.11, 14.12, and 15.2, or in Sections 15.5 and 15.6. This corresponds to parts of Proposition 9.2 on p94 of [8], and Proposition 3 on p7 of [263].

Let us begin with the approach in [8]. Let a ∈ I_{0} be given, with a ≠ 0, and let I(a) be the ideal in k generated by a. Under these conditions, there is a positive integer m such that

I_{0}^{m} ⊆ I(a) ⊆ I_{0}, (16.6.2)

as in Section 14.11. We may as well take m to be as small as possible, so that

I_{0}^{m−1} ⊈ I(a). (16.6.3)

Remember that I_{0}^{0} is interpreted as being equal to k, as in Section 12.10.

Let b be an element of I_{0}^{m−1} that is not in I(a). In particular, b ≠ 0, so that a/b is defined as an element of the field Q_{k} of fractions of k. Note that

b/a ∉ k, (16.6.4)

because b ∉ I(a). This means that

(16.6.5) b/a is not integral over k

in Q_{k}, because k is integrally closed in Q_{k}, by hypothesis.

Suppose for the sake of a contradiction that

(b/a) I_{0} ⊆ I_{0}. (16.6.6)

Let k[b/a] be the subring of Q_{k} generated by k and b/a, as in Section 16.4. If (16.6.6) holds, then we may consider I_{0} as a module over k[b/a], because I_{0} is an ideal in k. In fact, I_{0} would be faithful as a module over k[b/a], because I_{0} ≠ {0}, and Q_{k} is a field. We also have that I_{0} is finitely generated as a module over k, because k is Noetherian, by hypothesis. Under these conditions, we would get that b/a is integral over k, as in Section 16.4 again. This is a contradiction, which means that

(b/a) I_{0} ⊈ I_{0}. (16.6.7)

Remember that b ∈ I_{0}^{m−1}, so that

b I_{0} ⊆ I_{0}^{m} ⊆ I(a). (16.6.8)

This implies that (b/a) I_{0} ⊆ k. (16.6.9)

More precisely, (b/a) I_{0} is an ideal in k. It follows that

(b/a) I_{0} = k, (16.6.10)

because of (16.6.7), and the hypothesis that I_{0} be the unique maximal proper ideal in k.

This means that I_{0} is the same as the ideal in k generated by a/b, as desired.

16.7 Another argument using fractional ideals

Let k be an integral domain that is a local ring, with unique maximal proper ideal I_{0}. Suppose also that k is not a field, so that I_{0} ≠ {0}. We would like to consider another proof of (16.6.1) when k is Noetherian, of dimension one, and integrally closed, as on p8f of [263].

Remember that (k : I_{0})_{Q_{k}} is the set of x in the field Q_{k} of fractions of k such that x I_{0} ⊆ k, as in Section 15.7. This is a fractional ideal of k, because I_{0} ≠ {0}, as before. More precisely, if k is Noetherian, then (k : I_{0})_{Q_{k}} is finitely generated as a module over k, as in Section 11.7. This means that (k : I_{0})_{Q_{k}} is a fractional ideal of Q_{k} with respect to k, as in [263], in this case.

Of course, I_{0} (k : I_{0})_{Q_{k}} ⊆ k, (16.7.1)

by construction. In fact, I_{0} (k : I_{0})_{Q_{k}} is an ideal in k. We also have that

k ⊆ (k : I_{0})_{Q_{k}}, (16.7.2)

because I_{0} is an ideal in l. This implies that

I_{0} ⊆ I_{0} (k : I_{0})_{Q_{k}}. (16.7.3)

It follows that I_{0} (k : I_{0})_{Q_{k}} = I_{0} (16.7.4)

or I_{0} (k : I_{0})_{Q_{k}} = k, (16.7.5) because I_{0} is a maximal proper ideal in k. If (16.7.5) holds, then I_{0} is invertible as a fractional ideal of k, or equivalently in the sense mentioned in the remark on p9 of [263], as in Section 15.7. In this case, (16.6.1) holds, as in Section 15.9. This corresponds to the statement I on p8 of [263].

If (16.7.4) holds and k is Noetherian and integrally closed, then Statement II on p8 of [263] says that (k : I_{0})_{Q_{k}} = k. (16.7.6)

Of course, it suffices to show that

(k : I_{0})_{Q_{k}} ⊆ k, (16.7.7)

because of (16.7.2). To see this, let x ∈ (k : I_{0})_{Q_{k}} be given, so that

x I_{0} ⊆ I_{0}, (16.7.8)

by (16.7.4). This implies that x^{l} I_{0} ⊆ I_{0} (16.7.9)

for every positive integer l.

Let V_{n} be the submodule of Q_{k}, as a module over k, generated by 1, x, …, x^{n} for each positive integer n. Thus

V_{n} ⊆ V_{n+1} (16.7.10)

for every n ≥ 1, by construction. We also have that

V_{n} ⊆ (k : I_{0})_{Q_{k}} (16.7.11)

for each n, by (16.7.9) and the definition of (k : I_{0})_{Q_{k}}. Note that (k : I_{0})_{Q_{k}} is Noetherian as a module over k, because k is Noetherian, and (k : I_{0})_{Q_{k}} is finitely generated as a module over k, as in Section 9.7. This implies that

V_{n+1} = V_{n} (16.7.12)

when n is sufficiently large.

If (16.7.12) holds, then x^{n+1} ∈ V_{n}, and x is integral over k. This implies that x ∈ k, because k is integrally closed in Q_{k}, by hypothesis. This means that (16.7.7) holds, as desired. Alternatively, we could get that x is integral over k using (16.7.8) and a criterion for this in Section 16.4, as in Section 16.6. The approach discussed here, from p9 of [263], seems to be more direct in this case.

If k is Noetherian and has dimension one, then statement III on p8 of [263] says that (16.7.6) does not hold. This will be discussed in Section 16.9, after a preliminary result in the next section.

16.8 Powers in the denominator

Let k be an integral domain that is a local ring and not a field again, with unique maximal proper ideal I_{0} ≠ {0}. Also let x ∈ I_{0} with x ≠ 0 be given. Note that S = {x^{n} : n ∈ (Z_{+} ∪ {0})} (16.8.1) is a multiplicatively closed set in k, as in Section 12.6. Of course, 0 ∉ S, because x ≠ 0 and k is an integral domain.

The corresponding ring of fractions S^{−1} k may be identified with a subring of the field Q_{k} of fractions of k, as in Section 14.2. This subring consists of elements of Q_{k} of the form y/x^{n}, where y ∈ k, and n is a nonnegative integer.

Suppose that k has dimension one, in the sense of Section 14.8. This means that I_{0} is the unique nonzero proper prime ideal in k, as in Section 14.11. Let us check that S^{−1} k = Q_{k}, (16.8.2)

as in the first part of the proof of III on p9 of [263].

If (16.8.2) does not hold, then S^{−1} k is not a field. This would imply that there is a nonzero maximal proper ideal I_{1} in S^{−1} k. Observe that

x ∉ I_{1}, (16.8.3)

because x is invertible in S^{−1} k. It follows that

I_{0} ⊈ I_{1} ∩ k. (16.8.4)

Note that I_{1} is a prime ideal in S^{−1} k, which implies that I_{1} ∩ k is a prime ideal in k. Because I_{1} ≠ {0}, there is an element of I_{1} of the form y/x^{n}, where y ∈ k, y ≠ 0, and n is a nonnegative integer. It follows that y ∈ I_{1} ∩ k, so that I_{1} ∩ k ≠ {0}. This contradicts the fact that I_{0} is the unique nonzero proper prime ideal in k.

Alternatively, if z ∈ k and z ≠ 0, then we would like to show that there is a y ∈ k such that y z = x^{n} (16.8.5)

for some nonnegative integer n. This is clear when z is invertible in k, and so we may suppose that z is not invertible in k, so that the ideal I(z) in k generated by z is a proper ideal in k. Remember that the radical r(I(z)) of I(z) in k is equal to the intersection of all of the prime ideals in k that contain I(z), as in Section 12.10. This means that

r(I(z)) = I_{0}, (16.8.6)

because I(z) is a nonzero proper ideal in k, and I_{0} is the unique nonzero proper prime ideal in k. It follows that

x ∈ r(I(z)), (16.8.7)

so that (16.8.5) holds for some y ∈ k and n ≥ 0.

Equivalently, 1/z = y/x^{n} ∈ S^{−1} k. (16.8.8)

This implies (16.8.2). Note that the previous argument for (16.8.2) implies that 1/z can be expresed as in (16.8.8), so that (16.8.5) holds for some y ∈ k and n ≥ 0.

16.9 Using powers in the denominator

Let us return to the same notation and hypotheses as in Section 16.7. Thus k is an integral domain that is a local ring and not a field, with unique maximal proper ideal I_{0} ≠ {0}, as in the previous section as well. We would like to show that if k is Noetherian and has dimension one, then

(k : I_{0})_{Q_{k}} ≠ k, (16.9.1)

as in statement III on p8 of [263]. More precisely, the hypothesis that k be Noetherian is only used here to get that I_{0} be finitely generated as an ideal in k. Remember that k is automatically contained in (k : I_{0})_{Q_{k}}, so that (16.9.1) is the same as saying that (k : I_{0})_{Q_{k}} is not contained in k.

Let z ∈ k with z ≠ 0 be given, and let I(z) be the ideal in k generated by z. If x ∈ I_{0}, then x^{n} ∈ I(z) (16.9.2)

for some positive integer n, as in the previous section. Of course, this uses the hypothesis that k have dimension one. Equivalently, I_{0} is contained in the radical r(I(z)) of I(z). If I_{0} is finitely generated as an ideal in k, then it follows that I_{0}^{m} ⊆ I(z) (16.9.3)

for some positive integer m, as in Section 14.7.

Let us now take z ∈ I_{0}, with z ≠ 0. Remember that I_{0} ≠ {0}, by hypothesis. Let m be the smallest positive integer such that (16.9.3) holds. This implies that I_{0}^{m−1} ⊈ I(z), (16.9.4)

where I_{0}^{0} is interpreted as being equal to k, as usual. Note that I(z) ⊆ I_{0}, so that (16.9.4) holds automatically when m = 1.

Let y be an element of I_{0}^{m−1} that is not in I(z). Observe that

y I_{0} ⊆ I_{0}^{m} ⊆ I(z), (16.9.5)

by (16.9.3). This means that

(y/z) I_{0} ⊆ k, (16.9.6)

so that y/z ∈ (k : I_{0})_{Q_{k}}. (16.9.7)

However, y/z ∉ k, because y ∉ I(z). This implies (16.9.1), as desired.

16.10 Fractions and integral elements

Let k_{1} be a commutative ring with a multiplicative identity element 1 = 1_{k_{1}}, and let k be a subring of k_{1} that contains 1_{k_{1}}. Also let S be a multiplicatively closed set in k, as in Section 12.1. Note that S may be considered as a multiplicatively closed set in k_{1} as well. Thus S^{−1} k and S^{−1} k_{1} may be defined as in Section 12.1.

The obvious inclusion mapping from k into k_{1} leads to a ring homomorphism from S^{−1} k into S^{−1} k_{1} in a natural way, as in Section 12.11. The induced homomorphism from S^{−1} k into S^{−1} k_{1} is injective, because the inclusion mapping from k into k_{1} is injective, as before. Equivalently, if x ∈ k and r ∈ S, then x/r = 0 in S^{−1} k if and only if this holds in S^{−1} k_{1}, which can also be verified directly. Thus we may identify S^{−1} k with a subring of S^{−1} k_{1}.

Suppose that x ∈ k_{1} is integral over k, so that

x^{n} + a_{1} x^{n−1} + ⋯ + a_{n−1} x + a_{n} = 0 (16.10.1)

in k_{1} for some positive integer n and a_{1}, …, a_{n} ∈ k. If r ∈ S, then we get that

(x/r)^{n} + (a_{1}/r) (x/r)^{n−1} + ⋯ + (a_{n−1}/r^{n−1}) (x/r) + (a_{n}/r^{n}) = 0 (16.10.2)

in S^{−1} k. This implies that

(16.10.3) x/r ∈ S^{−1} k_{1} is integral over S^{−1} k.

In particular, if k_{1} is integral over k, then

(16.10.4) S^{−1} k_{1} is integral over S^{−1} k,

as in the second part of Proposition 5.6 on p61 of [8]. This also corresponds to part of the remark on p10 of [263].

Conversely, suppose that (16.10.3) holds for some x ∈ k_{1} and r ∈ S. This means that

(x/r)^{n} + (a_{1}/r_{1}) (x/r)^{n−1} + ⋯ + (a_{n−1}/r_{n−1}) (x/r) + (a_{n}/r_{n}) = 0 (16.10.5)

for some positiove integer n, a_{1}, …, a_{n} ∈ k, and r_{1}, …, r_{n} ∈ S. Put

t = r_{1} r_{2} ⋯ r_{n} ∈ S, (16.10.6)

and observe that (16.10.7) y = x t ∈ k_{1} is integral over k.

Thus x/r = y/(r t), (16.10.8)

with r t ∈ S.

Let C be the integral closure of k in k_{1}, as in Section 16.5, so that C is a subring of k_{1} that contains k. We may identify S^{−1} C with a subring of S^{−1} k_{1} that contains S^{−1} k, as before. Using (16.10.8), we get that

x/r ∈ S^{−1} C. (16.10.9)

Note that S^{−1} C is integral over S^{−1} k, as in (16.10.4). It follows that

(16.10.10) S^{−1} C is the integral closure of S^{−1} k in S^{−1} k_{1}.

This is Proposition 5.12 on p62 of [8]. This corresponds to the remark on p10 of [263] as well.

16.11 More on fractions, integral elements

Let k be an integral domain, let Q_{k} be the corresponding field of fractions, and let C be the integral closure of k in Q_{k}. Also let S be a multiplicatively closed set in k, with 0 ∉ S, so that S^{−1} k may be identified with a subring of Q_{k}, as in Section 14.2. Similarly, S^{−1} C may be identified with a subring of Q_{k}, which is the integral closure of S^{−1} k in Q_{k}, as in (16.10.10). Of course, S^{−1} Q_{k} may be identified with Q_{k}, and Q_{k} may be identified with the field of fractions of S^{−1} k.

In particular,

(16.11.1) if k is integrally closed, then S^{−1} k is integrally closed.

This reduces to the fact that S^{−1} C = S^{−1} k when C = k. This corresponds to part of the proof of Proposition 5.13 on p63 of [8].

Let ϕ be the obvious inclusion mapping from k into C, and let Φ be the obvious inclusion mapping from S^{−1} k into S^{−1} C. These are ring homomorphisms, and one may consider Φ as corresponding to ϕ as in Section 12.11.

Let W be C, considered as a module over k. We can define S^{−1} W as a module over S^{−1} k, as in Section 12.2. We may also consider S^{−1} C as a module over S^{−1} k, because S^{−1} k is a subring of S^{−1} C. This corresponds exactly to S^{−1} W, as a module over S^{−1} k, as in Section 12.11.

Similarly, let V be k, considered as a module over itself. We can define S^{−1} V as a module over S^{−1} k as in Section 12.2 again. This corresponds exactly to S^{−1} k as a module over itself, as before.

Let f be the obvious inclusion mapping from V into W, as modules over k. This leads to a homomorphism S^{−1} f from S^{−1} V into S^{−1} W, as modules over S^{−1} k, as in Section 12.2. It is easy to see that S^{−1} f corresponds exactly to Φ, with respect to the identifications of S^{−1} V and S^{−1} W with S^{−1} k and S^{−1} C, respectively, mentioned in the previous two paragraphs.

If I is a proper prime ideal in k, then S_{I} = k \ I is a multiplicatively closed subset of k that does not contain 0, as in Section 12.4. In particular, this holds when I is a proper maximal ideal in I.

Suppose that (16.11.2) S_{I}^{−1} k is integrally closed

for every maximal proper ideal I in k. This is the same as saying that

S_{I}^{−1} k = S_{I}^{−1} C (16.11.3)

for every maximal proper ideal I in k, because S_{I}^{−1} C corresponds to the integral closure of S_{I}^{−1} k in Q_{k}, as before. Equivalently, this means that S_{I}^{−1} f maps S_{I}^{−1} V onto S_{I}^{−1} W for every maximal proper ideal I in k, by the earlier remarks.

Under these conditions, f maps V onto W, as in Section 12.4. This means that k = C, so that k is integrally closed. This is part of Proposition 5.13 on p63 of [8].

Essentially the same point is addressed in the proof that (i) implies (ii) in Proposition 4 on p10 of [263]. Suppose that (16.11.2) holds for every maximal proper ideal I in k again, and let w ∈ C be given. This implies that

w ∈ S_{I}^{−1} C = S_{I}^{−1} k (16.11.4)

for every maximal proper ideal I in k.

Observe that {x ∈ k : x w ∈ k} (16.11.5)

is an ideal in k. If I is a maximal proper ideal in k, then (16.11.4) implies that there is an element of k \ I in (16.11.5). This means that (16.11.5) is not contained in I. It follows that (16.11.5) is equal to k, so that w ∈ k.

16.12 Dedekind domains and integral elements

Let k be a Noetherian integral domain. If I is a proper prime ideal in k, then S_{I} = k \ {0} is a multiplicatively closed set in k that does not contain 0, as before. Remember that k is a Dedekind domain if for every maximal proper ideal I ≠ {0} in k,

(16.12.1) k_{I} = S_{I}^{−1} k is a discrete valuation ring,

as in Section 15.11. This implies that k has dimension less than or equal to one, in the sense of Section 14.8, as before. More precisely, the formulation on p95 of [8] also asks that k not be a field, which would imply that k have dimension equal to 0.

Note that (16.12.1) implies that

(16.12.2) k_{I} is integrally closed,

as in Section 16.3. Of course, I = {0} is a maximal proper ideal in k if and only if k is a field, in which case k is obviously integrally closed. If k is a Dedekind domain, then it follows that k is integrally closed, as in the previous section. This corresponds to parts of Theorem 9.3 on p95 of [8], and Proposition 4 on p10 of [263].

Conversely, suppose that k has dimension less than or equal to one, and that k is integrally closed. Let I ≠ {0} be a maximal proper ideal in k, and observe that (16.12.2) holds, as in (16.11.1). We also have that k_{I} is a Noetherian integral domain, as in Sections 14.2 and 14.6. Remember that k_{I} is a local ring too, as in Section 12.4. It is easy to see that the unique maximal proper ideal in k_{I} is nonzero, because I ≠ {0}.

Under these conditions, k is not a field, because I ≠ {0}, so that k has dimension one. Remember that there is a one-to-one correspondence between the set of prime ideals in k that are contained in I and the set of proper prime ideals in k_{I}, as mentioned at the end of Section 12.13. It follows that k_{I} has dimension one as well.

Observe that (16.12.2) holds, because k is integrally closed, by hypothesis, as in (16.11.1). It follows that (16.12.1) holds, as in Section 16.6. This means that k is a Dedekind domain, as in Section 15.11. This corresponds to other parts of Theorem 9.3 on p95 of [8], and Proposition 4 on p10 of [263].

16.13 Fields, quotients, and integral elements

Let k_{1}, k_{2} be commutative rings with multiplicative identity elements 1_{k_{1}}, 1_{k_{2}}, respectively, and let k be a subring of k_{1} that contains 1_{k_{1}}. Also let ϕ be a ring homomorphism from k_{1} onto k_{2}, so that ϕ(1_{k_{1}}) = 1_{k_{2}}, and ϕ(k) is a subring of k_{2} that contains 1_{k_{2}}. If x ∈ k_{1} is integral over k, then it is easy to see that

(16.13.1) ϕ(x) is integral over ϕ(k) in k_{2}.

If k_{1} is integral over k, then it follows that

(16.13.2) k_{2} is integral over ϕ(k).

This corresponds to the first part of Proposition 5.6 on p61 of [8].

Suppose now that k_{1} is an integral domain, and that k is a subring of k that contains 1_{k_{1}} again. If k_{1} is integral over k, then

(16.13.3) k is a field if and only if k_{1} is a field.

This is Proposition 5.7 on p61 of [8].

Suppose first that k is a field, and let y ∈ k_{1} be given, with y ≠ 0. Because y is integral over k, by hypothesis, we have that

y^{n} + a_{1} y^{n−1} + ⋯ + a_{n−1} y + a_{n} = 0 (16.13.4)

for some positive integer n and a_{1}, …, a_{n} ∈ k. This implies that

y (y^{n−1} + a_{1} y^{n−2} + ⋯ + a_{n−1}) = −a_{n}. (16.13.5)

We may as well take n to be as small as possible, so that

y^{n−1} + a_{1} y^{n−2} + ⋯ + a_{n−1} ≠ 0. (16.13.6)

This means that a_{n} ≠ 0, because k_{1} is an integral domain, by hypothesis. It follows that a_{n} is invertible in k, and thus in k_{1}, because k is a field. This implies that y is invertible in k_{1}, with

y^{−1} = −a_{n}^{−1} (y^{n−1} + a_{1} y^{n−2} + ⋯ + a_{n−1}), (16.13.7)

as desired.

Conversely, suppose that k_{1} is a field, and let x ∈ k be given, with x ≠ 0. Under these conditions, x is invertible in k_{1}, and x^{−1} is integral over k. This means that x^{−m} + c_{1} x^{1−m} + ⋯ + c_{m−1} x^{−1} + c_{m} = 0 (16.13.8) for some positive integer m and c_{1}, …, c_{m} ∈ k. We can multiply both side by x^{m−1}, to get that

x^{−1} = −c_{1} − ⋯ − c_{m−1} x^{m−2} − c_{m} x^{m−1}. (16.13.9)

This implies that x^{−1} ∈ k, as desired.

16.14 Two corollaries, and another argument

Let k_{1} be a commutative ring with a multiplicative identity element 1 = 1_{k_{1}}, let k be a subring of k that contains 1_{k_{1}}, and suppose that k_{1} is integral over k. Suppose for the moment that I_{1} is a proper prime ideal in k_{1}, and note that

I = I_{1} ∩ k (16.14.1)

is a proper prime ideal in k. Under these conditions,

(16.14.2) I_{1} is a maximal proper ideal in k_{1} if and only if I is a maximal proper ideal in k,

as in Corollary 5.8 on p61 of [8].

To see this, put k_{2} = k_{1}/I_{1}, and let ϕ be the natural quotient homomorphism from k_{1} onto k_{2}. Of course, ϕ(k) may be identified with k/I, and k_{2} is an integral domain. We also have that k_{2} is integral over ϕ(k), because k_{1} is integral over k, as in (16.13.2). It follows that k_{2} is a field if and only if ϕ(k) is a field, as in (16.13.3). This is equivalent to (16.14.2), as desired.

Suppose now that I_{1}, I′_{1} are proper prime ideals in k_{1} such that

I_{1} ⊆ I′_{1}. (16.14.3)

If I_{1} ∩ k = I′_{1} ∩ k, (16.14.4)

then I_{1} = I′_{1}. (16.14.5)

This is Corollary 5.9 on p61 of [8], which corresponds to Lemma 2 on p14 of [263].

Let us begin with the proof in [8]. Let I be as in (16.14.4), which is a proper prime ideal in k. Thus S = k \ I is a multiplicatively closed set in k, as in Section 12.4. We may also consider S as a multiplicatively closed set in k_{1}, so that S^{−1} k and S^{−1} k_{1} may be defined as commutative rings as in Section 12.1. We may identify S^{−1} k with a subring of S^{−1} k_{1}, as in Section 16.10.

Remember that S^{−1} I may be considered as an ideal in S^{−1} k, and similarly S^{−1} I_{1}, S^{−1} I′_{1} may be considered as ideals in S^{−1} k_{1}, as in Section 12.13. We have seen that S^{−1} I is the same as the extension of I with respect to the natural ring homomorphism x ↦ x/1 from k into S^{−1} k, and similarly S^{−1} I_{1}, S^{−1} I′_{1} are the same as the extensions of I_{1}, I′_{1} with respect to the natural ring homomorphism from k_{1} in S^{−1} k_{1}, respectively. We also have that S^{−1} k is a local ring, with S^{−1} I as its unique maximal proper ideal, as in Section 12.4.

Of course, S^{−1} I_{1} ⊆ S^{−1} I′_{1}, (16.14.6)

because of (16.14.3). One can check that

(S^{−1} I_{1}) ∩ (S^{−1} k) = S^{−1} I, (16.14.7)

because I = I_{1} ∩ k, by construction. More precisely, it is easy to see that

S^{−1} I ⊆ (S^{−1} I_{1}) ∩ (S^{−1} k), (16.14.8)

and one can verify that

(S^{−1} I_{1}) ∩ (S^{−1} k) ⊆ S^{−1} I (16.14.9)

too. Similarly, (S^{−1} I′_{1}) ∩ (S^{−1} k) = S^{−1} I. (16.14.10)

Note that I_{1} and I′_{1} are disjoint from S, because I is as in (16.14.4). This implies that S^{−1} I_{1}, S^{−1} I′_{1} ≠ S^{−1} k_{1}, as in Section 12.13. In fact, S^{−1} I_{1} and S^{−1} I′_{1} are proper prime ideals in S^{−1} k_{1}, as before.

We also have that S^{−1} k_{1} is integral over S^{−1} k, because k_{1} is integral over k, by hypothesis, as in Section 16.10. It follows that S^{−1} I_{1}, S^{−1} I′_{1} are maximal proper ideals in S^{−1} k_{1}, because S^{−1} I is a maximal proper ideal in S^{−1} k, as in (16.14.2). This implies that

S^{−1} I_{1} = S^{−1} I′_{1}, (16.14.11)

by (16.14.6). We can get (16.14.5) from (16.14.11), because I_{1} and I′_{1} are prime ideals in k_{1} that are disjoint from S, as in Section 12.13.

Alternatively, let ϕ be the natural quotient homomorphism from k_{1} onto k_{2} = k_{1}/I_{1} again. Thus k_{2} is an integral domain that it integral over ϕ(k), as before. Suppose for the sake of a contradiction that (16.14.5) does not hold, and let x be an element of I′_{1} that is not in I_{1}, so that ϕ(x) ≠ 0. Note that ϕ(x) is integral over ϕ(x), which means that

ϕ(x)^{n} + a_{1} ϕ(x)^{n−1} + ⋯ + a_{n−1} ϕ(x) + a_{n} = 0 (16.14.12)

for some positive integer n and a_{1}, …, a_{n} ∈ ϕ(k). If we take n to be as small as possible, then we get that a_{n} ≠ 0, because k_{2} is an integral domain, as in the previous section.

Of course, ϕ(I′_{1}) is an ideal in k_{2}. It is easy to see that a_{n} ∈ ϕ(I′_{1}), because of (16.14.12). Thus there are y ∈ k and z ∈ I′_{1} such that

a_{n} = ϕ(y) = ϕ(z). (16.14.13)

In particular, ϕ(y − z) = 0, so that y − z ∈ I_{1}. It follows that y ∈ I′_{1}, because of (16.14.3). This means that y ∈ I′_{1} ∩ k, so that y ∈ I_{1} ∩ k, by (16.14.4). This implies that ϕ(y) = 0, contradicting the fact that a_{n} ≠ 0. This corresponds to the proof of Lemma 2 on p14 of [263].

16.15 Getting prime ideals in k_{1}

Let k_{1} be a commutative ring with a multiplicative identity element 1 = 1_{k_{1}} again, let k be a subring of k that contains 1_{k_{1}}, and suppose that k_{1} is integral over k. If I is a proper prime ideal in k, then there is a proper prime ideal I_{1} in k_{1} such that I_{1} ∩ k = I. (16.15.1)

This is Theorem 5.10 on p62 of [8].

As in the previous section, S = k \ I is a multiplicatively closed set in k, which may be considered as a multiplicatively closed set in k_{1} as well. Thus S^{−1} k and S^{−1} k_{1} may be defined as commutative rings, and we may identify S^{−1}k with a subring of S^{−1} k_{1}, as before. We also have that S^{−1} k_{1} is integral over S^{−1} k, because k_{1} is integral over k, as in Section 16.10.

Let α, α_{1} be the natural ring homomorphisms from k, k_{1} into S^{−1} k, S^{−1} k_{1}, respectively. The composition of α with the inclusion mapping from S^{−1} k into S^{−1} k_{1} is equal to the composition of the obvious inclusion mapping from k into k_{1} with α_{1}, as on p62 of [8]. This corresponds to some remarks in Sections 12.11 and 16.10.

Let I_{2} be a maximal proper ideal in S^{−1} k_{1}. Observe that

I_{2} ∩ (S^{−1} k) (16.15.2)

is a maximal proper ideal in S^{−1} k, as in (16.14.2). Remember that S^{−1} k is a local ring, with S^{−1} I as its unique maximal proper ideal, as in Section 12.4. Thus I_{2} ∩ (S^{−1} k) = S^{−1} I. (16.15.3)

We also have that I_{1} = α_{1}^{−1}(I_{2}) (16.15.4)

is a proper prime ideal in k_{1} because I_{2} is a proper prime ideal in S^{−1} k_{1}, as in Section 12.13. More precisely,

I_{1} ∩ S = ∅, (16.15.5)

as before. This means that I_{1} ∩ k is disjoint from S, so that

I_{1} ∩ k ⊆ I. (16.15.6)

In fact,

I_{1} ∩ k = α_{1}^{−1}(I_{2}) ∩ k = α^{−1}(I_{2} ∩ (S^{−1} k)) = α^{−1}(S^{−1} I), (16.15.7)

using the earlier remarks about α and α_{1} in the second step, and (16.15.3) in the third step. This implies that I ⊆ I_{1} ∩ k, so that (16.15.1) holds.

Suppose now that I_{k,1} ⊆ I_{k,2} ⊆ ⋯ ⊆ I_{k,n} (16.15.8)

16.15. GETTING PRIME IDEALS IN K_{1} 415

is a chain of proper prime ideals in k for some positive integer n,

I_{k_{1},1} ⊆ I_{k_{1},2} ⊆ ⋯ ⊆ I_{k_{1},m} (16.15.9)

is a chain of proper prime ideals in k_{1} for some positive integer m < n, and

I_{k_{1},j} ∩ k = I_{k,j} (16.15.10)

for each j = 1, …, m. Under these conditions, (16.15.9) can be extended to a chain I_{k_{1},1} ⊆ I_{k_{1},2} ⊆ ⋯ ⊆ I_{k_{1},n} (16.15.11)

of proper prime ideals in k_{1} that satisfies (16.15.10) for each j = 1, …, n. This is Theorem 5.11 on p62 of [8], which is called the “going-up theorem”.

To see this, we can use induction to reduce to the case where m = 1 and n = 2. Put k_{2} = k_{1}/I_{k_{1},1}, (16.15.12)

and let ϕ be the natural quotient homomorphism from k_{1} onto k_{2}. Note that ϕ(k) is isomorphic to k/I_{k,1} as a ring, and that k_{2} is integral over ϕ(k), because k_{1} is integral over k, as in Section 16.13. It is easy to see that ϕ(I_{k,2}) is a proper prime ideal in ϕ(k), because I_{k,2} is a proper prime ideal in k, and I_{k,1} is contained in I_{k,2}.

It follows that there is a proper prime ideal I_{k_{2},2} in k_{2} such that

I_{k_{2},2} ∩ ϕ(k) = ϕ(I_{k,2}), (16.15.13)

as in (16.15.1). Put I_{k_{1},2} = ϕ^{−1}(I_{k_{2},2}), (16.15.14)

which is a proper prime ideal in k_{1} that contains I_{k_{1},1}. We also have that

(16.15.15) I_{k_{1},2} ∩ k = ϕ^{−1}(I_{k_{2},2}) ∩ k = ϕ^{−1}(I_{k_{2},2} ∩ ϕ(k)) ∩ k
= ϕ^{−1}(ϕ(I_{k,2})) ∩ k = I_{k,2},

using the fact that I_{k,1} is contained in I_{k,2} in the last step.

Part V

Rings, modules, and field extensions Some properties of ideals

17.1 Sums, products, and intersections

Let k be a commutative ring with a multiplicative identity element, and let I_{1}, I_{2}, and I_{3} be ideals in k. Of course, I_{1} + I_{2} = {x_{1} + x_{2} : x_{1} ∈ I_{1}, x_{2} ∈ I_{2}} (17.1.1) is an ideal in k as well.

Remember that the product I_{1} I_{2} is the ideal in k generated by products of elements of I_{1} and I_{2}, as in Section 12.8. It is easy to see that the distributive law I_{1} (I_{2} + I_{3}) = I_{1} I_{2} + I_{1} I_{3} (17.1.2) holds, as on p6 of [8].

Note that I_{1} ∩ I_{2} + I_{1} ∩ I_{3} ⊆ I_{1} ∩ (I_{2} + I_{3}). (17.1.3) If I_{2} ⊆ I_{1} or I_{3} ⊆ I_{1}, (17.1.4) then one can check that I_{1} ∩ (I_{2} + I_{3}) = I_{1} ∩ I_{2} + I_{1} ∩ I_{3}. (17.1.5) This is the modular law mentioned on p6 of [8]. This corresponds to the modular identity in a lattice, as in Exercise 10 on p376 of [24], and L5 on p479 of [195].

We also have that (I_{1} + I_{2}) (I_{1} ∩ I_{2}) = I_{1} (I_{1} ∩ I_{2}) + I_{2} (I_{1} ∩ I_{2}) ⊆ I_{1} I_{2}, (17.1.6) as on p6 of [8]. Remember that I_{1} I_{2} ⊆ I_{1} ∩ I_{2}, as in Section 12.8. If I_{1} + I_{2} = k, (17.1.7) then we get that I_{1} ∩ I_{2} = I_{1} I_{2}, (17.1.8)

as on p6 of [8].

If (17.1.7) holds, then I_{1} and I_{2} are said to be coprime or comaximal as ideals in k, as on p7 of [8]. Equivalently, this means that there are x_{1} ∈ I_{1} and x_{2} ∈ I_{2} such that x_{1} + x_{2} = 1. (17.1.9)

Remember that the radical r(I) of an ideal I in k is defined in Section 12.10, and that I ⊆ r(I) automatically. If I_{1} and I_{2} are coprime in k, then it follows that their radicals r(I_{1}) and r(I_{2}) are coprime in k as well.

Conversely, if r(I_{1}) and r(I_{2}) are coprime in k, then I_{1} and I_{2} are coprime in k, as in Proposition 1.16 on p9 of [8]. To see this, remember that

r(I_{1} + I_{2}) = r(r(I_{1}) + r(I_{2})), (17.1.10)

as in Section 12.10. If r(I_{1}) and r(I_{2}) are coprime in k, then we get that

r(I_{1} + I_{2}) = k. (17.1.11)

This implies (17.1.7), as in Section 12.10.

17.2 Finitely many ideals

Let k be a commutative ring with a multiplicative identity element, and let I_{1}, …, I_{n} be finitely many ideals in k. If

(17.2.1) I_{j} and I_{l} are coprime when j ≠ l,

then

∏_{j=1}^{n} I_{j} = ⋂_{j=1}^{n} I_{j}, (17.2.2)

as in the first part of Proposition 1.10 on p7 of [8]. Of course, then n = 2 case corresponds to (17.1.8).

Suppose that n > 2, and that the analogous statement holds for n − 1. Put

I = ∏_{j=1}^{n−1} I_{j} = ⋂_{j=1}^{n−1} I_{j}. (17.2.3)

Because I_{j} + I_{n} = k (17.2.4)

for each j = 1, …, n − 1, by (17.2.1), there are x_{j} ∈ I_{j} and y_{j} ∈ I_{n} for each j = 1, …, n − 1 such that x_{j} + y_{j} = 1. (17.2.5)

This implies that

(17.2.6)  ∏_{j=1}^{n−1} x_{j} = ∏_{j=1}^{n−1}(1 − y_{j}) = 1 + an element of I_{n}.

Note that the left side is an element of I.

It follows that I + I_{n} = k, (17.2.7) as in the previous section. Thus

∏_{j=1}^{n} I_{j} = I I_{n} = I ∩ I_{n} = ⋂_{j=1}^{n} I_{j}, (17.2.8)

using the n = 2 case in the second step.

Let q_{j} be the natural homomorphism from k onto the quotient ring k/I_{j} for each j = 1, …, n. Using these homomorphisms, we get a ring homomorphism ϕ from k into the Cartesian product

∏_{j=1}^{n}(k/I_{j}). (17.2.9)

The second part of Proposition 1.10 on p7 of [8] says that

(17.2.10) ϕ is surjective

if and only if (17.2.1) holds.

If (17.2.10) holds, then for each j = 1, …, n there is an x_{j} ∈ k such that q_{j}(x_{j}) = 1 in k/I_{j}, and q_{l}(x_{j}) = 0 in k/I_{l} when j ≠ l. This means that 1 − x_{j} ∈ I_{j}, and x_{l} ∈ I_{l} when j ≠ l. It follows that

1 ∈ I_{j} + I_{l} (17.2.11)

when j ≠ l, so that (17.2.1) holds.

Conversely, suppose that (17.2.1) holds. If j ≠ l, then there are u_{j,l} ∈ I_{j} and v_{j,l} ∈ I_{l} such that u_{j,l} + v_{j,l} = 1, (17.2.12) because I_{j} + I_{l} = k. Let x_{j} be the product of v_{j,l} over l = 1, …, n with j ≠ l. This is the same as the product of (1 − u_{j,l}) over l = 1, …, n with j ≠ l. It follows that (17.2.13) x_{j} = 1 + an element of I_{j} for each j = 1, …, n, so that q_{j}(x_{j}) = 1 in k/I_{j}. We also have that x_{j} ∈ I_{l} when j ≠ l, so that q_{l}(x_{j}) = 0. One can use this to get that (17.2.10) holds, as desired.

Of course,

ker ϕ = ⋂_{j=1}^{n} I_{n}, (17.2.14)

by construction. Thus ϕ is injective if and only if ⋂_{j=1}^{n} I_{j} = {0}, as in the third part of Proposition 1.10 on p7 of [8].

17.3 Finite unions and intersections

Let k be a commutative ring with a multiplicative identity element, and let I_{1}, …, I_{n} be finitely many prime ideals in k. If I is an ideal in k such that

I ⊆ ⋃_{j=1}^{n} I_{j}, (17.3.1)

then I ⊆ I_{j} (17.3.2)

for some j. This is the first part of Proposition 1.11 on p8 of [8].

Equivalently, suppose that I ⊈ I_{j} (17.3.3)

for each j = 1, …, n. Under these conditions, we would like to show that

I ⊈ ⋃_{j=1}^{n} I_{j}. (17.3.4)

Of course, this is clear when n = 1. Suppose now that n > 1, and that the analogous statement holds for n − 1.

Using the induction hypothesis, we get that for each l = 1, …, n there is an x_{l} ∈ I such that x_{l} ∉ I_{j} (17.3.5)

when j ≠ l. If x_{l} ∉ I_{l} for some l, then (17.3.3) holds, as desired. Otherwise, we have that x_{l} ∈ I_{l} (17.3.6)

for each l = 1, …, n.

Put y_{l} = x_{1} x_{2} ⋯ x_{l−2} x_{l−1} x_{l+1} x_{l+2} ⋯ x_{n}, (17.3.7)

for each l = 1, …, n, which is to say the product of the x_{j}’s, 1 ≤ j ≤ n, except for j = l. Observe that y_{l} ∈ I, and that y_{l} ∈ I_{j} when j ≠ l, because of (17.3.6). We also have that y_{l} ∉ I_{l}, because of (17.3.5), and because I_{l} is a prime ideal.

It follows that

y = ∑_{l=1}^{n} y_{l} (17.3.8)

is an element of I that is not in I_{r} for any r = 1, …, n. This implies (17.3.4), as desired.

Now let I_{1}, …, I_{n} be finitely many ideals in k, and let I be a prime ideal in k. If

⋂_{j=1}^{n} I_{j} ⊆ I, (17.3.9)

then I_{l} ⊆ I (17.3.10)

for some l. In particular, if

I = ⋂_{j=1}^{n} I_{j}, (17.3.11)

then I = I_{l} (17.3.12)

for some l. This is the second part of Proposition 1.11 on p8 of [8].

To see this, suppose that I_{l} ⊈ I for any l. Thus, for each l = 1, …, n, there is an x_{l} ∈ I_{l} such that x_{l} ∉ I. This implies that

∏_{l=1}^{n} x_{l} ∈ ∏_{l=1}^{n} I_{l} ⊆ ⋂_{j=1}^{n} I_{j}, (17.3.13)

and that

∏_{l=1}^{n} x_{l} ∉ I, (17.3.14)

because I is a prime ideal. It follows that (17.3.9) does not hold.

If (17.3.11) holds, then (17.3.10) implies (17.3.12), because I ⊆ I_{l} automatically.

17.4 Irreducible ideals

Let k be a commutative ring with a multiplicative identity element, and let I be an ideal in k. Suppose that if I_{1}, I_{2} are ideals in k such that

I = I_{1} ∩ I_{2}, (17.4.1)

then I = I_{1} or I_{2}. (17.4.2)

Under these conditions, I is said to be irreducible as an ideal in k. Note that prime ideals in k are irreducible, as in the previous section.

If k is Noetherian, then

(17.4.3) every ideal in k may be expressed as the intersection of finitely many irreducible ideals in k,

as in Lemma 7.11 on p83 of [8]. Indeed, suppose for the sake of a contradiction that there are ideals in k that cannot be expressed as the intersection of finitely many irreducible ideals. Because k is Noetherian, there is an ideal I in k that cannot be expressed as the intersection of finitely many irreducible ideals in k, and which is maximal with respect to this property. In particular, I is reducible in k, in the sense that there are ideals I_{1} and I_{2} in k such that (17.4.1) holds, and (17.4.2) does not hold. It follows that I is a proper subset of each of I_{1} and I_{2}.

The maximality of I implies that each of I_{1} and I_{2} may be expressed as the intersection of finitely many irreducible ideals in k. This means that I may be expressed as the intersection of finitely many irreducible ideals in k, because of (17.4.1). This contradicts the hypothesis that I cannot be expressed in this way, as desired.

If I is an irreducible ideal in k, I ≠ k, and k is Noetherian, then

(17.4.4) I is a primary ideal in k,

as in Lemma 7.12 on p83 of [8]. Remember that I is a primary ideal in k if and only if k/I ≠ {0}, and every zero-divisor in k/I is nilpotent, as in Section 14.5. It is easy to see that I is irreducible in k if and only if {0} is irreducible in k/I.

If k is Noetherian, then k/I is Noetherian too, as in Proposition 6.6 on p76 of [8]. More precisely, if k is Noetherian as a commutative ring, then k/I is Noetherian as a module over k, as in Section 9.7. It is easy to see that this implies that k/I is Noetherian as a commutative ring, because any ideal in k/I may be considered as a submodule of k/I, as a module over k.

This permits us to reduce the earlier statement to the case where I = {0}, as in [8]. Thus we would like to show that if {0} is irreducible in k, k ≠ {0}, and k is Noetherian, then every zero-divisor in k is nilpotent.

Let x ∈ k be a zero-divisor, so that x y = 0 for some y ∈ k with y ≠ 0. If w ∈ k, then let I(w) be the ideal in k generated by w. Remember that the annihilator Ann(I(w)) of I(w) in k as defined in Section 12.8, and is an ideal in k. Note that Ann(I(w)) = {t ∈ k : t w = 0} (17.4.5)

for each ∈ k. It is easy to see that

Ann(I(x^{n})) ⊆ Ann(I(x^{n+1})) (17.4.6)

for every n ≥ 1.

If k is Noetherian, then there is a positive integer n_{0} such that

Ann(I(x^{n})) = Ann(I(x^{n_{0}})) (17.4.7)

for every n ≥ n_{0}. Let us check that

I(x^{n_{0}}) ∩ I(y) = {0} (17.4.8)

in this case. If z ∈ I(y), then x z = 0, because x y = 0, by hypothesis. Suppose that z ∈ I(x^{n_{0}}) too, so that z = t x^{n_{0}} for some t ∈ k. Under these conditions, we get that t x^{n_{0}+1} = x z = 0, (17.4.9)

so that t ∈ Ann(I(x^{n_{0}+1})). This implies that t ∈ Ann(I(x^{n_{0}})), by (17.4.7). It follows that z = t x^{n_{0}} = 0, so that (17.4.8) holds.

If {0} is irreducible as an ideal in k, then (17.4.8) implies that

I(x^{n_{0}}) = {0}, (17.4.10)

because y ≠ 0, by hypothesis. This means that x^{n_{0}} = 0, so that x is nilpotent, as desired.

17.5 More on primary ideals

Let k be a commutative ring with a multiplicative identity element. Remember that an ideal I ≠ k in k is said to be primary if for every x, y ∈ k with x y ∈ I, we have that x ∈ I, or y^{n} ∈ I for some positive integer n, as in Section 14.5. The radical r(I) of I in k was defined in Section 12.10, and we have seen that r(I) = k if and only if I = k. If I is a primary ideal in k, then r(I) is a proper prime ideal in k, as in Section 14.5. In this case, if I_{0} = r(I), then I is said to be I_{0}-primary, as before.

Let I_{0} be a proper prime ideal in k, and let I_{1}, …, I_{n} be finitely many I_{0}-primary ideals in k. Under these conditions,

(17.5.1) I = ⋂_{j=1}^{n} I_{j} is I_{0}-primary in k,

as in Lemma 4.3 on p51 of [8]. To see this, observe first that

r(I) = ⋂_{j=1}^{n} r(I_{j}) = I_{0}, (17.5.2)

where the first step is as in Section 12.10.

Suppose that x, y ∈ k, x y ∈ I, and y ∉ I. This implies that y ∉ I_{j} for some j, while x y ∈ I_{j}. It follows that x^{n} ∈ I_{j} for some positive integer n, because I_{j} is primary. This means that x ∈ r(I_{j}) = I_{0}. It follows that x^{m} ∈ I for some positive integer m, because of (17.5.2).

Let I_{0} be a proper prime ideal in k again, let I be a I_{0}-primary ideal in k, and let x ∈ k be given. Remember that

(I : x) = {t ∈ k : t x ∈ I} (17.5.3)

is an ideal in k, as in Section 12.8. Note that

I ⊆ (I : x), (17.5.4)

as before. Some properties of (I : x) are mentioned in Lemma 4.4 on p51 of [8], as follows.

Of course, if x ∈ I, then (I : x) = k. (17.5.5)

This means that r((I : x)) = k. (17.5.6)

More precisely, this works for any ideal I in k.

If x ∉ I, then (17.5.7) (I : x) is I_{0}-primary in k.

Let us first check that (I : x) ⊆ I_{0} (17.5.8)

in this case. If y ∈ (I : x), so that x y ∈ I, then y^{n} ∈ I for some positive integer n, because x ∉ I. This means that y ∈ r(I) = I_{0}, as desired. It follows from this and (17.5.4) that r((I : x)) = r(I) = I_{0}, (17.5.9)

because r(I_{0}) = I_{0}, as in Section 12.10.

Suppose that y, z ∈ k and y z ∈ (I : x), and let us show that z ∈ (I : x) or y^{n} ∈ (I : x) for some positive integer n. The second possibility is the same as saying that y is an element of (17.5.9). Suppose that y ∉ I_{0}, and let us verify that z ∈ (I : x). The hypothesis that y z ∈ (I : x) means that x z y ∈ I. This implies that x z ∈ I, because I is I_{0}-primary, and y ∉ I_{0}. Thus z ∈ (I : x). It follows that (I : x) is a primary ideal in k.

If x ∉ I_{0}, then (I : x) = I. (17.5.10)

More precisely, one can check that (I : x) ⊆ I in this case, because I is I_{0}- primary. Note that (17.5.10) implies (17.5.9).

17.6 Primary decompositions

Let k be a commutative ring with a multiplicative identity element, and let I be a proper ideal in k. A primary decomposition of I is an expression of I as the intersection of finitely many primary ideals I_{1}, …, I_{n} in k,

I = ⋂_{j=1}^{n} I_{j}, (17.6.1)

as on p51 of [8]. If I has a primary decomposition, then one may say that I is decomposable, as on p52 of [8].

If k is Noetherian, then every proper ideal in k is decomposable, as in Section 17.4. This corresponds to Theorem 7.13 on p83 of [8]. One might consider I = k as having a primary decomposition, with n = 0 in (17.6.1).

Let I be a proper ideal in k with a primary decomposition as in (17.6.1). One can reduce to the case where

(17.6.2) r(I_{1}), …, r(I_{n}) are distinct,

as mentioned on p52 of [8]. More precisely, if some of the I_{j}’s have the same radical, then their intersection is a primary ideal in k with the same radical, as in the previous section.

One can also reduce to the case where for each l = 1, …, n,

⋂_{j≠l} I_{j} ⊈ I_{l}, (17.6.3)

by simply dropping any of the primary ideals that are not needed to get (17.6.1), as on p52 of [8]. A primary decomposition (17.6.1) that satisfies (17.6.2) and (17.6.3) is said to be minimal, irredundant, reduced, or normal, etc., as in [8].

Let I be a proper ideal in k with primary decomposition as in (17.6.1) again. Also let x ∈ k be given, so that (I : x) may be defined as in (17.5.3), and similarly for (I_{j} : x). Observe that

(I : x) = ⋂_{j=1}^{n}(I_{j} : x). (17.6.4)

This implies that

r((I : x)) = ⋂_{j=1}^{n} r((I_{j} : x)), (17.6.5)

as in Section 12.10.

If x ∈ I_{j} for some j, then (I_{j} : x) = k, so that r((I_{j} : x)) = k, as in the previous section. Similarly, if x ∈ I, then (I : x) = k, and r((I : x)) = k. If x ∉ I, then

(17.6.6)  r((I : x)) = ⋂{r((I_{j} : x)) : 1 ≤ j ≤ n, x ∉ I_{j}}
= ⋂{r(I_{j}) : 1 ≤ j ≤ n, x ∉ I_{j}},

using (17.6.5) in the first step, and (17.5.9) in the second step.

If r((I : x)) is a proper prime ideal in k, then it follows that x ∉ I, and that

r((I : x)) = r(I_{l}) for some l, (17.6.7)

1 ≤ l ≤ n, as in Section 17.3. Conversely, if 1 ≤ l ≤ n and (17.6.3) holds, then there is an element x_{l} of

( ⋂_{j≠l} I_{j}) \ I_{l}. (17.6.8)

Under these conditions, we get that

r((I : x_{l})) = ⋂_{j=1}^{n} r((I_{j} : x)) = r(I_{l}), (17.6.9)

by (17.6.6).

If (17.6.1) is a minimal primary decomposition of I, then we get that

r(I_{1}), …, r(I_{n}) (17.6.10) are the same as the proper prime ideals in k of the form r((I : x)) for some x ∈ k. In particular, the collection of these proper prime ideals in k does not depend on the minimal primary decomposition (17.6.1) of I. This is Theorem 4.5 on p52 of [8], which is the first uniqueness theorem for minimal primary decompositions.

If x_{l} is an element of (17.6.8), then

(I : x_{l}) = (I_{l} : x_{l}), (17.6.11)

by (17.6.4), and because (I_{j} : x_{l}) = k when j ≠ l, as before. Note that (17.6.11) is r(I_{l})-primary in k, as in (17.5.7). This corresponds to the first remark on p52 of [8].

17.7 More on primary decompositions

Let k be a commutative ring with a multiplicative identity element, let I be a proper ideal in k, and suppose that (17.6.1) is a minimal primary decomposition of I in k. The prime ideals r(I_{j}), 1 ≤ j ≤ n, are said to belong to I or be associated with I, as on p52 of [8]. Note that

I ⊆ I_{l} ⊆ r(I_{l}) (17.7.1)

for each l = 1, …, n.

The prime ideals among those in (17.6.10) that are minimal with respect to inclusion are called the minimal or isolated prime ideals belonging to I, as on p52 of [8]. The other prime ideals among those in (17.6.10) are called the embedded prime ideals belonging to I.

Suppose that I_{0} is a proper prime ideal in k such that

I ⊆ I_{0}. (17.7.2)

This means that

⋂_{j=1}^{n} I_{j} ⊆ I_{0}, (17.7.3)

so that

⋂_{j=1}^{n} r(I_{j}) = r( ⋂_{j=1}^{n} I_{j}) ⊆ r(I_{0}) = I_{0}, (17.7.4)

using some basic properties of radicals of ideals, as in Section 12.10. This implies that r(I_{l}) ⊆ I_{0} (17.7.5)

for some l, as in Section 17.3. It follows that

(17.7.6) I_{0} contains a minimal prime ideal belonging to I,

because r(I_{l}) contains a minimal prime ideal belonging to I. This is the first part of Proposition 4.6 on p52 of [8].

Consider

(17.7.7) the set of all proper prime ideals in k that contain I.

Using (17.7.6), we get that every minimal element of (17.7.7) is a minimal prime ideal belonging to I. Conversely, we also get that every minimal prime ideal belonging to I is a minimal element of (17.7.7). Thus the minimal prime ideals belonging to I are the same as the proper prime ideals in k that contain I and are minimal with respect to inclusion. This is the second part of Proposition 4.6 on p52 of [8].

The first part of Proposition 4.7 on p53 of [8] states that

⋃_{l=1}^{n} r(I_{l}) = {x ∈ k : (I : x) ≠ I}. (17.7.8)

If x ∈ k is not an element of r(I_{l}) for some l, then

(I_{l} : x) = I_{l}, (17.7.9)

as in (17.5.10). If x is not an element of r(I_{l}) for any l, then it follows that

(I : x) = ⋂_{l=1}^{n}(I_{l} : x) = ⋂_{l=1}^{n} I_{l} = I, (17.7.10)

using (17.6.4) in the first step. This implies that the right side of (17.7.8) is contained in the left side. To get equality, we shall reduce to the case where I = {0}, as in [8], after some preliminary remarks in the next section.

17.8 Surjective ring homomorphisms

Let k and k̃ be commutative rings with multiplicative identity elements 1_{k}, 1_{k}, respectively. Also let ϕ be a ring homomorphism from k onto k̃. which implies that ϕ(1_{k}) = 1_{k}. If I is any ideal in k, then it is easy to see that ϕ(I) is an ideal in k̃. This is the same as the extension of I with respect to ϕ, as in Section 12.7. If Ĩ is an ideal in k̃, then ϕ^{−1}(Ĩ) is an ideal in k, which is the contraction of Ĩ with respect to ϕ, as before.

If I is an ideal in k, then

ϕ^{−1}(ϕ(I)) = I + ker ϕ. (17.8.1)

In particular, if ker ϕ ⊆ I, (17.8.2)

then I = ϕ^{−1}(ϕ(I)). (17.8.3)

428 CHAPTER 17. SOME PROPERTIES OF IDEALS If Ĩ is an ideal in k̃, then ker ϕ ⊆ ϕ^{−1}(Ĩ), (17.8.4) and ϕ(ϕ^{−1}(Ĩ)) = Ĩ. (17.8.5) It follows that I ↦ ϕ(I) (17.8.6) defines a one-to-one mapping fom the set of ideals in k that contain ker ϕ onto the set of ideals in k̃, with inverse given by Ĩ ↦ ϕ^{−1}(Ĩ). (17.8.7) This corresponds to Proposition 1.1 on p2 of [8], which is also mentioned on p9 of [8].

Note that ϕ^{−1}(Ĩ) is a proper ideal in k when Ĩ is a proper ideal in k̃. If I is a proper ideal in k that contains ker ϕ, then ϕ(I) is a proper ideal in k̃. If Ĩ is a prime ideal in k̃, then ϕ^{−1}(Ĩ) is a prime ideal in k, as in Section 12.7. If I is a prime ideal in k that contains ker ϕ, then one can check that (17.8.8) ϕ(I) is a prime ideal in k̃, using (17.8.3). This corresponds to a remark on p9 of [8].

If Ĩ is a primary ideal in k̃, then ϕ^{−1}(Ĩ) is a primary ideal in k, as in Section 14.5. If I is a primary ideal in k that contains ker ϕ, then one can verify that (17.8.9) ϕ(I) is a primary ideal in k̃, using (17.8.3) again. This corresponds to part of a remark near the beginning of the proof of Proposition 4.7 on p53 of [8].

If Ĩ is an ideal in k̃, then ϕ^{−1}(r(Ĩ)) = r(ϕ^{−1}(Ĩ)), (17.8.10) as in Section 12.10. Here r(·) is the radical of an ideal in k or k̃, as appropriate. If I is an ideal in k, then it is easy to see that ϕ(r(I)) ⊆ r(ϕ(I)), (17.8.11) as before. If I contains ker ϕ, then one can check that ϕ(r(I)) = r(ϕ(I)), (17.8.12) using (17.8.3).

Let I be an ideal in k again, and let x ∈ k be given. It is easy to see that ϕ((I : x)) ⊆ (ϕ(I) : ϕ(x)), (17.8.13) where (I : x) and (ϕ(I) : ϕ(x)) are the ideals in k and k̃, respectively, defined as in Section 17.5. If I contains ker ϕ, then one can verify that

ϕ((I : x)) = (ϕ(I) : ϕ(x)), (17.8.14)

using (17.8.3). Note that ker ϕ ⊆ I ⊆ (I : x) (17.8.15)

when I contains ker ϕ. In this case, if x ∈ k, then we get that

(17.8.16) (I : x) = I if and only if (ϕ(I) : ϕ(x)) = ϕ(I).

If A_{1}, A_{2} are subsets of k, then

ϕ(A_{1} ∩ A_{2}) ⊆ ϕ(A_{1}) ∩ ϕ(A_{2}). (17.8.17)

If A_{1}, A_{2} are subgroups of k, as a commutative group with respect to addition, with ker ϕ ⊆ A_{j} (17.8.18)

for j = 1, 2, then one can check that

ϕ(A_{1} ∩ A_{2}) = ϕ(A_{1}) ∩ ϕ(A_{2}). (17.8.19)

Similarly, if A_{1}, …, A_{n} are finitely many subgroups of k, as a commutative group with respect to addition, and if (17.8.18) holds for each j = 1, …, n, then

ϕ( ⋂_{j=1}^{n} A_{j}) = ⋂_{j=1}^{n} ϕ(A_{j}). (17.8.20)

This could also be obtained from the fact that ϕ^{−1}(ϕ(A)) = A when A is a subgroup of k, as a commutative group with respect to addition, that contains ker ϕ.

Let I be a proper ideal in k, and let I = ⋂_{j=1}^{n} I_{j} be a primary decomposition of I in k. Suppose that I contains ker ϕ, so that

ker ϕ ⊆ I_{j} (17.8.21)

for each j = 1, …, n. This implies that ϕ(I) is a proper ideal in k̃, and that

ϕ(I) = ⋂_{j=1}^{n} ϕ(I_{j}), (17.8.22)

as in (17.8.20). This is a primary decomposition of ϕ(I) in k̃, because ϕ(I_{j}) is a primary ideal in k̃ for each j = 1, …, n, as in (17.8.9). This corresponds to another part of a remark near the beginning of the proof of Proposition 4.7 on p53 of [8].

Note that ker ϕ is contained in r(I_{j}) for each j = 1, …, n, by (17.8.21). Suppose now that I = ⋂_{j=1}^{n} I_{j} is a minimal primary decomposition of I. One can check that (17.8.22) is a minimal primary decomposition of ϕ(I) under these conditions.

The analogue of (17.7.8) for ϕ(I) in k̃ is that

⋃_{l=1}^{n} r(ϕ(I_{l})) = {y ∈ k̃ : (ϕ(I) : y) ≠ ϕ(I)}. (17.8.23)

One can verify that this holds if and only if (17.7.8) holds.

In particular, we can take ϕ to be the natural quotient homomorphism from k onto k̃ = k/I. This permits us to reduce (17.7.8) to the case where I = {0}, as in the proof of Proposition 4.7 on p53 of [8]. This case will be discussed in Section 17.10.

17.9 Some remarks about zero-divisors

Let k be a commutative ring with a multiplicative identity element, and let D be the set of zero-divisors in k, which is to say the set of x ∈ k such that x y = 0 for some y ∈ k with y ≠ 0. If x ∈ k and I is an ideal in k, then (I : x) is the set of t ∈ k such that t x ∈ I, which is an ideal in k, as before. In particular,

({0} : x) = {t ∈ k : t x = 0}, (17.9.1)

so that (17.9.2) x ∈ D if and only if ({0} : x) ≠ {0}.

Alternatively, ({0} : x) is the same as the annihilator of the ideal in k generated by x, as in Section 12.8. We also have that

D = ⋃_{x≠0}({0} : x), (17.9.3)

as on p8 of [8]. More precisely, the union on the right is taken over all x ∈ k with x ≠ 0.

If E is any subset of k, then the radical r(E) of E in k may be defined by

(17.9.4) r(E) = {x ∈ k : x^{n} ∈ E for some n ∈ Z_{+}},

as on p9 of [8]. This is the same as in the definition of the radical of an ideal in k, as in Section 12.10, although this is not necessarily an ideal in k. It is easy to see that the radical of a union of a family of subsets of k is the same as the union of the radicals of the sets, as in [8]. Note that

E ⊆ r(E) (17.9.5)

for any E ⊆ k. We also have that

r(r(E)) = r(E) (17.9.6)

17.10. PRIMARY DECOMPOSITIONS OF {0} 431

for every E ⊆ k, as before.

If x ∈ k, then x ∈ r(D) if and only if

x^{n} y = 0 (17.9.7)

for some positive integer n and y ∈ k with y ≠ 0. We can take n as small as possible, to get that x ∈ D. This means that

r(D) = D, (17.9.8)

as in the proof of Proposition 1.15 on p9 of [8].

Using (17.9.3) and (17.9.8), we get that

D = r( ⋃_{x≠0}({0} : x)) = ⋃_{x≠0} r(({0} : x)). (17.9.9)

This is Proposition 1.15 on p9 of [8].

17.10 Primary decompositions of {0}

Let k be a commutative ring with a nonzero multiplicative identity element, and suppose that

⋂_{j=1}^{n} I_{j} = {0} (17.10.1)

is a minimal primary decomposition of {0} in k, as in Section 17.6. We would like to show that

⋃_{l=1}^{n} r(I_{l}) = {x ∈ K : ({0} : x) ≠ {0}}, (17.10.2)

which is the same as (17.7.8), with I = {0}. This will imply (17.7.8) for arbitrary I, as mentioned near the end of Section 17.8.

Let D be the set of zero-divisors in k, as in the previous section. Observe that (17.10.2) is equivalent to saying that

⋃_{l=1}^{n} r(I_{l}) = D, (17.10.3)

because of (17.9.2). This corresponds to the second part of Proposition 4.7 on p53 of [8].

If x ∈ k and x ≠ 0, then

r(({0} : x)) = ⋂{r(I_{j}) : 1 ≤ j ≤ n, x ∉ I_{j}}, (17.10.4)

as in (17.6.6). In particular, this implies that

r(({0} : x)) ⊆ I_{j} (17.10.5) for some j. It follows that

D ⊆ ⋃_{l=1}^{n} r(I_{l}), (17.10.6)

because of (17.9.9).

We also have that for each l = 1, …, n, there is an x_{l} ∈ k such that

r(I_{l}) = r(({0} : x_{l})), (17.10.7)

as in Section 17.6. More precisely, x_{l} ≠ 0, by the previous argument, or because r(I_{l}) ≠ k. This implies that

⋃_{l=1}^{n} r(I_{l}) ⊆ D, (17.10.8)

because of (17.9.9). Thus (17.10.3) holds, by (17.10.6) and (17.10.8).

Remember that the nilradical N of k is the same as the intersection of all the prime ideals in k, as in Section 12.5. Because of (17.10.1), we can use (17.7.6) with I = {0} to get that N is equal to the intersection of the minimal prime ideals belonging to {0}. This corresponds to a remark on p53 of [8].

17.11 Zero-divisors and prime ideals

Let k be a commutative ring with a nonzero multiplicative identity element. Also let Σ be the collection of all ideals I in k such that every element of I is a zero-divisor in k, as in Exercise 14 on p12 of [8]. One can check that every element of Σ is contained in a maximal element of Σ, with respect to inclusion, as in [8], using Zorn’s lemma or Hausdorff’s maximality principle. More precisely, one can verify that the union of any nonempty chain of elements of Σ, with respect to inclusion, is an element of Σ.

If I_{0} is a maximal element of Σ, then I_{0} is a prime ideal in k, as in [8]. To see this, suppose that x, y ∈ k, x y ∈ I_{0}, and y ∉ I_{0}, and let us show that x ∈ I_{0}. Remember that the set (I_{0} : x) of t ∈ k such that t x ∈ I_{0} is an ideal in k that contains I_{0}. We also have that y ∈ (I_{0} : x), by hypothesis, so that

I_{0} ≠ (I_{0} : x). (17.11.1)

It follows that there is a z ∈ (I_{0} : x) such that z is not a zero-divisor in k, by the maximality of I_{0}.

Let I_{1} be the ideal in k generated by I_{0} and x, so that every element of I_{1} is of the form a + b x for some a ∈ I_{0} and b ∈ k. If w ∈ I_{1}, then

w z ∈ I_{0}, (17.11.2)

because x z ∈ I_{0}, by construction. This implies that there is a t ∈ k such that t ≠ 0 and t w z = 0, (17.11.3) because every element of I_{0} is a zero-divisor in k. Note that t z ≠ 0, because z is not a zero-divisor in k. It follows that w is a zero-divisor in k.

Thus I_{1} ∈ Σ, which means that I_{1} = I_{0}, because I_{0} ⊆ I_{1} by construction, and I_{0} is a maximal element of Σ. This implies that x ∈ I_{0}, as desired.

If x ∈ k is a zero-divisor, then every element of the ideal in k generated by x is a zero-divisor, so that the ideal in k generated by x is an element of Σ. This implies that the ideal in k generated by x is contained in a maximal element of Σ, as before. It follows that the set D of zero-divisors in k is equal to the union of the maximal elements of Σ. In particular, this shows that D can be expressed as the union of a family of proper prime ideals in k, as in [8].

17.12 Saturated multiplicatively closed sets

Let k be a commutative ring with a nonzero multiplicative identity element 1_{k} = 1. Note that the set of all non-invertible elements of k is the same as the usion of all of the proper maximal ideals in k. In particular, this may be considered as the union of a family of proper prime ideals in k.

Let k_{1} be another commutative ring with a nonzero multiplicative identity element 1_{k_{1}}, and let ϕ be a ring homomorphism from k into k_{1} such that ϕ(1_{k}) = 1_{k_{1}}. Remember that the inverse image of a proper prime ideal in k_{1} under ϕ is a proper prime ideal in k, as in Section 12.7. This implies that

(17.12.1) {x ∈ k : ϕ(x) is not invertible in k_{1}}

can be expressed as the union of a family of proper prime ideals in k. More precisely, this is the inverse image under ϕ of the union of a family of proper prime ideals in k_{1}, as in the preceding paragraph.

Let S be a multiplicatively closed subset of k, as in Section 12.1. We say that S is saturated if for every x, y ∈ k, we have that

(17.12.2) x y ∈ S if and only if x, y ∈ S,

as in Exercise 7 on p44 of [8]. Note that k may be considered as a saturated multiplicatively closed subset of itself. If S is a saturated multiplicatively closed subset of k and 0 ∈ S, then S = k.

Observe that the group of invertible elements of k is a saturated multiplicativaly closed subset of k. If S is any saturated multiplicatively closed subset of k, then one can verify that S contains the group of invertible elements in k.

It is easy to see that the intersection of any nonempty family of multiplicatively closed subsets of k is a multiplicatively closed subset of k. Similarly, the intersection of any nonempty family of saturated multiplicatively closed subsets of k is a saturated multiplicatively closed subset of k.

If I is a proper prime ideal in k, then k \I is a multiplicatively closed subset of k, as in Section 12.4. In fact, one can check that k \ I is saturated. It follows that the complement in k of the union of a nonempty family of proper prime ideals is a saturated multiplicatively closed set. This is part of part (i) of Exercise 7 on p44 of [8].

Suppose that S is a saturated multiplicatively closed subset of k, and consider the corresponding ring of fractions S^{−1} k, as in Section 12.1. Let x ∈ k be given, and note that x/1 has a multiplicative inverse in S^{−1} k if and only if there are y ∈ k and t ∈ S such that

(x/1) (y/t) = (x y)/t (17.12.3)

is equal to 1/1 in S^{−1} k. This happens if and only if there is a v ∈ S such that

t v = x y v, (17.12.4)

as before. Under these conditions, we get that t v ∈ S, and thus that x, y ∈ S, because S is saturated.

This shows that

(17.12.5) S = {x ∈ k : x/1 is invertible in S^{−1} k}

in this case. This means that k \ S is as in (17.12.1), where ϕ is the natural ring homomorphism from k into k_{1} = S^{−1} k. It follows that k \ S can be expressed as the union of a family of proper prime ideals in k, as before. This is the other part of part (i) of Exercise 7 on p44 of [8].

Let D be the set of zero-divisors in k, as before. It is easy to see that the set S_{0} = k \ D (17.12.6)

of all non-zero-divisors in k is a multiplicatively closed subset of k. In fact, one can check that S_{0} is saturated as a multiplicatively closed subset of k, as in Exercise 9 on p44 of [8]. Using the remarks in the preceding paragraph, we get another way to see that D = k \ S_{0} (17.12.7)

can be expressed as the union of a family of proper prime ideals in k, as in the previous section. This is another part of Exercise 9 on p44 of [8].

Let S be any multiplicatively closed subset of k, so that the ring S^{−1} k of fractions of k with respect to S may be defined as in Section 12.1. Remember that for each x ∈ k, x/1 = 0 in S^{−1} k if and only if x t = 0 in k for some t ∈ S. This implies that the natural ring homomorphism from k into S^{−1} k is injective if and only if S ⊆ S_{0}. (17.12.8)

This corresponds to part (i) of Exercise 9 on p44 of [8]. The ring S_{0}^{−1} k is called the total ring of fractions of k, as in [8].

17.13 Saturations of ideals

Let k be a commutative ring with a multiplicative identity element, and let S be a multiplicatively closed subset of k, so that the ring S^{−1} k of fractions of k with respect to S may be defined as in Section 12.1. If I is an ideal in k, then S^{−1} I 17.13. SATURATIONS OF IDEALS 435 is an ideal in S^{−1} k, as in Section 12.13. This is the same as the extension of I with respect to the natural ring homomorphism from k into S^{−1} k, as before.

Let S(I) = (S^{−1} I)^{c} (17.13.1) be the contraction of S^{−1} I with respect to the natural ring homomorphism from k into S^{−1} k, as on p53 of [8]. This is the ideal in k obtained by taking the inverse image of S^{−1} I under the natural ring homomorphism from k into S^{−1} k, as in Section 12.7. This is the saturation of I in k with respect to S, as in Exercise 12 on p56 of [8].

Equivalently, S(I) = {x ∈ k : x/1 ∈ S^{−1} I}. (17.13.2) Thus x ∈ k is an element of S(I) if and only if x/1 = y/t (17.13.3) for some y ∈ I and t ∈ S. This is the same as saying that x t v = y v (17.13.4) for some v ∈ S, as in Section 12.1. It follows that S(I) = {x ∈ k : x r ∈ I for some r ∈ S}. (17.13.5)

If Ĩ is another ideal in k, then part (i) of Exercise 12 on p56 of [8] states that S(I) ∩ S(Ĩ) = S(I ∩ Ĩ). (17.13.6) This can be verified using (17.13.5), and the fact that S is multiplicatively closed. Alternatively, (S^{−1}I) ∩ (S^{−1}Ĩ) = S^{−1} (I ∩ Ĩ), (17.13.7) as in Section 12.2. One can get (17.13.6) from (17.13.7) and the analogous property for contractions of intersections, as in Section 12.9.

Part (ii) of Exercise 12 on p56 of [8] states that S(r(I)) = r(S(I)). (17.13.8) This can also be obtained using (17.13.5), or from the analogous statements for contractions and S^{−1} I, as in Sections 12.10 and 12.13.

Part (iii) of Exercise 12 on p56 of [8] states that (17.13.9) S(I) = k if and only if I ∩ S ≠ ∅. One can check this using (17.13.5) again, or the fact that S^{−1} I = S^{−1} k if and only if I intersects S, as in Section 12.13.

Let S_{1}, S_{2} be multiplicatively closed subsets of k, so that the set S_{1} S_{2} of products of elements of S_{1} and S_{2} is also a multiplicatively closed set in k. Under these conditions, part (iv) of Exercsie 12 on p56 of [8] states that

S_{1}(S_{2}(I)) = (S_{1} S_{2})(I). (17.13.10)

This can be verified using the analogue of (17.13.5) in the various relevant cases. Alternatively, one can use the ring isomorphism between (S_{1} S_{2})^{−1} k and an appropriate ring of fractions of S_{1}^{−1} k mentioned in Section 14.1.

17.14 Primary decompositions and fractions

Let k be a commutative ring with a multiplicative identity element, let S be a multiplicatively closed subset of k, and let I be a proper ideal in k. Suppose that I = ⋂_{j=1}^{n} I_{j} is a minimal primary decomposition of I, as in Section 17.6. We may reorder the I_{j}’s, if necessary, to get a nonnegative integer m ≤ n such that r(I_{j}) ∩ S = ∅ when j ≤ m (17.14.1)

and r(I_{j}) ∩ S ≠ ∅ when j > m. (17.14.2)

If m ≥ 1, then Proposition 4.9 on p54 of [8] states that

S^{−1} I = ⋂_{j=1}^{m}(S^{−1} I_{j}) (17.14.3)

and

S(I) = ⋂_{j=1}^{m} I_{j}, (17.14.4)

and that these are minimal primary decompositions.

To see this, observe first that

S^{−1} I = ⋂_{j=1}^{n}(S^{−1} I_{j}), (17.14.5)

as in Sections 12.2 and 12.13. If j > m, then

S^{−1} I_{j} = S^{−1} k, (17.14.6)

because of (17.14.2), as in Section 14.6. Thus (17.14.5) reduces to (17.14.3) when m ≥ 1, and otherwise S^{−1} I = S^{−1} k.

Remember that r(I_{j}) is a prime ideal in k for each j, as in Section 14.5. We also have that r(I_{j}) ≠ k, because I_{j} ≠ k, by hypothesis, as in Section 12.10. Note that r(S^{−1} I_{j}) = S^{−1} r(I_{j}) (17.14.7) for each j, as in Section 12.13.

If j ≤ m, then

(17.14.8) S^{−1} I_{j} is a primary ideal in S^{−1} k,

because of (17.14.1), as in Section 14.6. This implies that (17.14.3) is a primary decomposition of S^{−1} I in S^{−1} k when m ≥ 1.

If j ≤ m, then S^{−1} r(I_{j}) is a proper prime ideal in S^{−1} k, because of (17.14.1), as in Section 12.13. The S^{−1} r(I_{j})’s, j ≤ m, are distinct ideals in S^{−1} k too, as in Section 12.13, because the r(I_{j})’s are distinct ideals in k, by the minimality of the primary decomposition of I, as in Section 17.6. This is the same as saying that the r(S^{−1} I_{j})’s, j ≤ m, are disctinct ideals in S^{−1} k, by (17.14.7). This is the first of the two conditions required for minimality of the primary decomposition (17.14.3).

If j ≤ m, then S(I_{j}) = (S^{−1} I_{j})^{c} = I_{j}, (17.14.9)

because of (17.14.1), as in Section 14.6. One can use this to get the second condition required for the minimality of the primary decomposition (17.14.3), because of the analogous property of the primary decomposition of I.

Note that

S(I) = ⋂_{j=1}^{n} S(I_{j}), (17.14.10)

because of (17.13.6). If j > m, then

S(I_{j}) = (S^{−1} I_{j})^{c} = (S^{−1} k)^{c} = k, (17.14.11)

using (17.14.6) in the second step. If m ≥ 1, then we get that

S(I) = ⋂_{j=1}^{m} S(I_{j}) = ⋂_{j=1}^{m} I_{j}, (17.14.12)

using (17.14.9) in the second step. Of course, this is a primary decomposition of S(I) in k, and the minimality of this primary decomposition follows from the minimality of the primary decomposition of I in k. If m = 0, then S(I) = k.

17.15 The second uniqueness theorem

Let k be a commutative ring with a multiplicative identity element, let I be a proper ideal in k, and suppose that I = ⋂_{j=1}^{n} I_{j} is a minimal primary decomposition of I. Remember that the collection of proper prime ideals r(I_{1}), …, r(I_{n}) in k does not depend on the particular minimal primary decomposition of I, as in Section 17.6. These are the prime ideals in k that are said to belong to I, as in Section 17.7.

Let Σ be a collection of the prime ideals in k that belong to I. Suppose that if P′ is a prime ideal in k that belongs to I, P ∈ Σ, and P′ ⊆ P, then

P′ ∈ Σ. (17.15.1)

Under these conditions, Σ is said to be isolated, as on p54 of [8].

Put

S = k \ ( ⋃_{P∈Σ} P) = ⋂_{P∈Σ}(k \ P). (17.15.2)

This is a multiplicatively closed subset of k, because k \ P is a multiplicatively closed subset of k for every P ∈ Σ, as in Section 12.4.

Let P′ be a prime ideal in k that belongs to I and satisfies

P′ ∉ Σ. (17.15.3)

We would like to check that P′ ∩ S ≠ ∅, (17.15.4)

as on p54 of [8]. Equivalently, this means that

P′ ⊈ ⋃_{P∈Σ} P. (17.15.5)

Otherwise, if

P′ ⊆ ⋃_{P∈Σ} P, (17.15.6)

then P′ ⊆ P_{0} (17.15.7)

for some P_{0} ∈ Σ, as in Section 17.3. However, this would contradict (17.15.3), because Σ is supposed to isolated.

Suppose that Σ = {r(I_{l_{1}}), …, r(I_{l_{m}})} (17.15.8)

for some positive integer m, where l_{1} < ⋯ < l_{m} are positive integers less than or equal to n. In this case, we get that

S(I) = I_{l_{1}} ∩ ⋯ ∩ I_{l_{m}}, (17.15.9)

as in (17.14.12). Although we have not reordered the I_{j}’s here as in the previous section, they have essentially the same properties with respect to S as before, because of (17.15.4) and the definition (17.15.2) of S.

In particular, (17.15.9) does not depend on the particular minimal primary decomposition of I. This is Theorem 4.10 on p54 of [8], which is the second uniqueness theorem. Of course, (17.15.9) does depend on the choice of Σ.

Suppose that r(I_{l}) is minimal among the prime ideals belonging to I for some l ≤ n. In this case, Σ = {r(I_{l})} (17.15.10) is isolated in the sense defined earlier. If Σ = k \ r(I_{l}), as in (17.15.2), then we get that S(I) = I_{l}, (17.15.11)

as in (17.15.9). Thus I_{l} does not depend on the particular minimal primary decomposition of I, as in Corollary 4.11 on p54 of [8]. More precisely, I_{l} is uniquely determined by r(I_{l}).

The primary components I_{l} corresponding to minimal prime ideals r(I_{l}) belonging to I may be called isolated primary components, as in [8]. The collection of these isolated primary components does not depend on the particular minimal primary decomposition of I, as in the preceding paragraph.

Some properties of ideals, 2

18.1 Intersections of prime ideals

Let k be a commutative ring with a multiplicative identity element. Also let A be a nonempty set, and let I_{α} be an ideal in k for each α ∈ A. It is easy to see that

r( ⋂_{α∈A} I_{α}) ⊆ ⋂_{α∈A} r(I_{α}). (18.1.1)

If A has only finitely many elements, then

r( ⋂_{α∈A} I_{α}) = ⋂_{α∈A} r(I_{α}). (18.1.2)

This is mentioned in Section 12.10 when A has only two elements, and otherwise one can use a similar argument, or use the previous statement repeatedly.

If I_{α} is a prime ideal in k for every α ∈ A, then r(I_{α}) = I_{α} for each α, as before. This implies that

⋂_{α∈A} r(I_{α}) = ⋂_{α∈A} I_{α} ⊆ r( ⋂_{α∈A} I_{α}), (18.1.3)

so that (18.1.2) holds in this case too. More precisely, we get that

r( ⋂_{α∈A} I_{α}) = ⋂_{α∈A} r(I_{α}) = ⋂_{α∈A} I_{α} (18.1.4)

under these conditions.

If I is any ideal in k, then its radical r(I) is the same as the intersection of all of the prime ideals in k that contain I, as in Section 12.10. In particular, if r(I) = I, then I can be expressed as the intersection of a family of prime ideals in k. Conversely, if I can be expressed as the intersection of a family of prime ideals in k, then r(I) = I, as in (18.1.4). Of course, if I ≠ k, then we may as well consider only proper prime ideals in k. This corresponds to Exercise 9 on p11 of [8].

Suppose now that I is a proper ideal in k, and that I = ⋂_{j=1}^{n} I_{j} is a primary decomposition of I, as in Section 17.6. Note that

r(I) = ⋂_{j=1}^{n} r(I_{j}), (18.1.5)

as in (18.1.2). If r(I) = I, then it follows that

I = ⋂_{j=1}^{n} r(I_{j}). (18.1.6)

Remember that r(I_{j}) is a proper prime ideal in k for each j, because I_{j} is a primary ideal, as in Sections 14.5 and 17.5. Suppose that we have a minimal primary decomposition of I, so that the r(I_{j})’s are distinct, as in Section 17.6.

Let r(I_{l_{1}}), …, r(I_{l_{m}}) (18.1.7)

be the minimal elements with respect to inclusion among the ideals r(I_{j}), with l_{1} < ⋯ < l_{m}. Observe that

I = r(I_{l_{1}}) ∩ ⋯ ∩ r(I_{l_{m}}), (18.1.8)

because of (18.1.6). More precisely, this uses the fact that each of the r(I_{j})’s contains a minimal element. Thus (18.1.8) may be considered as another primary decomposition of I.

If any of the minimal elements in (18.1.7) contains the intersection of the others, then it contains one of the other minimal elements, as in Section 17.3. This is not possible, because of minimality. It follows that (18.1.8) is another minimal primary decomposition of I, as in Section 17.6.

Remember that the prime ideals corresponding to a minimal primary decomposition of I do not depend on the particular minimal primary decomposition of I, by the first uniqueness theorem, which was discussed in Section 17.6. Of course, the prime ideals (18.1.7) are the same as those associated to the minimal primary decomposition (18.1.8) of I, because the radical of the radical of an ideal is equal to the radical of the ideal, as in Section 12.10. This means that the minimal elements (18.1.7) include all of the r(I_{j})’s, as in Exercise 2 on p55 of [8].

Remember that the isolated primary components of a minimal primary decomposition of I do not depend on the particular minimal primary decomposition of I, as in Section 17.15. This implies that

I_{j} = r(I_{j}) (18.1.9)

for each j = 1, …, n.

18.2 Primary decompositions and (I : x)

Let k be a commutative Noetherian ring with a multiplicative identity element, and let I be a proper ideal in k. Remember that I has a primary decomposition, which one may take to be minimal, as in Section 17.6.

If x ∈ k, then (I : x) = {t ∈ k : t x ∈ I} is an ideal in k that contains I, as in Section 12.8. Under these conditions, the prime ideals in k that belong to I, as in Section 17.7, are the same as the proper prime ideals in k of the form (I : x) for some x ∈ k. This is Proposition 7.17 on p83 of [8].

Remember that the first uniquenss theorem, discussed in Section 17.6, is the analogous statement using the radicals of the ideals (I : x). However, the Noetherian condition was not needed for that result.

If (I : x) is a proper prime ideal in k for some x ∈ k, then

r((I : x)) = (I : x), (18.2.1)

as in Section 12.10. This implies that (I : x) is one of the prime ideals in k that belong to I, as in Section 17.6.

Conversely, we would like to show that each of the prime ideals in k that belong to I is of this form. To do this, the first step is to reduce to the case where I = {0}, by replacing k with k/I. This also uses the remarks in Section 17.8.

Thus we suppose now that ⋂_{j=1}^{n} I_{j} = {0} is a minimal primary decomposition of {0} in k. Put

A_{l} = ⋂_{j≠l} I_{j} (18.2.2)

for each l = 1, …, n, which is an ideal in k. The minimality of the primary decomposition implies that A_{l} ⊈ I_{l} (18.2.3)

for each l, as in Section 17.6. In particular, this means that A_{l} ≠ {0} for each l.

Note that A_{l} ∩ I_{l} = {0} (18.2.4)

for each l, by construction. Thus

A_{l} \ I_{l} = A_{l} \ (A_{l} ∩ I_{l}) = A_{l} \ {0} (18.2.5)

for each l. If x is an element of (18.2.5) for some l, then

r(({0} : x)) = r(I_{l}). (18.2.6)

as in (17.6.9). In particular, this means that

({0} : x) ⊆ r(I_{l}). (18.2.7)

18.3. PRIMARY DECOMPOSITIONS AND PRODUCTS 443

Let 1 ≤ l ≤ n be given, and note that r(I_{l}) is finitely-generated as an ideal in k, because k is Noetherian. It follows that r(I_{l})^{m} ⊆ I_{l} (18.2.8) for some positive integer m, as in Section 14.7. This implies that A_{l} r(I_{l})^{m} ⊆ A_{l} ∩ r(I_{l})^{m} ⊆ A_{l} ∩ I_{l} = {0}, (18.2.9) where the first step is as in Section 12.8.

Let m_{l} be the smallest positive integer such that A_{l} r(I_{l})^{m_{l}} = {0}. (18.2.10) Thus A_{l} r(I_{l})^{m_{l}−1} ≠ {0}. More precisely, if m_{l} = 1, then this reduces to the fact that A_{l} ≠ {0}, as before.

Let x be a nonzero element of A_{l} r(I_{l})^{m_{l}−1}. Observe that r(I_{l}) ⊆ ({0} : x), (18.2.11) by (18.2.10). We also have that (18.2.7) holds, because x is a nonzero element of A_{l}. It follows that r(I_{l}) = ({0} : x) (18.2.12) under these conditions.

18.3 Primary decompositions and products

Let k be an integral domain of dimension one, in the sense of Section 14.8. Also let I ≠ {0} be a proper ideal in k with a minimal primary decomposition I = ⋂_{j=1}^{n} I_{j}. Note that {0} ≠ I ⊆ I_{j} ⊆ r(I_{j}) (18.3.1) for each j = 1, …, n. This implies that (18.3.2) r(I_{j}) is a maximal proper ideal in k for each j, because r(I_{j}) is a proper prime ideal in k, and k is a one-dimensional integral domain. Remember that the r(I_{j})’s are distinct ideals in k, because of the minimality of the primary decomposition, as in Section 17.6.

It is easy to see that (18.3.3) the r(I_{j})’s are pairwise coprime in k, as in Section 17.1, because they are distinct maximal proper ideals in k. This implies that (18.3.4) the I_{j}’s are pairwise coprime in k too, as in Section 17.1 again. It follows that the product of the I_{j}’s is the same as their intersection, as in Section 17.2. This means that

I = ⋂_{j=1}^{n} I_{j} = ∏_{j=1}^{n} I_{j} (18.3.5)

in this case. This corresponds to the existence part of Theorem 9.1 on p93 of [8].

Suppose now that

I = ∏_{j=1}^{n} I_{j}, (18.3.6)

where I_{1}, …, I_{n} are finitely many primary ideals in k with distinct radicals. Under these conditions, (18.3.1) and thus (18.3.2) hold for each j, as before. This implies that (18.3.3) and (18.3.4) hold as well, as before. This means that the product of the I_{j}’s is the same as their intersection, as in Section 17.2 again. Thus (18.3.5) holds under these conditions as well.

Let 1 ≤ l ≤ n be given, and let us check that

I_{l} + ∏_{j≠l} I_{j} = k. (18.3.7)

This is the same as (17.2.7) when l = n, and the same argument can be used otherwise. This implies that

I_{l} + ⋂_{j≠l} I_{j} = k, (18.3.8)

because a product of ideals is contained in its intersection. It follows that ⋂_{j≠l} I_{j} is not contained in I_{l}, because I_{l} is a primary ideal in k, and thus proper. This means that

(18.3.9) the I_{j}’s form a minimal primary decomposition of I,

as in Section 17.6.

Using this, we get that the collection of the radicals of the I_{j}’s is uniquely determined, as in the first uniqueness theorem in Section 17.6. It is easy to see that r(I_{l}) is minimal among the r(I_{j})’s for each l, because k has dimension one, by hypothesis. It follows that

(18.3.10) the collection of I_{j}’s is uniquely determined

under these conditions, as in Section 17.15. Of course, the I_{j}’s can be reordered, without affecting their product. This corresponds to the uniqueness part of Theorem 9.1 on p93 of [8].

18.4 Products and Dedekind domains

Let k be an integral domain of dimension one, in the sense of Section 14.8, and suppose that k is also Noetherian. Any proper ideal I in k has a minimal primary decomposition I = ⋂_{j=1}^{n} I_{j}, as in Section 17.6. Suppose that I ≠ {0}, so that I_{j} ≠ {0} for each j, and I can be expressed as the product of the I_{j}’s, as in (18.3.5). Remember that the radical r(I_{j}) of I_{j} is a proper prime ideal in k for each j, as in Sections 14.5 and 17.5.

Suppose from now on in this section that k is a Dedekind domain. In this case, for each j there is a unique positive integer m_{j} such that

I_{j} = r(I_{j})^{m_{j}}, (18.4.1)

as in Section 15.14. It follows that

I = ⋂_{j=1}^{n} I_{j} = ∏_{j=1}^{n} I_{j} = ∏_{j=1}^{n} r(I_{j})^{m_{j}}. (18.4.2)

One could also allow I = k here, with n = 0, or m_{j} = 0 for each j. This corresponds to the existence part of Corollary 9.4 on p95 of [8].

Concerning uniqueness, suppose that I can be expressed as the product of finitely many proper prime ideals in k. This can be arranged into a product of positive powers of distinct proper prime ideals in k. Thus I can be expressed as in (18.3.6), where the I_{j}’s are positive powers of distinct proper prime ideals in k.

Remember that the radical of a positive power of a prime ideal in k is that prime ideal, as in Section 12.10. Thus the proper prime ideal in k corresponding to I_{j} is r(I_{j}) for each j. This means that I_{j} may be expressed as in (18.4.1) for some positive integer m_{j} for each j, so that I can be expressed as

I = ∏_{j=1}^{n} I_{j} = ∏_{j=1}^{n} r(I_{j})^{m_{j}}, (18.4.3)

where the r(I_{j})’s are distinct proper prime ideals in k.

More precisely, (18.3.1) and (18.3.2) hold for each j in this case, as before. It follows that I_{j} is a primary ideal in k for each j, as in Section 14.5. Using this, we get that the collection of r(I_{j})’s is uniquely determined, and in fact the collection of I_{j}’s is uniquely determined, as in the previous section. The m_{j}’s are also uniquely determined, as in Section 15.14 again. This corresponds to the uniqueness part of Corollary 9.4 on p95 of [8].

18.5 Artinian modules

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. A sequence V_{1}, V_{2}, V_{3}, … of submodules of V is said to be monotonically decreasing with respect to inclusion if

V_{j+1} ⊆ V_{j} (18.5.1) for each j, as usual. If every such sequence is eventually constant, then V is said to be Artinian as a module over k, as on p74 of [8]. This is equivalent to the condition that every nonempty collection of submodules of V have a minimal element, as in Proposition 6.1 on p74 of [8].

A number of basic properties of Artinian modules over k are analogous to properties of Noetherian modules over k, as mentioned on p75 of [8]. In particular, if V_{0} is a submodule of V, then one can check that V is Artinian if and only if (18.5.2) V_{0} and V/V_{0} are Artinian modules over k,

as in part (ii) of Proposition 6.3 on p75 of [8]. One can use this to get that

(18.5.3) the direct sum of finitely many Artinian modules over k is Artinian as well,

as in Corollary 6.4 on p76 of [8].

We say that k is Artinian as a commutative ring if it is Artinian as a module over itself, as on p76 of [8]. In this case, if V is a finitely-generated module over k, then (18.5.4) V is Artinian as a module over k,

as in Proposition 6.5 on p76 of [8]. This follows from the statements mentioned in the preceding paragraph, by expressing V as the quotient of the direct sum of finitely many copies of k, as a module over itself.

Let I be an ideal in k, and suppose for the moment that t · v = 0 for every t ∈ I and v ∈ V. This implies that V may be considered as a module over the quotient ring k/I. In this case, the submodules of V as a module over k are the same as the submodules of V as a module over k/I. It follows that V is Noetherian or Artinian as a module over k if and only if V has the same property as a module over k/I.

If k is Noetherian or Artinian as a ring, then k/I has the same property as a ring, as in Proposition 6.6 on p76 of [8]. More precisely, k is Noetherian or Artinian as a module over itself, by hypothesis. This implies that k/I is Noetherian or Artinian as a module over k, as appropriate, as before. Equivalently, this means that k/I has the same property as a module over itself, and thus as a ring, as in the preceding paragraph.

Suppose for the moment that k is a field, so that V is a vector space over k. If V has finite dimension as a vector space over k, then it is easy to see that

(18.5.5) V is Noetherian and Artinian as a module over k.

Otherwise, if V does not have finite dimension as a vector space over k, then there is an infinite sequence v_{1}, v_{2}, v_{3}, … of vectors in V that are linearly independent. One can use this to get that V is neither Noetherian nor Artinian as a module over k. This corresponds to part of Proposition 6.10 on p78 of [8].

18.6 Artinian and Noetherian rings

Let k be a commutative ring with a multiplicative identity element again, and let I_{1}, …, I_{n} be finitely many maximal proper ideals in k, which need not be distinct. Suppose that

∏_{j=1}^{n} I_{j} = {0}. (18.6.1)

Under these conditions, Corollary 6.11 on p78 of [8] says that k is Noetherian if and only if k is Artinian.

To see this, note that

∏_{j=1}^{l} I_{j} (18.6.2)

is an ideal in k for each l = 1, …, n, which may be interpreted as being equal to k when l = 0. Similarly,

∏_{j=1}^{l+1} I_{j} (18.6.3)

is an ideal in k for each l = 0, 1, …, n − 1, which is contained in (18.6.2). These ideals may be considered as modules over k, so that their quotient

( ∏_{j=1}^{l} I_{j})/( ∏_{j=1}^{l+1} I_{j}) (18.6.4)

is defined as a module over k. In fact, the quotient may be considered as a module over k/I_{l+1}, because the action of any element of I_{l+1} on the quotient is equal to 0, by construction.

Of course, k/I_{l+1} is a field, because I_{l+1} is a maximal proper ideal in k, by hypothesis. Thus the Netherian and Artinian conditions for (18.6.4), as a module over k/I_{l+1}, are equivalent to the quotient having finite dimension as a vector space over k/I_{l+1}, as in the previous section. It follows that the Noetherian and Artinian conditions for (18.6.4), as a module over k, are equivalent as well.

Suppose that k is Noetherian or Artinian as a ring, and thus as a module over itself. This implies that (18.6.2) has the same property, as a module over k, for each l. It follows that the quotient (18.6.4) has the same property, as a module over k, for each l. This means that (18.6.4) is Noetherian and Artinian, as a module over k, for each l, as in the previous paragraph.

Observe that (18.6.4) is the same as (18.6.2) when l = n − 1, because of (18.6.1). If (18.6.4) is Artinian, as a module over k, for each l, then one can check that k is Artinian. More precisely, one can check that (18.6.2) is Artinian for each l, using the characterization in (18.5.2). Similarly, if (18.6.4) is Noetherian, as a module over k, for each l, then k is Noetherian. It follows that k is Noetherian if and only if k is Artinian in this case, using the remarks in the preceding paragraph.

18.7 Commutative Artin rings

Let k be a commutative ring with a multiplicative identity element. If k is Artinian, as in Section 18.5, then one may also call k an Artin ring, as on p89 of [8]. Remember that every quotient of k is an Artin ring too in this case.

If k is an Artin ring and I is a proper prime ideal in k, then Proposition 8.1 on p89 of [8] states that I is a maximal proper ideal in k. In this case, k_{1} = k/I is an integral domain that is an Artin ring as well, and it suffices to show that k_{1} is a field. To see this, let x ∈ k_{1} with x ≠ 0 be given, and observe that the sequence of ideals in k_{1} generated by the powers of x decreases monotonically. Thus the sequence is eventually constant, because k_{1} is an Artin ring. This implies that x^{n} is an element of the ideal generated by x^{n+1} for some positive integer n, so that x^{n} = x^{n+1} y (18.7.1)

for some y ∈ k_{1}. This means that

x^{n} (x y − 1) = 0 (18.7.2)

in k_{1}, which implies that x y = 1, because k_{1} is an integral domain, and x ≠ 0. It follows that x is invertible in k_{1}, so that k_{1} is a field, as desired.

If k is an Artin ring, then Proposition 8.3 on p89 of [8] states that k has only finitely many maximal proper ideals. Of course, this is trivial when k = {0}, and so we may as well suppose that 1 ≠ 0 in k. Because k is Artinian, the collection C of all ideals in k that can be expressed as the intersection of finitely many maximal proper ideals in k has a minimal element. Suppose that

⋂_{j=1}^{n} I_{j} (18.7.3)

is such a minimal element, where I_{1}, …, I_{n} are maximal proper ideals in k. If I is any maximal proper ideal in k, then

I ∩ ( ⋂_{j=1}^{n} I_{j}) (18.7.4)

is another element of C. Note that (18.7.4) is contained in (18.7.3), which implies that they are equal, because of the minimality of (18.7.3). This means that

⋂_{j=1}^{n} I_{j} ⊆ I. (18.7.5)

It follows that I_{l} ⊆ I for some l, because I is a prime ideal in k, as in Section 17.3. More precisely, I = I_{l}, because I_{l} is a maximal proper ideal in k. Thus I_{1}, …, I_{n} are all of the maximal proper ideals in k, as desired.

A similar argument is used in the proof of Proposition 6 on p11 of [263], and we shall return to this in Section 18.10.

18.8. NILPOTENCY OF THE NILRADICAL 449

18.8 Nilpotency of the nilradical

Let k be a commutative ring with a multiplicative identity element. Remember that N denotes the nilradical of k, as in Section 12.5. If k is an Artin ring, then N^{m} = {0} (18.8.1) for some positive integer m, as in Proposition 8.4 on p89 of [8]. Another criterion for this was discussed in Section 14.7, which includes the case when k is Noetherian.

Of course, the sequence of powers of N decreases monotonically with respect to inclusion. If k is Artinian, then it follows that there is an ideal I_{1} in k such that I_{1} = N^{m} (18.8.2) for all sufficiently large m.

Suppose for the sake of a contradiction that I_{1} ≠ {0}. Let Σ be the collection of all ideals Ĩ in k such that I_{1} Ĩ ≠ {0}. (18.8.3) If l is a nonnegative integer, then I_{1} N^{l} = I_{1}, so that N^{l} ∈ Σ. In particular, Σ ≠ ∅. If k is Artinian, then Σ has a minimal element Ĩ_{0}.

Of course, I_{1} Ĩ_{0} ≠ {0}, because Ĩ_{0} ∈ Σ. Thus there is an x ∈ Ĩ_{0} whose product with an element of I_{1} is not zero. Let I(x) be the ideal in k generated by x, so that I(x) ∈ Σ. Clearly I(x) ⊆ Ĩ_{0}, because x ∈ Ĩ_{0}. This means that I(x) = Ĩ_{0}, (18.8.4) because Ĩ_{0} is minimal in Σ.

Observe that (I(x) N) I_{1} = I(x) (N I_{1}) = I(x) I_{1} ≠ 0, (18.8.5) so that I(x) N ∈ Σ. We also have that I(x) N ⊆ I(x), which implies that I(x) N = I(x), (18.8.6) because I(x) is minimal in Σ. It follows that x = x y (18.8.7) for some y ∈ N. This means that x = x y^{n} (18.8.8) for every positive integer n. However, y^{n} = 0 for some n, because y ∈ N, so that x = 0, which is a contradiction.

18.9 More on commutative Artin Rings

Let k be a commutative ring with a multiplicative identity element. Theorem 8.5 on p90 of [8] states that k is an Artin ring if and only if

(18.9.1) k is Noetherian and the dimension of k is equal to 0,

in the sense of Section 14.8. More precisely, the dimension of k is really only defined when k ≠ {0}, and otherwise this is trivial anyway.

Suppose that k is an Artin ring, and note that the dimension of k is zero, because proper prime ideals in k are maximal, as in Section 18.7. We have also seen that there are only finitely many maximal proper ideals I_{1}, …, I_{n} in k, which are all of the proper prime ideals in k in this case. Remember that the nilradical N of k is equal to the intersection of all of the prime ideals in k, as in Section 12.5. This means that

N = ⋂_{j=1}^{n} I_{j}. (18.9.2)

In particular, this implies that

∏_{j=1}^{n} I_{j} ⊆ N . (18.9.3)

Remember that N^{m} = {0} for some positive integer m, because k is Artinian, as in (18.8.1). It follows that

∏_{j=1}^{n} I_{j}^{m} = {0}, (18.9.4)

because of (18.9.3). Under these conditions, the hypothesis that k be Artinian implies that k is Noetherian, as in Section 18.6.

Suppose now that (18.9.1) holds, and let us show that k is Artinian. Of course, this is trivial when k = {0}, and so we may suppose that k ≠ {0}, so that {0} is a proper ideal in k. Because k is Noetherian, {0} has a primary decomposition in k, as in Section 17.6. This means that {0} has a minimal primary decomposition in k, as before.

Let I′_{1}, …, I′_{n} be the minimal prime ideals in k belonging to {0} with respect to this minimal promary decomposition of {0} in k, as in Section 17.7. These are the same as the proper prime ideals in k that are minimal with respect to inclusion, as before, and in particular this does not depend on the minimal primary decomposition of {0}. More precisely, normally one would consider the minimal proper prime ideals in k that contain the given ideal, and of course all ideals in k contain {0}.

If I_{0} is any proper prime ideal in k, then

I′_{j} ⊆ I_{0} (18.9.5) for some j, as in Section 17.7. As before, this was stated previously for proper prime ideals in k that contain the given ideal, which is {0} in this case. This implies that

N = ⋂_{j=1}^{n} I′_{j}. (18.9.6)

In particular,

∏_{j=1}^{n} I′_{j} ⊆ N , (18.9.7)

as before.

We also have that N^{m} = {0} for some positive integer m, because k is Noetherian, as in Section 14.7. Using this, we get that

∏_{j=1}^{n}(I′_{j})^{m} = {0}, (18.9.8)

as before. Note that I′_{1}, …, I′_{n} are maximal proper ideals in k, because they are proper prime ideals and k has dimension 0, by hypothesis. This means that the hypothesis that k be Noetherian implies that k is Artinian, as in Section 18.6 again.

18.10 Finite collections of prime ideals

Let k be a commutative ring with a nonzero multiplicative identity element, and suppose that ⋂_{j=1}^{n} I_{j} = {0} is a minimal primary decomposition of {0} in k, as in Section 17.6. If I_{0} is any prime ideal in k, then

r(I_{l}) ⊆ I_{0} (18.10.1)

for some l, as in Section 17.7. Remember that r(I_{j}) is a proper prime ideal in k for each j, as in Sections 14.5 and 17.5.

Suppose for the moment that k has dimension zero, in the sense of Section 14.8. If I_{0} is a proper prime ideal in k, then (18.10.1) implies that

r(I_{l}) = I_{0}. (18.10.2)

In particular, this means that

(18.10.3) there are only finitely many proper prime ideals in k

under these conditions.

If k is Noetherian, then {0} has a minimal primary decomposition in k, as in Section 17.6. Thus (18.10.3) holds when k is Noetherian and has dimension zero. Alternatively, k is Noetherian of dimension 0 if and only if k is an Artin ring, as in the previous section. If k is an Artin ring, then every proper prime ideal in k is a maximal proper ideal in k, and there are only finitely many maximal proper ideals in k, as in Section 18.7.

If k is Noetherian and I is an ideal in k, then

(18.10.4) k/I is Noetherian as a ring

too. More precisely, if k is Noetherian as a ring, then k is Noetherian as a module over itself, and

(18.10.5) k/I is Noetherian as a module over k

as well, as in Section 9.7. It is easy to see that

(18.10.6) a submodule of k/I as a module over k

is the same as

(18.10.7) a submodule of k/I as a module over itself.

One can use this to get that if k/I is Noetherian as a module over k, then k/I is Noetherian as a module over itself, and thus as a ring.

Suppose now that k is an integral domain of dimension one, in the sense of Section 14.8. Let x be a nonzero element of k, and let I(x) be the ideal in k generated by x. There is a natural one-to-one correspondence between the proper prime ideals in the quotient ring k/I(x) and the proper prime ideals in k that contain I(x), as in Section 17.8. Of course, an ideal in k contains I(x) if and only if it contains x. Using this correspondence, it is easy to see that

(18.10.8) k/I(x) has dimension zero,

because x ≠ 0, by hypothesis.

If k is Noetherian, then k/I(x) is Noetherian, as in (18.10.4), and it follows that (18.10.9) there are only finitely many proper prime ideals in k/I(x),

as before. This implies that

(18.10.10) there are only finitely many proper prime ideals in k that contain x,

as in the preceding paragraph.

Proposition 6 on p11 of [263] states that (18.10.10) holds in particular when k is a Dedekind domain, with a more direct proof in that case. This will be discussed further in the next section.

18.11 Some arguments for Dedekind domains

Let k be a Dedekind domain, as in Section 15.11, and let Q_{k} be the corresponding field of fractions. Also let x ∈ k with x ≠ 0 be given, and let I, I′ be ideals in k with x ∈ I ⊆ I′. (18.11.1)

Of course, I, I′ are integral ideals of k, and fractional ideals of k in particular, as in Sections 11.7 and 15.7. Note that I, I′ ≠ {0}, because of (18.11.1).

It follows that I, I′ are invertible as fractional ideals of k, as in Section 15.12. Remember that the inverse I^{−1} of I is equal to

(k : I) = (k : I)_{Q_{k}} = {y ∈ Q_{k} : y I ⊆ k}, (18.11.2)

as in Section 11.7, and similarly for the inverse (I′)^{−1} of I′. It follows that

k ⊆ (I′)^{−1} ⊆ I^{−1} ⊆ x^{−1} k, (18.11.3)

because of (18.11.1) and the fact that I′ ⊆ k. Note that I is the inverse of I^{−1}, by the definition of the inverse in Section 11.7.

Remember that k is Noetherian as a ring, because it is a Dedekind domain, by hypothesis, as in Section 15.11. This means that k is Noetherian as a module over itself, which implies that x^{−1} k is Notherian as a module over k. The ascending chain condition for submodules of x^{−1} k implies the ascending chain condition for ideals in k that contain x, because of the remarks in the preceding paragraph. This corresponds to the first part of the proof of Proposition 6 on p11 of [263]. Equivalently, this means that

(18.11.4) k/I(x) is an Artin ring,

as in Section 18.5, where I(x) is the ideal in k generated by x.

Suppose that I_{1}, I_{2}, I_{3}, … is an infinite sequence of proper prime ideals in k that each contain x. This implies that

⋂_{j=1}^{n} I_{j} (18.11.5)

is an ideal in k that contains x for each positive integer n, which defines a decreasing sequence of ideals in k, with respect to inclusion. It follows that there is a positive integer n_{0} such that

⋂_{j=1}^{n} I_{j} = ⋂_{j=1}^{n_{0}} I_{j} (18.11.6)

when n ≥ n_{0}, as in the previous paragraph. This means that

⋂_{j=1}^{n_{0}} I_{j} ⊆ I_{l} (18.11.7)

for all positive integers l. If l > n_{0}, then we get that

I_{j} ⊆ I_{l} (18.11.8)

for some j ≤ n_{0}, as in Section 17.3.

Remember that k has dimension less than or equal to one, because k is a Dedekind domain, as in Section 15.11. This implies that I_{j} is a maximal proper ideal in k for each j, because x ≠ 0, by hypothesis. Using this and (18.11.8), we get that each I_{l} is the same as I_{j} for some j ≤ n_{0}. This means that there are only finitely many proper prime ideals in k that contain x, because otherwise we could consider an infinite sequence of distinct proper prime ideals in k that contain x. This corresponds to the second part of the proof of Proposition 6 on p11 of [263].

18.12 Valuations on Dedekind domains

Let k be an integral domain, and let v be a valuation on k, as in Section 14.15. Remember that {x ∈ k : v(x) ≥ 0} (18.12.1)

is a subring of k, and that

{x ∈ k : v(x) > 0} (18.12.2)

is an ideal in (18.12.1). Of course, (18.12.2) is a proper subset of (18.12.1), because v(1) = 0, as before.

It is easy to see that (18.12.2) is in fact a prime ideal in (18.12.1). If k is a field, then (18.12.2) is a maximal proper ideal in (18.12.1), as in Section 14.14.

Now let k be a Dedekind domain, as in Section 15.11. In particular, k is an integral domain, and we let Q_{k} be the corresponding field of fractions. If I is a proper prime ideal in k, then S_{I} = k \ I is multiplicatively closed in k, as in Section 12.4, so that k_{I} = S_{I}^{−1} k may be defined as in Section 12.1. Suppose that I ≠ {0}, so that

(18.12.3) k_{I} is a discrete valuation ring,

as in Section 15.1, by definition of a Dedekind domain.

We also have that k_{I} is an integral domain, whose field of fractions may be identified with Q_{k}, as in Section 14.2. We may identify k_{I} with a subring of Q_{k}, as before. As in Section 15.1, the condition that k_{I} be a discrete valuation ring means that there is a discrete valuation v_{I} on Q_{k}, as in Section 14.14, such that k_{I} = {y ∈ Q_{k} : v_{I}(y) ≥ 0}. (18.12.4)

Remember that S_{I}^{−1} I is the unique maximal proper ideal in k_{I}, as in Section 12.4, and using notation mentioned near the beginning of Section 12.13. This means that

S_{I}^{−1} I = {y ∈ Q_{k} : v_{I}(y) > 0} = {y ∈ Q_{k} : v_{I}(y) ≥ 1}, (18.12.5)

as in Section 15.1.

It is easy to see that k ∩ (S_{I}^{−1} I) = I, (18.12.6) because I is a prime ideal in k. This corresponds to a broader statement in Section 12.13. Another related statement was mentioned in Section 14.6. It follows that I = {y ∈ k : v_{I}(y) > 0} = {y ∈ k : v_{I}(y) ≥ 1}. (18.12.7)

If x ∈ k, then v_{I}(x) ≥ 0 (18.12.8)

for every proper prime ideal I ≠ {0} in k, because of (18.12.4) and the fact that x ∈ k_{I}. If we also have that x ≠ 0, then

x ∈ I (18.12.9)

for only finitely many nonzero proper prime ideals I in k, as in the previous two sections. This means that

v_{I}(x) > 0 (18.12.10)

for only finitely many nonzero proper prime ideals I in k, because of (18.12.7). If x ∈ Q_{k} and x ≠ 0, then one can use the previous statement to get that

v_{I}(x) ≠ 0 (18.12.11)

for only finitely many nonzero proper prime ideals I in k. This corresponds to the corollary to Proposition 6 on p11 of [263].

If x is an invertible element of k, then

v_{I}(x) = 0 (18.12.12)

for every nonzero proper prime ideal I, because of (18.12.8) and the analogous statement for 1/x. If x ∈ k, x ≠ 0, and x is not invertible in k, then x is contained in a nonzero maximal proper ideal I in k, and I is a prime ideal in particular. This implies that (18.12.10) holds, as before.

18.13 Some remarks about ideals

Let k be a commutative ring with a multiplicative identity element, and let I be a proper prime ideal in k. Thus S_{I} = k \ I is a multiplicatively closed set in k, as in Section 12.4, so that k_{I} = S_{I}^{−1} k may be defined as in Section 12.1. If L is an ideal in k, then L may be considered as a module over k, so that S_{I}^{−1} L may be defined as a module over S_{I}^{−1} k, as in Section 12.2. This may be identified with a submodule of k_{I}, considered as a module over itself, as before. This means that S_{I}^{−1} L is an ideal in k_{I}, which is the same as the extension of L in k_{I} with respect to the natural ring homomorphism from k into k_{I}, as in Section 12.13.

Let L′ be another ideal in k with

L ⊆ L′, (18.13.1) so that S_{I}^{−1} L ⊆ S_{I}^{−1} L′. (18.13.2)

If S_{I}^{−1} L = S_{I}^{−1} L′ (18.13.3)

for every maximal proper ideal I in k, then

L = L′. (18.13.4)

Indeed, the quotient L′/L may be considered as a module over k, so that

S_{I}^{−1} (L′/L) (18.13.5)

may be defined as a module over k_{I}, as in Section 12.2. This is isomorphic to

(S_{I}^{−1} L′)/(S_{I}^{−1} L), (18.13.6)

as modules over k_{I}, as before. The hypothesis (18.13.3) is the same as saying that (18.13.6) is {0}, so that (18.13.5) is {0} too. If this holds for every maximal proper ideal I in k, then L′/L = {0}, (18.13.7)

as in Section 12.4. Of course, this is the same as saying that (18.13.4) holds.

18.14 Ideals in Dedekind domains

Let k be a Dedekind domain, as in Section 15.11, and suppose that k is not a field. This means that the maximal proper ideals in k are the same as the nonzero prime ideals in k, because k is an integral domain of dimension one. If I is a nonzero proper prime ideal in k, then we take S_{I} = k \ I and k_{I} = S_{I}^{−1} k again, so that k_{I} is a discrete valuation ring, as before. This leads to a discrete valuation v_{I} on the field Q_{k} of fractions of k, as in Section 18.12.

If L is an ideal in k, then S_{I}^{−1} L is an ideal in k_{I}, which is the same as the extension of L with respect to the natural ring homomorphism from k into k_{I}, as in the previous section. Remember that this homomorphism is injective, because k is an integral domain, as in Section 14.2. Thus

S_{I}^{−1} L ≠ {0} (18.14.1)

when L ≠ {0}. In this case, there is a unique nonnegative integer v_{I}(L) such that S_{I}^{−1} L = {y ∈ k_{I} : v_{I}(y) ≥ v_{I}(L)}, (18.14.2)

as in Section 15.1. Equivalently,

S_{I}^{−1}L = {y ∈ Q_{k} : v_{I}(y) ≥ v_{I}(L)}, (18.14.3)

because of (18.12.4).

Observe that (18.14.2) is the same as saying that

S_{I}^{−1} L = (S_{I}^{−1} I)^{v_{I}(L)}, (18.14.4)

because of (18.12.5), and where the right side is as in Section 12.10. This corresponds to the definition of v_{I}(L) on the bottom of p11 of [263]. In fact, v_{I}(L) is defined more broadly in [263], and we shall return to this in Section 19.3. One can check that

v_{I}(L) = min{v_{I}(y) : y ∈ S_{I}^{−1} L} = min{v_{I}(y) : y ∈ L}. (18.14.5)

If y ∈ L, then v_{I}(L) ≤ v_{I}(y). (18.14.6)

If y ≠ 0, then v_{I}(y) = 0 (18.14.7)

for all but finitely many nonzero proper prime ideals I in k, as in Section 18.12. This implies that v_{I}(L) = 0 (18.14.8)

for all but finitely many nonzero proper prime ideals I in k when L ≠ {0}, as mentioned at the bottom of p11 in [263].

One can show that L is uniquely determined by the v_{I}(L)’s, as I runs through all nonzero proper prime ideals in k. This will be discussed in the next section.

If x ∈ k, then let I(x) be the ideal in k generated by x. If x ≠ 0, then I(x) ≠ {0}, and it is easy to see that

v_{I}(I(x)) = v_{I}(x) (18.14.9)

for every nonzero proper prime ideal I in k. An extension of this is mentioned on p12 of [263], and this will be discussed in Section 19.4.

Thus I(x) is uniquely determined, as an ideal in k, by the v_{I}(x)’s, as I runs though all of the nonzero proper prime ideals in k. This means that x is determined up to multiplication by an invertible element of k by the v_{I}(x)’s.

Let L′ be another ideal in k, and suppose that

L ⊆ L′. (18.14.10)

This implies that S_{I}^{−1} L ⊆ S_{I}^{−1} L′ (18.14.11)

for every nonzero proper prime ideals I in k. Using this, it is easy to see that

v_{I}(L′) ≤ v_{I}(L) (18.14.12)

for all such I. The converse to this will also be discussed in the next section.

458 CHAPTER 18. SOME PROPERTIES OF IDEALS, 2

18.15 Another ideal in k

Let us continue with the same notation and hypotheses as in the previous section. Let L be a nonzero ideal in k again, and put

(18.15.1) L_{2} = {y ∈ k : v_{I}(y) ≥ v_{I}(L) for every nonzero proper

prime ideal I in k}.

One can check that this is an ideal in k, and that L ⊆ L_{2}. (18.15.2) We would like to show that L = L_{2}, (18.15.3) as mentioned near the top of p12 of [263]. In particular, this implies that (18.15.4) L is uniquely determined by the v_{I}(L)’s, where I runs through all nonzero proper prime ideals in k, as mentioned in the previous section.

In order to get (18.15.3), it suffices to verify that S_{I}^{−1} L = S_{I}^{−1} L_{2} (18.15.5) for every maximal proper ideal I in k, as in Section 18.13. These are the same as the nonzero proper prime ideals in k in this case, as mentioned at the beginning of the previous section. Of course, S_{I}^{−1} L ⊆ S_{I}^{−1} L_{2} (18.15.6) for all such I, because of (18.15.2). Thus it is enough to check that S_{I}^{−1} L_{2} ⊆ S_{I}^{−1} L (18.15.7) for all such I.

This is the same as saying that v_{I}(L_{2}) ≥ v_{I}(L) (18.15.8) for all nonzero proper prime ideals I in k, because of (18.14.2) and its analogue for L_{2}. Equivalently, this means that min{v_{I}(y) : y ∈ L_{2}} ≥ v_{I}(L) (18.15.9) for every nonzero proper prime ideals I in k, because of the analogue of (18.14.5) for L_{2}. This follows directly from the definition of L_{2}.

Let L′ be another nonzero ideal in k, and consider

(18.15.10) L′_{2} = {y ∈ k : v_{I}(y) ≥ v_{I}(L′) for every nonzero proper

prime ideal I in k}, 18.15. ANOTHER IDEAL IN K 459

as before. If (18.14.12) holds for every nonzero proper prime ideal I in k, then L_{2} ⊆ L′_{2}. (18.15.11) This implies that (18.14.10) holds, because of (18.15.3), and its analogue for L′.

There is another way to characterize L in terms of the v_{I}(L)’s, as in Proposition 7 on p12 of [263]. This will be discussed in Section 19.2. In particular, the remarks in the preceding paragraph follow easily from this other characterization.

Let x, y ∈ k be given, with x, y ≠ 0, so that the ideals I(x) and I(y) in k that they generate are nonzero as well. Suppose that v_{I}(y) ≤ v_{I}(x) (18.15.12) for all nonzero proper prime ideals I in k, so that v_{I}(I(y)) ≤ v_{I}(I(y)) (18.15.13) for all such I, as in (18.14.9). This implies that I(x) ⊆ I(y), (18.15.14) as before. If v_{I}(x) = v_{I}(y) (18.15.15) for every nonzero proper prime ideal I in k, then it follows that I(x) = I(y). (18.15.16)

Note that x/y ∈ Q_{k}, and that (18.15.12) is the same as saying that v_{I}(x/y) = v_{I}(x) − v_{I}(y) ≥ 0 (18.15.17) for all nonzero proper prime ideals I in k. Under these conditions, we have that x/y ∈ k, (18.15.18) because x ∈ I(y), as in (18.15.14). This implies that

(18.15.19) k = {w ∈ Q_{k} : v_{I}(w) ≥ 0 for every nonzero proper

prime ideal I in k},

because v_{I}(w) ≥ 0 for every such I when w ∈ k, as in (18.12.8).

Similarly, if v_{I}(x/y) = v_{I}(x) − v_{I}(y) = 0 (18.15.20) for every nonzero proper prime ideal I in k, then x/y, y/x ∈ k. (18.15.21) This means that (18.15.22) {w ∈ k : w is invertible in k}

= {w ∈ Q_{k} : v_{I}(w) = 0 for every nonzero proper

prime ideal I in k},

because v_{I}(w) = 0 for every such I when w ∈ k is invertible in k, as in (18.12.12).

Ideals and field extensions

19.1 Products of ideals in k

Let us continue as in Sections 18.14 and 18.15, so that k is a Dedekind domain that is not a field, and for each nonzero proper prime ideal I in k, we take S_{I} = k \ I and k_{I} = S_{I}^{−1} k. In this case, k_{I} is discrete valuation ring, which leads to a discrete valuation v_{I} on the field Q_{k} of fractions of k, as in Section 18.12. If L ≠ {0} is an ideal in k, then v_{I}(L) ≥ 0 may be defined as in Section 18.14.

Let L′ ≠ {0} be another ideal in k, and consider the product L L′ of L and L′, which is another ideal in k, as in Section 12.8. Note that

L L′ ≠ {0}, (19.1.1)

because k is an integral domain. We also have that

S_{I}^{−1}(L L′) = (S_{I}^{−1} L) (S_{I}^{−1} L′) (19.1.2)

for all nonzero proper prime ideals I in k, as ideals in k_{I}, as in Section 12.13. More precisely, this was obtained from the analogous statement for extensions of products of ideals, as in Section 12.9.

One can check that

v_{I}(L L′) = v_{I}(L) + v_{I}(L′) (19.1.3)

for all nonzero proper prime ideals I in k, using (19.1.2) and the remarks in Section 15.1. This is stated more broadly on p12 of [263], and we shall return to that in Section 19.4.

If I is any nonzero proper prime ideal in k, then

v_{I}(I) = 1, (19.1.4)

as in (18.12.5). Note that v_{I}(k) = 0. (19.1.5)

19.2. MORE ON IDEALS IN K 461

Let I′ ≠ I (19.1.6)

be another nonzero proper prime ideal in k. Remember that I′ is a maximal proper ideal in k, because k is an integral domain of dimension one, as in Section 15.11. This means that I′ ⊈ I. (19.1.7)

Thus S_{I}^{−1} I′ = k_{I}, (19.1.8)

as in Section 12.13. It follows that

v_{I}(I′) = 0 (19.1.9)

under these conditions.

Let I_{1}, …, I_{r} be finitely many distinct nonzero proper prime ideals in k, and let n_{1}, …, n_{r} be nonnegative integers. Remember that I_{j}^{n_{j}} may be defined as an ideal in k as in Section 12.10, so that

∏_{j=1}^{r} I_{j}^{n_{j}} = I_{1}^{n_{1}} ⋯ I_{r}^{n_{r}} (19.1.10)

may be defined as an ideal in k as in Section 12.8.

If I is a nonzero proper prime ideal in k, then

(19.1.11)  v_{I}( ∏_{j=1}^{r} I_{j}^{n_{j}}) = n_{l} when I = I_{l} for some l
= 0 otherwise.

This follows from (19.1.3), (19.1.4), (19.1.5), and (19.1.9).

19.2 More on ideals in k

Let us continue with the same notation and hypotheses as in the previous section, and let L be a nonzero ideal in k again. Remember that v_{I}(L) = 0 for all but finitely many nonzero proper prime ideals I in k, as in Section 18.14. Put

L_{1} = ∏_{I} I^{v_{I}(L)}, (19.2.1)

as mentioned at the top of p12 of [263]. Although the product is formally taken over all nonzero proper prime ideals I in k, this is interpreted as the product over any finite set of such ideals that includes all of those with

v_{I}(L) > 0. (19.2.2)

If I′ is any nonzero proper prime ideal in k, then

v_{I′}(L_{1}) = v_{I′}(L), (19.2.3)

as in (19.1.11). This implies that

L = L_{1}, (19.2.4)

as in Section 18.15.

This corresponds to the existence part of Proposition 7 on p12 of [263], and the uniqueness part follows from (19.1.11). This is stated more broadly in [263], and we shall return to this in Section 19.5. Remember that the existence and uniqueness of this type of expression for L was also discussed in Section 18.4, using another approach.

Let L′ ≠ {0} be another ideal in k, and suppose that

v_{I}(L′) ≤ v_{I}(L) (19.2.5)

for all nonzero proper prime ideals I in k. This implies that

L ⊆ L′, (19.2.6)

as in Sections 18.14 and 18.15. This could also be obtained from the expression for L as in (19.2.1), and its analogue for L′.

If x ∈ k and x ≠ 0, then the ideal I(x) in k generated by x is nonzero too. Remember that v_{I}(x) = 0 for all but finitely many nonzero proper prime ideals I in k, as in Section 18.12. Observe that

I(x) = ∏_{I} I^{v_{I}(x)}, (19.2.7)

as in (19.2.1), because v_{I}(I(x)) = v_{I}(x) for every nonzero proper prime ideal I in k, as in (18.14.9). This gives another way to look at the statements about I(x) mentioned in Section 18.15.

19.3 Fractional ideals of k

Let k be a Dedekind domain that is not a field again, and for each nonzero proper prime ideal I in k, take S_{I} = k \ I and k_{I} = S_{I}^{−1} k, as before. Remember that k_{I} is a discrete valuation ring, which leads to a discrete valuation v_{I} on the field Q_{k} of fractions of k, as in Section 18.12.

As in Section 11.7, a fractional ideal of k is a submodule M of Q_{k}, as a module over k, such that x M ⊆ k (19.3.1)

for some x ∈ k with x ≠ 0. If I is a nonzero proper prime ideal in k, then one can defined fractional ideals of k_{I} in the same way. Remember that the field of 19.3. FRACTIONAL IDEALS OF K 463

fractions of k_{I} may be identified with Q_{k}, as in Section 14.2, so that a fractional ideal of k_{I} may be considered as a submodule of Q_{k}, as a module over k_{I}.

If l is an integer, then

{y ∈ Q_{k} : v_{I}(y) ≥ l} (19.3.2)

is a fractional ideal of k_{I}. One can check that every nonzero fractional ideal of k_{I} is of this form, as in Section 15.7. Note that l is uniquely determined by (19.3.2), because v_{I} is a discrete valuation on Q_{k}, as in Sections 14.14 and 15.1. Remember that (19.3.2) is equal to k_{I} when l = 0, as in Sections 15.1 and 18.12.

Let M be a fractional ideal of k, so that S_{I}^{−1} M may be defined as a module over k_{I}, as in Section 12.2. We may consider S_{I}^{−1} M as a submodule of S_{I}^{−1} Q_{k}, as a module over k_{I}, and S_{I}^{−1} Q_{k} may be identified with Q_{k}, as a module over k_{I}, as in Section 14.2. Thus S_{I}^{−1} M may be considered as a submodule of Q_{k}, as a module over k_{I}, as before.

It is easy to see that S_{I}^{−1} M is a fractional ideal of k_{I}, as in Section 14.2. Suppose that M ≠ {0}, which implies that

S_{I}^{−1} M ≠ {0}. (19.3.3)

This implies that there is a unique integer v_{I}(M) such that

S_{I}^{−1} M = {y ∈ Q_{k} : v_{I}(y) ≥ v_{I}(M)}, (19.3.4)

as before. If M is an ordinary ideal in k, then this is the same as the definition of v_{I}(M) in Section 18.14.

Remember that S_{I}^{−1} I is the same as (19.3.2), with l = 1, as in Section 18.12. If l ≥ 0, then (S_{I}^{−1} I)^{l} (19.3.5)

may be defined as an ideal in k_{I}, as in Section 12.10, and it is easy to see that this is the same as (19.3.2). We also have that (19.3.2) is invertible as a fractional ideal of k_{I} for every integer l, with inverse

{y ∈ Q_{k} : v_{I}(y) ≥ −l}, (19.3.6)

as in Section 15.7. Thus in fact (19.3.2) is the same as (19.3.5) for every integer l, where (19.3.5) is defined in terms of inverses when l < 0. This means that (19.3.4) is the same as saying that

S_{I}^{−1} M = (S_{I}^{−1} I)^{v_{I}(M)}, (19.3.7)

which corresponds to the definition of v_{I}(M) on the bottom of p11 of [263]. If x ∈ k, x ≠ 0, is as in (19.3.1), then

v_{I}(y) ≥ −v_{I}(x) (19.3.8)

for every y ∈ M. One can check that

v_{I}(M) = min{v_{I}(y) : y ∈ S_{I}^{−1} M} = min{v_{I}(y) : y ∈ M}, (19.3.9) as in Section 18.14. In particular,

v_{I}(M) ≥ −v_{I}(x). (19.3.10)

Of course, v_{I}(M) ≤ v_{I}(y) (19.3.11)

for every y ∈ M.

If w ∈ Q_{k} and w ≠ 0, then v_{I}(w) = 0 for all but finitely many nonzero proper prime ideals I in k, as in Section 18.12. This implies that

v_{I}(M) = 0 (19.3.12)

for all but finitely many nonzero proper prime ideals I in k, because of (19.3.10) and (19.3.11). This is mentioned at the bottom of p11 of [263].

19.4 More on v_{I}(M)

Let us continue with the same notation and hypotheses as in the previous section. If w ∈ Q_{k} and w ≠ 0, then it is easy to see that

(19.4.1) w M is another nonzero fractional ideal of k.

We also have that v_{I}(w M) = v_{I}(w) + v_{I}(M). (19.4.2)

In particular, v_{I}(w k) = v_{I}(w), (19.4.3)

because v_{I}(k) = 0, as in Section 19.1. This is mentioned on p12 of [263].

Let M′ be another nonzero fractional ideal of k. Remember that there is an x ∈ k such that x ≠ 0 and (19.3.1) holds, and similarly there is an x′ ∈ k such that x′ ≠ 0 and x′ M′ ⊆ k. (19.4.4)

It follows that x x′ ≠ 0, because k is an integral domain, and

x x′ M, x x′ M′ ⊆ k, (19.4.5)

because M, M′ are submodules of Q_{k}, as a module over k.

If v_{I}(M) = v_{I}(M′) (19.4.6)

for all nonzero proper prime ideals I in k, then

M = M′. (19.4.7)

This follows from the analogous statement for nonzero ideals in k, as in Section 18.15, and (19.4.3), (19.4.5). Similarly, if

v_{I}(M′) ≤ v_{I}(M) (19.4.8)

19.4. MORE ON V_{I}(M) 465

for all nonzero proper prime ideals I in k, then

M ⊆ M′, (19.4.9)

as before.

Alternatively, if M is a nonzero fractional ideal of k, then it is easy to see that

(19.4.10) M_{2} = {y ∈ Q_{k} : v_{I}(y) ≥ v_{I}(M) for every nonzero proper

prime ideal I in k}

is a submodule of Q_{k}, as a module over k, with

M ⊆ M_{2}. (19.4.11)

One can show that M = M_{2}, (19.4.12)

using the same type of argument as in Section 18.15, as indicated near the top of p12 of [263]. This uses some remarks in Sections 12.2 and 12.4, as in Section 18.13.

If M, N are submodules of Q_{k}, as a module over k, then their product M N may be defined as a submodule of Q_{k}, as a module over k, as in Section 11.7. This includes products of ideals in k, as in Section 12.8. If M and N are fractional ideals of k, then it is easy to see that

(19.4.13) M N is a fractional ideal of k

too. If M, N ≠ {0}, then M N ≠ {0} (19.4.14)

as well.

If I is a nonzero proper prime ideal in k again, then

S_{I}^{−1}(M N) = (S_{I}^{−1} M) (S_{I}^{−1} N), (19.4.15)

as in Section 14.3. One can use this to get that

v_{I}(M N) = v_{I}(M) + v_{I}(N) (19.4.16)

for all nonzero fractional ideals M, N of k, as on p12 of [263].

Remember that every nonzero fractional ideal M of k is invertible, because k is a Dedekind domain, as in Section 15.12. The inverse M^{−1} satisfies

v_{I}(M^{−1}) = −v_{I}(M), (19.4.17)

because of (19.4.16).

Let I_{1}, …, I_{r} be finitely many distinct nonzero proper prime ideals in k, and let n_{1}, …, n_{r} be integers. We can define

I_{j}^{n_{j}} (19.4.18) as a nonzero fractional ideal of k, using inverses when n_{j} is negative. Thus

∏_{j=1}^{r} I_{j}^{n_{j}} = I_{1}^{n_{1}} ⋯ I_{r}^{n_{r}} (19.4.19)

may be defined as a nonzero fractional ideal of k as well. If I is a nonzero proper prime ideal in k, then

(19.4.20)  v_{I}( ∏_{j=1}^{r} I_{j}^{n_{j}}) = n_{l} when I = I_{l} for some l
= 0 otherwise,

as in Section 19.1.

19.5 More on fractional ideals

Let us continue with the same notation and hypotheses as in the previous two sections. Let M be a nonzero fractional ideal of k again, and remember that v_{I}(M) = 0 for all but finitely many nonzero proper prim ideals I in k, as in (19.3.12). Consider

M_{1} = ∏_{I} I^{v_{I}(M)}, (19.5.1)

as mentioned at the top og p12 of [263], which may be interpreted as the product over any finite set of nonzero proper prime ideals I in k that includes all of those with v_{I}(M) ≠ 0. (19.5.2)

If I′ is any nonzero proper prime ideal in k, then

v_{I′}(M_{1}) = v_{I′}(M), (19.5.3)

as in (19.4.20), so that M = M_{1}, (19.5.4)

as in the previous section. This corresponds to Proposition 7 on p12 of [263], as in Section 19.2.

If M and N are submodules of Q_{k}, as a module over k, then put

(M : N)_{Q_{k}} = {y ∈ Q_{k} : y N ⊆ M}, (19.5.5)

as on p11 of [263]. This is the same as in Section 14.3 when M = k, which corresponds to (k : N) in the notation of Section 11.7. As in Section 14.3, the subscript Q_{k} is included because similar notation was used in Section 12.8 for something else. It is easy to see that (19.5.5) is a submodule of Q_{k}, as a module over k, with (M : N)_{Q_{k}} N ⊆ M. (19.5.6)

If N is an invertible ideal of k, as in Section 11.7, then it follows that (M : N)_{Q_{k}} = (M : N)_{Q_{k}} N N^{−1} ⊆ M N^{−1}. (19.5.7) In this case, we also have that M N^{−1} N = M, (19.5.8) which implies that M N^{−1} ⊆ (M : N)_{Q_{k}}. (19.5.9) This means that (M : N)_{Q_{k}} = M N^{−1} (19.5.10) under these conditions. This was also mentioned in Section 11.7 when M = k, and it is implicit in a remark on p12 of [263]. Note that the remarks in this and the preceding paragraph can be used when k is any integral domain.

If M and N are nonzero fractional ideals of k, then they are invertible, because k is a Dedekind domain, as in Section 15.12. If I is a nonzero proper prime ideal in k, then we get that v_{I}((M : N)_{Q_{k}}) = v_{I}(M N^{−1}) = v_{I}(M) − v_{I}(N), (19.5.11) as on p12 of [263]. This uses (19.4.16), (19.4.17) in the second step.

If M, M′ are nonzero fractional ideals of k with M ⊆ M′, (19.5.12) then S_{I}^{−1} M ⊆ S_{I}^{−1} M′ (19.5.13) for every nonzero proper prime ideal I in k. This implies that v_{I}(M′) ≤ v_{I}(M), (19.5.14) by the definition of v_{I}(M).

If M and N are fractional ideals of k, then there is a y ∈ k such that y ≠ 0 and y M, y N ⊆ k, (19.5.15) as in (19.4.5). Using this, it is easy to see that (19.5.16) M + N is a fractional ideal of k, which is nonzero if M or N is nonzero. Of course, M, N ⊆ M + N, (19.5.17) so that v_{I}(M + N) ≤ v_{I}(M), v_{I}(N) (19.5.18) for every nonzero proper prime ideal I in k, as in (19.5.14), when M, N ≠ {0}. In fact, one can check that v_{I}(M + N) = min(v_{I}(M), v_{I}(N)), (19.5.19) as mentioned on p12 of [263].

19.6 Using some ideals in k

Let k be a Dedekind domain that is not a field, and take S_{I} = k \ I and k_{I} = S_{I}^{−1} k for each nonzero proper prime ideal I in k, as before. This leads to a discrete valuation v_{I} on the field Q_{k} of fractions of k, as in Section 18.12.

Let I_{1}, …, I_{l} (19.6.1)

be finitely many distinct nonzero proper prime ideals in k, and let

x_{1}, …, x_{l} ∈ Q_{k} (19.6.2)

and integers n_{1}, …, n_{l} be given. Under these conditions, there is an x ∈ Q_{k} such that v_{I_{j}}(x − x_{j}) ≥ n_{j} (19.6.3)

for each j = 1, …, l, and v_{I}(x) ≥ 0 (19.6.4)

for every nonzero proper prime ideal I in k that is not one of the ideals I_{1}, …, I_{l}. This is the approximation lemma on p12 of [263].

Suppose first that x_{1}, …, x_{l} ∈ k. (19.6.5)

If l = 1, then we can take x_{1} = x, and so we may as well suppose that l ≥ 2. It is easy to reduce to the case where all but one of x_{1}, …, x_{l} is equal to 0. We may as well suppose that x_{j} = 0 for j ≥ 2, (19.6.6)

by rearranging the I_{j}’s if necessary. We may as well suppose also that n_{j} ≥ 0 for each j.

Consider

L = I_{1}^{n_{1}} + ∏_{j=2}^{l} I_{j}^{n_{j}}, (19.6.7)

which is an ideal in k. We would like to check that

v_{I}(L) = 0 (19.6.8)

for every nonzero proper prime ideal I in k. Observe that

(19.6.9)  v_{I}(L) = min (v_{I}(I_{1}^{n_{1}}), v_{I}( ∏_{j=2}^{l} I_{j}^{n_{j}}))

= min (n_{1} v_{I}(I_{1}), ∑_{j=2}^{l} n_{j} v_{I}(I_{j})),

where the first step is as in (19.5.19), and the second step is as in Section 19.1.

Remember that v_{I}(I_{j}) ≥ 0 for each j, as in Section 18.14. If I = I_{1}, then (19.6.8) follows from the fact that

v_{I_{1}}(I_{j}) = 0 (19.6.10)

when j ≥ 2, because I_{1} ≠ I_{j}, as in Section 19.1. Similarly, if I ≠ I_{1}, then (19.6.8) follows from the fact that

v_{I}(I_{1}) = 0. (19.6.11)

Using (19.6.8), we get that L = k, (19.6.12)

as in Section 18.15. This means that x_{1} may be expressed as

x_{1} = x + y, (19.6.13)

where y ∈ I_{1}^{n_{1}}, and x ∈ ∏_{j=2}^{l} I_{j}^{n_{j}}. It is easy to see that x has the desired properties in this case.

We shall consider arbitrary elements x_{1}, …, x_{l} of Q_{k} in the next section. In fact, we shall see that the problem can be reduced to the one considered in this section. A corollary of this will also be discussed.

19.7 Reducing to the previous case

Let us continue with the same notation and hypotheses as in the previous section. If x_{1}, …, x_{l} ∈ Q_{k}, then each x_{j} may be expressed as

x_{j} = w_{j}/r, (19.7.1)

where w_{j} ∈ k, r ∈ k does not depend on j, and r ≠ 0. In this case, we would like to find x ∈ Q_{k} of the form

x = w/r, (19.7.2)

w ∈ k, with the desired properties. These properties may be restated as saying that v_{I_{j}}(w − w_{j}) ≥ n_{j} + v_{I_{j}}(r) (19.7.3)

for each j = 1, …, l, and v_{I}(w) ≥ v_{I}(r) (19.7.4)

for every nonzero proper prime ideal I in k that is not one of the ideals I_{1}, …, I_{l}.

Remember that v_{I}(r) = 0 for all but finitely many nonzero proper prime ideals I in k, because r ≠ 0, as in Section 18.12. Of course, (19.7.4) is the same as saying that v_{I}(w) ≥ 0 (19.7.5)

when v_{I}(r) = 0, and this condition holds automatically when w ∈ k. This means that there are only finitely many nonzero proper prime ideals I that are not one of the ideals I_{1}, …, I_{l}, and for which (19.7.4) is more restrictive than (19.7.5). These ideals may be added to the collection I_{1}, …, I_{l}, to get conditions like those considered in the first case. This shows that this second case may be obtained from the first case, using this larger collection of prime ideals in k.

Suppose now that

(19.7.6) k has only finitely many prime ideals.

Under these conditions, the corollary on p12 of [263] states that

(19.7.7) k is a principal ideal domain.

Remember that principal ideal domains are Dedekind domains, as in Section 15.11.

Remember also that ideals in k may be expressed as products of prime ideals, as in Section 19.2. In order to show that every ideal in k is principal, it suffices to show that every nonzero proper prime ideal I′ in k is principal. Let us take

I_{1} = I′, (19.7.8)

and let I_{2}, …, I_{l} be a list of all other nonzero proper prime ideals in k, without any repetitions.

Let x_{1} be an element of I_{1} such that

v_{I_{1}}(x_{1}) = 1. (19.7.9)

More precisely, there is an element of k_{I_{1}} = S_{I_{1}}^{−1} k with this property, because k_{I_{1}} is a discrete valuation ring. Any element of k_{I_{1}} with this property is an element of S_{I_{1}}^{−1} I_{1}, as in Section 18.12. Of course, the elements of S_{I_{1}} = k \ I_{1} are invertible in k_{I_{1}}, by construction, so that v_{I_{1}} = 0 on S_{I_{1}}. This leads to an element of I_{1} as in (19.7.9).

Let us take x_{j} = 1 for j ≥ 2, (19.7.10)

and n_{1} = 2, n_{j} = 1 for j ≥ 2. (19.7.11)

One can use the argument in the previous section to get x ∈ k such that

v_{I_{1}}(x) = 1, v_{I_{j}}(x) = 0 for j ≥ 2. (19.7.12)

This means that

v_{I_{1}}(x k) = 1, v_{I_{j}}(x k) = 0 for j ≥ 2. (19.7.13)

These are the same as the values of v_{I_{j}}(I_{1}), as in Section 19.1. This implies that I_{1} = x k, (19.7.14)

as in Section 18.15.

19.8 A result about linear independence

Let k be a field, and let Σ be a nonempty semigroup, with the semigroup operation expressed multiplicatively. A homomorphism from Σ into the multiplicative group k \ {0} may be called a character of Σ, as on p208 of [177].

Theorem 7 on p209 of [177], due to Artin, says that

(19.8.1) the characters on Σ of this type are linearly independent

in the space of all functions on Σ with values in k, as a vector space over k. This means that if χ_{1}, …, χ_{n} are finitely many distinct characters on Σ of this type, and if a_{1} χ_{1} + ⋯ + a_{n} χ_{n} = 0 (19.8.2)

for some a_{1}, …, a_{n} ∈ k, then a_{1} = ⋯ = a_{n} = 0. Of course, this is clear when n = 1.

Suppose for the sake of a contradiction that (19.8.2) holds for some n ≥ 2, and where a_{1}, …, a_{n} are not all 0. We may as well suppose also that n is as small as possible, which implies that

a_{j} ≠ 0 (19.8.3)

for each j.

Because χ_{1} ≠ χ_{2}, there is a z ∈ Σ such that

χ_{1}(z) ≠ χ_{2}(z). (19.8.4)

If x ∈ Σ, then a_{1} χ_{1}(x z) + ⋯ + a_{n} χ_{n}(x z) = 0, (19.8.5)

by hypothesis. This means that

a_{1} χ_{1}(z) χ_{1} + ⋯ + a_{n} χ_{n}(z) χ_{n} = 0 (19.8.6)

on Σ, because χ_{1}, …, χ_{n} are characters.

It follows that

a_{2} (χ_{2}(z) χ_{1}(z)^{−1} − 1) χ_{2} + ⋯ + a_{n} (χ_{n}(z) χ_{1}(z)^{−1} − 1) χ_{n} = 0 (19.8.7)

on Σ, by multiplying the left side of (19.8.6) by χ_{1}(z)^{−1}, and subtracting the left side of (19.8.2). The coefficient of χ_{2} is not zero, because of (19.8.3) and (19.8.4). Thus we can reduce the size of the relation, so that n is not minimal.

19.9 Algebraic elements and extensions

Let k be a field, and let X be an indeterminate. A formal polynomial in X with coefficients in k may be expressed as

f(X) = ∑_{j=0}^{n} f_{j} X^{j} (19.9.1)

for some nonnegative integer n, with f_{j} ∈ k for each j, as in Section 4.3. If

f_{n} ≠ 0, (19.9.2)

then the degree of f(X) is equal to n. If

f_{n} = 1, (19.9.3)

then f(X) is a monic polynomial in X of degree n.

Let k_{1} be a field that contains k as a subfield. Note that k_{1} may be considered as a commutative associative algebra over k. If x ∈ k_{1}, then

f(x) = ∑_{j=0}^{n} f_{j} x^{j} (19.9.4)

defines an element of k_{1}, as in Section 4.4. Remember that

f(X) ↦ f(x) (19.9.5)

defines a homomorphism from k[X] into k_{1}, as algebras over k, under these conditions. If f(x) = 0, (19.9.6)

then x is said to be a root of f(X) in k_{1}.

Of course, k_{1} may be considered as a vector space over k in particular. If k_{1} has finite dimension as a vector space over k, then k_{1} may be called a finite extension of k. The dimension of k_{1}, as a vector space over k, may be denoted

[k_{1} : k], (19.9.7)

and may be called the degree of k_{1} over k.

Let k_{2} be a field that contains k_{1} as a subfield, so that k_{2} contains k as a subfield as well. If k_{1} is a finite extension of k and k_{2} is a finite extension of k_{1}, then it is well known that k_{2} is a finite extension of k, with

[k_{2} : k] = [k_{2} : k_{1}] [k_{1} : k]. (19.9.8)

This corresponds to Corollary 1 on p433 of [24], Proposition 2 on p162 of [177], and Theorem 3 on p443 of [195].

An element of k_{1} is said to be algebraic over k if it is the root of a polynomial whose coefficients are in k and not all equal to 0, as on p421 of [24], p161 of [177], and p439 of [195]. If

(19.9.9) every element of k_{1} is algebraic over k,

then k_{1} is said to be algebraic as an extension of k, as on p161 of [177], and p456 of [195]. If k_{1} is a finite extension of k, then it is well known that k_{1} is algebraic over k, as in Theorem 8 on p430 of [24], Proposition 1 on p161 of [177], and mentioned on p443 of [195].

It is also well known that an algebraic element x of k_{1} over k is the root of a unique monic polynomial of minimal degree with coefficients in k, and that this polynomial is irreducible. This may be called the minimal polynomial of x over k, as on p423 of [24], and p440 of [195]. This is denoted Irr(x, k, X) on p161 of [177].

If x is any element of k_{1}, then let k(x) be the subfield of k_{1} generated by k and x. If x is algebraic over k, then

(19.9.10) k(x) is a finite extension of k,

with degree equal to the degree of the minimal polynomial of x over k. This corresponds to Theorem 7 on p429 of [24], Proposition 3 on p163 of [177], and a remark on p443 of [195].

Similarly, if x_{1}, …, x_{n} are finitely many elements of k_{1}, then k(x_{1}, …, x_{n}) denotes the subfield of k_{1} generated by k and x_{1}, …, x_{n}. If x_{1}, …, x_{n} are algebraic over k, then

(19.9.11) k(x_{1}, …, x_{n}) is a finite extension of k,

as in Proposition 5 on p165 of [177]. This also works when x_{1} is algebraic over k and x_{l} is algebraic over k(x_{1}, …, x_{l−1}) when l ≥ 2, as in Corollary 4 on p433 of [24].

If E is a finite extension of k, then E is generated by k and finitely many elements of E, as in Proposition 4 on p164 of [177].

19.10 Distinguished classes and the compositum

Let L be a field, and let k, E, and F be subfields of L, with

k ⊆ E, F. (19.10.1)

As on p163 of [177], (19.10.2) the compositum of E and F

is defined to be the smallest subfield of L that contains E and F. This may be denoted E F, as in [177], and of course

k ⊆ E F. (19.10.3)

One may consider E F as an extension of F, which may be called the translation or lifting of E, as an extension of k, to F, as on p164 of [177].

Similarly, the compositum of any nonempty family of subfields of L may be defined as the smallest subfield of L that contains all of the subfields in the family, as on p164 of [177].

Let x_{1}, …, x_{n} be finitely many elements of L, and suppose for the moment that E = k(x_{1}, …, x_{n}) (19.10.4) is the subfield of L generated by k and x_{1}, …, x_{n}, as in the previous section. One may say that E is finitely generated over k in this case, as on p164 of [177]. One can check that E F = F (x_{1}, …, x_{n}) (19.10.5)

under these conditions, as in [177]. Equivalently, the compositum of E and F is the subfield of L generated by F and x_{1}, …, x_{n}. Indeed, k(x_{1}, …, x_{n}) consists of quotients of polynomials in x_{1}, …, x_{n} with coefficients in k, where the denominator is nonzero, as in [177], and similarly for F (x_{1}, …, x_{n}).

Let C be a class of extensions of fields. We say that

(19.10.6) C is distinguished

if it satisfies the following three properties, as on p165 of [177].

Let k, E, F be fields, with

k ⊆ F ⊆ E. (19.10.7)

The first condition is that

(19.10.8) E is in C as an extension of k

if and only if (19.10.9) F is in C as an extension of k

and (19.10.10) E is in C as an extension of F.

Let k, E, F be subfields of a field L that satisfy (19.10.1). If (19.10.8) holds, then the second condition asks that

(19.10.11) E F is in C, as an extension of F.

Let k, E, F be subfields of a field L that satisfy (19.10.1) again. If (19.10.8) and (19.10.9) hold, then the third condition asks that

(19.10.12) E F is in C, as an extension of k.

This follows from the previous two conditions, as mentioned in [177].

Some examples of distinguished classes will be discussed in the next section, and some other examples will be considered later.

19.11 A couple of distinguished classes

Let us check that

(19.11.1) the class of finite extensions is distinguished,

as mentioned on p166 of [177]. Indeed, the first condition for a distinguished class follows from the remarks in Section 19.9. Concerning the second condition, let k, E, F be subfields of a field L with k ⊆ E, F, and suppose that E is a finite extension of k. This implies that E is generated by k and finitely many elements of E, so that the compositum E F of E and F is generated by F and the same finitely many elements of E, as in the previous section. It follows that

(19.11.2) E F is a finite extension of F,

as in Section 19.9, because the elements of E are algebraic over k, and thus over F.

We also have that

(19.11.3) the class of algebraic extensions is distinguished,

as in [177]. To see this, let k, E, F be fields, with k ⊆ F ⊆ E. If E is algebraic over k, then it is easy to see that F is algebraic over k, and that E is algebraic over F.

Conversely, suppose that F is algebraic over k, and that E is algebraic over F. Let α ∈ E be given, so that

a_{n} α^{n} + a_{n−1} α^{n−1} + ⋯ + a_{1} α + a_{0} = 0 (19.11.4)

for some a_{0}, a_{1}, …, a_{n} ∈ F that are not all equal to 0. Consider the subfield

F_{0} = k(a_{0}, a_{1}, …, a_{n}) (19.11.5)

of F generated by k and a_{0}, a_{1}, …, a_{n}. Note that

(19.11.6) F_{0} is a finite extension of k,

as in Section 19.9, because a_{0}, a_{1}, …, a_{n} are algebraic over k, by hypothesis. Of course, α is algebraic over F_{0}, by construction.

Consider the subfield F_{0}(α) of E generated by F_{0} and α. This is a finite extension of F_{0}, because α is algebraic over F_{0}. It follows that

(19.11.7) F_{0}(α) is a finite extension of k,

because F_{0} is a finite extension of k. This implies that α is algebraic over k. Thus E is algebraic over k, so that the class of algebraic extensions satisfies the first condition in the definition of a distinguished class.

Concerning the second condition in the definition of a distinguished class, let k, E, and F be subfields of a field L with k ⊆ E, F, and suppose that E is algebraic over k. In particular, this means that every element of E is algebraic over F. It follows that the subfield of E F generated by F and finitely many elements of E is finite over F, as in Section 19.9. Thus the elements of such a subfield of E F are algebraic over F. One can use this to get that

(19.11.8) E F is algebraic over F.

More precisely, E F is equal to the union of all of the subfields generated by F and finitely many elements of E. This is because this union is a subfield of E F, and it contains E and F by construction.

19.12 Some remarks about embeddings

Let k and L be fields, and let σ be an embedding of k into L. This means that σ is an injective homomorphism from k into L, as on p167 of [177]. Thus σ(k) is a subfield of L, and σ determines a field isomorphism from k onto σ(k).

Let E be an extension of k. We shall sometimes be interested embeddings τ of E into L that are extensions of σ, i.e., which are equal to σ on k. One may say that (19.12.1) τ is an embedding of E into L over σ

under these conditions, as on p167 of [177].

If k is a subfield of L, then we can take σ to be the identity mapping on k. In this case, one may say that

(19.12.2) τ is an embedding of E into L over k

when τ is equal to the identity mapping on k, as on p167 of [177].

If σ is an embedding of k into L and t ∈ k, then the notation

t^{σ} = σ(t) (19.12.3)

is sometimes used, as on p167 of [177]. Let X be an indeterminate, and let f(X) be a formal polynomial in X with coefficients in k, as in (19.9.1). Thus

f^{σ}(X) = ∑_{j=0}^{n} σ(f_{j}) X^{j} (19.12.4)

is a formal polynomial in X with coefficients in σ(k) ⊆ L, as on p115 of [177].

Let τ be an extension of σ to an embedding of E into L, as before. If x ∈ E, then f(x) is defined as an element of E as in Sections 4.4 and 19.9, and similarly f^{σ}(τ(x)) may be defined as an element of L. It is easy to see that

τ(f(x)) = f^{σ}(τ(x)) (19.12.5)

under these conditions. In particular, if x is a root of f(X) in E, then

(19.12.6) τ(x) is a root of f^{σ}(X) in L,

as on p167 of [177].

Let ρ be an embedding of E into itself over k, so that ρ is an injective homomorphism of E into itself that is equal to the identity mapping on k. In particular,

(19.12.7) ρ may be considered as a linear mapping from E into itself,

as a vector space over k, as on p167 of [177]. If E is a finite extension of k, then it follows that ρ(E) = E, (19.12.8) by a well-known result in linear algebra.

Let E′ be a subfield of E that contains k, and suppose that

ρ(E′) ⊆ E′. (19.12.9)

If E′ is a finite extension of k, then

ρ(E′) = E′, (19.12.10)

as in (19.12.8).

Let f(X) be a formal polynomial in X with coefficients in k. Note that ρ maps the set of roots of f(X) in E into itself, as before. If E′ is the subfield of E generated by k and the roots of f(X) in E, then (19.12.9) holds. If the coefficients of f(X) are not all 0, then there are only finitely many roots of f(X) in E, each of which is algebraic over k. This implies that E′ is a finite extension of k, as in (19.9.11), so that (19.12.10) holds under these conditions.

Lemma 1 on p167 of [177] states that (19.12.8) also holds when E is an algebraic extension of k. This is because every element of E is contained in a subfield E′ of E that is a finite extension of k that satisfies (19.12.9), as in the preceding paragraph.

19.13 Algebraically closed fields

Let k be a field again, and let X be an indeterminate. If every formal polynomial in X with coefficients in k of degree at least one has a root in k, then k is said to be algebraically closed, as on p169 of [177]. The term algebraically complete is also used for this, as on p437 of [24].

An algebraically closed field \overline{k} that contains k as a subfield is said to be an algebraic closure of k if

(19.13.1) \overline{k} is also algebraic over k.

Every field k has an algebraic closure, as in the corollary on p170 of [177].

Let L be an algebraically closed field, and let σ be an embedding of k into L. If E is an algebraic extension of k, then one may be interested in extensions of σ to embeddings of E into L, as on p170 of [177].

Suppose for the moment that

E = k(x) (19.13.2)

for some x ∈ E, which is to say that E is generated by k and x. Note that any extension of σ to an embedding τ from E into L is uniquely determined by τ(x).

Of course, if E is an algebraic extension of k, then x should be algebraic over k. Let f(X) be the minimal polynomial of x over k, as in Section 19.9. This leads to a polynomial f^{σ}(X) in X with coefficients in σ(k), as in (19.12.4). If τ is an embedding of E into L that is an extension of σ, then τ(x) is a root of f^{σ}(X) in L, as in (19.12.6).

In fact, every root of f^{σ}(X) in L occurs in this way, as on p171 of [177]. This implies that

(19.13.3) the number of extensions of σ to embeddings of E into L

is equal to (19.13.4) the number of distinct roots of f^{σ}(X) in L,

as in Proposition 8 on p171 of [177]. In particular, the number of these extensions is less than or equal to the degree of f^{σ}(X), which is the same as the degree of f(X). Of course, this also shows that there is such an extension, because L is algebraically closed.

If E is any algebraic extension of k, then Theorem 2 on p171 of [177] says that (19.13.5) there is an extension of σ to an embedding of E into L.

If E is also algebraically closed, and L is algebraic over σ(k), then any such extension of σ maps E onto L, as in [177].

If \overline{k} and \overline{k}′ are algebraic closures of k, then there is an isomorphism from \overline{k} onto \overline{k}′ that is equal to the identity mapping on k, as in the corollary on p172 of [177].

Let k_{2} be a field that contains k as a subfield, and let k_{1} be a subfield of k_{2} that contains k. If k_{1} is algebraic over k, and k_{2} is algebraic over k_{1}, then

(19.13.6) k_{2} is algebraic over k,

as in Corollary 4 on p444 of [195]. This is also part of Proposition 6 on p166 of [177].

If k_{1} is algebraic over k, and \overline{k_{1}} is an algebraic closure of k_{1}, then

(19.13.7) \overline{k_{1}} is an algebraic closure of k.

This follows from the fact that \overline{k_{1}} is algebraic over k, as in the preceding paragraph. This corresponds to a remark at the top of p174 of [177].

19.14 Splitting fields

Let k, k_{1} be fields, with k a subfield of k_{1}, and let X be an indeterminate. Also let f(X) be a formal polynomial in X with coefficients in k, as in Section 19.9, with degree at least one. Suppose that

(19.14.1) f(X) can be expressed as a product of linear factors as a formal polynomial in X with coefficients in k_{1},

and that

(19.14.2) k_{1} is generated as a field containing k by the roots of f(X) in k_{1}.

Under these conditions, k_{1} may be called a root field of f(X), as on p452 of [24], or a splitting field of f(X), as on p173 of [177], and p447 of [195]. It is well known that f(X) has a splitting field, as in Theorem 1 on p453 of [24], and the first part of Theorem 4 on p447 of [195].

Suppose that k_{2} is a field that contains k, and that f(X) can be expressed as a product of linear factors as a formal polynomial in X with coefficients in k_{2}. In this case,

(19.14.3) the subfield of k_{2} generated by k and the roots of f(X) in k_{2} is a splitting field of f(X).

In particular, one can take k_{2} to an algebraic closure of k, as mentioned on p174 of [177].

If E and K are splitting fields of f(X), then there is an isomorphism from E onto K that is the identity mapping on k, as in Theorem 2 on p454 of [24], the first part of Theorem 3 on p173 of [177], and the second part of Theorem 4 on p447 of [195]. If \overline{k} is an algebraic closure of k and

K ⊆ \overline{k}, (19.14.4)

then the second part of Theorem 3 on p173 of [177] says that any embedding of E into \overline{k} that is the identity mapping on k maps E onto K.

Let I be a nonempty set, and suppose that f_{j}(X) is a formal polynomial in X with coefficients in k and degree at least one for each j ∈ I. An extension k_{1} of k is said to be a splitting field of {f_{j}(X)}_{j∈I} if

(19.14.5) f_{j}(X) splits into a product of linear factors with coefficients in k_{1} for each j ∈ I,

and

(19.14.6) k_{1} is generated as a field containing k by the set of the roots of all of the f_{j}(X)’s, j ∈ I,

as on p174 of [177]. If I has only finitely many elements, then this is the same as a splitting field of the product of the f_{j}(X)’s, j ∈ I, as in the remark on p175 of [177].

Let \overline{k} be an algebraic closure of k. If j ∈ I, then let k_{j} be a splitting field of f_{j}(X) in \overline{k}. The subfield of \overline{k} generated by the subfields k_{j}, j ∈ I, is a splitting field of {f_{j}(X)}_{j∈I}, as mentioned on p174 of [177]. This is the smallest subfield of \overline{k} that contains k_{j} for each j ∈ I, which may be called the compositum of these subfields, as in Section 19.10.

Suppose that E and K are splitting fields of \{f_{j}(X)\}_{j∈I}, and let \overline{K} be an algebraic closure of K. Under these conditions, the corollary on p174 of [177] says that if σ is an embedding of E into \overline{K} that is equal to the identity mapping on k, then σ(E) = K. (19.14.7)

More precisely, for each j ∈ I, f_{j}(X) has a unique splitting field E_{j} in E, and a unique splitting field K_{j} in K. We also have that

σ(E_{j}) = K_{j} (19.14.8)

for each j ∈ I, as before. One can use this to get (19.14.7), because E and K are generated by the E_{j}’s and K_{j}’s, j ∈ I, as mentioned earlier.

19.15 Galois groups and normal extensions

Let k and K be fields, with k ⊆ K. An automorphism of K as a field is said to be an automorphism over k if it is equal to the identity mapping on k. Of course, the collection of automorphisms of K as a field is a group with respect to composition of mappings. The collection Gal(K/k) of automorphisms of K over k is a subgroup of this group, which is the Galois group of K over k.

This corresponds to the definition on p460 of [24]. The term Galois group is used the definition on p461 of [24] in the case where K is the root or splitting field of a polynomial with coefficients in k. In this case, one may also refer to this as the Galois group of the polynomial.

Galois groups of field extensions are mentioned on p190 of [177], in connection with Proposition 12 on p189 of [177], and again on p192 of [177]. The notation G = G(K/k) for the Galois group is used on p192 of [177]. The Galois group of a polynomial, as the Galois group of a splitting field of the polynomial, is mentioned on p199 of [177].

The Galois group of a field extension is defined on p458 of [195], using the notation Γ = Γ(K/k). The Galois group of a polynomial is defined on p450 of [195], as the Galois group of a splitting field of the polynomial.

Let k, K be fields, with k ⊆ K. There seem to be a few related ways in which the normality of K as an extension of k is sometimes defined.

Normality of K over k is defined on p459 of [195] to mean that for each u ∈ K with u ∉ k, there is an automorphism θ of K over k such that

θ(u) ≠ u. (19.15.1)

Equivalently, this means that

(19.15.2) k is exactly the set of points in K that are fixed by every element of the Galois group of K over k.

Let X be an indeterminate, and suppose now that K is an algebraic extension of k. Let us say that K is normal as an extension of k if

(19.15.3) for every irreducible formal polynomial f(X) in X with coefficients in k that has a root in K,

we have that

(19.15.4) f(X) can be expressed as a product of linear factors with coefficients in K.

This corresponds to the condition NOR 3 in Theorem 4 on p175 of [177]. Normality is defined in the same way on p468 of [24] when K is a finite extension of k.

If K is a finite extension of k, then Theorem 12 on p459 of [195] says that certain conditions on K are equivalent. Condition (ii) in that theorem is that K be normal over k in the sense of [195], as in (19.15.2). The definition of normality in the preceding paragraph corresponds to part of condition (iii) in that theorem. The other part of condition (iii) in that theorem is that K be separable over k, as in Section 20.4.

Part of Theorem 4 on p175 of [177] is that normality as in the condition NOR 3 is equivalent to the condition NOR 2, which says that

(19.15.5) K is the splitting field of a family of polynomials in k[X].

Theorem 14 on p468 of [24] is the analogous statement for finite extensions, i.e., if K is a finite extension of k, then K is normal over k if and only if

(19.15.6) K is the root field, or equivalently splitting field, of some polynomial with coefficients in k.

Condition (i) in Theorem 12 on p459 of [195] is that K be the splitting field of a polynomial with coefficients in k that is separable, as in the next section.

Let \overline{k} be an algebraic closure of k, and suppose that K is a subfield of \overline{k}. We can reduce to that case, using an embedding of K into \overline{k} over k, as in Section 19.13. Theorem 4 on p175 of [177] states that the conditions NOR2 and NOR 3 are equivalent to another condition NOR 1. This condition says that

(19.15.7) every embedding σ of K into \overline{k} over k is an automorphism of K.

Remember that σ is an embedding of K into \overline{k} over k when σ is the identity mapping on k, as in Section 19.12.

More on field extensions

20.1 Some properties of normal extensions

One can check that field extensions of degree 2 are normal, as mentioned on p175 of [177]. One can use this to get an example of fields k, E, and F with k ⊆ F ⊆ E such that F is normal over k, E is normal over F, and E is not normal over k, as in [177]. This means that the class of normal algebraic field extensions does not satisfy the “if” part of the first condition in the definition of a distinguished class, in Section 19.10.

Let k, E, and K be fields, with

k ⊆ E ⊆ K. (20.1.1)

Suppose that K is algebraic over k, which implies that E is algebraic over k, and that K is algebraic over E, as mentioned in Section 19.11. If K is normal over k, then the second part of Theorem 5 on p176 of [177] states that

(20.1.2) K is normal over E.

This is the second half of the “only if” part of the first condition in the definition of a distinguished class, for the class of normal algebraic field extensions. Indeed, if X is an indeterminate, and if K is the splitting field of a family of polynomials in k[X], then these polynomials may be considered as elements of E[X]. This means that K may be considered as the splitting field of a family of polynomials in E[X]. This corresponds to Exercise 5 on p470 of [24] in the case of finite extensions.

Alternatively, let \overline{E} be an algebraic closure of E, and remember that \overline{E} may be considered as an algebraic closure of k, as mentioned in Section 19.13. Suppose that K is a subfield of \overline{E}, and remember that we can reduce to this case using a suitable embedding, as in Section 19.13. If σ is any embedding of K into \overline{E} over E, then σ may be considered as an embedding of K into \overline{E} over k. The normality of K over k implies that

σ(K) = K, (20.1.3) and this implies that K is normal over E, as in the previous section. This argument is mentioned on p176 of [177].

Let k and L be fields, with k ⊆ L, and suppose that K, F are subfields of L that contain k. Thus the compositum K F of K and F may be defined as a subfield of L that contains k, as in Section 19.10. Suppose that K is algebraic over k, so that (20.1.4) K F is algebraic over F,

as in Section 19.11. If K is normal over k, then the first part of Theorem 5 on p176 of [177] says that (20.1.5) K F is normal over F.

This is the second condition in the definition of a distinguished class, for the class of normal algebraic field extensions.

As before, if K is the splitting field of a family of polynomials in k[X], then these polynomials may be considered as elements of F [X], and K F may be considered as the splitting field of this family of polynomials, as an extension of F. Alternatively, let \overline{F} be an algebraic closure of F. There is an embedding of K F into \overline{F} over F, because of (20.1.4), as in Section 19.13. Using this, we can reduce to the case where K F ⊆ \overline{F} . (20.1.6)

In particular, this means that K ⊆ \overline{F}.

One can get an algebraic closure \overline{k} of k that is a subfield of \overline{F} by taking the union of the subfields of \overline{F} that contain k and are algebraic over k, as in the proof of the corollary on p170 of [177]. Note that

K ⊆ \overline{k}, (20.1.7)

because k ⊆ K ⊆ \overline{F} and K is algebraic over k. Let σ be an embedding of K F into \overline{F} over F. Thus σ is equal to the identity mapping on F, and on k in particular, because k ⊆ F. This implies that

k = σ(k) ⊆ σ(K), (20.1.8)

and that σ(K) is algebraic over k.

It follows that σ(K) ⊆ \overline{k}. (20.1.9)

If K is normal over k, then we get that σ(K) = K, as in the previous section. This implies that σ(K F) = σ(K) σ(F) = K F, (20.1.10)

using the definition of the compositum in the first step. This means that K F is normal over F, as in the previous section again. This argument is also mentioned on p176 of [177].

20.2 More on normal extensions

Let k and L be fields with k ⊆ L, and let K_{1}, K_{2} be subfields of L that contain k, so that the compositum K_{1} K_{2} of K_{1} and K_{2} is a subfield of L that contains k. Suppose that K_{1} and K_{2} are algebraic over k, so that

(20.2.1) K_{1} K_{2} is algebraic over k,

as in Section 19.11. If K_{1} and K_{2} are normal over k, then

(20.2.2) K_{1} K_{2} is normal over k,

as in Theorem 5 on p176 of [177]. This is the third condition in the definition of a distinguished class, for the class of normal algebraic extensions. One way to see this is to use the fact that K_{1} and K_{2} are the splitting fields of families of polynomials with coefficients in k, as in Section 19.15, so that K_{1} K_{2} is the splitting field of the union of two such families.

Alternatively, let \overline{k} be an algebraic closure of k. There is an embedding of K_{1} K_{2} into \overline{k} over k, because of (20.2.1), as in Section 19.13. We can use this to reduce to the case where K_{1} K_{2} ⊆ \overline{k}, (20.2.3)

so that K_{1}, K_{2} ⊆ \overline{k}. Let σ be any embedding of K_{1} K_{2} into \overline{k} over k. Note that K_{1}, K_{2} ⊆ K_{1} K_{2}, and that

σ(K_{1}) = K_{1}, σ(K_{2}) = K_{2}, (20.2.4)

because K_{1} and K_{2} are normal over k, as in Section 19.15. This implies that

σ(K_{1} K_{2}) = σ(K_{1}) σ(K_{2}) = K_{1} K_{2}, (20.2.5)

so that (20.2.2) holds, as before. This argument is mentioned on p176 of [177].

Let K_{1} and K_{2} be subfields of L that contain k again, so that K_{1} ∩ K_{2} is a subfield of L that contains k as well. Clearly

(20.2.6) K_{1} ∩ K_{2} is algebraic over k

when K_{1} or K_{2} is algebraic over k. If K_{1} and K_{2} are algebraic and normal over k, then (20.2.7) K_{1} ∩ K_{2} is normal over k,

as in Theorem 5 on p176 of [177]. Indeed, let f(X) be an irreducible formal polynomial in an indeterminate X with coefficients in k that has a root in K_{1} ∩ K_{2}. If K_{1} and K_{2} are normal over k, then f(X) can be expressed as a product of linear factors with coefficients in K_{1}, and as a product of linear factors with coefficients in K_{2}, as in Section 19.15. One can check that these two expressions are the same, so that f(X) can be expressed as a product of linear factors with coefficients in K_{1} ∩ K_{2}. This implies (20.2.7), as before.

Alternatively, if \overline{k} is an algebraic closure of k again, then we can reduce to the case where (20.2.3) holds, as before. Let σ be any embedding of K_{1} ∩ K_{2} into \overline{k} over k, and let τ be an extension of σ to an embedding of K_{1} K_{2} into \overline{k}, as in Section 19.13. Observe that

σ(K_{1} ∩ K_{2}) = τ(K_{1} ∩ K_{2}) = τ(K_{1}) ∩ τ(K_{2}). (20.2.8)

If K_{1} and K_{2} are normal over k, then τ(K_{1}) = K_{1} and τ(K_{2}) = K_{2}, as in Section 19.15, so that σ(K_{1} ∩ K_{2}) = K_{1} ∩ K_{2}. (20.2.9)

This implies (20.2.7), as on p176 of [177].

20.3 Separable polynomials

Let k be a field, and let X be an indeterminate. Also let f(X) be a formal polynomial in X with coefficients in k, as in Section 19.9. Suppose that f(X) has degree n for some positive integer n, so that the coefficient of X^{j} in f(X) is equal to 0 when j > n, and is not zero when j = n. We say that f(X) is separable over k when

(20.3.1) f(X) has n distinct roots in some splitting field,

and thus in every splitting field, as on p465 of [24], p178 of [177], and p448 of [195]. Of course, f(X) has at most n roots in any extension of k, by a standard argument.

Equivalently, this means that

(20.3.2) f(X) has no multiple roots in the splitting field.

Otherwise, one may say that f(X) is inseparable. These terms are used a bit differently on p121 of [52], and we shall say more about that in a moment.

Let f′(X) be the formal derivative of f(X), as in Section 4.6. It is well known and not difficult to show that

(20.3.3) a root of f(X) is a multiple root if and only if it is also a root of f′(X),

as on p465 of [24], Proposition 1 on p131 of [177], and p454 of [195]. This implies that f(X) is separable if and only if

(20.3.4) f(X) and f′(X) have no common factors,

as in Theorem 11 on p466 of [24], and Proposition 8 on p454 of [195]. Equivalently, f(X) is separable if and only if

(20.3.5) the greatest common divisor of f(X) and f′(X) is equal to 1.

It is well known that the greatest common divisor of f(X) and f′(X) can be obtained using the Euclidean algorithm in k[X], as in [24, 195]. More precisely, the condition (20.3.5) holds when f(X) and f′(X) are considered as formal polynomials in X with coefficients in k if and only if it holds when they are considered as formal polynomials with coefficients in a larger field, as in [24, 195].

Suppose that f(X) is irreducible as a formal polynomial in X with coefficients in k. In this case, (20.3.5) holds if and only if

f′(X) ≠ 0, (20.3.6)

as in Corollary 1 on p466 of [24].

If k has characteristic 0, then (20.3.6) holds automatically, so that f(X) is separable. This corresponds to Corollary 2 on p466 of [24], the first part of Proposition 9 on 178 of [177], and Proposition 9 on p454 of [195]. Separability of a formal polynomial f(X) is defined on p121 of [52] to mean (20.3.6), but it seems to be used only for irreducible polynomials.

Let f(X) and g(X) be formal polynomials in X with coefficients in k, and suppose that (20.3.7) f(X) g(X) is separable.

Under these conditions, one can check that

(20.3.8) f(X) and g(X) are separable.

20.4 Separable elements and extensions

Let k_{1} be a field, and let k be a subfield of k_{1}. An algebraic element x of k_{1} over k is said to be separable over k if

(20.4.1) the minimal polynomial of x over k is separable,

as on p456 of [195]. This means that

(20.4.2) the coefficients of the derivative of the minimal polynomial of x are not all 0,

as in the previous section. This holds automatically when k has characteristic 0, as before.

If k_{1} is an algebraic extension of k, then separability of x ∈ k_{1} is defined in an equivalent way on p121 of [52]. Observe that

(20.4.3) the derivative of the minimal polynomial of x over k is not equal to 0 at x

in this case, because x is not a multiple root of its minimal polynomial over k, as in [52].

If an element x of an extension of k is a root of a separable polynomial with coefficients in k, then (20.4.4) x is separable over k, as mentioned on p178 of [177]. In this case, this polynomial can be expressed as the product of the minimal polynomial of x over k and another polynomial with coefficients in k, so that the minimal polynomial of x over k is separable, as in the previous section.

A finite extension k_{1} of k is said to be separable over k on p465 of [24] if

(20.4.5) every x ∈ k_{1} is the root of a separable polynomial with coefficients in k.

This is the same as saying that

(20.4.6) every element of k_{1} is separable over k

in the previous sense, as in the preceding paragraph. This is how separability of k_{1} over k is defined on p121 of [52]. Separability of k_{1} over k is also defined in this way on p456 of [195] when k_{1} is any algebraic extension of k.

Separability of finite extensions is defined another way on p178 of [177], and we shall say more about that in the next section. Separability of an algebraic element x of an extension of k is defined in [177] to mean that the subfield k(x) of the extension generated by k and x is separable over k in this sense. It is mentioned afterwards that this is equivalent to the separability of the minimal polynomial of x over k. Theorem 7 on p180 of [177] states that a finite extension of k is separable if and only if every element of the extension is separable over k.

Let E be a field that contains k as a subfield, and let F be a subfield of E that contains k. If x ∈ E is separable over k, then

(20.4.7) x is separable over F

as well, as mentioned on p180 of [177]. Indeed, if x is the root of a separable polynomial with coefficients in k, then this polynomial may also be considered as having coefficients in F, and it is separable as a polynomial with coefficients in F.

Suppose that k_{1} is a finite extension of k, and that

(20.4.8) k_{1} is generated by k and finitely many elements of k_{1} that are separable over k.

Under these conditions, it is remarked on p181 of [177] that an argument in the proof of Theorem 7 on the previous page shows that k_{1} is separable.

If k_{1} is an algebraic extension of k, then separability of k_{1} is defined on p181 of [177] to mean that

(20.4.9) every finitely-generated subextension of k_{1} is separable over k.

Note that this implies that every element of k_{1} is separable over k, in the sense defined on p178 of [177].

Suppose that k_{1} is an algebraic extension of k that is generated by a family of elements {x_{j}}_{j∈I}. If x_{j} is separable over k for each j ∈ I, then Theorem 8 on p181 of [177] states that

(20.4.10) k_{1} is separable over k.

20.5 Separable extensions and embeddings

Let k be a field, and let E be an algebraic extension of k. Also let L be an algebraically closed field, and let σ be an embedding of k into L. We would like to consider extensions of σ to embeddings of E into L, as on p176 of [177]. The existence of such an extension follows from Theorem 2 on p171 of [177], as mentioned in Section 19.13.

Note that any extension of σ to an embedding of E into L maps E onto a subfield of L that is algebraic over σ(k). Because of this, we shall suppose that

(20.5.1) L is algebraic over σ(k),

so that (20.5.2) L is an algebraic closure of σ(k),

as in [177].

Let L′ be another algebraically closed field, and let τ be an embedding of k into L′. Suppose that

(20.5.3) L′ is an algebraic closure of τ(k),

as before. Under these conditions, there is a field isomorphism λ from L onto L′ such that λ = τ ∘ σ^{−1} on σ(k), (20.5.4)

as in [177]. This uses the second part of Theorem 2 on p171 of [177], which was also mentioned in Section 19.13.

Let S_{σ}, S_{τ} be the sets of all extensions of σ, τ to embeddings of E into L, L′, respectively, as in [177]. One can use λ to get a one-to-one mapping from S_{σ} onto S_{τ}, as on p177 of [177]. In particular, this means that the cardinalities of S_{σ} and S_{τ} are the same. Thus this cardinality depends only on k and E, and it is denoted [E : k]_{s}, (20.5.5)

and called the separable degree of E over k, as in [177].

If F is a subfield of E that contains k, then it is easy to see that F is algebraic over k, and that E is algebraic over F, because E is algebraic over k. This means that [E : F]_{s} and [F : k]_{s} (20.5.6)

may be defined in the same way as before. Under these conditions, the first part of Theorem 6 on p177 of [177] says that

[E : k]_{s} = [E : F]_{s} [F : k]_{s}. (20.5.7)

More precisely, if σ is an embedding of k into L, then σ has [F : k]_{s} distinct extensions to embeddings of F into L, and each of these has [E : F]_{s} distinct extensions to embeddings of E into L. Of course, any extension of σ to an embedding of E into L may be considered as an extension to E of an extension of σ to an embedding of F into L.

If E is a finite extension of k, then the second part of Theorem 6 on p177 of [177] states that the separable degree of E over k is finite, with

[E : k]_{s} ≤ [E : k]. (20.5.8)

This follows from Proposition 8 on p171 of [177] when E is generated by a single element, as in Section 19.13.

Suppose that E is a finite extension of k, and that F is a subfield of E that contains k again. The corollary on p178 of [177] states that

[E : k]_{s} = [E : k] (20.5.9)

if and only if [E : F]_{s} = [E : F] (20.5.10)

and [F : k]_{s} = [F : k]. (20.5.11)

If E is a finite extension of k, then separability of E over k is defined on p178 of [177] to mean that (20.5.9) holds. This is equivalent to the other formulations of separability of finite extensions discussed in the previous section, as mentioned earlier.

20.6 Separable extensions are distinguished

Remember that the notion of a distinguished class of field extensions is defined on p165 of [177], as mentioned in Section 19.10. Theorem 9 on p181 of [177] states that

(20.6.1) the class of separable algebraic extensions is distinguished.

To see this, let k, E, F be fields with k ⊆ F ⊆ E. Suppose for the moment that E is a separable algebraic extension of k, which implies that E is algebraic over F and that F is algebraic over k, as mentioned in Section 19.11. Note that every element of F is separable over k, so that F is separable over k. We also have that every element of E is separable over F, as in Section 20.4, so that E is separable over F.

Conversely, suppose that F is a separable algebraic extension of k, and that E is a separable algebraic extension of F. This implies that E is an algebraic extension of k, as in Section 19.11. If E is a finite extension of k, then F is a finite extension of k and E is a finite extension of F, so that

[F : k]_{s} = [F : k] and [E : F]_{s} = [E : F], (20.6.2) as in the previous section. One can use this to get (20.5.9) from (20.5.7). This means that E is a separable extension of k, as before.

Otherwise, let α ∈ E be given. Note that α is separable over F, because E is separable over F, by hypothesis. This means that α is the root of a separable polynomial with coefficients in F. Let a_{0}, a_{1}, …, a_{n} ∈ F be the coefficients of this polynomial, and consider the subfield

F_{0} = k(a_{0}, a_{1}, …, a_{n}) (20.6.3)

of F generated by k and these coefficients. This is a finite extension of k, as in Section 19.9, because a_{0}, a_{1}, …, a_{n} are algebraic over k, by hypothesis.

We also have that F_{0} is separable over k, because a_{0}, a_{1}, …, a_{n} are separable over k, by hypothesis. Of course, α is separable over F_{0}, by construction. This means that the subfield F_{0}(α) of E generated by F_{0} and α is separable over F_{0}. It follows that (20.6.4) F_{0}(α) is separable over k,

because of the earlier argument for finite separable extensions. This implies that α is separable over k, so that E is separable over k.

This shows that separable extensions satisfy the first condition in the definition of a distinguished class. To get the second condition, let k, E, and F be subfields of a field L with k ⊆ E, F, and suppose that E is separable algebraic over k. Remember that the compositum E F of E and F is algebraic over k, as in Section 19.11.

Every element of E is separable over k, by hypothesis, and thus over F. This implies that E F is separable over F, as in Section 20.4.

20.7 Minimal normal extensions

Let k be a field, and let E be a finite extension of k. Also let \overline{E} be an algebraic closure of E, and remember that \overline{E} may be considered as an algebraic closure of k, as in Section 19.13. Note that E is finitely generated as a field extension of k, as mentioned in Section 19.9. One can get a finite normal extension of k in \overline{E} that contains E using the splitting field of finitely many polynomials over k whose zeros include a set of generators of E as an extension of k.

If E is also separable over k, then one can use finitely many separable polynomials over k, to get a finite normal extension of k in \overline{E} that contains E and is separable too. This uses the fact that an extension of k is separable if it is generated by elements that are separable over k, as in Section 20.4. This is related to the first part of Exercise 1 on p461 of [195].

The intersection of all of the normal algebraic extensions of k in \overline{E} that contain E is a normal algebraic extension of k, as in Section 20.2. This is the smallest normal algebraic extension of k in \overline{E} that contains E, as on p182 of [177]. This is a finite extension of k, as before.

Let σ_{1}, …, σ_{n} be a list of the distinct embeddings of E into \overline{E} over k, so that n = [E : k]_{s}, as in Section 20.5. Consider the compositum

K = σ_{1}(E) σ_{2}(E) ⋯ σ_{n}(E) (20.7.1)

of the subfields σ_{1}(E), σ_{2}(E), …, σ_{n}(E) of \overline{E}, as in Section 19.10. This is a finite extension of k, because the class of finite extensions is distinguished, as in Section 19.11.

In fact, (20.7.2) K is a normal extension of k,

as on p182 of [177]. To see this, let τ be any embedding of K into \overline{E} over k. Observe that τ ∘ σ_{1}, τ ∘ σ_{2}, …, τ ∘ σ_{n} (20.7.3)

are distinct embeddings of E into \overline{E} over k. It follows that these embeddings are the same as σ_{1}, σ_{2}, …, σ_{n}, possibly rearranged, so that

τ(K) = K. (20.7.4)

This implies (20.7.2), as in Section 19.15.

Remember that σ_{1}, …, σ_{n} may be extended to embeddings of any algebraic extension of E, as in Section 19.13. One can use this to get that any normal algebraic extension of k in \overline{E} that contains E also contains σ_{1}(E), …, σ_{n}(E). This means that such an extension contains K. Thus K is the smallest normal algebraic extension of k in \overline{E} that contains E, as on p182 of [177].

If E is separable over k, then

(20.7.5) K is separable over k,

as on p182 of [177], because separable algebraic extensions form a distinguished class. This is related to the second part of Exercise 1 on p461 of [195].

20.8 Galois extensions

Let k be a field, and let K be an algebraic extension of k. Remember that normality of K as an extension of k is defined as in Section 19.15. If K is normal and separable over k, then K is said to be a Galois extension of k, as on p192 of [177].

If K is any extension of k, then normality of K over k is defined on p459 of [195] to mean that

(20.8.1) k = {u ∈ K : θ(u) = u for every θ ∈ Gal(K/k)},

as mentioned in Section 19.15. Here Gal(K/k) is the Galois group of K over k, as before. If K is a finite extension of k, then condition (iii) in Theorem 12 on p459 of [195] is the same as saying that K is normal as an extension of k as in Section 19.15, and separable, so that K is a Galois extension. Condition (ii) in the theorem is that K be normal as an extension of k as defined in [195], and part of the theorem is that these two conditions are equivalent in this case.

Let K be an algebraic extension of k again. If K is a Galois extension of k, then the first part of Theorem 1 on p192 of [177] states that (20.8.1) holds. The proof of this will be mentioned in Section 21.7, as well as a related statement for normal extensions that may not be separable.

The proof in [195] that (20.8.1) implies that K is a Galois extension of k when K is a finite extension of k seems to work as well when K is an algebraic extension of k. This is related to Exercise 3 on p461 of [195].

Let \overline{K} be an algebraic closure of K. This may be considered as an algebraic closure of k, as mentioned in Section 19.13. Every element of the Galois group of K over k may be considered as an embedding of K into \overline{K} over k. As in Section 19.15, normality of K over k in the sense that we are using here is equivalent to the condition that every embedding of K into \overline{K} over k correspond to an element of the Galois group of K over k in this way. This corresponds to a remark on p192 of [177].

More precisely, let S_{0} be the set of embeddings of K into \overline{K} over k, which is to say the set of embeddings of K into \overline{K} that are equal to the identity mapping on k, as in Section 19.12. This corresponds to S_{σ} in Section 20.5, with E = K, L = \overline{K}, and σ equal to the identity mapping on k, considered as an embedding of k into \overline{K}. Thus Gal(K/k) ⊆ S_{0}, (20.8.2)

and normality of K over k is equivalent to

Gal(K/k) = S_{0}, (20.8.3)

as in the preceding paragraph.

Suppose from now on in this section that K is a finite extension of k, and let #A denote the cardinality of a set A. In this case, we get that Gal(K/k) is finite, with # Gal(K/k) ≤ #S_{0} = [K : k]_{s} ≤ [K : k], (20.8.4)

where the second and third steps are as in Section 20.5. We also get that

# Gal(K/k) = #S_{0} = [K : k]_{s} (20.8.5)

if and only if (20.8.3) holds, which means that K is normal over k, as before. Observe that # Gal(K/k) = [K : k] (20.8.6)

if and only if (20.8.5) holds and

[K : k]_{s} = [K : k]. (20.8.7)

This means that (20.8.6) holds if and only if K is normal and separable over k, or equivalently Galois over k, as in Theorem 12 on p459 of [195].

Remember that K is normal over k if and only if k is the root field, or equivalently splitting field, of a polynomial with coefficients in k, as in Section 19.15. Another part of Theorem 12 on p459 of [195] states that K is Galois over k if and only if K is the splitting field of a separable polynomial with coefficients in k. The fact that the splitting field of a separable polynomial with coefficients in k is a separable extension of k also follows from a statement mentioned in Section 20.4. The corollary on p469 of [24] says that K is the root field of a separable polynomial with coefficients in k when K is normal and separable. Theorems 12 and 13 on p469 of [24] state that root fields of separable polynomials with coefficients in k have some of the properties of finite Galois extensions mentioned earlier.

20.9 Separable extensions and the trace

Let k be a field, and let E be a finite separable extension of k. Also let \overline{E} be an algebraic closure of E, which may be considered as an algebraic closure of k, as in Section 19.13. One could start with an algebraic closure \overline{k} of k, as on p210 of [177]. In this case, one could reduce to the case where E is a subfield of \overline{k} that contains k, using an embedding of E into \overline{k}, as in Section 19.13, and as mentioned on p211 of [177]. Of course, this means that \overline{k} may be considered as an algebraic closure of E.

Put r = [E : k]_{s} = [E : k], (20.9.1)

and let σ_{1}, …, σ_{r} be the distinct embeddings of E into \overline{E} over k. If x ∈ E, then the trace of x from E to k is defined by

Tr_{k}^{E}(x) = ∑_{j=1}^{r} σ_{j}(x), (20.9.2)

as on p210 of [177]. More precisely, the trace is initially defined in a slightly different way for any finite extension of k in [177], and then it is mentioned that it is equal to 0 when the extension is not separable.

Although (20.9.2) may be defined initially as an element of \overline{E}, it is in fact an element of the smallest normal algebraic extension K of k in \overline{E} that contains E, as in Section 20.7. It is easy to see that (20.9.2) is invariant under the Galois group of K over k, because any element of the Galois group of K over k simply permutes the terms in the sum on the right side of (20.9.2). This implies that

Tr_{k}^{E}(x) ∈ k, (20.9.3)

as in the previous section, because K is Galois over k, as in Section 20.7. This corresponds to part of the first part of Theorem 8 on p210 of [177].

Clearly Tr_{k}^{E} is a homomorphism from E into k, as commutative groups with respect to addition, as in Theorem 8 on 210 of [177]. In fact, Tr_{k}^{E} is a linear mapping from E into k, as vector spaces over k, as mentioned on p211 of [177].

Let F be a subfield of E that contains k. Note that F is a finite separable extension of k, and that E is a finite separable extension of F, because separable algebraic extensions form a distinguished class, as in Section 20.6. Of course, \overline{E} may be considered as an algebraic closure of F as well. Thus Tr_{F}^{E} may be defined as a mapping from E into F, and Tr_{k}^{F} may be defined as a mapping from F into k, as before. It is not too difficult to show that

Tr_{k}^{E} = Tr_{k}^{F} ∘ Tr_{F}^{E}, (20.9.4)

as in the second part of Theorem 8 on p210 of [177].

Suppose for the moment that

E = k(α) (20.9.5)

for some α ∈ E, so that E is generated by k and α. Let X be an indeterminate, and let f(X) = X^{n} + a_{n−1} X^{n−1} + ⋯ + a_{0} (20.9.6)

be the minimal polynomial of α over k, as in Section 19.9. Under these conditions, we have that Tr_{k}^{k(α)}(α) = −a_{n−1}, (20.9.7)

as in the third part of Theorem 8 on p210 of [177].

Of course, −a_{n−1} is the same as the sum of the roots of f(X), with their appropriate multiplicaities. In this case, f(X) has no multiple roots, because of separability, as in Section 20.4. The embeddings of k(α) over k into an algebraic closure correspond exactly to the roots of f(X), as in Section 19.13. This means that Tr_{k}^{k(α)}(α) is the same as the sum of the roots of f(X), as in (20.9.7).

Let E be any finite separable extension of k again. The first part of Theorem 9 on p211 of [177] states that

Tr_{k}^{E}(x) ≠ 0 (20.9.8)

for some x ∈ E. To see this, we consider E \ {0} as a commutative group with respect to multiplication, and σ_{1}, …, σ_{r} as group homomorphisms from E \ {0} into \overline{E} \ \{0\}. These homomorphisms are linearly independent as functions on E \\{0\} with values in \overline{E}, as in Section 19.8, because they are distinct on E \\{0\}, by hypothesis. This implies in particular that their sum is nonzero on E \ {0}.

If x ∈ k, then Tr_{k}^{E}(x) = r · x, (20.9.9)

which is the sum of r x’s in k, as mentioned in the remark on p212 of [177]. This implies (20.9.8) when x ≠ 0 and k has characteristic 0, and when r is not a multiple of the characteristic of k, if it is positive.

If x ∈ E, then y ↦ x y (20.9.10)

defines a linear mapping from E into itself, as a vector space over k. The trace of this linear mapping may be defined as an element of k in the usual way, because E has finite dimension as a vector space over k, by hypothesis. It is well known that the trace of this linear mapping is equal to Tr_{k}^{E}(x), as in Exercise 4 on p351 of [177].

20.10 Traces and integral elements

Let k_{0} be an integral domain, and let k = Q_{k_{0}} be its field of fractions. Also let E be a finite extension of k, and let \overline{E} be an algebraic closure of E. Suppose that (20.10.1) x ∈ E is integral over k_{0},

so that it satisfies a monic polynomial equation with coefficients in k_{0}, as in Section 16.3. This polynomial may be expressed as the product of the minimal polynomial of x over k and another polynomial with coefficients in k. It follows that the roots of the minimal polynomial of x over k satisfy the same monic polynomial equation with coefficients in k_{0} as x does, so that they are integral over k_{0} too.

It is well known and easy to see that the coefficients of the minimal polynomial of x over k may be expressed as sums of products of its roots. This implies that

(20.10.2) the coefficients of the minimal polynomial of x over k are integral over k_{0},

as in Section 16.5. This corresponds to part of the corollary on p240 of [177]. Note that all rings in Chapter IX of [177] are supposed to be commutative, as mentioned at the beginning of that chapter, and that the term entire ring is used for integral domains on p61 of [177].

Suppose now that E is a finite separable extension of k. Under these conditions, (20.10.3) Tr_{k}^{E}(x) is integral over k_{0}

as well. This corresponds to another part of the corollary on p240 of [177]. This corresponds to remarks in the proofs of Proposition 5.17 on p64 of [8] and Proposition 8 on p13 of [263] too.

Although (20.10.3) can be obtained from (20.10.2), it can be verified more directly from the definition (20.9.2) of the trace, as follows. If σ_{1}, …, σ_{r} are the distinct embeddings of E into \overline{E} over k, as in the previous section, then σ_{j}(x) satisfies the same polynomial equations as x for each j. In particular, this means that (20.10.4) σ_{j}(x) is integral over k_{0}

for each j. It follows that their sum is integral over k_{0}, as in Section 16.5 again.

20.11 Traces and integral closures

Let k_{0} be an integral domain, and let k = Q_{k_{0}} be its field of fractions again. Also let E be a finite separable extension of k, and let \overline{E} be an algebraic closure of E. Suppose that (20.11.1) k_{0} is integrally closed, so that k_{0} is integrally closed in k, as in Section 16.3. Let

(20.11.2) k_{1} = {x ∈ E : x is integral over k_{0}}

be the integral closure of k_{0} in E, as in Section 16.5.

Under these conditions, Proposition 5.17 on p64 of [8] says that there is a basis v_{1}, …, v_{r} for E as a vector space over k such that

k_{1} ⊆ k_{0} v_{1} + ⋯ + k_{0} v_{r}. (20.11.3)

This is related to Proposition 13.14 on p298 of [92], Exercise 5 on p252 of [177], and Proposition 8 on p13 of [263], and we shall say more about that in a moment.

Of course, every element of E is algebraic over k, as mentioned in Section 19.9. This implies that every x ∈ E satisfies a polynomial equation

a_{n} x^{n} + a_{n−1} x^{n−1} + ⋯ + a_{0} = 0 (20.11.4)

with coefficients in k_{0} and a_{n} ≠ 0, because k is the field of fractions of k_{0}, by a standard argument. One can multiply both sides of the equation by a_{n}^{n−1} to get that a_{n} x is integral over k_{0}, so that

a_{n} x ∈ k_{1}. (20.11.5)

We can use this to get a basis u_{1}, …, u_{r} for E, as a vector space over k, with

u_{1}, …, u_{r} ∈ k_{1}, (20.11.6)

as in [8].

Let Tr_{k}^{E} be as in Section 20.9, and note that

Tr_{k}^{E}(x y) (20.11.7)

defines a symmetric bilinear form on E, as a vector space over k. This bilinear form is also nondegenerate on E, because Tr_{k}^{E} is nonzero on some elements of E, as before.

One can use this to get a basis v_{1}, …, v_{r} of E as a vector space over k such that

(20.11.8)  Tr_{k}^{E}(u_{j} v_{l}) = 1 when j = l
= 0 when j ≠ l.

as in Corollary 1 on p212 of [177]. More precisely, one can use (20.11.7) to identify E with its dual space, as a vector space over k, in which case v_{1}, …, v_{r} corresponds to the dual basis associated to u_{1}, …, u_{r}.

If x ∈ E, then x can be expressed in a unique way as a linear combination of v_{1}, …, v_{r} with coefficients in k, and in fact

x = ∑_{j=1}^{r} Tr_{k}^{E}(x u_{j}) v_{j}, (20.11.9)

because of (20.11.8). Remember that k_{1} is a subring of E, as in Section 16.5. If x ∈ k_{1}, then x u_{j} ∈ k_{1} for each j, and

Tr_{k}^{E}(x u_{j}) ∈ k_{1} (20.11.10)

for each j, as in (20.10.3). This implies (20.11.3).

Suppose now that we also have that

(20.11.11) k_{0} is Noetherian

as a commutative ring, and thus as a module over itself. The right side of (20.11.3) may be considered as a module over k_{0}, which is freely generated by v_{1}, …, v_{r}. This module is isomorphic to the direct sum of r copies of k_{0}, and is thus Noetherian as a module over k_{0}, as in Section 9.7.

Note that k_{1} may be considered as a module over k_{0}, because k_{0} is a subring of k_{1}. It follows that

(20.11.12) k_{1} is finitely generated as a module over k_{0},

because of (20.11.3). This corresponds to Proposition 13.14 on p298 of [92], Exercise 5 on p252 of [177], and Proposition 8 on p13 of [263]. More precisely, an integral domain is said to be normal on p118 of [92] if it is integrally closed in the corresponding field of fractions.

20.12 Dedekind domains and integral closures

Let k_{0} be an integral domain, let k = Q_{k_{0}} be its field of fractions, and let E be a finite extension of k. Suppose that k_{0} is integrally closed, as in (20.11.1), and let k_{1} be the integral closure of k_{0} in E, as in (20.11.2). Suppose also that k_{0} is Noetherian, as in (20.11.11), and consider the condition (20.11.12), which corresponds to the hypothesis (F) on p13 of [263]. This condition holds when

(20.12.1) E is separable, as a finite extension of k,

as in the previous section.

If (20.11.12) holds, then we get that

(20.12.2) k_{1} is Noetherian, as a module over k_{0},

because of (20.11.11), as in Section 9.7. This implies that

(20.12.3) k_{1} is Noetherian as a commutative ring,

which is to say that k_{1} is Noetherian as a module over itself.

Suppose now that k_{0} is a Dedekind domain, which includes the conditions that k_{0} be a Noetherian integral domain, as in Section 15.11. This implies that k_{0} be integrally closed as well, as in Section 16.12.

Of course, k_{1} is an integral domain too. Note that k_{1} is integrally closed in E, as in Section 16.5. It is easy to see that E is the field of fractions of k_{1}, using (20.11.5). This means that

(20.12.4) k_{1} is integrally closed,

as in Section 16.3.

Remember that k_{0} has dimension less than or equal to one, in the sense of Section 14.8, because k_{0} is a Dedekind domain, as in Section 15.11. One can check that (20.12.5) k_{1} has dimension less than or equal to one,

because k_{1} is integral over k_{0}. This uses the results mentioned in Section 16.14. If (20.11.12) holds, so that (20.12.3) holds, then it follows that

(20.12.6) k_{1} is a Dedekind domain,

as in Section 16.12. This corresponds to Proposition 9 on p13 of [263], and it is related to Theorem 9.5 on p96 of [8]. This also works if one requires Dedekind domains to not be fields, or equivalently to have dimension equal to one, as mentioned in Section 15.11. More precisely, if k_{0} is not a field, then

(20.12.7) k_{1} is not a field,

because k_{0} = k ∩ k_{1}, and k is a field.

A remark on p14 of [263] states that (20.12.6) holds without the additional hypothesis (20.11.12). Part of Theorem 11.13 on p267 of [92], the Krull–Akizuki theorem, states that if k_{0} is a one-dimensional Noetherian integral domain with field of fractions k = Q_{k_{0}}, and if E is a finite extension of k, then the integral closure k_{1} of k_{0} in E is Noetherian. This will be discussed further in Section 25.8.

20.13 Some remarks about algebras

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. If we also have a mapping from A × A into A that is bilinear over k, then we may say that A is an algebra in the strict sense over k. If a, b ∈ A, then let us denote the image of (a, b) under this mapping as a b. Of course, if (a b) c = a (b c) (20.13.1)

for all a, b, c ∈ A, then A is said to be an associative algebra over k. If

a b = b a (20.13.2)

for all a.b ∈ A, then A is said to be commutative as an algebra over k. An element e = e_{A} of A is said to be the multiplicative identity element if

a e = e a = a (20.13.3) for every a ∈ A. It is easy to see that a multiplicative identity element in A is unique when it exists.

An associative algebra over k is a ring in particular, at least if one does not require that a ring have a multiplicative identity element. Sometimes the convention that rings have multiplicative identity elements is used, and that associative algebras do not necessarily have them, although sometimes associative algebras are asked to have multiplicative identity elements too. Note that a ring may be considered as an associative algebra over Z.

Let A be an algebra over k in the strict sense, and let A_{0} be a submodule of A, as a module over k. If x y ∈ A_{0} (20.13.4)

for every x, y ∈ A_{0}, then A_{0} is said to be a subalgebra of A. This implies that A_{0} is an algebra over k in the strict sense too, with respect to the restriction of multiplication on A to A_{0}.

Let B be another algebra over k in the strict sense. A homomorphism ϕ from A into B as modules over k is said to be an algebra homomorphism if

ϕ(x y) = ϕ(x) ϕ(y) (20.13.5)

for all x, y ∈ A, as usual. If A, B have multiplicative identity elements e_{A}, e_{B}, respectively, then one may want to include the condition that

ϕ(e_{A}) = ϕ(e_{B}). (20.13.6)

If A has a multiplicative identity element e_{A} and ϕ(A) = B, then ϕ(e_{A}) is the multiplicative identity element in B. If ϕ is an algebra homomorphism from A into B, C is another algebra over k in the strict sense, and ψ is an algebra homomorphism from B into C, then ψ ∘ ϕ is an algebra homomorphism from A into C.

A one-to-one algebra homomorphism ϕ from A onto B is said to be an algebra isomorphism. In this case, the inverse mapping ϕ^{−1} is an algebra isomorphism from B onto A.

Of course, k may be considered as a module over itself, and as an algebra over itself. If A has a multiplicative identity element e_{A}, then

t ↦ t e_{A} (20.13.7)

is an algebra homomorphism from k into A.

The center is defined by

Z(A) = {x ∈ A : x y = y x for all y ∈ A}. (20.13.8)

This is a submodule of A, as a module over k. If A is associative, then Z(A) is a subalgebra of A.

Suppose now that A is a ring with a multiplicative identity element e_{A}. If A is an algebra over k, then (20.13.7) may be considered as a ring homomorphism from k into A that sends the multilicative identity element 1_{k} in k to e_{A}, and which maps k into the center of A. Sometimes an associative algebra over k is defined in this way, using a ring homomorphism from k into A that sends 1_{k} to e_{A} and maps k into Z(A). This means that scalar multiplication by t ∈ k on A is defined using multiplication by the image of t in A.

Let A be any algebra over k in the strict sense again, and let X_{1}, …, X_{n} be n distinct commuting indeterminates for some positive integer n. The space A[[X_{1}, …, X_{n}]] of formal power series in X_{1}, …, X_{n} with coefficients in A may be considered as an algebra in the strict sense over k as well, with multiplication defined as in Section 4.3. The space A[X_{1}, …, X_{n}] of formal polynomials in X_{1}, …, X_{n} with coefficients in A is a subalgebra of A[[X_{1}, …, X_{n}]], as before. One may consider A as a subalgebra of A[X_{1}, …, X_{n}] as before too.

20.14 A result of Emmy Noether

Let A be a commutative ring with a multiplicative identity element. One may say that A is reduced if 0 is the only nilpotent element of A, as on p33 of [92]. Note that the term “ring” is normally interpreted in [92] as being a commutative ring with a multiplicative identity element, as on p11 of [92]. Of course, an integral domain is clearly reduced.

Let k_{00} be a field, and suppose now that A is a commutative associative algebra over k_{00} with a multiplicative identity element. If A is finitely generated as an algebra over k_{00}, and reduced as a commutative ring, then A may be called an affine algebra over k_{00}, as on p35 of [92]. One may also call A an affine ring in this case, as in [92].

Suppose now that A is an affine domain, which is to say an affine ring that is also an integral domain. Let Q_{A} be the field of fractions of A, and let L be a finite extension of Q_{A}. If T is the integral closure of A in L, then a well-known result of Emmy Noether states that

(20.14.1) T is finitely generated as a module over A.

This corresponds to Corollary 13.13 on p297 of [92].

In the notation of Section 20.12, we would like to take k_{0} = A, so that k = Q_{k_{0}} = Q_{A}, and E = L. This means that k_{1} = T, so that (20.14.1) corresponds to (20.11.12). It follows that k_{1} = T is Noetherian as a module over k_{0} = A under these conditions, and thus as a ring, as before. This corresponds to part of the second remark on p13 of [263].

The proof of Noether’s result will be discussed beginning in Section 24.10.

20.15 Some remarks about multiple roots

Let k be a field, and let X be an indeterminate. Also let g(X) be a polynomial in X with coefficients in k, and suppose that

f(X) = (X − α)^{m} g(X) (20.15.1) for some α ∈ k and m ∈ Z_{+}. If

g(α) ≠ 0, (20.15.2)

then m is said to be the multiplicity of α as a root of f(X), as on p131, 178 of [177]. We say that α is a multiple root of f(X) when m > 1, and otherwise α is a simple root of f(X) when m = 1, as in [177].

Suppose that k has positive characteristic p. It is well known and not difficult to see that the binomial coefficient \binom{p}{l} is a multiple of p when 1 ≤ l ≤ p − 1. This implies that (x + y)^{p} = x^{p} + y^{p} (20.15.3)

for every x, y ∈ k, as on p131 of [177].

Of course, (x y)^{p} = x^{p} y^{p} for all x, y ∈ k, so that

x ↦ x^{p} (20.15.4)

is a homomorphism from k into itself, as a field. The kernel of this homomorphism is trivial, so that (20.15.4) is injective.

If r is a positive integer, then it follows that

x ↦ x^{p^{r}} (20.15.5)

is an injective homomorphism from k into itself, as a field, as on p132 of [177]. Let c ∈ k be given, and consider

X^{p^{r}} − c. (20.15.6)

If this polynomial has a root a ∈ k, so that

a^{p^{r}} = c, (20.15.7)

then X^{p^{r}} − c = X^{p^{r}} − a^{p^{r}} = (X − a)^{p^{r}}, (20.15.8)

as a formal polynomial in X with coefficients in k. This means that a is the only root of (20.15.6), as on p132 of [177].

If (20.15.9) (20.15.4) maps k onto itself,

then k is said to be perfect, as on p190 of [177]. Any field of characteristic 0 is said to be perfect too, as in [177]. An equivalent formulation is mentioned on p121 of [52]. Note that (20.15.9) holds when k has only finitely many elements, because (20.15.4) is injective.

If f(X) is a formal polynomial in X with coefficients in k and f′(X) = 0, then f(X) can be expressed as a linear combination of powers of X^{p} with coefficients in k, as in Proposition 2 on p131 of [177]. If k is perfect, then it follows that f(X) can be expressed as the pth power of a formal polynomial in X with coefficients in k, as mentioned on p121 of [52]. In particular, this implies that f(X) is reducible when the degree of f(X) is positive, as in [52].

502 CHAPTER 20. MORE ON FIELD EXTENSIONS

If k is perfect and f(X) is irreducible and nonconstant, then it follows that f′(X) ≠ 0. This means that (20.15.10) f(X) is separable, as in Section 20.3. This implies that (20.15.11) algebraic elements of extensions of k are separable, so that (20.15.12) algebraic extensions of k are separable, when k is perfect.

Part VI

Rings, fields, modules, and derivations More on field extensions, 2

21.1 More on separable elements

Let k be a field, let \overline{k} be an algebraic closure of k, let α be an element of \overline{k}, and let f(X) be the minimal polynomial of α over k. If k has characteristic 0, then all of the roots of f(X) are simple, so that f(X) is separable as a polynomial over k, as in Section 20.3. If k has positive characteristic p, then there is a nonnegative integer μ such that

(21.1.1) every root of f(X) has multiplicity p^{μ},

as in Proposition 9 on p178 of [177]. In this case, we have that

[k(α) : k] = p^{μ} [k(α) : k]_{s}, (21.1.2)

and that (21.1.3) α^{p^{μ}} is separable over k

as in [177].

To see this, let α_{1}, …, α_{r} be a list of the distinct roots of f(X) in \overline{k}, and let m_{l} be the multiplicity of α_{l} in f(X) for each l = 1, …, r. Note that

f(X) = ∏_{l=1}^{r}(X − α_{l})^{m_{l}}, (21.1.4)

because f(X) is a monic polynomial by hypothesis, as in Section 19.9.

If 1 ≤ j ≤ r, then there is a unique isomorphism σ_{j} from k(α) onto k(α_{j}) over k such that σ_{j}(α) = α_{j}, (21.1.5)

as on p179 of [177]. This can be extended to an automorphism of \overline{k} over k, as in Section 19.13, and we let σ_{j} denote this extension as well.

If f^{σ_{j}}(X) is the formal polynomial associated to f(X) and σ_{j} as in Section 19.12, then f^{σ_{j}}(X) = f(X), (21.1.6) because σ_{j} is the identity mapping on k, and f(X) has coefficients in k. One can use this to get that

f(X) = ∏_{l=1}^{r}(X − σ_{j}(α_{l}))^{m_{l}}, (21.1.7)

as on p179 of [177]. If α has multiplicity m in f(X), then it follows that

m_{j} = m, (21.1.8)

as in [177].

Let f′(X) be the formal derivative of f(X), as in Section 4.6. Remember that f(X) has no multiple roots when f′(X) ≠ 0, because f(X) is irreducible, as in Section 20.3. In particular, this holds when k has characteristic 0, as before.

Suppose that k has positive characteristic p. If

f′(X) = 0, (21.1.9)

then the coefficient of X^{l} in f(X) is 0 unless l is an integer multiple of p. This means that f(X) = g(X^{p}) (21.1.10)

for some formal monic polynomial g(X) with coefficients in k, as on p179 of [177]. It is easy to see that g(X) is irreducible as a formal polynomial with coefficients in k, because f(X) is irreducible. If g′(X) ≠ 0, then we can stop.

Remember that [k(α) : k] is equal to the degree of f, as mentioned in Section 19.9. Note that g(α^{p}) = f(α) = 0, (21.1.11)

so that [k(α^{p}) : k] is equal to the degree of g. Of course,

k(α^{p}) ⊆ k(α). (21.1.12)

It follows that [k(α) : k(α^{p})] = p, (21.1.13)

as mentioned on p179 of [177], using a standard properties of degrees of extensions mentioned in Section 19.9.

We can repeat the process as needed to get a nonnegative integer μ and an irreducible formal monic polynomial h(X) with coefficients in k such that

f(X) = h(X^{p^{μ}}) (21.1.14)

and h′(X) ≠ 0, (21.1.15)

as on p179 in [177]. Thus h(X) has no multiple roots, as before. Clearly

h(α^{p^{μ}}) = f(α) = 0, (21.1.16) and [k(α^{p^{μ}}) : k] is equal to the degree of h. Note that (21.1.3) follows from (21.1.16). We also have that

k(α^{p^{μ}}) ⊆ k(α), (21.1.17)

and [k(α) : k(α^{p^{μ}})] = p^{μ}, (21.1.18)

as on p179 of [177].

If 1 ≤ l ≤ r, then h(α_{l}^{p^{μ}}) = f(α_{l}) = 0. (21.1.19)

It follows that h(X) can be expressed as the product of X − α_{l}^{p_{μ}} and a formal polynomial in X with coefficients in \overline{k} that is not equal to 0 at α_{l}^{p^{μ}}, because h(X) does not have multiple roots. This means that f(X) can be expressed as the product of X^{p^{μ}} − α_{l}^{p^{μ}} = (X − α_{l})^{p^{μ}} (21.1.20)

and a formal polynomial that is a linear combination of powers of X^{p^{μ}}, and which is not equal to 0 at α_{l}. Thus the multiplicity of α_{l} as a root of f(X) is equal to p^{μ}, as in (21.1.1).

The degree of f(X) is equal to p^{μ} r, so that the degree of h(X) is equal to r. Note that α_{1}^{p^{μ}}, …, α_{r}^{p^{μ}} (21.1.21)

are distinct roots of h(X), and that these are all of the roots of h(X), because the degree of h(X) is equal to r, as in [177].

Remember that [k(α) : k]_{s} is the number of embeddings of k(α) into \overline{k} over k, as in Section 20.5. This is equal to the number of distinct roots of f(X), by Proposition 8 on p171 of [177], which was mentioned in Section 19.13. This implies (21.1.2), because [k(α) : k] is equal to the degree of f(X), as mentioned in Section 19.9. Note that

[k(α^{p^{μ}}) : k]_{s} = [k(α^{p^{μ}}) : k], (21.1.22)

which is the degree of h(X), and that

[k(α) : k]_{s} = [k(α^{p^{μ}}) : k]_{s}, (21.1.23)

as on p179 of [177].

21.2 More on algebraic extensions

Let k be a field of positive characteristic p, and let α be an algebraic element of an extension of k. If there is a nonnegative integer n such that

α^{p^{n}} ∈ k, (21.2.1)

then α is said to be purely inseparable over k, as on p186 of [177].

Let E be an algebraic extension of k, and consider the following four conditions on E. The first condition is that

[E : k]_{s} = 1, (21.2.2)

where the left side is as in Section 20.5. The second condition is that

(21.2.3) every α ∈ E is purely inseparable over k.

The third condition is that for every α ∈ E, the minimal polynomial of α over k is of the form X^{p^{n}} − a (21.2.4)

for some nonnegative integer n and a ∈ k. The fourth condition is that there be a family {α_{j}}_{j∈I} of generators of E as an extension of k such that

(21.2.5) α_{j} is purely inseparable over k

for each j ∈ I.

These four conditions are equivalent, as in [177]. To see this, suppose for the moment that (21.2.2) holds, and let α ∈ E be given. Remember that k(α) is the subfield of E generated by k and α, and observe that

[k(α) : k]_{s} = 1, (21.2.6)

as in Section 20.5.

Let \overline{E} be an algebraic closure of E, which may be considered as an algebraic closure of k, as in Section 19.13. Also let X be an indeterminate, and let f(X) be the minimal polynomial of α over k. Remember that [k(α) : k]_{s} is the same as the number of embeddings of k(α) into \overline{E} over k, as in Section 20.5, and that this is the same as the number of distinct roots of f(X) in \overline{E}, as in Section 19.13. Thus (21.2.6) is the same as saying that f(X) has only one root in \overline{E}, which is α.

Put m = [k(α) : k], (21.2.7)

which is the same as the degree of f(X). The remarks in the preceding paragraph imply that f(X) = (X − α)^{m}, (21.2.8)

as a formal polynomial in X with coefficients in k(α). Let us express m as

m = p^{n} r, (21.2.9)

where n is a nonnegative integer, and r is a positive integer that is not a multiple of p. Observe that

f(X) = (X − α)^{p^{n} r} = (X^{p^{n}} − α^{p^{n}})^{r}. (21.2.10)

It follows that

f(X) = X^{p^{n} r} − r · α^{p^{n}} X^{p^{n} (r−1)} + · · · . (21.2.11)

This implies that

r α^{p^{n}} ∈ k, (21.2.12)

because the coefficients of f(X) are in k. This means that (21.2.1) holds, because r is not a multiple of p.

Suppose now that α ∈ E is purely inseparable over k, so that

α^{p^{n_{0}}} ∈ k (21.2.13)

for some nonnegative integer n_{0}. Let f(X) be the minimal polynomial of α over k again, and let m be the degree of f(X). Of course, α is a root of

X^{p^{n_{0}}} − α^{p^{n_{0}}} = (X − α)^{p^{n_{0}}} , (21.2.14)

so that X^{p^{n_{0}}} − α^{p^{n_{0}}} is equal to the product of f(X) and another monic formal polynomial in X with coefficients in k. It follows that α is the only root of f(X).

If m is as in (21.2.9), then we get that (21.2.1) holds, as before. The irreducibility of f(X) implies that r = 1, and that

f(X) = X^{p^{n}} − α^{p^{n}}. (21.2.15)

This shows that (21.2.3) implies the third condition mentioned earlier. The third condition clearly implies (21.2.3), which implies the fourth condition.

Suppose that E satisfies the fourth condition mentioned earlier, and let f_{j}(X) be the minimal polynomial of α_{j} over k for each j ∈ I. If j ∈ I, then there is a nonnegative integer n_{j} such that

α_{j}^{p^{n_{j}}} ∈ k, (21.2.16)

as in (21.2.5). This implies that

X^{p^{n_{j}}} − α_{j}^{p^{n_{j}}} = (X − α_{j})^{p^{n_{j}}} (21.2.17)

is equal to the product of f_{j}(X) and another monic formal polynomial in X with coefficients in k, as before. In fact, if we take n_{j} to be as small as possible, then we get that f_{j}(X) is equal to (21.2.17), as before. However, for the moment we only need the fact that α_{j} is the only root of f_{j}(X) in \overline{E}.

Any embedding of E into \overline{E} over k sends α_{j} to a root of f_{j}(X). This means that such an embedding sends α_{j} to itself for each j, because it is the only root of f_{j}(X). It follows that such an embedding is equal to the identity mapping on E. This implies (21.2.2).

If E satisfies these conditions, then E is said to be purely inseparable as an extension of k, as on p187 of [177].

21.3 Maximal separable subextensions

Let k be a field of positive characteristic p. Proposition 10 on p187 of [177] states that

(21.3.1) the class of purely inseparable extensions of k is distinguished,

in the sense described in Section 19.10. The first condition in the definition of a distinguished class can be obtainsd using (21.2.2) and a property of the separable degree on Section 20.5. The second condition in the definition of a distinguished class can be obtained from the fourth condition in the previous section.

Let E be an algebraic extension of k, and let E_{0} be the compositum of all of the subfields F of E that contain k and that are separable over k, as in Section 19.10. Proposition 11 on p187 of [177] states that

(21.3.2) E_{0} is separable over k,

and that (21.3.3) E is purely inseparable over E_{0}.

More precisely, (21.3.2) holds because E_{0} is generated by elements that are separable over k, as in Section 20.4. In fact, E_{0} consists of all of the elements of E that are separable over k, as in [177]. If α ∈ E, then

α^{p^{n}} ∈ E_{0} (21.3.4)

for some nonnegative integer n, as in (21.1.3), which implies (21.3.3).

If E is both separable and purely inseparable over k, then Corollary 1 on p188 of [177] states that E = k. (21.3.5)

Indeed, if α ∈ E, then the minimal polynomial of α over k is of the form (21.2.4) for some n ≥ 0, as before. However, if E is separable over k, then α is separable over k, so that the formal derivative of this polynomial should not be 0, as in Sections 20.3 and 20.4. This means that n = 0, so that α ∈ k.

If r is a nonnegative integer, then put

E^{p^{r}} = {x^{p^{r}} : x ∈ E}. (21.3.6)

This is a subfield of E, because x ↦ x^{p^{r}} is a homomorphism from E into itself as a field, as in Section 20.15.

Consider the condition that

(21.3.7) the compositum of E^{p} and k is equal to E,

as in Corollary 4 on p188 of [177]. If r is a positive integer, the one can use (21.3.7) to get that

(21.3.8) the compositum of E^{p^{r}} and k is equal to E, as mentioned in [177].

Suppose that E is a finite extension of k, and let E_{0} be the maximal subfield of E that contains k and is separable over k, as before. Remember that E is generated by k and finitely many elements α_{1}, …, α_{n}, as mentioned in Section 19.9. We also have that for each j = 1, …, n there is a nonnegative integer m_{j} such that α_{j}^{p^{m_{j}}} ∈ E_{0}, (21.3.9)

as in (21.3.4). This implies that there is a nonnegative integer m such that

α_{j}^{p^{m}} ∈ E_{0} (21.3.10)

for each j = 1, …, m. It follows that

E^{p^{m}} ⊆ E_{0} (21.3.11)

under these conditions, as in [177].

If (21.3.7) holds, then (21.3.8) holds with r = m, and we get that

E = E_{0}. (21.3.12)

This means that E is separable over k, as in Corollary 4 on p188 of [177].

If E is any algebraic extension of k again, then let F be the compositum of E^{p} and k, so that F is a subfield of E that contains k. Clearly

(21.3.13) E is purely inseparable over F,

as mentioned in [177].

Suppose that E is separable over k. It is easy to see that

(21.3.14) E is separable over F.

This is part of the fact that the class of separable extensions is distinguished, as in Section 20.6.

Combining (21.3.13) and (21.3.14), we get that

E = F, (21.3.15)

as in (21.3.5). This means that (21.3.7) holds, so that (21.3.8) holds, as in Corollary 4 on p188 of [177].

Suppose that k is perfect, so that k^{p} = k, as in Section 20.15. This implies that E is separable over k, as before. This means that (21.3.8) holds, as in the preceding paragraph. This is the same as saying that the compositum of E^{p} and k^{p} is equal to E in this case, so that

E^{p} = E. (21.3.16)

Thus E is perfect too, as in the corollary on p190 of [177].

If k is perfect, then it is easy to see that k^{p^{r}} = k for every positive integer r. In this case, if E is purely inseparable over k, then it follows that E = k. One can use also this and Proposition 12 on p189 of [177] to get that any algebraic extension over k is separable, as in the corollary on p190 of [177].

21.4 Some additional corollaries

Let k be a field of positive characteristic p again, and let K be a normal algebraic extension of k. If K_{0} is the maximal separable extension of k contained in K, then Corollary 2 on p188 of [177] says that

(21.4.1) K_{0} is normal over k.

To see this, let \overline{K} be an algebraic closure of K, which may be considered as an algebraic closure of k, as in Section 19.13. Also let σ_{0} be an embedding of K_{0} into \overline{K} over k. There is an extension of σ_{0} to an embedding σ of K into \overline{K} over k, as in Section 19.13.

Because K is normal over k, we have that σ(K) = K, as in Section 19.15. It is easy to see that (21.4.2) σ(K_{0}) is separable over k,

because K_{0} is separable over k. This implies that

σ(K_{0}) ⊆ K_{0}, (21.4.3)

because σ(K_{0}) ⊆ σ(K) = K, (21.4.4)

and because K_{0} is the maximal separable extension of k contained in K. Similarly, (21.4.5) σ^{−1}(K_{0}) is separable over k,

and σ^{−1}(K_{0}) ⊆ σ^{−1}(K) = K, (21.4.6)

so that σ^{−1}(K_{0}) ⊆ K_{0}. (21.4.7)

It follows that σ_{0}(K_{0}) = σ(K_{0}) = K_{0}, (21.4.8)

so that (21.4.1) holds, as in Section 19.15.

If E is a finite extension of k, then

[E : k]_{i} = \frac{[E : k]}{[E : k]_{s}} (21.4.9)

is called the inseparable degree or degree of inseparability of E over k, as on p180 of [177]. This is a nonnegative integer power of p, as in Corollary 1 on p180 of [177]. This follows from (21.1.2) when E is generated by a single element, as an extension of k. Otherwise, one can repeat the process, using basic properties of the ordinary and separable degrees of finite extensions, as in Sections 19.9 and 20.5. This can also be defined for extensions of fields of characteristic 0, in which case it is equal to 1.

Note that E is separable over k if and only if

[E : k]_{i} = 1, (21.4.10)

as in Corollary 2 on p180 of [177]. This uses the characterization of separability mentioned in Section 20.5. Similarly, E is purely inseparable over k if and only if [E : k]_{i} = [E : k], (21.4.11)

as in (21.2.2).

Let F be a subfield of E that contains k. Corollary 3 on p180 of [177] states that [E : k]_{i} = [E : F]_{i} [F : k]_{i}, (21.4.12)

which follows from the analogous properties of the ordinary and separable degrees.

Now let E, F be finite extensions of k such that E is separable over k, and F is purely inseparable over k. Suppose that E and F are subfields of a larger field, so that their compositum E F is a subfield of that larger field, as in Section 19.10. Under these conditions, Corollary 3 on p188 of [177] states that

[E F : F] = [E : k] = [E F : k]_{s} (21.4.13)

and [E F : E] = [F : k] = [E F : k]_{i}. (21.4.14)

We also have that (21.4.15) E F is separable over F

and (21.4.16) E F is purely inseparable over E,

as mentioned in [177].

Of course, [E F : k]_{s} = [E F : F]_{s} [F : k]_{s} (21.4.17)

and [E F : k]_{s} = [E F : E]_{s} [E : k]_{s}, (21.4.18)

as in Section 20.5. It follows from (21.4.17) that

[E F : k]_{s} = [E F : F]_{s} = [E F : F], (21.4.19)

using the hypothesis that F be purely inseparable over k in the first step, and (21.4.15) in the second step. One can get the second step in (21.4.13) using (21.4.18), (21.4.16), and the hypothesis that E be separable over k.

Similarly, [E F : k]_{i} = [E F : F]_{i} [F : k]_{i} (21.4.20)

and [E F : k]_{i} = [E F : E]_{i} [E : k]_{i}, (21.4.21)

21.5. SOME REMARKS ABOUT GAL(K/K) 513

as in (21.4.12). Using (21.4.21), we get that [E F : k]_{i} = [E F : E]_{i} = [E F : E], (21.4.22) because E is separable over k by hypothesis, and because of (21.4.16). The second step in (21.4.14) can be obtained from (21.4.20), (21.4.15), and the hypothesis that F be purely inseparable over k.

21.5 Some remarks about Gal(K/k)

Let k and K be fields, with k ⊆ K, and remember that Gal(K/k) is the Galois group of automorphisms of K over k, as in Section 19.15. If L is a subfield of K that contains k, then (21.5.1) Gal(K/L) is a subgroup of Gal(K/k).

Consider (21.5.2) K_{1} = {u ∈ K : θ(u) = u for every θ ∈ Gal(K/k)}, which is a subfield of K that contains k. Observe that Gal(K/K_{1}) = Gal(K/k). (21.5.3)

If K_{1} = k, then K is normal over k in the sense defined on p459 of [195], as mentioned in Section 19.15. Note that (21.5.4) K_{1} = {u ∈ K : θ(u) = u for every θ ∈ Gal(K/K_{1})}, because of (21.5.3). This means that K is automatically normal over K_{1} in the sense of [195].

Suppose from now on in this section that K is algebraic over k. Let \overline{K} be an algebraic closure of K, which may be considered as an algebraic closure of k, as in Section 19.13. Let L be a subfield of K that contains k again. If θ ∈ Gal(K/k), then the restriction of θ to L may be considered as an embedding of L into \overline{K}. If L is normal over k, then it follows that θ(L) = L, (21.5.5) as in Section 19.15.

Suppose that k has positive characteristic p. If L is a subfield of K that contains k, and if L is separable over k, then (21.5.6) θ(L) is separable over k for every θ ∈ Gal(K/k). Let K_{0} be the maximal separable extension of k contained in K, as in Section 21.3. It is easy to see that θ(K_{0}) = K_{0} (21.5.7) for every θ ∈ Gal(K/k).

21.6 Separability of K over K_{1}

Let k be a field of positive characteristic p, and let K be an algebraic extension of k. If K_{1} is as in (21.5.2), then we would like to show that

(21.6.1) K is separable over K_{1}.

This corresponds to part of the conclusion of Proposition 12 on p189 of [177]. Although the proposition is stated with the additional hypothesis that K be normal over k, this does not appear to be needed for this part of the conclusion.

If K is a finite extension of k, then K is a finite extension of K_{1}. In this case, one could use Theorem 12 on p459 of [195] to get (21.6.1). More precisely, condition (ii) in that theorem is that K be normal over K_{1} in the sense of [195], which corresponds to (21.5.4). Condition (iii) in that theorem is that (21.6.1) hold, and that K be normal in the sense used here, as in [24, 177]. Part of the theorem is that these conditions are equivalent for finite extensions.

In fact, the proof in [195] that condition (ii) implies condition (iii) seems to work as well when K is an algebraic extension of k, as mentioned in Section 20.8. This is related to Exercise 3 on p461 of [195], as before.

The argument in [177] also begins with the case where K is finite over k, with a suitable reduction afterwards. Let α ∈ K be given. One would like to find a maximal collection σ_{1}, …, σ_{r} of the Galois group Gal(K/k) of K over k such that σ_{1}(α), …, σ_{r}(α) (21.6.2)

are distinct elements of K. If K is a finite extension of k, then

[K : k]_{s} < ∞, (21.6.3)

as in Section 20.5. This means that Gal(K/k) has only finitely many elements, so that it is easy to find σ_{1}, …, σ_{r} ∈ Gal(K/k) as before.

However, if σ ∈ Gal(K/k), then σ(α) is a root of the minimal polynomial of α over k, and of course there are only finitely many roots of this polynomial. One can use this to get σ_{1}, …, σ_{r} ∈ Gal(K/k) as before, without asking that Gal(K/k) have only finitely many elements. Note that α should be one of the elements of K in the list (21.6.2), because otherwise one could also include the identity mapping on K. Equivalently, one may as well take the identity mapping on K to be one of the elements of Gal(K/k) being used.

Let X be an indeterminate, and consider the polynomial

∏_{j=1}^{r}(X − σ_{j}(α)). (21.6.4)

Note that α is a root of this polynomial, as in the preceding paragraph, and that the roots of this polynomial are distinct, by construction. Although the coefficients of this polynomial may be taken initially to be in K, one can check that they are in K_{1}, because the set of roots of the polynomial is invariant under Gal(K/k), by construction. This implies that α is separable over K_{1}.

The argument in [195] is somewhat analogous, although it considers any irreducible polynomial with coefficients in K_{1} and a root in K, and shows that it should be a multiple of a polynomial of the same type as in the previous paragraph.

Let K_{0} be the maximal separable extension of k contained in K, as in Section 21.3, and consider the compositum K_{0} K_{1} of K_{0} and K_{1}, which is another subfield of K that contains k. Remember that K is purely inseparable over K_{0}, as in Section 21.3, which implies that K is purely inseparable over K_{0} K_{1}. We also have that K is separable over K_{0} K_{1}, because of (21.6.1). It follows that

K = K_{0} K_{1}, (21.6.5)

as in Section 21.3. This is another part of Proposition 12 on p189 of [177].

21.7 Normality and separability

Let k be a field of positive characteristic p, and let K be a normal algebraic extension of k, as in Section 19.15. If K_{1} is as in (21.5.2) again, then

(21.7.1) K_{1} is purely inseparable over k,

as in Proposition 12 on p189 of [177].

To see this, let α ∈ K_{1} be given. Let \overline{K} be an algebraic closure of K, which may be considered as an algebraic closure of k, as in Section 19.13. Remember that k(α) is the subfield of K generated by k and α, and let τ be an embedding of k(α) into \overline{K} over k. This can be extended to an embedding of K into \overline{K} over k, as in Section 19.13, and we let τ denote this extension as well. Note that

τ(K) = K, (21.7.2)

because K is normal over k, as in Section 19.15.

Thus τ ∈ Gal(K/k), so that

τ(α) = α, (21.7.3)

because α ∈ K_{1}, by hypothesis. This implies that

(21.7.4) τ is the identity mapping on k(α).

This means that [k(α) : k]_{s} = 1, (21.7.5)

as in Section 20.5. It follows that α is purely inseparable over k, as in Section 21.2. This shows that (21.7.1) holds.

If K is also a separable extension of k, then K_{1} is separable over k, and (21.7.1) implies that K_{1} = k, (21.7.6) as in Section 21.3. Alternatively, one can use the argument in the previous paragraphs, and observe that (21.7.5) is the same as saying that [k(α) : k] = 1 in this case. This implies that α ∈ k, so that (21.7.6) holds. This works as well for fields of characteristic 0, which corresponds to the first part of Theorem 1 on p192 of [177], as in Section 20.8.

Let K_{0} be the maximal separable extension of k that is contained in K, as in Section 21.3. Observe that K_{0} ∩ K_{1} is separable over k, and purely inseparable over k, because of (21.7.1). This implies that

K_{0} ∩ K_{1} = k, (21.7.7)

as in Section 21.3. This is another part of Proposition 12 on p189 of [177].

21.8 Primitive elements of finite extensions

Let k be a field, and let E be a finite extension of k. If α is an element of E such that

(21.8.1) the subfield k(α) of E generated by k and α is equal to E,

then α is said to be a primitive element of E over k, as on p186 of [177]. If there is a primitive element α of E over k, then

(21.8.2) there are only finitely many subfields F of E with k ⊆ F,

as in Theorem 14 on p185 of [177]. To see this, let X be an indeterminate, and let f(X) be the minimal polynomial of α over k, as in Section 19.9. Similarly, if F is a subfield of E that contains k, then α may be considered as an algebraic element of E over F, and we let g_{F} (X) be the minimal polynomial of α over F.

We would like to first check that there are only finitely many possible polynomials g_{F} (X), over all such F. Of course, f(X) may be considered as a polynomial with coefficients in F, so that f(X) may be expressed as the product of g_{F} (X) and another polynomial in X with coefficients in F.

Let \overline{E} be an algebraic closure of E, which may be considered as an algebraic closure of k, as in Section 19.13. More precisely, for the moment we simply want f(X) to be a product of linear factors, as a formal polynomial in X with coefficients in \overline{E}. This means that g_{F} (X) may be expressed in the same way, using only linear factors that are linear factors of f(X). Of course, the roots of g_{F} (X) are roots of f(X) as well. It follows that there are only finitely many possible polynomials g_{F} (X), because g_{F} (X) is monic, by construction.

We would like to show that the mapping

F ↦ g_{F} (X) (21.8.3)

is injective. Let F_{0} = F_{0}(g_{F} (X)) (21.8.4) be the subfield of F generated by k and the coefficients of g_{F} (X). In fact, we shall show that F_{0} = F. (21.8.5)

Note that g_{F} (X) is irreducible as a polynomial in X with coefficients in F_{0}, because g_{F} (X) is irreducible as a polynomial in X with coefficients in F, by construction. Of course, E is the same as the subfields F_{0}(α), F (α) of E generated by F_{0}, F, respectively, and α. This implies that

[E : F_{0}] = [E : F], (21.8.6)

as in Section 19.9. More precisely, both sides of the equation are equal to the degree of g_{F} (X), because g_{F} (X) is also the minimal polynomial of α over F_{0}.

Remember that [E : F_{0}] = [E : F_{1}] [F_{1} : F_{0}], (21.8.7)

as in Section 19.9. It follows that from this and (21.8.6) that

[F : F_{0}] = 1, (21.8.8)

so that (21.8.5) holds.

21.9 Existence of primitive elements

Let k be a field, and let E be a finite extension of k again. If (21.8.2) holds, then (21.9.1) there is a primitive element of E over k.

This is another part of Theorem 14 on p185 of [177], which also corresponds to Exercise 7 on p465 of [195].

Of course, if k has only finitely many elements, then E has only finitely many elements. In this case, it is well known that the multiplicative group of nonzero elements of E is a cyclic group. Any generator of E \ {0} with respect to multiplication is a primitive element of E over k in particular.

Otherwise, suppose that k has infinitely may elements, and let α, β ∈ E be given. If c ∈ k, then consider the subfield

k(α + c β) (21.9.2)

of E generated by k and α + c β. Using (21.8.2), we get that there are c_{1}, c_{2} ∈ k such that c_{1} ≠ c_{2} k(α + c_{1} β) = k(α + c_{2} β). (21.9.3)

Of course, the subfield (21.9.3) of E contains α+c_{1} β and α+c_{2} β, so that it contains (c_{1} − c_{2}) β as well. This implies that this subfield contains β, because c_{1} − c_{2} is a nonzero element of k. It follows that this subfield contains c_{1} β, c_{2} β, because c_{1}, c_{2} ∈ k, and thus this subfield contains α too. This shows that (21.9.3) is the same as

(21.9.4) the subfield of E generated by k and α, β.

In particular, this means that (21.9.4) is generated by one of its elements and k.

Remember that E can be generated by finitely many elements and k, as in Section 19.9. One can repeat the argument in the previous two paragraphs to get that E is generated by one element and k.

21.10 Separability and primitive elements

Let k be a field, and let E be a finite extension of k that is separable over k. Under these conditions, another part of Theorem 14 on p185 of [177] states that (21.9.1) holds. To show this, we may as well suppose that k has infinitely many elements, as in the previous section.

Let α, β ∈ E be given, and consider the subfield k(α, β) of E generated by α, β, and k. Let \overline{E} be an algebraic closure of E again, which may be considered as an algebraic closure of k, as in Section 19.13. Note that k(α, β) is separable over k, as in Section 20.4.

Let σ_{1}, …, σ_{n} be the distinct embeddings of k(α, β) into \overline{E} over k. Observe that σ_{j}(α) ≠ σ_{l}(β) or σ_{j}(β) ≠ σ_{l}(β) (21.10.1)

when j ≠ l, because σ_{j} ≠ σ_{l}. Let X be an indeterminate, and consider the polynomial

P (X) = ∏_{j≠l}(σ_{j}(α) + σ_{j}(β) X − σ_{l}(α) − σ_{l}(β) X), (21.10.2)

as on p186 of [177]. Each of the linear factors in the product on the right side is nonzero, because of (21.10.1). This means that P (X) is nonzero as a polynomial in X.

It follows that P (X) has only finitely many roots in k, so that

P (c) ≠ 0 (21.10.3)

for some c ∈ k, because k has infinitely many elements. Observe that

(21.10.4)  P (c) = ∏_{j≠l}(σ_{j}(α) + c σ_{j}(β) − σ_{l}(α) − c σ_{l}(β))
= ∏_{j≠l}(σ_{j}(α + c β) − σ_{l}(α + c β)).

Thus (21.10.3) is the same as saying that

σ_{j}(α + c β) ≠ σ_{l}(α + c β) (21.10.5)

when j ≠ l.

Consider the subfield k(α + c β) of E generated by α + c β and k, and note that k(α + c β) ⊆ k(α, β). (21.10.6)

21.11. MORE ON GALOIS EXTENSIONS 519 Using (21.10.5), we get that the restrictions of σ_{1}, …, σ_{n} to k(α + c β) are distinct embeddings of k(α + c β) into \overline{E}. This implies that n ≤ [k(α + c β) : k], (21.10.7) as in Section 20.5.

We also have that n = [k(α, β) : k], (21.10.8) because k(α, β) is separable over k, as in Section 20.5 again. Thus (21.10.7) is the same as saying that [k(α, β) : k] ≤ [k(α + c β) : k]. (21.10.9) Combining this with (21.10.6), we obtain that k(α, β) = k(α + c β). (21.10.10)

One can use (21.10.10) to get that (21.9.1) holds, as in the previous section.

21.11 More on Galois extensions

Let k, F, K be fields with k ⊆ F ⊆ K. (21.11.1) Remember that the Galois group Gal(K/k) is the group of automorphisms of K over k, as in Section 19.15. Of course, Gal(K/F) can be defined in the same way, and it is easy to see that (21.11.2) Gal(K/F) is a subgroup of Gal(K/k).

Suppose that K is a Galois extension of k, as in Section 20.8, so that K is an algebraic extension of k that is normal and separable. Under these conditions, (21.11.3) K is a Galois extension of F, as in Theorem 1 on p192 of [177]. Indeed, K is algebraic over F, because K is algebraic over k, as mentioned in Section 19.11. We also have that K is normal over F, because K is normal over k, as in Section 20.1. The separability of K over F follows from the separability of K over k as well, as in Section 20.6.

Another part of Theorem 1 on p192 of [177] says that F ↦ Gal(K/F) (21.11.4) is injective as a mapping from the set of fields F that satisfy (21.11.1) into the set of subgroups of Gal(K/k). Note that (21.11.5) F = {v ∈ K : θ(v) = v for every θ ∈ Gal(K/F)}, because of (21.11.3), as in Sections 20.8 and 21.7. This implies that (21.11.4) is injective.

Suppose that K also has finite degree over k. This means that

(21.11.6) Gal(K/k) is a finite group,

as in Section 20.8. Using the injectivity of (21.11.4), we get that

(21.11.7) there are only finitely many fields F that satisfy (21.11.1).

This is a simplification of part of Corollary 4 on p193 of [177]. This corresponds to Exercise 5 on p474 of [24].

Now let E be a finite separable extension of k, and let \overline{E} be an algebraic closure of E. There is a finite normal separable extension K of k in \overline{E} that contains E, as in Section 20.7. Thus (21.11.7) holds, so that

(21.11.8) there are only finitely may subfields F of E that contain k

in particular, as in Corollary 4 on p193 of [177]. This corresponds to Exercise 7 on p474 of [24] when k has characteristic 0, and to Exercise 4 on p464 of [195].

One can use (21.11.8) to get that there is a primitive element of E over k, as in Section 21.9. This corresponds to Exercise 8 on p465 of [195].

Alternatively, one can use the existence of a primitive element of E over k, as in Section 21.10, to get (21.11.8), as in Section 21.8. This is mentioned on p194 of [177], after the proof of Corollary 4.

21.12 Some fields of rational functions

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let X_{1}, …, X_{n} be n distinct commuting indeterminates. If V is a module over k, then the space V [X_{1}, …, X_{n}] of formal polynomials in X_{1}, …, X_{n} with coefficients in V may be defined as a module over k as in Section 4.3. Similarly, if n ≥ 2, then the space V [X_{1}, …, X_{n−1}] of formal polynomials in X_{1}, …, X_{n−1} with coefficients in V may be defined as a module over k, as well as the space

(V [X_{1}, …, X_{n−1}])[X_{n}] (21.12.1)

of formal polynomials in X_{n} with coefficients in V [X_{1}, …, X_{n−1}]. There is a one-to-one correspondence between V [X_{1}, …, X_{n}] and (21.12.1), in which a polynomial in X_{1}, …, X_{n} is expressed as a polynomial in X_{n} whose coefficients are polynomials in X_{1}, …, X_{n−1}. More precisely, this defines an isomorphism between these two modules over k.

If A is an algebra over k in the strict sense, then A[X_{1}, …, X_{n}] may be defined as an algebra over k in the strict sense, as in Section 20.13. If n ≥ 2, then A[X_{1}, …, X_{n−1}] and (A[X_{1}, …, X_{n−1}])[X_{n}] may be defined as algebras over k in the strict sense as well. The mapping described in the preceding paragraph defines an algebra isomorphism from A[X_{1}, …, X_{n}] onto (A[X_{1}, …, X_{n−1}])[X_{n}].

If k is an integral domain and X is an indeterminate, then it is well known and easy to see that (21.12.2) k[X] is an integral domain

too. It follows that

(21.12.3) k[X_{1}, …, X_{n}] is an integral domain

for each n, using induction and the isomorphism mentioned in the previous paragraph.

If k is a field, then k(X_{1}, …, X_{n}) (21.12.4)

is defined as the field Q_{k[X_{1},...,X_{n}]} of fractions of k[X_{1}, …, X_{n}], as in Sections 11.4 and 12.1. The elements of (21.12.4) may be called formal rational functions in X_{1}, …, X_{n} with coefficients in k.

Let k be a field of characteristic p for some prime number p, and let X be an indeterminate. Consider the field k(X) of formal rational functions in X with coefficients in k, as in the preceding paragraph. The subfield

k(X^{p}) (21.12.5)

of k(X) generated by k and X^{p} consists of ratios of polynomials in X^{p} with coefficients in k, where the denominator is nonzero as a formal polynomial.

If f(X) ∈ k(X), then it is easy to see that

f(X)^{p} ∈ k(X^{p}), (21.12.6)

using a basic fact mentioned in Section 20.15. This implies that k(X) is purely inseparable over k(X^{p}), as in Section 21.2. Of course, (21.12.6) is the same as saying that k(X)^{p} ⊆ k(X^{p}), (21.12.7)

where the left side is as in Section 21.3. More precisely,

k(X)^{p} = k^{p}(X^{p}), (21.12.8)

the subfield of k(X) generated by k^{p} and X^{p}. This is equal to k(X^{p}) when k is perfect, so that k^{p} = k, as in Section 20.15.

If f(X) is a nonzero element of k[X], then

1/f(X) = f(X)^{p−1}/f(X)^{p}. (21.12.9)

This implies that every element of k(X) may be expressed as the ratio of an element of k[X] and a nonzero element of k(X)^{p}. One can use this to get that the monomials X^{j}, (21.12.10)

0 ≤ j ≤ p − 1, form a basis for k(X) as a vector space over k(X^{p}). It follows that [k(X) : k(X^{p})] = p. (21.12.11)

Note that X is a primitive element of k(X) over k(X^{p}), as in Section 21.8.

Let Y be another indeterminate that commutes with X, and consider the field k(X, Y) (21.12.12)

of formal rational functions in X, Y with coefficients in k, as in (21.12.4). The subfield k(X^{p}, Y^{p}) (21.12.13)

of (21.12.12) generated by k and X^{p}, Y^{p} consists of ratios of polynomials in X^{p}, Y^{p} with coefficients in k, where the denominator is nonzero as a formal polynomial.

If f(X, Y) ∈ k(X, Y), then

f(X, Y)^{p} ∈ k(X^{p}, Y^{p}), (21.12.14)

as before. Thus k(X, Y) is purely inseparable over k(X^{p}, Y^{p}), as in Section 21.2 again. Equivalently, (21.12.14) means that

k(X, Y)^{p} ⊆ k(X^{p}, Y^{p}), (21.12.15)

where the left side is as in Section 21.3. In fact,

k(X, Y)^{p} = k^{p}(X^{p}, Y^{p}), (21.12.16)

the subfield of k(X, Y) generated by k^{p}, X^{p}, and Y^{p}. This is the same as k(X^{p}, Y^{p}) when k is perfect, as before.

If f(X, Y) is a nonzero element of k[X, Y], then

1/f(X, Y) = f(X, Y)^{p−1}/f(X, Y)^{p}, (21.12.17)

so that every element of k(X, Y) may be expressed as the ratio of an element of k[X, Y] and a nonzero element of k(X, Y)^{p}. One can use this to get that the monomials X^{j} Y^{l}, (21.12.18)

0 ≤ j, l ≤ p − 1, form a basis for k(X, Y) as a vector space over k(X^{p}, Y^{p}), as before. This means that

[k(X, Y) : k(X^{p}, Y^{p})] = p^{2}, (21.12.19)

as in part (i) of Exercise 3 on p191 of [177].

One can show that k(X, Y) does not have a primitive element over k(X^{p}, Y^{p}). This is the same as saying that there are infinitely many subfields of k(X, Y) that contain k(X^{p}, Y^{p}), as in Sections 21.8 and 21.9. This corresponds to part (ii) of Exercise 3 on p191 of [177].

21.13 Derivations on algebras

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, as in Section 20.13. A homomorphism δ from A into itself, as a module over k, is said to be a derivation on A if

δ(x y) = δ(x) y + x δ(y) (21.13.1)

for all x, y ∈ A, as on p4 of [144], and Definition 2 on p2 of [264]. This is related to Definition 4 on p167 of [153], and we shall say more about this in Section 23.2. Derivations on rings are mentioned on p266 of [177], and another version will be discussed in Section 23.3.

Remember that the space Hom_{k}(A, A) of all homomorphisms from A into itself, as a module over k, is a module over k with respect to pointwise addition and scalar multiplication. It is easy to see that the space Der(A) = Der_{k}(A) of all derivations on A is a submodule of Hom_{k}(A, A), as a module over k. If δ_{1}, δ_{2} ∈ Der(A), then one can check that

δ_{1} ∘ δ_{2} − δ_{2} ∘ δ_{1} ∈ Der(A), (21.13.2)

as mentioned on p4 of [144] and p173 of [153], and as in Example (iv) on p2 of [264].

Suppose for the moment that A is an associative algebra over k. If a ∈ A, then one can verify that δ_{a}(x) = a x − x a (21.13.3)

defines a derivation on A, as in Exercise 1 on p174 of [153]. This may be called an inner derivation on A. If a, b ∈ A, then one can also check that

δ_{a} ∘ δ_{b} − δ_{b} ∘ δ_{a} = δ_{a b−b a} (21.13.4)

on A, as in Exercise 1 on p174 of [153]. In fact, if δ is any derivation on A, then

δ ∘ δ_{a} − δ_{a} ∘ δ = δ_{δ(a)} (21.13.5)

on A.

Suppose for the moment that A is a commutative associative algebra over k. If a, x, y ∈ A and δ is a derivation on A, then

a δ(x y) = a δ(x) y + x a δ(y). (21.13.6)

This means that x ↦ a δ(x) (21.13.7)

is a derivation on A as well.

Let A be any algebra over k in the strict sense again, and let A_{0} be a subalgebra of A, as in Section 20.13. If δ is a derivation on A and

δ(A_{0}) ⊆ A_{0}, (21.13.8) then the restriction of δ to A_{0} is a derivation on A_{0}.

Let X_{1}, …, X_{n} be n distinct commuting indeterminates for some positive integer n, and remember that the space A[[X_{1}, …, X_{n}]] of formal power series in X_{1}, …, X_{n} with coefficients in A may be considered as an algebra over k in the strict sense, as in Sections 4.3 and 20.13. Similarly, the space A[X_{1}, …, X_{n}] of formal polynomials in X_{1}, …, X_{n} with coefficients in A is a subalgebra of A[[X_{1}, …, X_{n}]]. The corresponding formal partial differentiation operators

∂_{l} = ∂/∂X_{l} (21.13.9)

defined on A[[X_{1}, …, X_{n}]] in Section 4.6 for l = 1, …, n are derivations on A[[X_{1}, …, X_{n}]], as an algebra over k in the strict sense, as before. These operators map A[X_{1}, …, X_{n}] into itself, and thus define derivations on A[X_{1}, …, X_{n}] too, as an algebra over k in the strict sense.

If δ is any derivation on A, then

(21.13.10) ker δ is a subalgebra of A.

If A has a multiplicative identity element e_{A}, then it is easy to see that

δ(e_{A}) = 0. (21.13.11)

If x, y ∈ A satisfy x y = e_{A}, (21.13.12)

then it follows that δ(x) y + x δ(y) = 0. (21.13.13)

If A is associative and x has a multiplicative inverse in A, then we get that

δ(x^{−1}) = −x^{−1} δ(x) x^{−1}. (21.13.14)

In particular, if A is associative, x has a multiplicative inverse in A, and δ(x) = 0, then (21.13.14) implies that

δ(x^{−1}) = 0. (21.13.15)

More precisely, if x, y satisfy (21.13.12) and δ(x) = 0, then

x δ(y) = 0, (21.13.16)

by (21.13.13). If z ↦ x z has trivial kernel in A, then it follows that

δ(y) = 0. (21.13.17)

21.14 Derivations on rings and fields

Let A be a ring. Thus A is an associative algebra over the integers, so that the notion of a derivation on A may be defined as in the previous section.

Equivalently, a mapping δ from A into itself is a derivation on A as a ring if it is a homomorphism from A into itself as a commutative group with respect to addition, and (21.13.1) holds for every x, y ∈ A.

Suppose for the moment that A is a commutative ring. If x ∈ A, n ≥ 2 is an integer, and δ is a derivation on A, then

δ(x^{n}) = n · x^{n−1} δ(x). (21.14.1)

Let k be a commutative ring with a multiplicative identity element 1_{k}. If A is an associative algebra over k, and thus a ring in particular, then a mapping δ from A into itself is a derivation on A as an algebra over k if and only if δ is a derivation on A as a ring and δ is a homomorphism from A into itself as a module over k. This is the same as saying that δ is a derivation on A as a ring and that δ(t x) = t δ(x) (21.14.2)

for every t ∈ k and x ∈ A, because a derivation on A as a ring is a homomorphism from A into itself as a commutative group with respect to addition.

If A has a multiplicative identity element e_{A}, and δ is a derivation on A as a ring, then δ(e_{A}) = 0, as in (21.13.11). If A is an associative algebra over k, and δ is a derivation on A as an associative algebra over k, then it follows that

δ(t e_{A}) = 0 (21.14.3)

for every t ∈ k. Conversely, if δ is a derivation on A as a ring, and (21.14.3) holds for every t ∈ k, then δ is a derivation on A as an associative algebra over k. Indeed, if t ∈ k and x ∈ A, then

δ(t x) = δ((t e_{A}) x) = δ(t e_{A}) x + (t e_{A}) δ(x) = t δ(x). (21.14.4)

In particular, the notion of a derivation on k as a ring may be defined in this way. If δ is a derivation on k as a ring, then

δ(1_{k}) = 0, (21.14.5)

as in (21.13.11). Of course, k may be considered as an associative algebra over itself. If δ is a derivation on k as an associative algebra over itself, then (21.14.5) implies that δ(t) = 0 (21.14.6)

for every t ∈ k, as in (21.14.3).

Suppose that k is an integral domain, and let Q_{k} be the corresponding field of fractions, as in Sections 11.4 and 12.1. If δ is a derivation on k as a ring, then one can check that δ has a unique extension to a derivation on Q_{k} as a ring. More precisely, if u, v ∈ k and v ≠ 0, then one would like to put

δ(u/v) = (v δ(u) − u δ(v))/v^{2}, (21.14.7)

as on p267 of [177]. This is related to Theorem 13 on p170 of [153], and to some remarks on p174 of [153].

Now let k be a field, and let X_{1}, …, X_{n} be n distinct commuting indeterminates for some positive integer n. The partial differentiation operators (21.13.9) are derivations on k[X_{1}, …, X_{n}], as before. These differentiation operators can be extended uniquely to derivations on the field k(X_{1}, …, X_{n}) of formal rational functions in X_{1}, …, X_{n} with coefficients in k, as in the preceding paragraph.

21.15 Some more derivations

Let k be a commutative ring with a multiplicative identity element, let A be an algebra over k in the strict sense, and let I be a submodule of A, as a module over k. If for every a ∈ A and x ∈ I we have that

a x, x a ∈ A, (21.15.1)

then I is said to be a two-sided ideal in A. In this case, let A/I be the quotient of A by I, as a module over k, and let q_{A/I} be the corresponding quotient mapping from A onto A/I. We would like to define multiplication on A/I so that q_{A/I}(a) q_{A/I}(b) = q_{A/I}(a b) (21.15.2)

for every a, b ∈ A. One can check that the right side depends only on q_{A/I}(a), q_{A/I}(b), so that A/I is also an algebra over k in the strict sense in this way.

Let δ be a derivation on A, and suppose that

δ(I) ⊆ I. (21.15.3)

This implies that there is a unique homomorphism δ_{A/I} from A/I into itself, as a module over k, such that

δ_{A/I}(q_{A/I}(a)) = q_{A/I}(δ(a)) (21.15.4)

for every a ∈ A. It is easy to see that δ_{A/I} is a derivation on A/I as an algebra over k in the strict sense under these conditions.

Now let k be a field of positive characteristic p, and let X be an indeterminate. The formal differentiation operator ∂ = ∂/∂X on k[X] is a derivation on k[X] as an algebra over k, as in Section 21.13. If a ∈ k, then

∂(X^{p} − a) = p X^{p−1} = 0. (21.15.5)

If I(a) is the ideal in k[X] generated by X^{p} − a, then it follows that

∂(I(a)) ⊆ I(a). (21.15.6)

This means that ∂ induces a derivation ∂_{k[X]/I(a)} on the quotient k[X]/I(a) as an algebra over k, as in the preceding paragraph.

Let E be an extension of k that is generated by k and an element α of E with α^{p} = a. (21.15.7)

Thus X^{p} − a = (X − α)^{p}, (21.15.8)

as in Section 20.15. Suppose that α is not in k, so that X^{p} − a is the minimal polynomial of α over k, as in Section 21.2. This means that E is isomorphic to k[X]/I(a), as an extension of k. This leads to a derivation δ_{α} on E, as an algebra over k, corresponding to ∂_{k[X]/I(a)} on k[X]/I(a).

By construction, δ_{α}(α^{n}) = n · α^{n−1} (21.15.9)

for n = 1, …, p, because α corresponds to the image of X in k[X]/I(a). Of course, the right side is equal to 0 when n = p. One could also simply define δ_{α} directly on E in this way, and verify that it is a derivation on E, as an algebra over k. This corresponds to some remarks on p175 of [153], and to Case 3 on p268 of [177].

Some more rings and fields

22.1 Semigroups with cancellation

Let Σ be a semigroup. if for every a, x, x′ ∈ Σ with

a x = a x′ (22.1.1)

we have that x = x′, (22.1.2)

then we say that the left cancellation law holds in Σ, as on p24 of [151]. Similarly, if for every a, x, x′ ∈ Σ with x a = x′ a (22.1.3)

we have that (22.1.2) holds, then we say that the right cancellation law holds in Σ. Of course, groups satisfy both cancellation laws, and the two cancellation laws are the same when Σ is commutative. If Σ satisfies either cancellation law, then every sub-semigroup of Σ has the same property.

Let us say that a ring A is an integral domain in the strict sense if for every a, b ∈ A with a, b ≠ 0 we have that

a b ≠ 0. (22.1.4)

This is the way that the term “integral domain” is defined on p53 of [151], and we use this slightly different terminology to distinguish this from the other definition, where A is also asked to be commutative and to have a nonzero multiplicative identity element. This is the same as saying that

A \ {0} (22.1.5)

is a sub-semigroup of A, as a semigroup with respect to multiplication, as in [151].

If A is an integral domain in the strict sense and a, b, c ∈ A satisfy a ≠ 0 and a b = a c, (22.1.6) then it is easy to see that b = c, (22.1.7)

as on p54 of [151]. Similarly, (22.1.7) holds when

b a = c a. (22.1.8)

Each of these conditions is necessary for A to be an integral domain in the strict sense, as mentioned in [151]. In particular, if A is an integral domain in the strict sense, then (22.1.5) satisfies the left and right cancellation laws.

Let Σ be a semigroup again, and suppose that Σ has an identity element e_{Σ}. The set of elements of Σ with an inverse in Σ may be denoted

G(Σ). (22.1.9)

This is a sub-semigroup of Σ that contains e_{Σ} and is a group, as on p24 of [151].

Suppose from now on in this section that Σ is a commutative semigroup with an identity element e_{Σ}, with the semigroup operation expressed multiplicatively, and that Σ satisfies the cancellation law. Let us say that x, y ∈ Σ are associates if x = a y, y = b x (22.1.10)

for some a, b ∈ Σ, as on p114 of [151]. Of course, this is related to the analogous notion for elements of a commutative ring with a multiplicative identity element, and integral domains in particular, as in Section 15.3. If (22.1.10) holds, then

a b x = a y = x, (22.1.11)

so that a b = e_{Σ}, (22.1.12)

by the cancellation law. This means that

a, b ∈ G(Σ), (22.1.13)

with b = a^{−1}.

Let us say that an element of Σ is irreducible if it is not invertible in Σ, and if it cannot be expressed as the product of two noninvertible elements of Σ, as on p115 of [151]. This is the same as the definition of irreducibility for nonzero elements of an integral domain in Section 15.3, with the condition of noninvertibility included.

Let us say that an element p of Σ is prime if for every a, b ∈ Σ such that a b is a multiple of p in Σ, we have that either a or b is a multiple of p in Σ, as on p116 of [151]. This is the same as the definition of a nonzero prime element of an integral domain in Section 15.3, without including the condition that the element be noninvertible.

If p is a noninvertible prime element of Σ, then

(22.1.14) p is irreducible in Σ,

for the same reasons as in Section 15.3. Indeed, if p = y z for some y, z ∈ Σ, then y or z is a multiple of p in Σ, and thus an associate of p, as before. This implies that the other one is invertible in Σ, because of the cancellation law.

22.2 Gaussian semigroups

Let Σ be a commutative semigroup with an identity element e_{Σ}, with the semigroup operation expressed multiplicatively, and for which the cancellation law holds. We say that Σ is Gaussian if

(22.2.1) every noninvertible element of Σ can be expressed as a product of finitely many irreducible elements of Σ,

and

(22.2.2) expresions of noninvertible elements of Σ as products of finitely many irreducible elements of Σ are essentially unique,

as in Definition 1 on p115 of [151]. Essential uniqueness of this expression means that any other expression with these properties has the same number of factors, and can be obtained by permuting the factors of the other expression and multiplying them by invertible elements of Σ.

An integral domain is said to be Gaussian if the corresponding semigroup of nonzero elements is Gaussian, as in Definition 1 on p115 of [151]. In this case, the integral domain is also said to be a unique factorization domain, or a factorial ring, as on p14 of [92], and p71 of [177]. It is well known that Z is a unique factorization domain, and that the ring k_{0}[X] of polynomials in an indeterminate X with coefficients in a field k_{0} is a unique factorization domain. It is also well known that principal ideal domains are unique factorization domains, as mentioned in Section 15.3. In particular, Z is a principal ideal domain, and k_{0}[X] is a principal ideal domain for any field k_{0}.

Consider the following condition:

(22.2.3) there is no infinite sequence {a_{j}}_{j=1}^{∞} of noninvertible elements of Σ such that for each j, a_{j} is the product of a_{j+1} and a noninvertible element of Σ.

This corresponds to the condition A on p116 of [151]. One can show that this holds when Σ is Gaussian, as in [151].

Condition B on p116 of [151] states that

(22.2.4) every irreducible element of Σ is prime.

One can check that this holds when Σ is Gaussian, as in Exercise 3 on p118 of [151].

It is well known that (22.2.3) implies (22.2.1), as on p117 of [151]. One can also use (22.2.4) to get (22.2.2), as in [151].

Let k be an integral domain, and suppose now that Σ = k \ {0}. In this case, (22.2.3) is the same as saying that

(22.2.5) there is no infinite sequence of strictly increasing principal ideals in k.

This corresponds to Condition A’ on p121 of [151]. Equivalently, this means that any increasing sequence of principal ideals in k is eventually constant. In particular, this holds when k is Noetherian.

Some of the previous statements are discussed for integral domains on p14 of [92].

22.3 Greatest common divisors

Let Σ be a commutative semigroup with an identity element e_{Σ}, with the semigroup operation expressed multiplicatively, and for which the cancellation law holds. An element z of Σ is said to be a greatest common divisor of x, y ∈ Σ if

(22.3.1) x and y are multiples of z in Σ,

and if for every w ∈ Σ such that x and y are multiples of w in Σ, we have that

(22.3.2) z is a multiple of w in Σ.

This corresponds to Definition 2 on p118 of [151]. This was mentioned in Section 15.3 for integral domains.

If z is a greatest common divisor for x, y in Σ, then

(22.3.3) any other greatest common divisor of x, y in Σ is an associate of z,

as on p119 of [151]. This was also mentioned in Section 15.3 for integral domains. Condition C on p119 of [151] states that

(22.3.4) every pair of elements x, y of Σ has a greatest common divisor in Σ.

This holds when Σ is Gaussian, as on p118 of [151].

One can show that (22.3.4) implies (22.2.4), as on p120 of [151]. It follows that Σ is Gaussian when (22.2.3) and (22.3.4) hold, as in Theorem 1 on p120 of [151].

If k is a principal ideal domain, then k \ {0} satisfies (22.3.4), as in Section 15.3. This was used in the proof that k is a unique factorization domain, as in Theorem 2 on p122 of [151].

22.4 More on unique factorization domains

Let k be a unique factorization domain. It is well known that if k is also a Dedekind domain, then

(22.4.1) k is a principal ideal domain.

Of course, principal ideal domains are unique factorization domains, as mentioned in Sections 15.3 and 22.2. Principal ideal domains are Dedekind domains too, as in Section 15.11.

In order to show that (22.4.1) holds, let I be a proper ideal in k, with I ≠ {0}. Let x be a nonzero element of I, which is not invertible in k, because I ≠ k, by hypothesis. Thus x may be expressed as a product of finitely many nonzero prime elements of k. If

(22.4.2) I is a prime ideal in k,

then at least one of these nonzero prime elements y of k is an element of I. This implies that (22.4.3) I contains the ideal I(y) in k generated by y.

Note that (22.4.4) I(y) is a proper prime ideal in k,

because y is a prime element of k, and that I(y) ≠ {0}, because y ≠ 0. Suppose that (22.4.5) k has dimension one,

as in Section 14.8. This means that

(22.4.6) I(y) is a maximal proper ideal in k,

because k is an integral domain, by hypothesis, so that {0} is a proper prime ideal in k. It follows that I = I(y). (22.4.7)

If k is a Dedekind domain, then k has dimension less than or equal to one, as in Section 15.11. If the condition that k not be a field is included in the definition of a Dedekind domain, then k has dimension equal to one. Of course, if k is a field, then k is automatically a principal ideal domain, and so we may as well suppose that k has dimension one. It follows that all proper prime ideals in k are principal ideals, as in the preceding paragraph.

Remember that any proper nonzero ideal in k may be expressed as a product of proper nonzero prime ideals in k when k is a Dedekind domain, as in Sections 18.4 and 19.2. This means that any proper nonzero ideal in k may be expressed as a product of principal ideals in k, as in the preceding paragraph. It follows that every proper nonzero ideal in k is a principal ideal, because a product of principal ideals in k is a principal ideal as well.

Remember that an integral domain k is said to be integrally closed if it is integrally closed in its field Q_{k} of fractions, as in Section 16.3. If k is a unique factorization domain, then it is well known

(22.4.8) k is integrally closed,

as mentioned in Section 16.3. This also corresponds to Proposition 6 on p240 of [177].

Suppose that k is a Noetherian integral domain of dimension less than or equal to one. If k is a unique factorization domain, then (22.4.8) holds, and it follows that k is a Dedekind domain, as in Section 16.12. This implies that (22.4.1) holds, as before.

22.5 Some remarks about polynomials

Let k be a commutative ring with a multiplicative identity element. Remember that z ∈ k is said to be nilpotent if z^{n} = 0 in k for some positive integer n, as in Section 12.5. The set N = N (k) (22.5.1)

of all nilpotent elements of k is called the nilradical of k, and is an ideal in k, as before. If z ∈ N, then 1 − z is invertible in k, as in Section 13.1. If y ∈ k is invertible in k, then one can use the previous statement to get that

(22.5.2) y − z is invertible in k,

as in the second part of Exercise 1 on p10 of [8].

Let X be an indeterminate, and consider the corresponding ring k[X] of formal polynomials in X with coefficients in k, as in Section 4.3. Let

f(X) = ∑_{j=0}^{n} f_{j} X^{j} (22.5.3)

be given. The first part of Exercise 2 on p11 of [8] asks one to show that

(22.5.4) f(X) is invertible in k[X]

if and only if (22.5.5) f_{0} is invertible in k

and f_{j} ∈ N for each j = 1, …, n (22.5.6)

when n ≥ 1. The “if” part can be obtained from the remarks in the preceding paragraph.

Suppose that (22.5.4) holds, and let

g(X) = ∑_{l=0}^{m} g_{l} X^{l} (22.5.7)

be the multiplicative inverse of f(X) in k[X]. Note that f_{0} g_{0} = 1, which implies (22.5.5). If n ≥ 1, then it is suggested in [8] that one show that

f_{n}^{r+1} g_{m−r} = 0 (22.5.8)

for r = 0, 1, …, m, using induction on r. This implies that f_{n}^{m+1} = 0 when r = m, because g_{0} is invertible in k. Thus f_{n} X^{n} is nilpotent in k[X]. It follows that (22.5.9) f(X) − f_{n} X^{n} is invertible in k[X],

as before. One can repeat the process to get that (22.5.6) holds when n ≥ 1.

534 CHAPTER 22. SOME MORE RINGS AND FIELDS

The second part of Exercise 2 on p11 of [8] asks one to show that (22.5.10) f(X) is nilpotent in k[X] if and only if f_{j} ∈ N (k) for each j = 0, 1, …, n. (22.5.11) The “if” part follows from the fact that the nilradical N (k[X]) of k[X] is an ideal in k[X]. Conversely, it is easy to see that (22.5.10) implies that f_{0} and f_{n} are nilpotent in k. In particular, this implies that (22.5.12) f(X) − f_{n} X^{n} is nilpotent in k[X], and one can repeat the process to get (22.5.11).

The third part of Exercise 2 on p11 of [8] says that (22.5.13) f(X) is a zero-divisor in k[X] if and only if (22.5.14) there is an a ≠ 0 in k such that a f(X) = 0. Of course, the “if” part holds automatically. Conversely, if (22.5.13) holds, then there is a g(X) ∈ k[X] such that g(X) ≠ 0 and f(X) g(X) = 0. (22.5.15) We can take g(X) as in (22.5.7), with g_{m} ≠ 0. We may also suppose that m ≥ 1, because otherwise we get (22.5.14).

As in [8], we can take m to be as small as possible, so that the product of f(X) with a nonzero element of k[X] of degree strictly less than m is nonzero. Using (22.5.15), we get that f_{n} g_{m} = 0. (22.5.16) Note that f(X) (f_{n} g(X)) = 0, (22.5.17) by (22.5.15), and that the degree of f_{n} g(X) is strictly less than m. It follows that f_{n} g(X) = 0, (22.5.18) by the minimality of m, as in [8]. It is suggested in [8] that one use induction to show that f_{n−r} g(X) = 0 (22.5.19) for each r = 0, 1, …, n.

In particular, (22.5.19) implies that f_{n−r} g_{m} = 0 (22.5.20) for r = 0, 1, …, n. This means that f(X) g_{m} = 0, as in (22.5.14).

22.6 The content of a polynomial

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let X_{1}, …, X_{n} be n distinct commuting indeterminates. Also let f(X) = f(X_{1}, …, X_{n}) be a formal polynomial in X_{1}, …, X_{n} with coefficients in k, as in Section 4.3. The content of f(X) may be denoted

c(f(X)) = Content(f(X)), (22.6.1)

and is defined to be the ideal in k generated by the coefficients of f(X), as in Exercise 2 on p99 of [8] when n = 1, and Exercise 3.4 on p109 of [92]. A variant of this is defined on p126 of [177] when n = 1 and k is a unique factorization domain, and we shall return to this in Section 22.10.

Remember that the radical r(I) of an ideal I in k is defined in Section 12.10, and is the same as the intersection of all of the prime ideals in k that contain I. If g(X) = g(X_{1}, …, X_{n}) is another element of k[X_{1}, …, X_{n}], then the first part of part a of Exercise 3.4 on p109 of [92] states that

(22.6.2) Content(f(X) g(X)) ⊆ Content(f(X)) Content(g(X))
⊆ r(Content(f(X) g(X))).

This is a well-known version of Gauss’ lemma, as in [92]. Of course, the product of the ideals Content(f(X)), Content(g(X)) is the ideal in k defined as in Section 12.8. Note that the first inclusion in (22.6.2) follows immediately from the definitions.

To get the second inclusion, we may as well suppose that

Content(f(X) g(X)) ≠ k. (22.6.3)

Let I be a proper prime ideal in k such that

Content(f(X) g(X)) ⊆ I. (22.6.4)

It suffices to show that

(22.6.5) Content(f(X)) Content(g(X)) ⊆ I.

Note that the quotient k/I is an integral domain, because I is a proper prime ideal in k. Using the corresponding quotient mapping from k onto k/I, we get formal polynomials f_{k/I}(X), g_{k/I}(X) in X_{1}, …, X_{n} with coefficients in k/I. The formal polynomial in X_{1}, …, X_{n} with coefficients in k/I associated to f(X) g(X) is the same as f_{k/I}(X) g_{k/I}(X). This implies that

f_{k/I}(X) g_{k/I}(X) = 0 (22.6.6)

in (k/I)[X_{1}, …, X_{n}], because of (22.6.4).

Observe that (k/I)[X_{1}, …, X_{n}] is an integral domain, because k/I is an integral domain, as in Section 21.12. Thus (22.6.6) implies that

f_{k/I}(X) = 0 or g_{k/I}(X) = 0 (22.6.7) in (k/I)[X_{1}, …, X_{n}]. This means that

(22.6.8) Content(f(X)) ⊆ I or Content(g(X)) ⊆ I.

It follows that (22.6.5) holds.

If Content(f(X)) = k, (22.6.9)

then f(X) is said to be primitive, as on p109 of [92], and in the fourth part of Exercise 2 on p11 of [8] when n = 1. The fourth part of Exercise 2 on p11 of [8] states that (22.6.10) f(X) g(X) is primitive

if and only if (22.6.11) f(X) and g(X) are primitive.

More precisely, this is stated in [8] for n = 1, and it can be obtained from (22.6.2) for all n. This uses the fact that if the radical of an ideal I in k is equal to k, then I = k, as mentioned in Section 12.10.

Suppose that n = 1, and let us mention another way to show that (22.6.11) implies (22.6.10). Suppose that (22.6.4) holds for some proper ideal I in k, and let f_{k/I}(X), g_{k/I}(X) be as before. Thus (22.6.6) holds, as before. If g(X) is primitive, then g_{k/I}(X) ≠ 0 (22.6.12)

in (k/I)[X]. This implies that there is an a ≠ 0 in k/I such that

a f_{k/I}(X) = 0 (22.6.13)

in (k/I)[X], as in (22.5.14). However, if f(X) is primitive, then f_{k/I}(X) is primitive as a polynomial with coefficients in k/I. In this case, (22.6.13) does not hold, and we get that (22.6.10) holds.

22.7 Some remarks about prime elements

Let k be an integral domain, and let p be a prime element of k. Remember that p is not invertible in k, by definition, as in Section 15.3. If x ∈ k, then let I(x) be the ideal in k generated by x.

Note that I(p^{j+1}) ⊆ I(p^{j}) (22.7.1)

and I(p^{j+1}) ≠ I(p^{j}) (22.7.2)

for each j, because p is nonzero and not invertible in k, by hypothesis. More precisely, if p^{j} ∈ I(p^{j+1}) for some j, then p^{j} = p^{j+1} x for some x ∈ k. This implies that p x = 1, because p ≠ 0 and k is an integral domain.

One can check that

⋂_{j=1}^{∞} I(p^{j}) (22.7.3)

is a prime ideal in k. We would like to have that

⋂_{j=1}^{∞} I(p^{j}) = {0}. (22.7.4)

It is easy to see that (22.7.4) holds when k is a unique factorization domain. More precisely, this holds when every nonzero noninvertible element of k can be expressed as the product of finitely many noninvertible irreducible elements of k. This means that (22.7.4) holds when k is Noetherian, and in fact when increasing sequences of principal ideals in k are eventually constant, as in Section 22.2.

We can also use an argument like one in the proof of Proposition 2 on p7 of [263], which was discussed in Section 15.6. This is somewhat simpler here, because k is an integral domain, as in the remark after the proof of Proposition 2 on p7 of [263]. Suppose that y ≠ 0 is an element of (22.7.3). This means that for each j ≥ 1 there is an x_{j} ∈ k such that

y = p^{j} x_{j}. (22.7.5)

Note that x_{j} = p x_{j+1} (22.7.6)

for each j, because k is an integral domain. Thus

I(x_{j}) ⊆ I(x_{j+1}) (22.7.7)

for each j.

If increasing sequences of principal ideals in k are eventually constant, then

I(x_{l}) = I(x_{l+1}) (22.7.8)

for some l. This means that x_{l+1} = t_{l} x_{l} (22.7.9)

for some t_{l} ∈ k. It follows that

x_{l+1} = t_{l} p x_{l+1}, (22.7.10)

because of (22.7.6). This implies that

t_{l} p = 1, (22.7.11)

because k is an integral domain. This is a contradication, because p is not invertible in k, by hypothesis.

Consider the quotient ring

k_{0} = k/( ⋂_{j=1}^{∞} I(p^{j})). (22.7.12)

This is an integral domain, because (22.7.3) is a proper prime ideal in k. If q_{0} is the corresponding quotient homomorphism from k onto k_{0}, then one can verify that (22.7.13) q_{0}(p) is a prime element of k_{0}.

If x ∈ k, then I_{0}(q_{0}(x)) = q_{0}(I(x)) (22.7.14)

is the ideal in k_{0} generated by q_{0}(x). Of course,

⋂_{j=1}^{∞} I_{0}(q_{0}(p)^{j}) = ⋂_{j=1}^{∞} I_{0}(q_{0}(p^{j})) = ⋂_{j=1}^{∞} q(I(p^{j})). (22.7.15)

Suppose that x ∈ k has the property that q_{0}(x) is an element of (22.7.15). Thus q_{0}(x) ∈ q_{0}(I(p^{j})) for each j, which implies that x ∈ I(p^{j}) for each j. This means that x is an element of (22.7.3), so that q_{0}(x) = 0. It follows that

⋂_{j=1}^{∞} I_{0}(q_{0}(p)^{j}) = {0}. (22.7.16)

22.8 Prime elements and valuations

Let us continue with the same notation and hypotheses as at the beginning of the previous section, and suppose now that (22.7.4) holds. If x ∈ k and x ≠ 0, then put v_{p}(x) = max{j ≥ 0 : x ∈ I(p^{j})}, (22.8.1)

where the maximum is attained because of (22.7.4). It is easy to see that

v_{p}(x y) = v_{p}(x) + v_{p}(y) (22.8.2)

for all x, y ∈ k \ {0}, because p is prime in k.

We also have that

v_{p}(x + y) ≥ min(v_{p}(x), v_{p}(y)) (22.8.3)

when x + y ≠ 0. Thus v_{p} is a valuation on k, as in Section 14.15. We may put v_{p}(0) = +∞, as before.

Let Q_{k} be the field of fractions of k, as usual. We can extend v_{p} to Q_{k} by putting v_{p}(y/z) = v_{p}(y) − v_{p}(z) (22.8.4)

for every y, z ∈ k \ {0}, as in Section 14.15. This corresponds to some remarks on p126 of [177], except that the value at 0 is defined to be −∞ in [177]. Note that v_{p}(Q_{k} \ {0}) = Z, (22.8.5)

because v_{p}(p^{j}) = j for every integer j, by construction. This means that v_{p} is a discrete valuation on Q_{k}, as in Section 14.14.

22.9. UNIQUE FACTORIZATION DOMAINS AND V_{P} 539

Remember that S_{I(p)} = k \ I(p) (22.8.6)

is multiplicatively closed in k, as in Section 12.4, because I(p) is a proper prime ideal in k. Thus the corresponding ring of fractions

k_{I(p)} = S_{I(p)}^{−1} k (22.8.7)

may be defined as in Section 12.1. This may be identified with the subring of Q_{k} consisting of elements of the form x/r, with x ∈ k and r ∈ S_{I(p)}, as in Section 14.2. Using this identification, we have that

k_{I(p)} = {w ∈ Q_{k} : v_{p}(w) ≥ 0}. (22.8.8)

It follows that (22.8.9) k_{I(p)} is a discrete valuation ring

with respect to v_{p}, as in Sections 15.1 and 15.2.

Note that k_{I(p)} is a local ring, with unique maximal proper ideal S_{I(p)}^{−1} I(p), as in Section 12.4. We also have that I(p^{j}) = I(p)^{j} for each positive integer j, and that (S_{I(p)}^{−1} I(p))^{j} = S_{I(p)}^{−1} I(p)^{j} = S_{I(p)}^{−1} I(p^{j}), (22.8.10)

where the first step is as in Section 12.13. If y/r ∈ S_{I(p)}^{−1} I(p^{j}) for some y ∈ k, r ∈ S_{I(p)}, and j ≥ 1, then there is a t ∈ S_{I(p)} such that

y t ∈ I(p^{j}). (22.8.11)

This implies that y ∈ I(p^{j}), (22.8.12)

because t ∉ I(p) and p is a prime element of k. It follows that

⋂_{j=1}^{∞}(S_{I(p)}^{−1} I(p))^{j} = ⋂_{j=1}^{∞} S_{I(p)}^{−1} I(p^{j}) = {0}, (22.8.13)

because of (22.7.4). Of course, S_{I(p)}^{−1} I(p) is the same as the ideal in k_{I(p)} generated by p = p/1, as an element of k_{I(p)}. This shows that k_{I(p)} satisfies the conditions in Section 15.5, which is another way to look at (22.8.9).

22.9 Unique factorization domains and v_{p}

Let k be an integral domain. If x, y ∈ k, then put

x ∼ y (22.9.1)

when y = u x for some invertible element u in k. It is easy to see that this defines an equivalence relation on k. This means that x ∼ y when x, y are associates in k, because k is an integral domain, as in Section 15.3. This is the same as saying that the ideals in k generated by x and y are the same, as before.

If x ∼ y, then x is irreducible in k if and only if y is irreducible in k. Similarly, if p, p̃ ∈ k and p ∼ p̃, then p is prime in k if and only if p̃ is prime in k. In this case, I(p^{j}) = I(p^{j}) (22.9.2)

for each j ≥ 1, where I(x) is the ideal in k generated by x ∈ k, as before. This means that

⋂_{j=1}^{∞} I(p^{j}) = ⋂_{j=1}^{∞} I(p^{j}). (22.9.3)

If this is equal to {0}, then we have that

v_{p} = v_{p′} (22.9.4)

on k, where v_{p}, v_{p′} are as in the previous section.

Let P be the set of all equivalence classes of prime elements of k with respect to (22.9.1). Let us choose exactly one element from each of these equivalence classes to get a set P of prime elements of k, as on p72 of [177]. Of course, if k is a field, then there are no prime elements in k.

Suppose that every nonzero noninvertible element of k can be expressed as the product of finitely many noninvertible irreducible elements of k. Remember that this holds when k is Noetherian, and more precisely when every increasing sequence of principal ideals in k is eventually constant, as in Section 22.2. If p is a prime element of k, then (22.7.4) holds, as before, so that v_{p} may be defined as in (22.8.1). If x is an invertible element of k, then it is easy to see that

v_{p}(x) = 0 (22.9.5)

for every prime p in k. If x ∈ k, x ≠ 0, and

v_{p}(x) > 0 (22.9.6)

for some prime p in k, then x is a multiple of p in k.

Let x ∈ k with x ≠ 0 be given. If x is not invertible, then x can be expressed as the product of finitely many noninvertible irreducible elements of k, by hypothesis. If (22.9.6) holds for some prime p in k, then one of the finitely many irreducible elements in k just mentioned is a multiple of p in k, because x is a multiple of p. This means that one of these irreducible elements is equal to p times an invertible element of k.

It follows that (22.9.6) can hold for primes p in only finitely many of the equivalence classes mentioned earlier. Thus (22.9.6) can hold for only finitely many elements p of P.

Suppose now that k is a unique factorization domain. If k is not a field, then it is easy to see that k has at least one prime element. If x ∈ k and x ≠ 0, then

x = u(x) ∏_{{p∈P \atop vp(x)≠0}} p^{v_{p}(x)}, (22.9.7)

where u(x) is an invertible element of k, as on p72 of [177]. More precisely, the product in the second part of the right side is interpreted as being equal to 1 when (22.9.5) holds for every p ∈ P.

Let Q_{k} be the field of fractions corresponding to k, and remember that v_{p} may be extended to Q_{k} for each prime p in k, as in the previous section. If w ∈ Q_{k} and w ≠ 0, then v_{p}(w) ≠ 0 (22.9.8)

can hold for primes p in only finitely many of the equivalence classes mentioned earlier, because of the analogous statement for nonzero elements of k. Observe that

w = u(w) ∏_{{p∈P \atop vp(w)≠0}} p^{v_{p}(w)}, (22.9.9)

where u(w) is an invertible element of k, as in the previous paragraph.

22.10 Another type of content

Let k be a unique factorization domain, and let Q_{k} be the corresponding field of fractions. Also let X be an indeterminate, and let f(X) = ∑_{j=0}^{n} f_{j} X^{j} be a formal polynomial in X with coefficients in Q_{k}. If f(X) ≠ 0 and p is a prime element of k, then put

ord_{p} f(X) = min{v_{p}(f_{j}) : f_{j} ≠ 0}, (22.10.1)

as on p126 of [177]. If f(X) = 0, then ord_{p} f(X) is defined to be −∞ is [177]. Although one might take it to be +∞ to be compatible with defining v_{p}(0) to be +∞, an advantage of taking it to be −∞ will be mentioned in a moment.

If f(X) ≠ 0 and u is an invertible element of k, then one may call

u p^{ord_{p}f(X)} (22.10.2)

a p-content of f(X), as on p126 of [177]. If f(X) = 0, then one might interpret this as being equal to 0. This could correspond to taking ord_{p} f(X) = −∞ in this case, particularly in the classical siutation where k = Z.

If f(X) ≠ 0, then ord_{p} f(X) ≠ 0 (22.10.3)

can hold for only primes p in only finitely many of their equivalence classes, because of the analogous statement for nonzero elements of Q_{k} in the previous section. Let P and P be as in the previous section too. The content of f(X) is defined in [177] by

(22.10.4) cont(f(X)) = ∏_{{p∈P \atop ordp f(X)≠0}} p^{ord_{p} f(X)}.

The product on the right is interpreted as being equal to 1 when ord_{p} f(X) = 0 for every p, as before.

More precisely, cont(f(X)) is only considered to be defined up to multiplication by an invertible element of k. Using this, cont(f(X)) does not depend on the particular choices of primes from their equivalence classes. One might interpret (22.10.4) as being equal to 0 when f(X) = 0, as before.

If f(X) ≠ 0 and f(X) has coefficients in k, then

ord_{p} f(X) ≥ 0 (22.10.5)

for every prime p in k. This means that cont(f(X)) is an element of k, although defined only up to multiplication by an invertible element of k. Under these conditions, (22.10.6) Content(f(X)) ⊆ I(cont(f(X))), where the left side is the content of f(X) as an ideal in k as in Section 22.6, and the right side is the ideal in k generated by cont(f(X)). This is because

f_{j} ∈ I(cont(f(X))) (22.10.7)

for each j, by construction.

Suppose that Content(f(X)) ⊆ I(x), (22.10.8) the ideal in k generated by some x ∈ k. This is the same as saying that

f_{j} ∈ I(x) (22.10.9)

for each j, so that v_{p}(f_{j}) ≥ v_{p}(x) (22.10.10) for each prime p in k when f_{j} ≠ 0. It follows that

ord_{p} f(X) ≥ v_{p}(x) (22.10.11)

for every prime p in k, so that

cont(f(X)) ∈ I(x), (22.10.12)

and thus I(cont(f(X))) ⊆ I(x). (22.10.13) If Content(f(X)) = I(x), (22.10.14) then we get that (22.10.15) Content(f(X)) = I(cont(f(X))), because of (22.10.6). In particular, this holds when k is a principal ideal domain.

If f(X) ≠ 0 has coefficients in Q_{k} again and b ∈ Q_{k}, b ≠ 0, then it is easy to see that (22.10.16) cont(b f(X)) = b cont(f(X)), as mentioned on p126 of [177]. This permits us to express f(X) as

f(X) = cont(f(X)) f_{1}(X), (22.10.17)

where f_{1}(X) ∈ k[X] and cont(f_{1}(X)) = 1, as in [177]. This also corresponds to Lemma 2 on p85 of [24].

22.11 Another version of Gauss’ lemma

Let k be a unique factorization domain, and let Q_{k} be the corresponding field of fractions again. Also let X be an indeterminate, and let f(X) = ∑_{j=0}^{n} f_{j} X^{j} and g(X) = ∑_{l=0}^{m} g_{l} X^{m} be formal polynomials in X with coefficients in Q_{k}. If f(X), g(X) ≠ 0, then another version of Gauss’ lemma states that

(22.11.1) cont(f(X) g(X)) = cont(f(X)) cont(g(X)),

as on p127 of [177]. This corresponds to part of Lemma 3 on p86 of [24] as well. Let us express f(X) as in (22.10.17), and similarly

g(X) = cont(g(X)) g_{1}(X), (22.11.2)

where f_{1}(X), g_{1}(X) ∈ k[X], and

(22.11.3) cont(f_{1}(X)) = cont(g_{1}(X)) = 1.

Thus (22.11.4) f(X) g(X) = cont(f(X)) cont(g(X)) f_{1}(X) g_{1}(X), and (22.11.1) is the same as saying that

cont(f_{1}(X) g_{1}(X)) = 1. (22.11.5)

This means that we can reduce to the case where f(X), g(X) ∈ k[X] and cont(f(X)) = cont(g(X)) = 1, so that we want to show that

cont(f(X) g(X)) = 1. (22.11.6)

This formulation of Gauss’ lemma corresponds to Lemma 1 on p85 of [24], and to Lemma 2 on p125 of [151].

Let p be a prime element of k. We would like to show that

ord_{p}(f(X) g(X)) = 0. (22.11.7)

By hypothesis, ord_{p} f(X) = ord_{p} g(X) = 0. (22.11.8) Let j_{0} be the smallest integer such that 0 ≤ j_{0} ≤ n, f_{j_{0}} ≠ 0, and f_{j_{0}} is not a multiple of p in k. Similarly, let l_{0} be the smallest integer such that 0 ≤ l_{0} ≤ m, g_{l_{0}} ≠ 0, and g_{l_{0}} is not a multiple of p in k. The coefficient of X^{j_{0}+l_{0}} in f(X) g(X) is the sum of f_{j} g_{l} (22.11.9) with j+l = j_{0}+l_{0}, 0 ≤ j ≤ n, and 0 ≤ l ≤ m, as in Section 4.3. By construction, (22.11.9) is a multiple of p in k when j < j_{0} or l < l_{0}. It is not a multiple of p in k when j = j_{0} and l = l_{0}. This implies that the coefficient of X^{j_{0}+l_{0}} in f(X) g(X) is not a multiple of p in k, so that (22.11.7) holds.

One may say that f(X) ∈ k[X] is primitive when cont(f(X)) = 1. This is equivalent to the formulation on p85 of [24], and on p124 of [151]. One can check that this is the same as saying that Content(f(X)) is not contained in a proper principal ideal in k. This corresponds to the second type of primitivity for polynomials with coefficients in a unique factorization domain mentioned in part c of Exercise 3.4 on p109 of [92].

22.12 Unique factorization in k[X]

Let k_{0} be a field, and let X be an indeterminate. It is well known that k_{0}[X] is a principal ideal domain, and a unique factorization domain in particular, as mentioned in Section 22.2. This corresponds to Theorems 11, 14 on p81, 82 of [24], Theorem 3 and its corollary on p120, 121 of [177], and Theorems 20, 24 on p115, 118 of [195].

If k is an integral domain, then it is easy to see that the product of nonzero elements of k[X] is a constant polynomial only when each of the factors is a constant. In particular, the only invertible elements of k[X] are constant polynomials that are invertible as elements of k. This could also be obtained from the characterization of invertibility in k[X] for any commutative ring k with a multiplicative identity element mentioned in Section 22.5.

Let k be a unique factorization domain, and let Q_{k} be the corresponding field of fractions. Suppose that f(X) ≠ 0 in k[X] can be expressed as

f(X) = g(X) h(X), (22.12.1)

with g(X), h(X) ∈ Q_{k}[X]. Of course, g(X), h(X) ≠ 0, so that they may be expressed as

(22.12.2) g(X) = cont(g(X)) g_{1}(X), h(X) = cont(h(X)) h_{1}(X),

where g_{1}(X), h_{1}(X) ∈ k[X] and cont(g_{1}(X)) = cont(h_{1}(X)) = 1, as in Section 22.10. It follows that

f(X) = cont(f(X)) g_{1}(X) h_{1}(X), (22.12.3)

because of (22.11.1). This corresponds to part of Lemma 3 on p86 of [24], and to the corollary to Gauss’ lemma on p127 of [177].

If f(X) is reducible in Q_{k}[X], then it follows that

(22.12.4) f(X) is reducible in k[X].

We also have that (22.12.4) holds when

(22.12.5) cont(f(X)) is not invertible in k and f(X) is not a constant polynomial,

as in Section 22.10. It is easy to see that an element of k is reducible in k if and only if it is reducible as an element of k[X].

Suppose that f(X) ≠ 0 can be expressed as in (22.12.1), with g(X), h(X) in k[X]. If g(X) and h(X) are not constants, then

(22.12.6) f(X) is reducible in Q_{k}[X].

If cont(f(X)) is invertible in k, then cont(g(X)) and cont(h(X)) are invertible in k, because they are in k, and their product is equal to cont(f(X)), as in (22.11.1). Of course, this is simpler when g(X) or h(X) is a constant. This implies that (22.12.6) holds when f(X) is reducible in k[X] and cont(f(X)) is invertible in k.

It is well known that

(22.12.7) k[X] is a unique factorization domain

under these conditions, as in Lemma 4 on p86 of [24], Theorem 4 on p126 of [151], and Theorem 10 on p127 of [177]. The irreducible elements of k[X] correspond to either irreducible elements of k, or nonzero elements a(X) of k[X] such that cont(a(X)) is invertible in k and a(X) is irreducible in Q_{k}[X], as in the previous two paragraphs. Of course, k[X] is a subring of Q_{k}[X], which is a unique factorization ring, as before. One can use this to get the analogous properties of k[X], by standard arguments.

If X_{1}, …, X_{n} are n distinct commuting indeterminates for some positive integer n, then

(22.12.8) k[X_{1}, …, X_{n}] is a unique factorization domain

as well. This can be obtained from (22.12.7) and the isomorphism between k[X_{1}, …, X_{n}] and (k[X_{1}, …, X_{n−1}])[X_{n}] mentioned in Section 21.12 when n is greater than or equal to 2. This corresponds to Theorem 16 on p87 of [24], a remark on p127 of [151], and to the corollary on p128 of [177].

22.13 Gauss’ lemma and Dedekind domains

Let k be a Dedekind domain, as in Section 15.11, and let X be an indeterminate. If f(X), g(X) are nonzero formal polynomials in X with coefficients in k, then

(22.13.1) Content(f(X) g(X)) = Content(f(X)) Content(g(X)),

where the content of a polynomial is as in Section 22.6. This is another version of Gauss’ lemma, as in Exercise 2 on p99 of [8].

Suppose for the moment that k is a principal ideal domain. In this case, Content(f(X)) is the same as the ideal in k generated by cont(f(X)), as in Section 22.10. Using this, one can obtain (22.13.1) from the version of Gauss’ lemma discussed in Section 22.11.

If k is any commutative ring with a multiplicative identity element, and f(X), g(X) are primitive, in the sense that their contents are equal to k, then the content of f(X) g(X) is equal to k as well, as in Section 22.6. If k is a principal ideal domain, then one can also obtain (22.13.1) by reducing to this case.

Remember that

(22.13.2) Content(f(X) g(X)) ⊆ Content(f(X)) Content(g(X)),

as in Section 22.6. In the definition of a Dedekind domain on p95 of [8], k is not a field, and we shall use this here, although (22.13.1) is trivial otherwise. Let I be a proper maximal ideal in k, which is the same as saying that I ≠ {0} is a proper prime ideal in k, because k has dimension one, as in Section 15.11. This implies that S_{I} = k \ I is multiplicatively closed in k, as in Section 12.4, and we take k_{I} = S_{I}^{−1} k, as in Section 12.1. This is a discrete valuation ring, as in Section 15.11 again.

We would like to check that

(22.13.3) S_{I}^{−1}(Content(f(X) g(X))) = S_{I}^{−1}(Content(f(X)) Content(g(X)))

for all such I, following the suggestion in [8]. This will imply (22.13.1), as in Section 12.4. Note that

(22.13.4) S_{I}^{−1}(Content(f(X)) Content(g(X)))

= (S_{I}^{−1} Content(f(X))) (S_{I}^{−1} Content(g(X))),

as in Section 12.13. Thus it suffices to verify that

S_{I}^{−1}(Content(f(X) g(X))) (22.13.5)

= (S_{I}^{−1} Content(f(X))) (S_{I}^{−1} Content(g(X)))

for all such I.

Let f_{I}(X), g_{I}(X), and (f g)_{I}(X) be the formal polynomials in X with coefficients in k_{I} corresponding to f(X), g(X), and f(X) g(X), respectively. It is easy to see that

(22.13.6) Content(f_{I}(X)) = S_{I}^{−1} Content(f(X)),

and similarly for g_{I}(X) and (f g)_{I}(X). More precisely, Content(f_{I}(X)) is the content of f_{I}(X) as a polynomial with coefficients in k_{I}, which is the ideal in k_{I} generated by the coefficients of f_{I}(X), as in Section 22.6.

Of course, (f g)_{I}(X) = f_{I}(X) g_{I}(X), (22.13.7)

as formal polynomials in X with coefficients in k_{I}. This means that

(22.13.8) S_{I}^{−1}(Content(f(X) g(X))) = Content((f g)_{I}(X))
= Content(f_{I}(X) g_{I}(X)),

as ideals in k_{I}.

This shows that (22.13.5) is the same as saying that

(22.13.9) Content(f_{I}(X) g_{I}(X)) = Content(f_{I}(X)) Content(g_{I}(X)),

as ideals in k_{I}. Remember that k_{I} is a discrete valuation ring, and thus a principal ideal domain, as in Section 15.1. The analogous statement for principal ideal domains was mentioned earlier, and so we get that (22.13.9) holds for all maximal proper ideals in k.

22.14 Some remarks about fractions

Let k be a commutative ring with a multiplicative identity element, and let S be a multiplicatively closed subset of k, as in Section 12.1. Thus the corresponding ring of fractions S^{−1} k may be defined as before. Let us suppose that 0 ∉ S, to avoid trivialities.

Remember that f(x) = x/1 defines a homomorphism from k into S^{−1} k, as in Section 12.1. If I_{1} is an ideal in k, then S^{−1} I_{1} is the same as the extension of I_{1} in S^{−1} k with respect to f, as in Section 12.13. Let x_{1} ∈ k be given, and let I(x_{1}) be the ideal in k generated by x_{1}. It is easy to see that

(22.14.1) S^{−1} I(x_{1}) is the same as the ideal in S^{−1} k generated by x_{1}/1.

Suppose for the moment that k is a principal ring, in the sense that 1 ≠ 0 in k, and every ideal in k is a principal ideal, as on p58 of [177]. Exercise 4 on p73 of [177] asks one to show that

(22.14.2) S^{−1} k is a principal ring

as well. Note that the multiplicative identity element 1/1 in S^{−1} k is nonzero in this case, as in Section 12.1.

Let I_{2} be an ideal in S^{−1} k, and let

I_{2}^{c} = f^{−1}(I_{2}) (22.14.3)

be its contraction in k with respect to f, as in Section 12.7. If k is a principal ring, then there is an x_{0} ∈ k such that I_{2}^{c} is generated by x_{0}, as an ideal in k. Under these conditions, one can check that

(22.14.4) I_{2} is generated by x_{0}/1,

as an ideal in S^{−1} k. More precisely, this can be obtained from the fact that I_{2} is the same as the extension of I_{2}^{c} with respect to f, as in Section 12.13, or verified in essentially the same way.

If I_{1} is a proper prime ideal in k that is disjoint from S, then S^{−1} I_{1} is a proper prime ideal in S^{−1} k, as in Section 12.13. Of course, if I_{1} ∩ S ≠ ∅, then S^{−1} I_{1} = k.

Let p be a prime element of k, so that the ideal I(p) in k generated by p is a proper prime ideal in k, as in Section 15.3. If a multiple of p in k is contained in S, so that I(p) ∩ S ≠ ∅, (22.14.5)

then p/1 is invertible in S^{−1} k. This is the same as saying that S^{−1} I(p) = S^{−1} k.

Otherwise, if I(p) ∩ S = ∅, (22.14.6)

then (22.14.7) S^{−1} I(p) is a proper prime ideal in S^{−1} k, as before. This means that

(22.14.8) p/1 is a prime element of S^{−1} k

when (22.14.6) holds. This corresponds to part of Exercise 5 on p73 of [177].

Suppose that k is an integral domain, so that S^{−1} k is an integral domain too, as in Section 14.2. If k is a unique factorization domain, then Exercise 5 on p73 of [177] asks one to show that

(22.14.9) S^{−1} k is a unique factorization domain,

where the primes in S^{−1} k correspond to the primes p in k such that (22.14.6) holds. This is fairly straightforward, using the remarks in the previous two paragraphs.

22.15 Some more remarks about fractions

Let k be a commutative ring with a multiplicative identity element, and let S be a multiplicatively closed subset of k. Remember that the dimension of k may be defined as in Section 14.8, so that the dimension of S^{−1} k may be defined in the same way. It is easy to see that

(22.15.1) the dimension of S^{−1} k is less than or equal to the dimension of k.

This uses the correspondence between the proper prime ideals in S^{−1} k and the prime ideals in k that do not intersect S mentioned in Section 12.13.

Suppose again that 0 ∉ S. If k is an integral domain, then S^{−1} k is an integral domain, which can be identified with a subring of the field Q_{k} of fractions of k, as in Section 14.2. The field of fractions of S^{−1} k may be identified with Q_{k}, as before.

If k is a Dedekind domain, then the first part of Exercise 1 on p99 of [8] asks one to show that (22.15.2) S^{−1} k is a Dedekind domain

or S^{−1} k is a field. (22.15.3)

Remember that the definition of a Dedekind domain on p95 of [8] asks that k have dimension one, so that k should not be a field.

In order for k to be a Dedekind domain, k should be Noetherian, as in Section 15.11. This implies that S^{−k} k is Noetherian, as in Section 14.6.

Similarly, if k is a Dedekind domain, then the dimension of k is less than or equal to one, or equal to one using the definition in [8]. This implies that the dimension of S^{−1} k is less than or equal to one, as in (22.15.1). If the dimension of S^{−1} k is equal to 0, then (22.15.3) holds, because S^{−1} k is an integral domain, as in Section 14.8. Thus we may as well suppose that the dimension of S^{−1} k is equal to one.

One way to get (22.15.2) is to use the the definition of a Dedekind domain in Section 15.11. This means that if I_{2} is a proper nonzero prime ideal in S^{−1} k, then one would like to verify that the localization

(S^{−1} k)_{I_{2}} (22.15.4)

of S^{−1} k at I_{2} is a discrete valuation ring. This is the ring

((S^{−1} k) \ I_{2})^{−1} (S^{−1} k) (22.15.5)

of fractions of S^{−1} k with respect to the multiplicative closed subset

(S^{−1} k) \ I_{2} (22.15.6)

of S^{−1} k, as in Section 12.4. As is Section 12.13, there is a proper nonzero prime ideal I_{1} in k such that S ∩ I_{1} = ∅ (22.15.7)

and I_{2} = S^{−1} I_{1}. (22.15.8)

Of course, k_{I_{1}} = (k \ I_{1})^{−1} k (22.15.9)

is a discrete valuation ring, because k is a Dedekind domain, by hypothesis. Thus it suffices to check that (22.15.4) is isomorphic to (22.15.9). Note that

S ⊆ k \ I_{1}, (22.15.10)

by (22.15.7). Remember that we can indentify S^{−1} k, (22.15.5), and (22.15.9) with subrings of Q_{k}, as in Section 14.2. It is easy to see that (22.15.5) and (22.15.9) correspond to the same subrings of Q_{k}.

Alternatively, in order to obtain (22.15.2), it suffices to verify that

(22.15.11) S^{−1} k is integrally closed,

as in Section 16.12. Remember that k is integrally closed, because k is a Dedekind domain, as in Section 16.12 again. This implies (22.15.11), as in Section 16.11.

We can also use the characterization of Dedekind domains in terms of invertibility of fractional ideals, as in Sections 15.12 and 15.13. Remember that the field of fractions of S^{−1} k may be identified with Q_{k}, as in Section 14.2. Thus a fractional ideal M of S^{−1} k may be considered as a submodule of Q_{k}, as a module over S^{−1} k, for which there is an x ∈ k and an r ∈ S such that x ≠ 0 and (x/r) M ⊆ S^{−1} k, (22.15.12)

as in Section 11.7. Observe that M is finitely generated as a module over S^{−1} k, because S^{−1} k is Noetherian, as before. Let y_{1}, …, y_{l} be finitely many elements of M that generate M, as a module over S^{−1} k.

Consider the submodule of Q_{k} generated by y_{1}, …, y_{l} as a module over k. This is a fractional ideal of k, as in Section 11.7. Suppose that M ≠ {0}, so that M_{0} ≠ {0}. It follows that M_{0} is invertible as a fractional ideal of k, because k is a Dedekind domain, as in Section 15.12. This means that there is a submodule N_{0} of Q_{k}, as a module over k, such that

M_{0} N_{0} = k, (22.15.13)

as in Section 11.7.

Note that S^{−1} M_{0}, S^{−1} N_{0} may be considered as submodules of Q_{k}, as a module over S^{−1} k, as in Section 14.2. Of course,

S^{−1} M_{0} = M, (22.15.14)

by construction. It follows that

M (S^{−1} N_{0}) = S^{−1} k, (22.15.15)

as in Section 14.3. This means that M is invertible as a fractional ideal of S^{−1} k. This implies (22.15.2), as in Section 15.13.

More on derivations

23.1 Some more properties of derivations

Let k be a commutative ring with a multiplicative identity element, let A be an algebra in the strict sense over k, and let δ be a derivation on A, as in Section 21.13. If n is a positive integer, then it is well known that the Leibniz formula

δ^{n}(x y) = ∑_{j=0}^{n} \binom{n}{j} · δ^{j}(x) δ^{n−j}(y) (23.1.1)

holds for all x, y ∈ A, where \binom{n}{j} is the usual binomial coefficient. This can be verified using induction, as mentioned on p174 of [153].

Let p be a prime number, and suppose for the moment that

p · x = 0 (23.1.2)

for every x ∈ A. Note that this happens when p · 1 = 0 in k. Remember that \binom{p}{l} is a multiple of p when 1 ≤ l ≤ p − 1, as mentioned in Section 20.15. In this case, we get that δ^{p}(x y) = δ^{p}(x) y + x δ^{p}(y) (23.1.3)

for every x, y ∈ A, by (23.1.1). This means that

(23.1.4) δ^{p} is a derivation on A,

as on p174 of [153].

Suppose for the rest of the section that A is an associative algebra over k. If a, b ∈ A, then put L_{a}(x) = a x, R_{b}(x) = x b (23.1.5)

for every x ∈ A. These are the left and right multiplication operators associated to a, b, respectively. Observe that

L_{a} ∘ R_{b} = R_{b} ∘ L_{a} (23.1.6) on A, by associativity. If n is a positive integer, then it follows that

(L_{a} + R_{b})^{n} = ∑_{j=0}^{n} \binom{n}{j} · L_{a}^{j} ∘ R_{b}^{n−j}, (23.1.7)

by the binomial theorem.

Equivalently,

(L_{a} + R_{b})^{n} = ∑_{j=0}^{n} \binom{n}{j}L_{a^{j}} ∘ R_{b^{n−j}}. (23.1.8)

Remember that δ_{a} = L_{a} − R_{a} (23.1.9)

is the inner derivation on A associated to a, as in Section 21.13. Thus

δ_{a}^{n} = ∑_{j=0}^{n} \binom{n}{j} · L_{a^{j}} ∘ R_{(−a)^{n−j}} (23.1.10)

for every n ≥ 1, as in (23.1.8).

If (23.1.2) holds, then we get that

δ_{a}^{p} = L_{a^{p}} + R_{(−a)^{p}} = δ_{a^{p}}. (23.1.11)

More precisely, R_{(−a)^{p}} = −R_{a^{p}} automatically when p is odd, and a^{2} = −a^{2} when p = 2. This corresponds to part of Exercise 1 on p174 of [153].

23.2 Another type of derivation

Let k be a commutative ring with a multiplicative identity element, let B be an algebra over k in the strict sense, and let A be a subalgebra of B. Also let δ be a homomorphism from A into B, as modules over k. If

δ(x y) = δ(x) y + x δ(y) (23.2.1)

for every x, y ∈ A, then δ is said to be a derivation from A into B. This corresponds to Definition 4 on p167 of [153]. Of course, this is the same as in Section 21.13 when A = B.

The space Hom_{k}(A, B) of all homomorphisms from A into B, as modules over k, is a module over k with respect to pointwise addition and scalar multiplication. The space Der(A, B) = Der_{k}(A, B) of all derivations from A into B is a submodule of Hom_{k}(A, B), as a module over k.

Note that the restriction to A of a derivation from B into itself is a derivation from A into B. If δ is a derivation from A into B and A_{0} is a subalgebra of A, then the restriction of δ to A_{0} is a derivation from A_{0} into B.

Suppose for the moment that B is an associative algebra over k. If b ∈ B, then δ_{b}(x) = b x − x b (23.2.2) defines a derivation from B into itself, as in Section 21.13. Note that

b ↦ δ_{b} (23.2.3)

defines a homomorphism from B into the space Der(B) = Der(B, B) of all derivations from B into itself, as modules over k. The restriction of δ_{b} to A may be considered as an inner derivation on A, as a derivation from A into B.

Suppose for the moment that B is a commutative associative algebra over k. If b ∈ B and δ is a derivation from A into B, then it is easy to see that

x ↦ b δ(x) (23.2.4)

is a derivation from A into B too, as in Section 21.13.

If δ is a derivation from A into B, then

(23.2.5) ker δ is a subalgebra of A,

as in Section 21.13. If B has a multiplicative identity element e that is contained in A, then δ(e) = 0, (23.2.6)

as before. If x, y ∈ A and x y = e, then

δ(x) y + x δ(y) = 0. (23.2.7)

If B is associative and x has a multiplicative inverse in A, then it follows that

δ(x^{−1}) = −x^{−1} δ(x) x^{−1}. (23.2.8)

This implies that δ(x^{−1}) = 0 (23.2.9)

when δ(x) = 0, as before.

Let d be a formal symbol, which is taken to commute with all elements of B and k, and satisfy d^{2} = 0, as in Section 5.4. The algebra B[d] of dual numbers associated to B consists of formal expressions of the form b_{1} + b_{2} d, with b_{1}, b_{2} ∈ B, as before. This corresponds to the direct sum of two copies of B, as a module over k. One can define multiplication on B[d] as in Section 5.4, to get an algebra in the strict sense over k. This is an associative algebra over k when B is associative, and a commutative algebra over k when B is commutative. We may consider B as a subalgebra of B[d], as before. If B has a multiplicative identity element e_{B}, then this is the multiplicative identity element of B[d] too.

Let δ be a homomorphism from A into B, as modules over k. Note that

x ↦ x + δ(x) d (23.2.10)

defines an injective homomorphism from A into B[d], as modules over k. One can check that δ is a derivation from A into B if and only if

(23.2.11) (23.2.10) is an algebra homomorphism from A into B[d].

This corresponds to Theorem 12 on p169 of [152].

23.3 A related notion of derivation

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. A module V over k is said to be a two-sided module over A if it is both a left and right module over A, and the actions of A on V on the left and right commute with each other. This will be discussed further in Section 35.1. Of course, if A is commutative, then a one-sided module over A may be considered as a two-sided module over A.

Let δ be a homomorphism from A into V, as modules over k. If

δ(x y) = δ(x) · y + x · δ(y) (23.3.1)

for every x, y ∈ A, then δ is said to be a crossed homomorphism or derivation, as on p168 of [50]. This is dicussed further in Section 35.3. If A is commutative, and V is a module over A, considered as a two-sided module over A, then this corresponds to the definition on p385 of [92].

Suppose for the moment that

(23.3.2) B is an associative algebra over k that contains A as a subalgebra,

and that (23.3.3) e_{A} is the multiplicative identity element of B.

In this case,

(23.3.4) B may be considered as a two-sided module over A.

Observe that a derivation from A into B as in the preceding paragraph is the same as in the previous section under these conditions.

If δ is a V -valued derivation on A, then it is easy to see that

δ(e_{A}) = 0, (23.3.5)

as mentioned in [50]. Note that

(23.3.6) ker δ is a subalgebra of A,

as before. If x, y ∈ A satisfy x y = e_{A}, then

δ(x) · y + x · δ(y) = 0. (23.3.7)

If x has a multiplicative inverse in A, then

δ(x^{−1}) = −x^{−1} · δ(x) · x^{−1}. (23.3.8)

In particular, δ(x^{−1}) = 0 when δ(x) = 0, as before.

The space Hom_{k}(A, V) of all homomorphisms from A into V, as modules over k, is a module over k, with respect to pointwise addition and scalar multiplication, as usual. The space Der(A, V) = Der_{k}(A, V) of all V -valued derivations on A is a submodule of Hom_{k}(A, V), as a module over k.

23.4. MORE ON V -VALUED DERIVATIONS 555

If v ∈ V, then δ_{v}(x) = v · x − x · v (23.3.9)

defines a homomorphism from A into V, as modules over k. One can check that

(23.3.10) δ_{v} is a derivation on A,

as a V -valued function on A. This is called a principal crossed homomorphism or inner derivation, as on p169 of [50]. This is discussed further in Section 35.4.

If B is as in (23.3.2) and (23.3.3), then this is the same as an inner derivation on A, as a derivation from A into B, as in the previous section.

23.4 More on V -valued derivations

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V be a module over k that is a two-sided module over A. If A_{0} is a subalgebra of A that contains e_{A}, then V may be considered as a two-sided module over A_{0} as well. If δ is a derivation from A into V, then

(23.4.1) the restriction of δ to A_{0} is a derivation from A_{0} into V,

as a two-sided module over A_{0}.

Let U, W be modules over k that are two-sided modules over A. A homomorphism ϕ from U into W, as modules over k, is considered to be a homomorphism from U into W as two sided-modules over A, if ϕ is a homomorphism from U into W, as both left and right modules over A. In this case, if δ is a U-valued derivation on A, then it is easy to see that

(23.4.2) ϕ ∘ δ is a W -valued derivation on A.

Suppose that A is commutative, and that V is a one-sided module over A, considered as a two-sided module over A, as before. If a ∈ A and δ is a V -valued derivation on A, then x ↦ a · δ(x) (23.4.3)

defines a V -valued derivation on A too. This may be considered as a version of (23.4.2), with U = W = V and ϕ(v) = a · v.

Similarly, if δ is a derivation from A into itself and v ∈ V, then

x ↦ δ(x) · v (23.4.4)

is a V -valued derivation on A. This corresponds to taking U = A, as a module over itself, W = V, and ϕ(z) = z · v for z ∈ A in (23.4.2).

Let B be as in (23.3.2) and (23.3.3) again, and suppose that B is commutative. If b ∈ B and δ is a derivation from A into B, then x ↦ b δ(x) is a derivation from A into B as well, as in the previous section. This may be considered as another version of (23.4.2), with U = W = B, and ϕ(z) = bz.

23.5 Some algebras from modules

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V be a module over k that is a two-sided module over A. The Cartesian product C_{A,V} = A × V of A and V is a module over k, with respect to coordinatewise addition and scalar multiplication. We can define multiplication on C_{A,V} by

(a_{1}, v_{1}) (a_{2}, v_{2}) = (a_{1} a_{2}, a_{1} · v_{2} + v_{1} · a_{2}) (23.5.1)

for every a_{1}, a_{2} ∈ A and v_{1}, v_{2} ∈ V. One can check that C_{A,V} is an associative algebra over k. Clearly e_{C_{A,V}} = (e_{A}, 0) (23.5.2)

is the multiplicative identity element in C_{A,V}.

By construction, a ↦ (a, 0) is an injective algebra homomorphism from A into C_{A,V}. Similarly, the obvious coordinate projection from C_{A,V} onto A is an algebra homomorphism, with kernel equal to {0} × V. If we identify A with A × {0} and V with {0} × V, then V corresponds to a submodule of C_{A,V}, as a two-sided module over A.

If δ is a homomorphism from A into V, as modules over k, then

(a, 0) ↦ (0, δ(a)) (23.5.3)

is a homomorphism from A × {0} into {0} × V ⊆ C_{A,V}, as modules over k. It is easy to see that δ is a derivation from A into V if and only if (23.5.3) is a derivation from A × {0} into C_{A,V}, as in Section 23.2.

If v ∈ V, then

(0, v) (x, 0) − (x, 0) (0, v) = (0, v · x − x · v) (23.5.4)

for every x ∈ A. This shows that the inner derivation (23.3.9) from A into V corresponds to

(23.5.5) the inner derivation from A × {0} into C_{A,V} associated to (0, v)

as in Section 23.2, using the correspondence between V -valued derivations on A and derivations from A × {0} into C_{A,V} mentioned in the preceding paragraph.

If δ is a homomorphism from A in V again, as modules over k, then

x ↦ (x, δ(x)) (23.5.6)

defines an injective homomorphism from A into C_{A,V}, as modules over k. One can verify that δ is a derivation from A into V, in the sense of Section 35.3, if and only if

(23.5.7) (23.5.6) is an algebra homomorphism from A into C_{A,V} .

This is analogous to Theorem 12 on p169 of [153], which was mentioned in Section 23.2.

Let W be another module over k that is a two-sided module over A, so that C_{A,W} may be defined as before. If ϕ is a homomorphism from V into W, as two-sided modules over A, then

(a, v) ↦ (a, ϕ(v)) (23.5.8)

defines a homomorphism from C_{A,V} into C_{A,W}, as algebras over k.

Suppose for the moment that A is commutative, and that V is a one-sided module over A, considered as a two-sided module over A. In this case,

(23.5.9) C_{A,V} is commutative

as well.

If V = A, considered as a two-sided module over itself, then C_{A,V} = C_{A,A} corresponds to the algebra A[d] of dual numbers associated to A, as in Section 5.4. Similarly, suppose that B is as in (23.3.2) and (23.3.3), and that V = B, considered as a two-sided module over A. Let B[d] be the algebra of dual numbers associated to B, as in Section 5.4 again. Under these conditions, C_{A,V} = C_{A,B} corresponds to the subalgebra of B[d] consisting of elements of the form a + b d, (23.5.10)

with a ∈ A and b ∈ B.

23.6 Some more derivations on polynomials

Let k be a commutative ring with a mutiplicative identity element, let A be a commutative associative algebra over k with a multiplicative identity element e_{A}, and let X_{1}, …, X_{n} be n distinct commuting indeterminates for some positive integer n. Remember that the space A[X_{1}, …, X_{n}] of formal polynomials in X_{1}, …, X_{n} with coefficients in A is a commutative associative algebra over k, with e_{A} as its multiplicative identity element, as in Section 4.3.

Suppose for the moment that V_{0} is a module over k that is also a module over A. Under these conditions, the space V_{0}[X_{1}, …, X_{n}] of formal polynomials in X_{1}, …, X_{n} with coefficients in V_{0} is a module over k that may be considered as a module over A[X_{1}, …, X_{n}], as in Section 4.3.

Now let V be a module over k that is a module over A[X_{1}, …, X_{n}], considered as a two-sided module over A[X_{1}, …, X_{n}]. Also let

v_{1}, …, v_{n} ∈ V (23.6.1)

be given. If f(X) = f(X_{1}, …, X_{n}) ∈ A[X_{1}, …, X_{n}], then put

L_{j,v_{j}}(f(X)) = (∂_{j}f(X)) · v_{j} (23.6.2)

for each j = 1, …, n. Observe that

(23.6.3) L_{j,v_{j}} defines a derivation from A[X_{1}, …, X_{n}] into V, as in Section 23.4, because ∂_{j} = ∂/∂X_{j} is a derivation on A[X_{1}, …, X_{n}], as mentioned in Section 21.13.

Clearly L_{j,v_{j}}(a) = 0 (23.6.4)

for each j when a ∈ A, so that a is a constant polynomial in X_{1}, …, X_{n}. We also have that

(23.6.5)  L_{j,v_{j}}(e_{A} X_{l}) = v_{j} when j = l
= 0 otherwise.

It is easy to see that L_{j,v_{j}} is uniquely determined by these properties, because A[X_{1}, …, X_{n}] is generated as an algebra over k by A and e_{A} X_{1}, …, e_{A} X_{n}.

We may consider V as a module over A, and thus a two-sided module over A, because A is a subalgebra of A[X_{1}, …, X_{n}]. Let δ_{A} be a derivation from A into V. If

f(X) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} X^{α} (23.6.6)

is an element of A[X_{1}, …, X_{n}], then put

δ̂_{A}(f(X)) = ∑_{α∈(Z_{+}∪{0})^{n}} δ_{A}(f_{α}) X^{α}. (23.6.7)

This defines an element of V, because f_{α} = 0 for all but finitely many α, so that the right side reduces to a finite sum. One can check that

(23.6.8)  δ̂_{A} is a derivation from A[X_{1}, …, X_{n}] into V.

Of course, δ̂_{A} = δ_{A} on A, (23.6.9)

by construction. Observe that

δ̂_{A}(e_{A} X_{j}) = δ_{A}(e_{A}) X_{j} = 0 (23.6.10)

for each j. As before, δ̂_{A} is uniquely determined on A[X_{1}, …, X_{n}] by these properties.

It follows that

(23.6.11) δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}} is a derivation from A[X_{1}, …, X_{n}] into V,

with

δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}} = δ_{A} on A, (23.6.12)

and

(δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}})(e_{A} X_{l}) = v_{l} (23.6.13)

for each l = 1, …, n. As usual, δ̂_{A} +∑_{j=1}^{n} L_{j,v_{j}} is uniquely determined by these properties.

If δ is any derivation from A[X_{1}, …, X_{n}] into V, then the restriction δ_{A} of δ to A may be considered as a derivation from A into V, as a two-sided module over A, as in Section 23.4. Using δ_{A}, we can define δ̂_{A} on A[X_{1}, …, X_{n}] as before. Put v_{j} = δ(e_{A} X_{j}) (23.6.14)

for each j = 1, …, n, so that L_{j,v_{j}} may be defined as in (23.6.2). Under these conditions, we have that

δ = δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}}, (23.6.15)

as in the preceding paragraph. In particular, if A = k, then δ_{A} = 0, so that δ̂_{A} = 0, and

δ = ∑_{j=1}^{n} L_{j,v_{j}}. (23.6.16)

23.7 Some extensions of derivations

Let k be a field, and let E be an extension of k. Also let A be a subalgebra of E, as an algebra over k, that contains k. Suppose that δ_{A} is a derivation from A into E, as algebras over k. Let ξ_{1}, …, ξ_{n} ∈ E be given for some positive integer n, and consider the subalgebra

A_{1} = A[ξ_{1}, …, ξ_{n}] (23.7.1)

of E, as an algebra over k, generated by A and ξ_{1}, …, ξ_{n}. If η_{1}, …, η_{n} ∈ E, then we may be interested in an extension δ_{A_{1}} of δ_{A} to a derivation from A_{1} in to E such that δ_{A_{1}}(ξ_{j}) = η_{j} (23.7.2)

for each j = 1, …, n, as on p170 of [153]. A version of this is mentioned on p267 of [177] too. Note that δ_{A_{1}} is unique when it exists, as in [153, 177].

Let X_{1}, …, X_{n} be n distinct commuting indeterminates, and let

f(X) = f(X_{1}, …, X_{n}) (23.7.3)

be a formal polynomial in X_{1}, …, X_{n} with coefficients in A. Of course, f(X) may be considered as a formal polynomial in X_{1}, …, X_{n} with coefficients in E, so that if x = (x_{1}, …, x_{n}) is an n-tuple of elements of E, then

f(x) = f(x_{1}, …, x_{n}) (23.7.4)

may be defined as an element of E, as in Section 4.4. Note that f(x) ∈ A when x ∈ A^{n}, and that f(x) ∈ A_{1} when x ∈ (A_{1})^{n}.

Put f_{δ_{A}}(X) = δ̂_{A}(f(X)), (23.7.5) where the right side is as in (23.6.7). More precisely, we can take

V = E[X_{1}, …, X_{n}] (23.7.6)

in the previous section, or define this directly in the same way, to get a formal polynomial in X_{1}, …, X_{n} with coefficients in E.

Let δ_{1} be a derivation from A_{1} into E such that

δ_{1} = δ_{A} on A. (23.7.7)

If x ∈ (A_{1})^{n}, then one can check that

δ_{1}(f(x)) = f_{δ_{A}}(x) + ∑_{j=1}^{n}(∂_{j}f)(x) δ_{1}(x_{j}). (23.7.8)

Suppose that δ_{1}(ξ_{j}) = η_{j} (23.7.9)

for each j = 1, …, n. Put ξ = (ξ_{1}, …, ξ_{n}), so that

δ_{1}(f(ξ)) = f_{δ_{A}}(ξ) + ∑_{j=1}^{n}(∂_{j}f)(ξ) η_{j}, (23.7.10)

by (23.7.8). If f(ξ) = 0, (23.7.11)

then we get that

f_{δ_{A}}(ξ) + ∑_{j=1}^{n}(∂_{j}f)(ξ) η_{j} = 0. (23.7.12)

This gives some necessary conditions for the existence of δ_{1}.

Put I(ξ) = {f(X) ∈ A[X_{1}, …, X_{n}] : f(ξ) = 0}. (23.7.13)

This is an ideal in A[X_{1}, …, X_{n}], as an algebra over k, because

f(X) ↦ f(ξ) (23.7.14)

is a homomorphism from A[X_{1}, …, X_{n}] into E, as algebras over k. More precisely, this is a homomorphism from E[X_{1}, …, X_{n}] into E, as in Section 4.4.

In order to check that (23.7.12) holds for every f(X) ∈ I(ξ), it suffices to verify (23.7.12) for a set of generators of I(ξ), as an ideal in A[X_{1}, …, X_{n}]. This can be seen using the facts that partial differentiation is a derivation on A[X_{1}, …, X_{n}], as mentioned in Section 21.13, and that δ̂_{A} is a derivation from A[X_{1}, …, X_{n}] into E[X_{1}, …, X_{n}], as in the previous section. This is the way that the necessary conditions for the existence of a derivation on A_{1} as before are stated in Theorem 14 on p172 of [153], and in the remarks preceding Theorem 7 on p267 of [177].

23.8 Existence of the extension

Let us continue with the same notation and hypotheses as in the previous section. More precisely, suppose now that (23.7.12) holds for every element f(X) of the ideal (23.7.13), and let us show how to find a derivation δ_{A_{1}} from A_{1} into E that is equal to δ_{A} on A and satisfies (23.7.2). This corresponds to the sufficiency of the conditions in Theoram 14 on p172 of [153], and to Theorem 7 on p267 of [177].

Note that (23.7.14) defines a homomorphism from A[X_{1}, …, X_{n}] onto A_{1}, as algebras over k, by the definition (23.7.1) of A_{1}. The kernel of this homomorphism is (23.7.13), by construction. We would like to define δ_{A_{1}} as a mapping from A_{1} into E by putting

δ_{A_{1}}(f(ξ)) = f_{δ_{A}}(ξ) + ∑_{j=1}^{n}(∂_{j}f)(ξ) η_{j} (23.8.1)

for every f(X) ∈ A[X_{1}, …, X_{n}]. This mapping is well defined, because of the hypothesis that (23.7.12) holds when f(X) ∈ I(ξ).

It is easy to see that δ_{A_{1}} is a homomorphism from A_{1} into E, as modules over k. We also have that δ_{A_{1}} = δ_{A} on A, (23.8.2)

by taking f(X) to be a constant polynomial in (23.8.1). One can check that

(23.8.3) δ_{A_{1}} is a derivation on A_{1},

using the facts that partial differentation is a derivation on A[X_{1}, …, X_{n}], and that δ̂_{A} is a derivation from A[X_{1}, …, X_{n}] into E[X_{1}, …, X_{n}], as before.

Let A_{2} be the subfield of E generated by A_{1}. There is a unique extension of δ_{A_{1}} to a derivation δ_{A_{2}} from A_{2} into E, for the same reasons as mentioned in Section 21.14. This corresponds to Theorem 13 on p170 of [153], as mentioned on p174 of [153]. This is included directly in the statement and proof of Theorem 7 on p267 of [177].

23.9 The n = 1 case

Let k be a field, let K be an extension of k, and let E be an extension of K. Also let δ_{K} be a derivation from K into E, where K and E are considered as algebras over k. More precisely, one might consider K and E as rings initially, or equivalently as algebras over Z, instead of algebras over a subfield of K. Note that the kernel of δ_{K} is a subfield of K under these conditions, as in Section 23.2. Thus we may as well take k to be any subfield of K that is contained in the kernel of δ_{K}.

Let ξ be an element of E that is algebraic over K, so that the subfield K(ξ) of E generated by K and ξ is a finite extension of k, as in Section 19.9. This is the same as the subring K[ξ] of E generated by K and ξ. If η ∈ E, then we may be interested in an extension δ_{K(ξ)} of δ_{K} to a derivation from K(ξ) into E such that δ_{K(ξ)}(ξ) = η, (23.9.1)

as in the previous two sections.

Let X be an indeterminate, and let f(X) = ∑_{j=0}^{n} f_{j} X^{j} be a formal polynomial in X with coefficients in K. Also let f′(X) be the formal derivative of f(X), and put

f_{δ_{K}} (X) = δ̂_{K}(f(X)) = ∑_{j=0}^{n} δ_{K}(f_{j}) X^{j}, (23.9.2)

as in Sections 23.6 and 23.7. In order to have a derivation δ_{K(ξ)} from K(ξ) into E as in the preceding paragraph, it is necessary and sufficent that

f_{δ_{K}} (ξ) + f′(ξ) η = 0 (23.9.3)

when f(ξ) = 0, as in the previous two sections. More precisely, it is necessary and sufficient that this hold when f(X) is the minimal polynomial of ξ over K.

Suppose for the moment that ξ is separable over K, so that f′(ξ) ≠ 0, as in Section 20.4. This implies that (23.9.3) holds if and only if

η = −f_{δ_{A}}(ξ)/f′(ξ). (23.9.4)

Note that the right side is an element of K(ξ). If δ_{K} = 0 on K, then f_{δ_{K}} (X) = 0, so that f_{δ_{K}} (ξ) = 0. This corresponds to some remarks on p175 of [153], and to Case 1 on p268 of [177].

Suppose that k has positive characteristic p, that ξ is not an element of K, and that ξ is purely inseparable over K. This means that

f(X) = X^{p^{m}} − a (23.9.5)

for some positive integer m and a ∈ K, as in Section 21.2. It follows that f′(X) = 0, so that (23.9.3) reduces to

f_{δ_{K}} (ξ) = 0. (23.9.6)

Observe that f_{δ_{K}} (X) = δ_{K}(1) X^{p^{m}} − δ_{K}(a) = −δ_{K}(a), (23.9.7)

so that (23.9.6) is the same as saying that

δ_{K}(a) = 0. (23.9.8)

If this condition holds, that (23.9.3) holds for every η. This corresponds to some more remarks on p175 of [153], and to Case 3 on p268 of [177]. A version of this was mentioned in Section 21.15.

23.10 Some derivations on power series

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let X_{1}, …, X_{n} be n distinct commuting indeterminates. Also let B be an algebra over k in the strict sense, let A be a subalgebra of B, and let let δ_{A} be a derivation from A into B. If

f(X) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} X^{α} (23.10.1)

is a formal power series in X_{1}, …, X_{n} with coefficients in A, then put

δ̂_{A}(f(X)) = ∑_{α∈(Z_{+}∪{0})^{n}} δ_{A}(f_{α}) X^{α}. (23.10.2)

This is an element of B[[X_{1}, …, X_{n}]], and one can verify that

(23.10.3) δ̂_{A} is a derivation from A[[X_{1}, …, X_{n}]] into B[[X_{1}, …, X_{n}]].

Note that δ̂_{A}(A[X_{1}, …, X_{n}]) ⊆ B[X_{1}, …, X_{n}] (23.10.4)

and δ̂_{A} = δ_{A} on A. (23.10.5)

Suppose now that δ_{A} is a derivation from A into B[[X_{1}, …, X_{n}]]. One can define (23.10.2) as a formal power series in X_{1}, …, X_{n} with coefficients in B, as follows. If γ ∈ (Z_{+} ∪ {0})^{n}, then we would like to define the coefficient of X^{γ} in (23.10.2), as an element of B. Observe that there are only finitely many α ∈ (Z_{+} ∪ {0})^{n} such that α + β = γ (23.10.6)

for some β ∈ (Z_{+} ∪ {0})^{n}. The coefficient of X^{γ} in (23.10.2) is defined to be

(23.10.7) the sum of the coefficient of X^{β} in δ_{A}(f_{α}) over all such α.

One can check that this satisfies (23.10.3) and (23.10.5), as before. If f(X) is an element of A[X_{1}, …, X_{n}], then (23.10.2) defines an element of B[[X_{1}, …, X_{n}]] more directly, because all but finitely many terms in the sum on the right are equal to 0. If δ_{A} is a derivation from A into B[X_{1}, …, X_{n}], then (23.10.4) holds as well.

Now let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V_{0} be a module over k that is a two-sided module over A. The space V_{0}[[X_{1}, …, X_{n}]] of formal power series in X_{1}, …, X_{n} with coefficients in V_{0} is a module over k and a left and right module over A[[X_{1}, …, X_{n}]], as in Section 4.3. One can check that the actions of A[[X_{1}, …, X_{n}]] on V_{0}[[X_{1}, …, X_{n}]] on the left and right commute with each other, so that V_{0}[[X_{1}, …, X_{n}]] is a two-sided module over A[[X_{1}, …, X_{n}]].

Let δ_{A} be a derivation from A into V_{0}. If f(X) ∈ A[[X_{1}, …, X_{n}]], then

δ̂_{A}(f(X)) ∈ V_{0}[[X_{1}, …, X_{n}]] (23.10.8)

may be defined as in (23.10.2). One can verify that

(23.10.9) δ̂_{A} is a derivation from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]],

as before. Clearly

δ̂_{A}(A[X_{1}, …, X_{n}]) ⊆ V_{0}[X_{1}, …, X_{n}] (23.10.10)

and (23.10.5) holds.

If δ_{A} is a derivation from A into V_{0}[[X_{1}, …, X_{n}]] and f(X) is an element of A[[X_{1}, …, X_{n}]], then (23.10.2) may be interpreted as a formal power series in X_{1}, …, X_{n} with coefficients in V_{0} in the same way as before. One can check that this satisfies (23.10.5) and (23.10.9), as usual. If f(X) ∈ A[X_{1}, …, X_{n}], then (23.10.9) defines an element of V_{0}[[X_{1}, …, X_{n}]] more directly, as before. If δ_{A} is a derivation from A into V_{0}[X_{1}, …, X_{n}], then (23.10.10) holds too.

23.11 A property of some derivations

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V_{0} be a two-sided module over A, and let X_{1}, …, X_{n} be n distinct commuting indeterminates for some positive integer n, so that V_{0}[[X_{1}, …, X_{n}]] is a two-sided module over A[[X_{1}, …, X_{n}]], as before. Let N be a nonnegative integer, and let f(X) ∈ A[[X_{1}, …, X_{n}]] be as in (23.10.1). Remember that f(X) is said to vanish to order N for some nonnegative integer N if f_{α} = 0 for every multi-index α of length n with |α| ≤ N, as in Section 4.5. This may be considered as holding trivially when N = −1, as before.

Let δ be a derivation from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]]. If an element f(X) of A[[X_{1}, …, X_{n}]] vanished to order N, then

(23.11.1) δ(f(X)) vanishes to order N − 1.

This is easy to see when f(X) is of the form

f(X) = a(X) X^{β}, (23.11.2)

with a(X) ∈ A[[X_{1}, …, X_{n}]] and |β| ≥ N + 1. Otherwise, one can check that f(X) may be expressed as the sum of finitely many elements of A[[X_{1}, …, X_{n}]] of this form.

This implies that δ is uniquely determined by its restriction to A[X_{1}, …, X_{n}]. It follows that

(23.11.3) δ is uniquely determined by its restriction to A and δ(e_{A} X_{1}), …, δ(e_{A} X_{n}).

23.12. MORE ON DERIVATIONS ON A[[X_{1}, …, X_{N}]] 565

Similarly, let δ_{0} be a derivation from A[X_{1}, …, X_{n}] into V_{0}[[X_{1}, …, X_{n}]], considered as a two-sided module over A[X_{1}, …, X_{n}]. If f(X) ∈ A[X_{1}, …, X_{n}] vanishes to order N, then

(23.11.4) δ_{0}(f(X)) vanishes to order N − 1.

This is a bit easier in this case.

If f(X) ∈ A[[X_{1}, …, X_{n}]] is as in (23.10.1), then we would like to put

δ̂_{0}(f(X)) = ∑_{α∈(Z_{+}∪{0})^{n}} δ_{0}(f_{α} X^{α}). (23.11.5)

Note that (23.11.6) δ_{0}(f_{α} X^{α}) vanishes to order |α| − 1

for each multi-index α, as before. This implies that for each multi-index γ, the coefficient of X^{γ} in δ_{0}(f_{α} X^{α}) is equal to 0 for all but finitely many α. Using this, we can define (23.11.5) as an element of V_{0}[[X_{1}, …, X_{n}]]. Of course,

δ̂_{0} = δ_{0} on A[X_{1}, …, X_{n}], (23.11.7)

by construction. One can verify that

(23.11.8) δ̂_{0} is a derivation from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]].

If A is commutative, and V_{0} is a one-sided module over A considered as a twosided module over A, then this could also be obtained from the description of derivations on A in Section 23.6, and some of the remarks in the previous section. An analogous argument when A is not commutative will be mentioned in Section 23.15.

23.12 More on derivations on A[[X_{1}, …, X_{n}]]

Let k be a commutative ring with a multiplicative identity element, let A be a commutative associative algebra over k with a multiplicative identity element, and let X_{1}, …, X_{n} be n distinct commuting indeterminates for some positive integer n again. Also let V_{0} be a module over A, so that

(23.12.1) V_{0}[[X_{1}, …, X_{n}]] is a module over A[[X_{1}, …, X_{n}]],

that may be considered as a two-sided module over A[[X_{1}, …, X_{n}]]. Similarly, V_{0}[X_{1}, …, X_{n}] may be considered as a two-sided module over A[X_{1}, …, X_{n}], as in Section 23.6.

Let v_{1}(X), …, v_{n}(X) ∈ V_{0}[[X_{1}, …, X_{n}]] (23.12.2)

be given. If f(X) ∈ A[[X_{1}, …, X_{n}]], then put

L_{j,v_{j}(X)}(f(X)) = (∂_{j}f(X)) · v_{j}(X) (23.12.3) for each j = 1, …, n. Note that

(23.12.4) L_{j,v_{j}(X)} defines a derivation from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]],

as in Section 23.4. If v_{j}(X) ∈ V_{0}[X_{1}, …, X_{n}], (23.12.5)

then L_{j,v_{j}(X)}(A[X_{1}, …, X_{n}]) ⊆ V_{0}[X_{1}, …, X_{n}]. (23.12.6)

We also have that L_{j,v_{j}(X)}(a) = 0 (23.12.7)

and

(23.12.8)  L_{j,v_{j}(X)}(e_{A} X_{l}) = v_{j}(X) when j = l
= 0  otherwise.

It is easy to see that L_{j,v_{j}(X)} is uniquely determined on A[[X_{1}, …, X_{n}]] by (23.12.4), (23.12.7), and (23.12.8), using the remarks in the previous section.

If δ_{A} is a derivation from A into V_{0}[[X_{1}, …, X_{n}]], then we get a derivation δ̂_{A} from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]], as in Section 23.10. This derivation is equal to δ_{A} on A, and satisfies

δ̂_{A}(e_{A} X_{j}) = δ_{A}(e_{A}) X_{j} = 0 (23.12.9)

for each j, by construction. It is easy to see that δ̂_{A} is uniquely determined on A[[X_{1}, …, X_{n}]] by these properties, using the remarks in the previous section, as before.

Thus

(23.12.10)  δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}(X)} is a derivation from A[[X_{1}, …, X_{n}]]
into V_{0}[[X_{1}, …, X_{n}]]

such that

δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}(X)} = δ_{A} on A (23.12.11)

and

(δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}(X)})(e_{A} X_{l}) = v_{l}(X) (23.12.12)

for each l = 1, …, n. This derivation is uniquely determined on A[[X_{1}, …, X_{n}]] by (23.12.11) and (23.12.12), because of the remarks in the previous section, as 23.13. SOME REMARKS ABOUT CENTRALIZERS 567

usual. If δ is any derivation from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]], then it follows that

δ = δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}(X)} (23.12.13)

with δ_{A} equal to the restriction of δ to A, and v_{j}(X) = δ(e_{A} X_{j}) (23.12.14) for each j. If A = k, then δ_{A} = 0, so that δ̂_{A} = 0.

23.13 Some remarks about centralizers

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense. If E is a subset of A, then the centralizer of E in A may be defined by (23.13.1) C_{A}(E) = {a ∈ A : a x = x a for every x ∈ E}. This is a submodule of A, as a module over k. If A is associative, then (23.13.2) C_{A}(E) is a subalgebra of A. If A has a multiplicative identity element e_{A}, then e_{A} ∈ C_{A}(E). (23.13.3) Note that C_{A}(A) is the same as the center of A, as in Section 20.13. If E_{1}, E_{2} are subsets of A, then E_{1} ⊆ C_{A}(E_{2}) (23.13.4) if and only if E_{2} ⊆ C_{A}(E_{1}). (23.13.5)

Let B be an algebra over k in the strict sense, and suppose that A is a subalgebra of B. If E is a subset of B, then the centralizer of E in A may be defined as in (23.13.1). Equivalently, C_{A}(E) = C_{A}^{B}(E) = C_{B}(E) ∩ A. (23.13.6) This is a submodule of A, as a module over k, as before. If B is associative, then (23.13.2) holds. If A has a multiplicative identity element e_{A} that is also the multiplicative identity element of B, then (23.13.3) holds. Note that C_{A}(E_{1} ∪ E_{2}) = C_{A}(E_{1}) ∩ C_{A}(E_{2}) (23.13.7) for all E_{1}, E_{2} ⊆ B.

Let δ be a derivation from A into B, and suppose for the moment that a ∈ C_{A}(A ∪ δ(A)) = C_{A}(A) ∩ C_{A}(δ(A)). (23.13.8)

If y ∈ A, then δ(a y) = δ(a) y + a δ(y) = δ(a) y + δ(y) a (23.13.9) and δ(a y) = δ(y a) = δ(y) a + y δ(a). (23.13.10) This implies that δ(a) y = y δ(a), (23.13.11) so that δ(a) ∈ C_{B}(A). (23.13.12) This is the same as saying that A ⊆ C_{A}({δ(a)}). (23.13.13) If A = B, then we get that δ(C_{A}(A)) ⊆ C_{A}(A). (23.13.14)

Suppose now that A is an associative algebra over k with a multiplicative identity element e_{A}, and let V be a two-sided module over A. If E is a subset of V, then the centralizer of E in A may be defined by (23.13.15) C_{A}(E) = C_{A}^{V} (E) = {a ∈ A : a · v = v · a for every v ∈ E}. This is a submodule of A, as a module over k, and a subalgebra of A. Indeed, if a_{1}, a_{2} ∈ C_{A}(E) and v ∈ E, then

(23.13.16) (a_{1} a_{2}) · v = a_{1} · (a_{2} · v) = a_{1} · (v · a_{2})
= (a_{1} · v) · v_{2} = (v · a_{1}) · a_{2} = v · (a_{1} a_{2}).

Note that (23.13.3) holds, and that this version is the same as the previous one when B is an associative algebra over k that contains A as a subalgebra, and for which e_{A} is the multiplicative identity element, so that B may be considered as a two-sided module over A.

Let δ be a derivation from A into V, and suppose that a ∈ C_{A}^{A}(A) ∩ C_{A}^{V} (δ(A)). (23.13.17) This implies that δ(a) · y = y · δ(a) (23.13.18) for every y ∈ A, as in (23.13.11), so that A ⊆ C_{A}^{V} ({δ(a)}), (23.13.19) as in (23.13.13). If v ∈ V satisfies y · v = v · y (23.13.20) for every y ∈ A, then v is said to be an invariant element of V, as on p170 of [50], and as mentioned in Section 35.6. This is the same as saying that A ⊆ C_{A}^{V} ({v}). (23.13.21) Thus we get that δ(a) is an invariant element of V when (23.13.17) holds.

23.14. MORE ON CENTRALIZERS 569

23.14 More on centralizers

Let k be a commutative rng with a multiplicative identity element, let B be an associative algebra over k, and let A be a subalgebra of B. If b ∈ C_{B}(A) (23.14.1) and δ is a derivation from A into B, then one can check that x ↦ b δ(x) (23.14.2) and x ↦ δ(x) b (23.14.3) are also derivations from A into B. This was mentioned in Section 23.2 when B is commutative. Of course, (23.14.2) and (23.14.3) are the same when b ∈ C_{B}(δ(A)), (23.14.4) and in particular when B is commutative.

Suppose that B has a multiplicative identity element that is contained in A, so that B may be considered as a two-sided module over A. If b ∈ B, then z ↦ b z (23.14.5) is a homomorphism from B into itself, as a right module over A or B, and z ↦ z b (23.14.6) is a homomorphism from B into itsef, as a left module over A or B. If (23.14.1) holds, then (23.14.5) is a homomorphism from B into itself, as a left module over A, and (23.14.6) is a homomorphism from B into itself, as a right module over A. Of course, (23.14.2) and (23.14.3) are the same as the compositions of δ with (23.14.5) and (23.14.6), respectively. Under these conditions, the fact that (23.14.2) and (23.14.3) are derivations from A into B also follows from a remark in Section 23.4.

Now let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V be a two-sided module over A. If a ∈ A, then v ↦ a · v (23.14.7) is a homomorphism from V into itself, as a right module over A, and v ↦ v · a (23.14.8) is a homomorphism from V into itself, as a left module over A. If a ∈ C_{A}(A), (23.14.9) then (23.14.7) is a homomorphism of V into itself, as a left module over A, and (23.14.8) is a homomorphism of V into tiself, as a right module over A. If δ is a derivation from A into V, then it follows that

x ↦ a · δ(x) (23.14.10)

and x ↦ δ(x) · a (23.14.11)

are derivations from A into V as well, as in Section 23.4. This was mentioned previously when A is commutative and V is a one-sided module over A, considered as a two-sided module over A, in which case (23.14.10) and (23.14.11) are the same.

If v_{0} ∈ V, then y ↦ y · v_{0} (23.14.12)

is a homomorphism from A into V, as left modules over A, and

y ↦ v_{0} · y (23.14.13)

is a homomorphism from A into V, as right modules over A. If v_{0} is an invariant element of V, as in the previous section, then (23.14.12) and (23.14.13) are the same, so that we get a homomorphism from A into V, as two-sided modules over A. In this case, if δ is a derivation from A into itself, then

x ↦ δ(x) · v_{0} = v_{0} · δ(x) (23.14.14)

is a derivation from A into V, as in Section 23.4. This was mentioned earlier when A is commutative and V is a one-sided module over A, considered as a two-sided module over A.

Let X_{1}, …, X_{n} be n distinct commuting indeterminates for some positive integer n, and suppose now that V is a two-sided module over A[X_{1}, …, X_{n}]. If 1 ≤ j ≤ n and v_{j} is an invariant element of V, as a two-sided module over A[X_{1}, …, X_{n}], then

L_{j,v_{j}}(f(X)) = (∂_{j}f(X)) · v_{j} = v_{j} · (∂_{j}f(X)) (23.14.15)

is a derivation from A[X_{1}, …, X_{n}] into V, as in the preceding paragraph. Similarly, if V is a two-sided module over A[[X_{1}, …, X_{n}]] and v_{j} is invariant as an element of V, as a two-sided module over A[[X_{1}, …, X_{n}]], then (23.14.15) is a derivation from A[[X_{1}, …, X_{n}]] into V.

23.15 Derivations and partial derivatives

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let X_{1}, …, X_{n} be n distinct commuting indeterminates for some positive integer n. If V_{0} is a two-sided module over A, then V_{0}[X_{1}, …, X_{n}] and V_{0}[[X_{1}, …, X_{n}]] may be considered as two-sided modules over A[X_{1}, …, X_{n}] and A[[X_{1}, …, X_{n}]], respectively, as before. If u(X) ∈ V_{0}[[X_{1}, …, X_{n}]], then

a(X) · u(X) = u(X) · a(X) (23.15.1)

when a(X) = a(X_{1}, …, X_{n}) ∈ A[[X_{1}, …, X_{n}]] has coefficients in the centralizer C_{A}^{V_{0}}(V_{0}) of V_{0} in A. This means that

(23.15.2) a(X) is in the centralizer of V_{0}[[X_{1}, …, X_{n}]] in A[[X_{1}, …, X_{n}]],

as in Section 23.13. Similarly, if a(X) ∈ A[X_{1}, …, X_{n}] has coefficients in C_{A}^{V_{0}}(V_{0}), then

(23.15.3) a(X) is in the centralizer of V_{0}[X_{1}, …, X_{n}] in A[X_{1}, …, X_{n}].

Let V be a module over k that is a two-sided module over A[X_{1}, …, X_{n}], suppose that

(23.15.4) e_{A} X_{j} is an element of the centralizer of V in A[X_{1}, …, X_{n}]

for each j = 1, …, n. Let δ be a derivation from A[X_{1}, …, X_{n}] into V, and observe that (23.15.5) v_{j} = δ(e_{A} X_{j}) is an invariant element of V

for each j = 1, …, n, as in Section 23.13, because e_{A} X_{j} is also an element of the center of A[X_{1}, …, X_{n}].

Thus (23.14.15) defines a derivation from A[X_{1}, …, X_{n}] into V. Let δ_{A} the restriction of δ to A, which is a derivation from A into V, as a two-sided module over A. This can be extended to a derivation δ̂_{A} from A[X_{1}, …, X_{n}] into V, as in Section 23.6. It is easy to see that

δ = δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}} (23.15.6)

on A[X_{1}, …, X_{n}], as before. If V = V_{0}[[X_{1}, …, X_{n}]], then one can use this to get an extension of δ to a derivation from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]], as mentioned in Section 23.11.

Similarly, if δ is a derivation from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]], then

(23.15.7) v_{j}(X) = δ(e_{X} X_{j}) is an invariant element of V_{0}[[X_{1}, …, X_{n}]]

for each j = 1, …, n, as in Section 23.13. This implies that

L_{j,v_{j}(X)}(f(X)) = (∂_{j}f(X)) · v_{j}(X) = v_{j}(X) · (∂_{j}f(x)) (23.15.8)

is a derivation from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]], as in the previous section. Let δ_{A} be the restriction of δ to A, which is a derivation from A into V_{0}[[X_{1}, …, X_{n}]], as a two-sided module over A. This can be extended to a derivation δ̂_{A} from A[[X_{1}, …, X_{n}]] into V_{0}[[X_{1}, …, X_{n}]], as in Section 23.10. One can check that

δ = δ̂_{A} + ∑_{j=1}^{n} L_{j,v_{j}(X)} (23.15.9)

on A[[X_{1}, …, X_{n}]], as in Section 23.12. More precisely, this equation holds on A, and at e_{A} X_{l} for each l = 1, …, n, by construction. This implies that this equation holds on A[X_{1}, …, X_{n}], because both sides of the equation are derivations. It follows that this equation holds on A[[X_{1}, …, X_{n}]], as in Section 23.11.

More on field extensions, 3

24.1 More on formal polynomials

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. Also let I be a nonempty set, and let {T_{j}}_{j∈I} be a family of distinct commuting indeterminates indexed by I. Let us say that a function α on I with values in the set Z_{+} ∪ {0} of nonnegative integers is a multi-index associated to I if

(24.1.1) α(j) = 0 for all but finitely many j ∈ I.

In this case, we put

|α| = ∑_{j∈I} α(j), (24.1.2)

and the corresponding formal monomial

T^{α} = ∏_{j∈I} T_{j}^{α(j)} (24.1.3)

has degree |α|. If j_{1}, …, j_{n} are finitely many distinct elements of I and α(j) = 0 unless j = j_{l} for some 1 ≤ l ≤ n, then

|α| = ∑_{l=1}^{n} α(j_{l}), (24.1.4)

and we may express T^{α} as

T^{α} = ∏_{l=1}^{n} T_{j_{l}}^{α(j_{l})}. (24.1.5)

Let Mult(I) (24.1.6) be the set of all multi-indices associated to I. This is a commutative semigroup with respect to addition. A formal polynomial in the T_{j}’s, j ∈ I, with coefficients in V may be expressed as

f(T) = ∑_{α∈Mult(I)} f_{α} T^{α}, (24.1.7)

where f_{α} ∈ V for all α, and

(24.1.8) f_{α} = 0 for all but finitely many α ∈ Mult(I).

The space V [{T_{j}}_{j∈I}] of these formal polynomials may be defined as the space

c_{00}(Mult(I), V) (24.1.9)

of all V -valued functions on Mult(I) with finite support. This is a module over k with respect to pointwise addition and scalar multiplication, which corresponds to termwise addition and scalar multiplication of formal sums as in (24.1.7). This is basically the same as in Section 4.3 when I = {1, …, n} for some positive integer n. As before, we may identify V with the submodule of V [{T_{j}}_{j∈I}] consisting of f(T) as in (24.1.7) such that f_{α} = 0 when α ≠ 0.

Let A be an algebra over k in the strict sense, as in Section 20.13, let f(T) in A[{T_{j}}_{j∈I}] be as in (24.1.7), and let

g(T) = ∑_{β∈Mult(I)} g_{β} T^{β} (24.1.10)

be another element of A[{T_{j}}_{j∈I}]. If γ ∈ Mult(I), then put

h_{γ} = ∑_{{α,β∈Mult(I) \atop α+β=γ}} f_{α} g_{β}. (24.1.11)

It is easy to see that the sum on the right has only finitely many terms, and thus defines an element of A. We also have that

h_{γ} = 0 (24.1.12)

for all but finitely many γ ∈ Mult(I), so that

h(T) = ∑_{γ∈Mult(I)} h_{γ} T^{γ} (24.1.13)

is an element of A[{T_{j}}_{j∈I}]. Under these conditions, put

f(T) g(T) = h(T). (24.1.14)

It is easy to see that

(24.1.15) A[{T_{j}}_{j∈I}] is an algebra in the strict sense over k with respect to this definition of multiplication. We may consider A as a subalgebra of A[{T_{j}}_{j∈I}], as before. If A is commutative or associative, then A[{T_{j}}_{j∈I}] has the same property. If A has a multiplicative identity element e_{A}, then the corresponding formal polynomial is the multiplicative identity element in A[{T_{j}}_{j∈I}].

In particular, k[{T_{j}}_{j∈I}] is a commutative associative algebra over k with a multiplicative identity element. Note that polynomials in arbitrary sets of commuting indeterminates with coefficients in a commutative ring with a multiplicative identity element are defined on p111 of [177]. Similarly, V [{T_{j}}_{j∈I}] may be considered as a module over k[{T_{j}}_{j∈I}]. If A is an associative algebra with a multiplicative identity element, and V is a left or right module over A, then V [{T_{j}}_{j∈I}] may be considered as a left or right module over A[{T_{j}}_{j∈I}]. If V is a two-sided module over A, then V [{T_{j}}_{j∈I}] may be considered as a two-sided module over A[{T_{j}}_{j∈I}].

24.2 More on polynomial functions

Let k be a commutative ring with a multiplicative identity element, and let I be a nonempty set. If V is a module over k, then let V^{I} = c(I, V) be the space of V -valued functions on I, which is a module over k with respect to coordinatewise addition and scalar multiplication. If v ∈ V^{I} and j ∈ I, then v_{j} ∈ V is the jth coordinate of v.

If t ∈ k^{I} and α is a multi-index associated to I, then

t^{α} = ∏_{j∈I} t_{j}^{α(j)} (24.2.1)

defines an element of k. More precisely, t_{j}^{α(j)} is interpreted as being the multiplicative identity element 1 when α(j) = 0, as usual, so that the right side reduces to a finite product in k. Equivalently, if j_{1}, …, j_{n} are finitely many distinct elements of I and α(j) = 0 unless j = j_{l} for some l, then

t^{α} = ∏_{l=1}^{n} t_{j_{l}}^{α(j_{l})}. (24.2.2)

If β is another multi-index associated to I, then

t^{α+β} = t^{α} t^{β}. (24.2.3)

Let {T_{j}}_{j∈I} be a family of distinct commuting indeterminates indexed by I, and let V be a module over k again. If f(T) is a formal polynomial in the T_{j}’s with coefficients in V as in (24.1.7), and t ∈ k^{I}, then

f(t) = ∑_{α∈Mult(I)} f_{α} t^{α} (24.2.4)

defines an element of V, where all but finitely many terms in the sum are equal to 0. This is the same as in Section 4.4 when I = {1, …, n} for some positive integer n. Note that f(T) ↦ f(t) (24.2.5)

is a homomorphism from V [{T_{j}}_{j∈I}] into V, as modules over k, as before.

Let A be an algebra over k in the strict sense, so that A[{T_{j}}_{j∈I}] is an algebra over k in the strict sense as well, as in the previous section. If f(T), g(T) are elements of A[{T_{j}}_{j∈I}], h(T) = f(T) g(T), and t ∈ k^{I}, then one can check that h(t) = f(t) g(t), (24.2.6)

as in Section 4.4.

Let V be a module over k again, and let f(T) ∈ k[{T_{j}}_{j∈I}] and g(T) in V [{T_{j}}_{j∈I}] be given, so that h(T) = f(T) g(T) is defined as an element of V [{T_{j}}_{j∈I}], as in the previous section. If t ∈ k^{I}, then f(t) is defined as an element of k, g(t) and h(t) are defined as elements of V, and (24.2.6) holds, as in Section 4.4.

Let A be an associative algebra over k with a multiplicative identity element e_{A}, and suppose that V is a left or right module over A. If f(T), g(T) are in A[{T_{j}}_{j∈I}], V [{T_{j}}_{j∈I}], as appropriate, then h(T) = f(T) g(T) is defined as an element of V [{T_{j}}_{j∈I}], as in the previous section. If t ∈ k^{I}, then f(t), g(t) are defined as elements of A, V, as appropriate, h(t) ∈ V, and (24.2.6) holds, as in Section 4.4.

Suppose that a ∈ A^{I} has commuting coordinates, so that

a_{j} a_{l} = a_{l} a_{j} (24.2.7)

for every j, l ∈ I. If α is a multi-index associated to I, then

a^{α} = ∏_{j∈I} a_{j}^{α(j)} (24.2.8)

defines an element of A, where a_{j}^{α(j)} is interpreted as being equal to e_{A} when α(j) = 0, so that this reduces to a finite product in A. As before, if j_{1}, …, j_{n} are finitely many distinct elements of II and α(j) = 0 unless j = j_{l} for some l, then

a^{α} = ∏_{l=1}^{n} a_{j_{l}}^{α(j_{l})}. (24.2.9)

If β is another multi-index associated to I, then

a^{α+β} = a^{α} a^{β}. (24.2.10)

If f(T) is a formal polynomial in the T_{j}’s with coefficients in k as in (24.1.7), then

f(a) = ∑_{α∈Mult(I)} f_{α} a^{α} (24.2.11)

24.3. SOME EXAMPLES WITH I = Z_{+} 577

defines an element of A, where all but finitely many terms in the sum are equal to 0. If g(T) ∈ k[{T_{j}}_{j∈I}] and h(T) = f(T) g(T), then

h(a) = f(a) g(a), (24.2.12)

as in Section 4.4. Thus f(T) ↦ f(a) is an algebra homomorphism from k[{T_{j}}_{j∈I}] into A, as before.

24.3 Some examples with I = Z_{+}

Let k be a field, and let T_{1}, T_{2}, T_{3}, … be an infinite sequence of distinct commuting indeterminates. Also let

I_{n} = I(T_{1}, …, T_{n}) (24.3.1)

be the ideal in k[{T_{j}}_{j=1}^{∞}] generated by T_{1}, …, T_{n} for each positive integer n. It is easy to see that I_{n} ⊆ I_{n+1} (24.3.2)

and I_{n} ≠ I_{n+1} (24.3.3)

for each n. Thus k[{T_{j}}_{j=1}^{∞}] is not Noetherian, as in Example 6 on p75 of [8]. In this case,

I = ⋃_{n=1}^{∞} I_{n} (24.3.4)

is the same as the ideal in k[{T_{j}}_{j=1}^{∞}] generated by all of the T_{j}’s. It is easy to see that this is not a finitely-generated ideal in k[{T_{j}}_{j=1}^{∞}]. This ideal is the same as the kernel of the homomorphism f(T) ↦ f(0) from k[{T_{j}}_{j=1}^{∞}] onto k.

Let L be the ideal in A = k[{T_{j}}_{j=1}^{∞}] generated by

T_{1}, T_{2}^{2}, T_{3}^{3}, …, T_{l}^{l}, …, (24.3.5)

as in the first example on p91 of [8]. Thus B = A/L is a commutative associative algebra over k with a multiplicative identity element. Note that

L ⊆ I. (24.3.6)

Let q be the natural quotient mapping from A onto B. Using (24.3.6), we get that f(T) ↦ f(0) induces a homomorphism from B onto k, as algebras over k.

If f(T) ∈ I, then one can check that

f(T)^{m} ∈ L (24.3.7)

when m is sufficiently large. This means that

(24.3.8) q(f(T)) is nilpotent in B.

In fact, q(I) is the nilradical of B, as in Section 12.5. Note that q(I) is a maximal proper ideal in B. One can verify that q(I) is the only proper prime ideal in B, as mentioned in [8].

We also have that B is not Noetherian. This can be seen by showing that q(I) is not finitely generated as an ideal in B, as mentioned in [8]. Alternatively, one can check that q(I_{n}) ≠ q(I_{n+1}) for each n.

24.4 Algebraic independence

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let I be a nonempty set. Also let V be a left module over A, and let v_{j} be an element of V for each j ∈ I. The family {v_{j}}_{j∈I} is said to be linearly independent in V over A if for every family {a_{j}}_{j∈I} of elements of A with a_{j} = 0 for all but finitely many j ∈ I,

∑_{j∈I} a_{j} · v_{j} = 0 (24.4.1)

implies that a_{j} = 0 for every j ∈ I, as on p81 of [177]. Sometimes we may wish to consider the linear independence of a subset E of V, which is the same as the linear independence of a family of elements of E in which every element of E occurs once. Of course, there are analogous notions for right modules over A.

Suppose now that A is a commutative associative algebra over k. A family a = {a_{j}}_{j∈I} of elements of A is said to be algebraically independent in A over k if the corresponding family

a^{α}, α ∈ Mult(I), (24.4.2)

of monomials in A is linearly independent over k, as on p112 of [177].

If {T_{j}}_{j∈I} is a family of distinct commuting indeterminates indexed by I, then the family of T_{j}’s is algebraically independent in k[{T_{j}}_{j∈I}] over k, by construction. This corresponds to a remark on p113 of [177].

We might as well ask that t ↦ t e_{A} be an injective mapping from k into A, because otherwise e_{A} is not linearly independent as an element of A over k. This means that k corresponds to a subring of A that contains e_{A}.

As before, we may sometimes wish to consider the algebraic independence of a subset E of A over k. This is the same as the algebraic indepence of a family of elements of E in which every element of E occurs once.

Algebraic independence of finitely many elements of a commutative ring with a nonzero multiplicative identity element relative to a subring that contains the multiplicative identity element is defined beginning on p3 of [153]. This is equivalent to the previous definition, with k taken to be the subring of the initial ring.

Let {T_{j}}_{j∈I} be a family of distinct commuting indeterminates again, and let a = {a_{j}}_{j∈I} be any family of elements of A indexed by I. Remember that

f(T) ↦ f(a) (24.4.3) defines an algebra homomorphism from k[{T_{j}}_{j∈I}] into A, as in Section 24.2. This homomorphism maps k[{T_{j}}_{j∈I}] onto the subalgebra of A generated by e_{A} and the a_{j}’s, j ∈ I. This homomorphism is injective exactly when the a_{j}’s are algebraically independent in A over k.

Note that algebraically independent elements of a commutative ring with a nonzero multiplicative identity element relative to a subring that contains the multiplicative identity element are said to be “indeterminates” relative to the subring on p4 of [153]. This is related to some remarks on p119 of [177].

24.5 More on algebraic independence

Let k be a commutative ring with a multiplicative identity element, let I be a nonempty set, and let {X_{j}}_{j∈I} be a family of distinct commuting indeterminates indexed by I. Also let I_{0} be a nonempty subset of I, and let V be a module over k. We can identify V [{X_{j}}_{j∈I_{0}}] with a submodule of V [{X_{j}}_{j∈I}], as a module oer k, in an obvious way. This corresponds to extending a multi-index α associated to I_{0} to a multi-index associated to I by putting α = 0 on I \ I_{1}.

Similarly, if A is an algebra over k in the strict sense, then A[{X_{j}}_{j∈I_{0}}] may be identified with a subalgebra of A[{X_{j}}_{j∈I}], as an algebra over k in the strict sense.

If k is an integral domain, then

(24.5.1) k[{X_{j}}_{j∈I}] is an integral domain.

This follows from a remark in Section 21.12 when I has only finitely many elements. This means that k[{X_{j}}_{j∈I_{0}}] is an integral domain when I_{0} has only finitely many elements. It is easy to get (24.5.1) from this.

Suppose from now on in this section that k is a field. As in Section 21.12,

k({X_{j}}_{j∈I}) (24.5.2)

may be defined as the field Q_{k[{X_{j}}_{j∈I}]} of fractions of k[{X_{j}}_{j∈I}]. The elements of (24.5.2) may be called formal rational functions in the X_{j}’s, j ∈ I, with coefficients in k, as before.

We shall primarily be concerned here with algebraic independence in a field K that contains a field k as a subfield, as on p253 of [177]. Algebraic independence of an arbitrary subset E of K relative to k is defined beginning on p151 of [153] to mean that every finite subset of E is algebraically independent. This is equivalent to the previous definition as well.

If a = {a_{j}}_{j∈I} is a family of algebraically independent elements of K over k, then we get an injective homomorphism from k[{X_{j}}_{j∈I}] into K, as algebras over k, as before. This leads to an injective homomorphism from k({X_{j}}_{j∈I}) into K, as algebras over k. This is how algebraic independence in K over k is defined on p561 of [92]. An equivalent description is mentioned on p562 of [92], which is essentially the same as in the previous section.

A nonempty subset E of K is said to be algebraically dependent over k if it is not algebraically independent over k, as on p152 of [153]. This means that E has a nonempty finite subset that is algebraically dependent over k, as in [153].

Let E be a subset of K, and let k(E) be the subfield of K generated by k and E. An element x of K is said to be algebraically dependent on E over k if x is algebraic over k(E), as in Section 19.9. This corresponds to Definition 1 on p152 of [153]. This means that x is algebraically dependent on a finite subset of E, as mentioned in [153].

It is not too difficult to show that a nonempty subset E of K is algebraically dependent over k if and only if there is an x ∈ E such that

(24.5.3) x is algebraically dependent on E \ {x} over k.

This is Theorem 5 on p152 of [153].

24.6 Transcendence bases

Let K be a field, and let k be a subfield of K. A subset B of K is said to be a transcendence basis for K over k if

(24.6.1) B is algebraically independent over k

and (24.6.2) every x ∈ K is algebraically dependent on B,

as on p562 of [92], and p154 of [153]. Equivalently, this means that (24.6.1) holds, and that B is maximal with respect to inclusion, as on p253 of [177].

Let k_{1} be a subfield of K, and let E_{1} be a subset of K_{1} such that K_{1} is generated as a field by k_{1} and E_{1}. If

(24.6.3) every element of E_{1} is algebraic over k_{1},

then (24.6.4) every element of K is algebraic over k_{1}.

Indeed, if y ∈ K, then there are finitely many elements x_{1}, …, x_{n} of E such that y is an element of the subfield k_{1}(x_{1}, …, x_{n}) of K generated by k_{1} and x_{1}, …, x_{n}. Remember that k_{1}(x_{1}, …, x_{n}) is a finite extension of k_{1} under these conditions, because of (24.6.3), as mentioned in Section 19.9. This implies that every element of k_{1}(x_{1}, …, x_{n}) is algebraic over k_{1}, as mentioned in Section 19.9 too.

Suppose that E_{0} ⊆ K and

(24.6.5) K is generated as a field by k and E_{0}.

In order to verify that (24.6.2) holds for some B ⊆ K, it is enough to check that

(24.6.6) every x ∈ E_{0} is algebraically dependent on B, by the remarks in the preceding paragraph. More precisely, this works when

(24.6.7) K is generated as a field by k, B, and E_{0}.

One can use Zorn’s lemma or Hausdorff’s maximality principle to get that K has a transcendence basis over k, as in the Basis theorem on p154 of [153], and as mentioned on p253 of [177]. More precisely, suppose that B_{0} ⊆ K is algebraically independent over k, and that K is generated as a field by k, B_{0}, and E_{0} ⊆ K. Under these conditions, there is a transcendence basis B of K over k such that B_{0} ⊆ B ⊆ B_{0} ∪ E_{0}. (24.6.8)

This corresponds to Exercise 1 on p156 of [153], and to the second part of Theorem 1 on p254 of [177].

Of course, it suffices to get a maximal set B that satisfies (24.6.1) and (24.6.8). This is much simpler when E_{0} has only finitely many elements. This is also fairly simple when E_{0} is countably infinite.

It is well known that any two transcendence bases for K over k have the same cardinality, as in Theorem A1.1 on p562 of [92], the Basis theorem on p154 of [153], and Theorem 1 on p254 of [177]. More precisely, the proof in [92] is only given when the two transcendence bases are finite, and the proof is [177] is given when one of the transcendence bases is finite. The cardinality of a transcendence base for K over k is called the transcendence degree of K over k, as on p155 of [153]. This may also be described as the largest cardinality of an algebraically independent subset of K over k, as on p253 of [177].

If K has a transcendence basis B over k such that K is equal to the subfield k(B) generated by k and B, then K is said to be a purely transcendental extension of k, as on p155 of [153]. This means that K is isomorphic to the field of rational functions in some family of distinct commuting indeterminates with coefficients in k, as on p561 of [92].

24.7 A lemma about polynomials

Let k be a field, let r be a positive integer, and let X_{1}, …, X_{r} be r distinct commuting indeterminates. Put

T = k[X_{1}, …, X_{r}], (24.7.1)

and let f(X) be a nonconstant element of T. Let e be a positive integer, and put x′_{j} = X_{j} − X_{r}^{e^{j}} (24.7.2)

for j = 1, …, r − 1.

We would like to show that if e is large enought, then f(X) can be expressed as a nonzero constant times a monic polynomial in X_{r}, whose coefficients can be expressed as polynomials in x′_{1}, …, x′_{r−1} with coefficients in k. This is immediate when r = 1, suitably interpreted, and so we suppose that r ≥ 2.

This corresponds to part (a) of Lemma 13.2 on p286 of [92], due to Nagata. More precisely, this formulation is mentioned at the beginning of the proof in [92].

If α = (α_{1}, …, α_{r}) is a multi-index of length r, then the corresponding monomial X^{α} may be expressed as

X^{α} = X_{1}^{α_{1}} ⋯ X_{r}^{α_{r}} = ( ∏_{j=1}^{r−1}(x′_{j} + X_{r}^{e^{j}})^{α_{j}}) X_{r}^{α_{r}}. (24.7.3)

This can be expanded into a polynomial in X_{r} with coefficients that are polynomials in x′_{1}, …, x′_{r−1}. The term of highest degree in X_{r} is of the form

X_{r}^{d(α,e)}, (24.7.4)

where

d(α, e) = ∑_{j=1}^{r−1} α_{j} e^{j} + α_{r}. (24.7.5)

Of course, f(X) may be expressed as a polynomial in x′_{1}, …, x′_{r−1} and X_{r}, which may be considered as a polynomial in X_{r} whose coefficients are polynomials in x′_{1}, …, x′_{r−1}. Indeed, f(X) is by definition a linear combination of monomials as in (24.7.3) with coefficients in k. Because f(X) is not constant, by hypothesis, the coefficient of (24.7.3) in f(X) is nonzero for at least one nonzero α.

Suppose that max(α_{1}, …, α_{r−1}, α_{r}) < e (24.7.6)

for each of the finitely many multi-indices α for which the coefficient of X^{α} in f(X) is not zero. This implies that

(24.7.7) the corresponding degrees d(α, e) are distinct,

because (24.7.5) is the expansion of d(α, e) with respect to the base e, as in [92]. In particular, there is exactly one such multi-index

α^{max} (24.7.8)

corresponding to the maximal value of d(α, e). Note that

d(α^{max}, e) > 0, (24.7.9)

because d(α, e) > 0 when α ≠ 0.

It follows that f(X) can be expressed as a polynomial in X_{r} whose coefficients are polynomials in x′_{1}, …, x′_{r−1}, where the term of highest degree in X_{r} is a nonzero constant times X_{r}^{d(α^{max},e)}. (24.7.10)

This may also be considered as a nonzero constant times a monic polynomial in X_{r}, whose coefficients are polynomials in x′_{1}, …, x′_{r−1}.

An related statement will be discussed in Section 24.14.

24.8 Noether’s normalization theorem

Let k be a field, and let A be a commutative algebra over k with a multiplicative identity element. We may as well suppose that k is a subalgebra of A, with the multiplicative identity element of k as the multiplicative identity element of A.

Suppose that

(24.8.1) A is generated by k and finitely many elements x_{1}, …, x_{n}

for some positive integer n, as an algebra over k. If x_{1}, …, x_{n} are algebraically independent over k, then we can stop.

Otherwise, x_{1}, …, x_{n} satisfy a nontrivial polynomial equation with coefficients in k, as in Section 24.4. Let e be a positive integer, and put

w_{j} = x_{j} − x_{n}^{e^{j}} (24.8.2)

for j = 1, …, n − 1. Consider the subalgebra

k[w_{1}, …, w_{n−1}] (24.8.3)

of A generated by k and w_{1}, …, w_{n−1}. If e is large enough, then

(24.8.4) x_{n} satisfies a monic polynomial equation with coefficients in k[w_{1}, …, w_{n−1}].

This follows from the remarks in the previous section.

This means that

(24.8.5) x_{n} is integral over k[w_{1}, …, w_{n−1}],

as in Section 16.3. If n = 1, then (24.8.3) is interpreted as being equal to k. It follows that (24.8.6) A is integral over k[w_{1}, …, w_{n−1}],

as in Section 16.5.

We can repeat the process as needed until we get y_{1}, …, y_{r} ∈ A for some r ≤ n such that

(24.8.7) y_{1}, …, y_{r} are algebraically independent over k

and (24.8.8) A is integral over k[y_{1}, …, y_{r}],

the subalgebra of A generated by k and y_{1}, …, y_{r}. This corresponds to Theorem 6 on p260 of [177]. This also uses the transitivity property of integrality discussed in Section 16.5.

This statement corresponds to Exercise 16 on p69 of [8] as well, and we shall say more about this in Section 24.14.

Suppose that A is an integral domain, and let Q_{A} be the corresponding field of fractions, as usual. Note that the subfield k(x_{1}, …, x_{n}) of Q_{A} generated by k and x_{1}, …, x_{n} is the same as Q_{A}. Let k(y_{1}, …, y_{r}) be the subfield of Q_{A} generated by k and y_{1}, …, y_{r}.

Observe that (24.8.9) Q_{A} is algebraic over k(y_{1}, …, y_{r}),

because of (24.8.8). This means that

(24.8.10) y_{1}, …, y_{r} is a transcendence basis for Q_{A}.

Of course, (24.8.9) implies that

(24.8.11) Q_{A} is a finite extension of k(y_{1}, …, y_{r}),

because Q_{A} is finitely generated as an extension of k(y_{1}, …, y_{r}), as in Section 19.9.

24.9 Some remarks about some extensions

Let k be a field of positive characteristic p, and let K be a field that contains k as a subfield. Also let I be a nonempty set, and let {a_{j}}_{j∈I} be a family of elements of K indexed by I. If q = p^{n} for some positive integer n and

(24.9.1) {a_{j}^{q}}_{j∈I} is algebraically independent over k,

then (24.9.2) {a_{j}}_{j∈I} is algebraically independent over k.

Indeed, if the a_{j}’s satsify a nontrivial polynomial equation with coefficients in k, then one can get a nontrivial polynomial equation for the a_{j}^{q}’s by taking the qth power. This uses a remark in Section 20.15.

Similarly, let k′ be a subfield of K that contains k, and suppose that

(k′)^{q} ⊆ k. (24.9.3)

If (24.9.2) holds, then

(24.9.4) {a_{j}}_{j∈I} is algebraically independent over k′,

for essentially the same reasons as before.

Let k_{00} be a field of positive characteristic p, and let k_{00}(y_{1}, …, y_{r}) be a field that contains k_{00} as a subfield, and which is generated by k_{00} and elements y_{1}, …, y_{r} that are algebraically independent over k_{00}. Also let L′ be a finite extension of k_{00}(y_{1}, …, y_{r}) that is purely inseparable over k_{00}(y_{1}, …, y_{r}), as in Section 21.2. Remember that L′ is generated by k_{00}(y_{1}, …, y_{r}) and finitely many elements w_{1}, …, w_{l} of L′, as in Section 19.9. This implies that there is a positive integer n such that if q = p^{n}, then

(L′)^{q} ⊆ k_{00}(y_{1}, …, y_{r}). (24.9.5)

More precisely, it is enough to take n large enough so that

w_{1}^{q}, …, w_{l}^{q} ∈ k_{00}(y_{1}, …, y_{r}). (24.9.6)

Because of (24.9.6), each of w_{1}^{q}, …, w_{l}^{q} may be expressed as rational functions of y_{1}, …, y_{r} with coefficients in k_{00}. Let E_{00} be a nonempty finite subset of k_{00} that contains the coefficients of such rational functions. Let us take L′′ to be an extension of L′ with the following properties. First, L′′ contains qth roots of the elements of E_{00}. Second, there are z_{1}, …, z_{r} ∈ L′′ such that

z_{j}^{q} = y_{j} (24.9.7)

for each j = 1, …, r.

More precisely, we can take L′′ to be generated as a field by L′, the qth roots of the elements of E_{00}, and z_{1}, …, z_{r}. This implies that L′′ is a finite extension of L′, as in Section 19.9. Let k′_{00} be the subfield of L′′ generated by k and the qth roots of the elements of E_{00}. Thus k′_{00} is a finite extension of k_{00}, and

(k′_{00})^{q} ⊆ k_{00}. (24.9.8)

It follows that

(24.9.9) z_{1}, …, z_{r} are algebraically independent over k′_{00},

as before.

By construction, L′′ = k′_{00}(z_{1}, …, z_{r}), (24.9.10)

the subfield of L′′ generated by k′_{00} and z_{1}, …, z_{r}. Consider the subalgebra

k′_{00}[z_{1}, …, z_{r}] (24.9.11)

of L′′, as an algebra over k′_{00}, generated by z_{1}, …, z_{r}. This is isomorphic to the algebra of formal polynomials in r distinct commuting indeterminates with coefficients in k′_{00}, because of (24.9.9). Of course, the field of fractions of (24.9.11) is isomorphic to (24.9.10). Note that

(k′_{00}[z_{1}, …, z_{r}])^{q} ⊆ k_{00}[y_{1}, …, y_{r}]. (24.9.12)

We also have that k′_{00}[z_{1}, …, z_{r}] is a unique factorization domain, as in Section 22.12. This implies that k′_{00}[z_{1}, …, z_{r}] is integrally closed, as mentioned in Section 16.3. Of course, k′_{00}[z_{1}, …, z_{r}] is integral over k_{00}[y_{1}, …, y_{r}], because of (24.9.12). It follows that

(24.9.13) k′_{00}[z_{1}, …, z_{r}] is the integral closure of k_{00}[y_{1}, …, y_{r}] in k_{00}(z_{1}, …, z_{r}).

One can check that

(24.9.14) k′_{00}[z_{1}, …, z_{r}] is finitely generated as a module over k_{00}[y_{1}, …, y_{r}].

This is related to part of the proof of Corollary 13.13 on p297 of [92]. This will be discussed further beginning in the next section, and in particular this part will be used in Section 24.12.

586 CHAPTER 24. MORE ON FIELD EXTENSIONS, 3

24.10 A result mentioned earlier

Let k_{00} be a field, and let A be a commutative associative algebra over k_{00} with a multiplicative identity element. We may as well suppose that k_{00} is a subalgebra of A, with the multiplicative identity element of k_{00} as the multiplicative identity element in A, as before. Suppose that A is an affine domain over k_{00}, as in Section 20.14, so that A is an integral domain that is finitely generated as an algebra over k_{00}. Let Q_{A} be the field of fractions of A, as usual.

Let L be a finite extension of Q_{A}, and let T be the integral closure of A in L, as in Section 16.5. A well-known result of Emmy Noether states that (24.10.1) T is finitely generated as a module over A, as mentioned in Section 20.14. This corresponds to Corollary 13.13 on p297 of [92], and we would like to start discussing the proof in this section.

Note that A is not asked to be integrally closed here, so that the case where L = Q_{A} is already interesting. Even in this case, other extension fields are considered in the argument, as mentioned in [92].

As in Section 24.8, there are finitely many elements y_{1}, …, y_{r} of A such that (24.10.2) y_{1}, …, y_{r} are algebraically independent over k_{00} and (24.10.3) A is integral over k_{00}[y_{1}, …, y_{r}], the subalgebra of A generated by k_{00} and y_{1}, …, y_{r}. Note that (24.10.4) every element of T is integral over k_{00}[y_{1}, …, y_{r}], because of (24.10.3), as in Section 16.5. Of course, every element of L that is integral over k_{00}[y_{1}, …, y_{r}] is automatically integral over A, and thus an element of T. It follows that (24.10.5) T is the integral closure of k_{00}[y_{1}, …, y_{r}] in L.

Suppose that we can show that (24.10.6) T is finitely generated as a module over k_{00}[y_{1}, …, y_{r}]. This would imply (24.10.1), because k_{00}[y_{1}, …, y_{r}] ⊆ A, by construction. This means that we can reduce to the case where A = k_{00}[y_{1}, …, y_{r}], (24.10.7) as in [92]. More precisely, if k_{00}(y_{1}, …, y_{r}) is the subfield of Q_{A} generated by k and y_{1}, …, y_{r}, then (24.10.8) Q_{A} is a finite extension of k_{00}(y_{1}, …, y_{r}), 24.11. REDUCING TO NORMAL EXTENSIONS 587 as in (24.8.11). This implies that (24.10.9) L is a finite extension of k_{00}(y_{1}, …, y_{r}), because L is a finite extension of Q_{A}, by hypothesis.

Remember that (24.10.2) means that k_{00}[y_{1}, …, y_{r}] is isomorphic to the algebra of formal polynomials in r distinct commuting indeterminates with coefficients in k_{00}, as in Section 24.4. Hilbert’s basis theorem implies that (24.10.10) k_{00}[y_{1}, …, y_{r}] is a Noetherian ring, as in Section 29.7. It follows that (24.10.11) every finitely-generated module over k_{00}[y_{1}, …, y_{r}] is Noetherian, as in Section 9.7.

Note that (24.10.12) k_{00}[y_{1}, …, y_{r}] is integrally closed. Indeed, (24.10.13) k_{00}[y_{1}, …, y_{r}] is a unique factorization domain, as in Section 22.12. This implies (24.10.12), as mentioned in Section 16.3.

If L is separable as an extension of Q_{A}, then one can get (24.10.6) from (24.10.10) and (24.10.12), as in Section 20.11. In particular, this holds when Q_{A} has characteristic 0, as in Sections 20.3 and 20.4.

24.11 Reducing to normal extensions

Let us continue with the same notation and hypotheses as in the previous section. Let L_{0} be another finite extension of Q_{A}, and let (24.11.1) T_{0} be the integral closure of k_{00}[y_{1}, …, y_{r}] in L_{0}. Suppose that we can show that (24.11.2) T_{0} is finitely generated as a module over k_{00}[y_{1}, …, y_{r}]. This would imply that (24.11.3) T_{0} is Noetherian as a module over k_{00}[y_{1}, …, y_{r}], as in (24.10.11). If L ⊆ L_{0}, (24.11.4) then T ⊆ T_{0}, (24.11.5) and (24.10.6) would follow from (24.11.3).

588 CHAPTER 24. MORE ON FIELD EXTENSIONS, 3

In fact, we can choose L_{0} so that (24.11.4) holds and (24.11.6) L_{0} is a finite normal extension of Q_{A}, as in Section 20.7. This means that we can reduce to the case where (24.11.7) L is a normal extension of Q_{A}, as on p297 of [92].

Put (24.11.8) L_{1} = {u ∈ L : θ(u) = u for every θ ∈ Gal(L/Q_{A})}, so that L_{1} is a subfield of L that contains Q_{A}, as in Section 21.5. Note that (24.11.9) L is separable over L_{1}, as in Section 21.6, because L is a finite extension of Q_{A}, and thus an algebraic extension of Q_{A}. If (24.11.7) holds and Q_{A} has positive characteristic, then (24.11.10) L_{1} is purely inseparable over Q_{A}, as in Section 21.7. If (24.11.7) holds and Q_{A} has characteristic 0, then L_{1} = Q_{A}, as in Section 20.8, because L is separable over Q_{A}.

Let (24.11.11) T_{1} be the integral closure of k_{00}[y_{1}, …, y_{r}] in L_{1}, and note that T_{1} is integrally closed, as in Section 16.5. We also have that (24.11.12) T is the integral closure of T_{1} in L. Indeed, every element of T is integral over T_{1}, because of (24.10.4). The integral closure of T_{1} in L is integral over k_{00}[y_{1}, …, y_{r}], and thus contained in T, as in Section 16.5.

Suppose that we can show that (24.11.13) T_{1} is finitely generated as a module over k_{00}[y_{1}, …, y_{r}]. This would imply that (24.11.14) T_{1} is Noetherian as a module over k_{00}[y_{1}, …, y_{r}], because of (24.10.11). Using this, it would follow that (24.11.15) T_{1} is Noetherian as a commutative ring, because k_{00}[y_{1}, …, y_{r}] ⊆ L_{1}, by construction.

Under these conditions, we would get that (24.11.16) T is finitely generated as a module over T_{1}, as in Section 20.11, because of (24.11.9), (24.11.12), and (24.11.15). This would imply (24.10.6), because of (24.11.13).

This corresponds to the last part of the proof of Corollary 13.13 on p297 of [92]. The argument to get (24.11.13) will be discussed in the next section.

24.12. THE PURELY INSEPARABLE CASE 589

24.12 The purely inseparable case

Let us continue with the same notation and hypotheses as in the previous two sections. More precisely, we use the reduction to the case where (24.10.7) holds, so that Q_{A} = k_{00}(y_{1}, …, y_{r}) (24.12.1) in particular. We may as well suppose that k_{00} has positive characteristic p, because the characteristic 0 case can be obtained as in Section 24.10.

Let L′ be a finite extension of (24.12.1) such that (24.12.2) L′ is purely inseparable over (24.12.1). Also let (24.12.3) T′ be the integral closure of k_{00}[y_{1}, …, y_{r}] in L′. We would like to show that (24.12.4) T′ is finitely generated as a module over k_{00}[y_{1}, …, y_{r}]. This would imply (24.11.13), because of (24.11.10).

Let L′′ be another finite extension of (24.12.1), and let (24.12.5) T′′ be the integral closure of k_{00}[y_{1}, …, y_{r}] in L′′. Suppose that we can show that (24.12.6) T′′ is finitely generated as a module over k_{00}[y_{1}, …, y_{r}]. This would imply that (24.12.7) T′′ is Noetherian as a module over k_{00}[y_{1}, …, y_{r}], because of (24.10.11). If L′ ⊆ L′′, (24.12.8) then T′ ⊆ T′′, (24.12.9) and (24.12.4) would follow from (24.12.7). Of course, this is the same as at the beginning of the previous section.

In fact, we can take L′′ to be as in Section 24.9. Remember that (24.12.10) L′′ is a finite extension of L′, and thus of (24.12.1), which could also be seen using (24.9.10). We also have that T′′ = k′_{00}[z_{1}, …, z_{r}] (24.12.11) in the earlier notation, as in (24.9.13). This means that (24.12.6) is the same as (24.9.14), which can be verified directly, as before.

24.13 Some remarks about nonzero polynomials

Let k be a field, and let X_{1}, …, X_{r} be r distinct commuting indeterminates for some positive integer r. Also let f(X) = f(X_{1}, …, X_{r}) be a formal polynomial in X_{1}, …, X_{r} with coefficients in k, and suppose that

f(X) ≠ 0 (24.13.1)

as an element of k[X_{1}, …, X_{r}]. If k has infinitely many elements, then it is well known that there is an x = (x_{1}, …, x_{r}) ∈ k^{r} (24.13.2)

such that f(x) = f(x_{1}, …, x_{r}) ≠ 0, (24.13.3)

as in Theorem 10 on p112 of [151].

If r = 1, then this follows from the fact that the number of roots of a polynomial in one variable in k is less than or equal to the degree of the polynomial. Suppose that r ≥ 2, and that the analogous statement holds for r − 1. We can express f(X) as

f(X_{1}, …, X_{r−1}, X_{r}) = ∑_{j=0}^{n} B_{j}(X_{1}, …, X_{r−1}) X_{r}^{j} (24.13.4)

for some nonnegative integer n, where B_{j}(X_{1}, …, X_{r−1}) is a formal polynomial in X_{1}, …, X_{r−1} with coefficients in k. We may as well suppose that

B_{n}(X_{1}, …, X_{r−1}) ≠ 0, (24.13.5)

by taking n to be as small as possible.

Using the induction hypothesis, we get that there are x_{1}, …, x_{r−1} ∈ k such that B_{n}(x_{1}, …, x_{r−1}) ≠ 0. (24.13.6)

This implies that

f(x_{1}, …, x_{r−1}, X_{r}) = ∑_{j=0}^{n} B_{j}(x_{1}, …, x_{r−1}) X_{r}^{j} (24.13.7)

is a nonzero formal polynomial in X_{r} with coefficients in k. It follows that there is an x_{r} ∈ k such that (24.13.3) holds, as before.

The same conclusion holds when the degree of f(X_{1}, …, X_{r}) as a formal polynomial in X_{l} is strictly less than the number of elements of k for each l = 1, …, r, as in Exercise 2 on p113 of [151].

Suppose now that r ≥ 2, and that f(X) is homogeneous of degree d ≥ n, so that (24.13.8) B_{j}(X_{1}, …, X_{r−1}) is homogeneous of degree d − j for each j, as a formal polynomial in X_{1}, …, X_{r−1} with coefficients in k. It follows that

f(X_{1}, …, X_{r−1}, 1) = ∑_{j=0}^{n} B_{j}(X_{1}, …, X_{r−1}) ≠ 0, (24.13.9)

as a formal polynomial in X_{1}, …, X_{r−1} with coefficients in k. Under the same conditions as before, we get that there are x_{1}, …, x_{r−1} ∈ k such that

f(x_{1}, …, x_{r−1}, 1) = ∑_{j=0}^{n} B_{j}(x_{1}, …, x_{r−1}) ≠ 0. (24.13.10)

24.14 Another lemma about polynomials

Let k be a field, let r be a positive integer, let X_{1}, …, X_{r} be r distinct commuting indeterminates, and put T = k[X_{1}, …, X_{r}], as in Section 24.7. Also let f(X) be a nonconstant element of T, let a_{1}, …, a_{r−1} be elements of k, and put

x′′_{j} = X_{j} − a_{j} X_{r} (24.14.1)

for j = 1, …, r−1. We would like to show that if k has infinitely many elements, then we can choose a_{1}, …, a_{r−1} so that f(X) can be expressed as a nonzero constant times a monic polynomial in X_{r}, whose coefficients can be expressed as polynomials in x′′_{1}, …, x′′_{r−1} with coefficients in k. This corresponds to part (c) of Lemma 13.2 on p286 of [92] and its proof, due to Noether. This is related to part of the argument outlined in Exercise 16 on p69 of [8] too, and we shall say more about that in a moment.

The statement is clear when r = 1, suitably interpreted, and so we may suppose that r ≥ 2. If α = (α_{1}, …, α_{r}) is a multi-index of length r, then

X^{α} = X_{1}^{α_{1}} ⋯ X_{r}^{α_{r}} = ( ∏_{j=1}^{r−1}(x′′_{j} + a_{j} X_{r})) X_{r}^{α_{r}}. (24.14.2)

Put |α| = ∑_{j=1}^{r} α_{j}, as in Section 4.3. We can expand (24.14.2) into a polynomial in X_{r} of degree less than or equal to |α| with coefficients that are polynomials in x′′_{1}, …, x′′_{r−1} with coefficients in k. The coefficient of X_{r}^{|α|} is equal to

∏_{j=1}^{r−1} a_{j}^{α_{j}}. (24.14.3)

If l is a nonnegative integer, then let f_{l}(X) be the part of f(X) that is homogeneous of degree l, consisting of the terms that are multiples of X^{α} with |α| = l. Using the remarks in the preceding paragraph, f_{l}(X) may be expressed as a polynomial in X_{r} of degree less than or equal to l with coefficients that are polynomials in x′′_{1}, …, x′′_{r−1} with coefficients in k. The coefficient of X_{r}^{l} is equal to f_{l}(a_{1}, …, a_{r−1}, 1). (24.14.4)

Let l_{0} be the largest nonnegative integer such that

f_{l_{0}}(X) ≠ 0. (24.14.5)

If k has infinitely many elements, then there are a_{1}, …, a_{r−1} ∈ k such that

f_{l_{0}}(a_{1}, …, a_{r−1}, 1) ≠ 0, (24.14.6)

as in the previous section. Note that

f(X) = ∑_{l=0}^{l_{0}} f_{l}(X). (24.14.7)

One can use this to express f(X) as f(a_{1}, …, a_{r−1}, 1) times a monic polynomial in X_{r} of degree l_{0} whose coefficients are polynomials in x′′_{1}, …, x′′_{r−1} with coefficients in k.

If k has infinitely many elements, then one can use this to get another proof of Noether’s normalization theorem, as in Section 24.8. In this argument, instead of (24.8.2), one can take w_{j} = x_{j} − a_{j} x_{n} (24.14.8)

with a_{j} ∈ k for j = 1, …, n − 1. This corresponds to Exercise 16 on p69 of [8]. More precisely, in this case one can choose y_{1}, …, y_{r} to be linear combinations of x_{1}, …, x_{n} with coefficients in k, as mentioned in [8].

24.15 Extending some homomorphisms

Let B be a commutative ring with a multiplicative identity element 1 = 1_{B}, and let A be a subring of B that contains 1_{B}. Also let f be a ring homomorphism from A into L such that f(1_{B}) is the multiplicative identity element 1_{L} in L. Suppose that (24.15.1) B is integral over A,

as in Section 16.5. Under these conditions, there is an extension of f to a ring homomorphism from B into L, as in Exercise 2 on p67 of [8].

Note that f(A) is an integral domain. This means that ker f is a proper prime ideal in A. Using (24.15.1), we get that there is a proper prime ideal I_{B} such that A ∩ I_{B} = ker f, (24.15.2)

as in Section 16.15. This corresponds to the hint mentioned in [8].

Of course, B/I_{B} is an integral domain too. Let q_{B} be the natural quotient mapping from B onto B/I_{B}. The kernel of the restriction of q_{B} to A is equal to ker f, because of (24.15.2). Thus f leads to an embedding of q_{B}(A) into L that sends q_{B}(1_{B}) to 1_{L}.

Let E be the field of fractions of B/I_{B}, and let k be the subfield of E generated by q_{B}(A), which corresponds to the field of fractions of q_{B}(A). The embedding of q_{B}(A) into L mentioned in the preceding paragraph has a unique extension to an embedding of k into L.

By hypothesis, every element of B is integral over A, as in Section 16.3. This implies that every element of B/I_{B} is integral over q_{B}(A). In particular,

(24.15.3) every element of B/I_{B} is algebraic over k,

as an element of E, as in Section 19.9. It follows that

(24.15.4) E is algebraic over k,

using some of the remarks in Section 19.9.

If L is algebraically closed, then the embedding of k into L mentioned earlier can be extended to an embedding of E into L, as in Section 19.13.

The composition of q_{B} with this embedding of E into L is a homomorphism from B into L that is equal to f on A, by construction.

Part VII

Some rings, modules, and polynomials Composition series and chain conditions

25.1 Composition series of modules

Let A be a ring with a multiplicative identity element e_{A}, and let M be a left or right module over A. A sequence {M_{j}}_{j=0}^{n} of submodules of M is said to be a chain if M = M_{0} ⊇ M_{1} ⊇ ⋯ ⊇ M_{n} = {0} (25.1.1)

and M_{j} ≠ M_{j−1} for each j ≥ 1, so that the inclusions are strict. The length of this chain is defined to be n.

A composition series of M is defined to be a maximal chain, so that no additional submodules may be inserted into the sequence of modules. This is equivalent to saying that each of the quotients

M_{j−1}/M_{j} (25.1.2)

is simple, in the sense that it has no nonzero proper submodules. This corresponds to a remark on p76 of [8], and on p72 of [92].

Let length(M) be the length of the shortest composition series of M, if there is one, and put length(M) = +∞ otherwise. This is called the length of M, as on p77 of [8], p72 of [92], and p104 of [177]. Note that a composition series of M is called a simple finite filtration of M in [177].

Suppose that {M_{j}}_{j=0}^{n} is a composition series of M of minimal length, and that N is a submodule over M. Put

N_{j} = N ∩ M_{j} (25.1.3)

596 CHAPTER 25. COMPOSITION SERIES AND CHAIN CONDITIONS

for each j = 0, 1, 2, …, n, so that N_{0} = N, N_{n} = {0}, and N_{j} ⊆ N_{j−1} when j ≥ 1. Note that N_{j−1}/N_{j} may be identified with a submodule of M_{j−1}/M_{j} for each j ≥ 1. Because M_{j−1}/M_{j} is simple, by hypothesis, we get that for each j ≥ 1, either N_{j−1}/N_{j} = M_{j−1}/M_{j}, (25.1.4) or N_{j−1}/N_{j} = {0}, so that N_{j} = N_{j−1}. (25.1.5)

This leads to a composition series of N, by dropping any repeated terms.

It follows that (25.1.6) length(N) ≤ length(M).

More precisely, if (25.1.5) holds for some j, then

(25.1.7) length(N) < length(M).

Otherwise, if (25.1.4) holds for each j ≥ 1, then one can check that N_{j} = M_{j} for each j, so that N = M. This corresponds to the first part of the proof of Proposition 6.7 on p77 of [8].

The second part of the proof of Proposition 6.7 on p77 of [8] states that if {M′_{j}}_{j=0}^{k} is any chain of submodules of M, then

k ≤ length(M). (25.1.8)

Indeed, (25.1.9) length(M′_{j}) < length(M′_{j−1})

for each j ≥ 1, as in (25.1.7). This implies (25.1.8), because M′_{0} = M and the length of M′_{k} = {0} is equal to 0. In particular, the length of any composition series of M is less than or equal to length(M), and thus it is equal to length(M), by the definition of length(M).

If a chain of submodules of M is not maximal, then one can insert an additional submodule into the chain to get a chain of submodules of larger length. This can only be done finitely many times, because of (25.1.8). It follows that any chain of submodules of M can be extended to a composition series of M, as in Proposition 6.7 on p77 of [8]. These statements correspond to parts of Theorem 2.13 on p72 of [92] as well.

25.2 More on composition series

Let A be a ring with a multiplicative identity element e_{A} again, and let M be a left or right module over A. It is well known that

(25.2.1) M has a composition series

if and only if

(25.2.2) the submodules of M satisfy both the ascending and descending chain conditions,

so that M is both Noetherian and Artinian. This corresponds to Proposition 6.8 on p77 of [8], and to the first part of Theorem 2.13 on p72 of [92]. The “only if” part follows easily from (25.1.8).

To get the “if” part, remember that every nonempty collection of submodules of M has a maximal element, as in Section 9.7. If M ≠ {0}, then one can use this to get a maximal proper submodule M_{1} of M. Similarly, if M_{1} ≠ {0}, then one can get a maximal proper submodule M_{2} of M_{1}. One can repeat the process as long as possible, to get finite or infinite decreasing sequence of submodules of M. The descending chain condition implies that this sequence is finite, so that one gets a composition series for M.

If M′ is a submodule of M, then (25.2.1) holds if and only if

(25.2.3) M′ and M/M′ have composition series.

This can be obtained from the equivalence of (25.2.1) and (25.2.2) and standard results about chain conditions, and it can also be verified more directly, as follows. If M′ and M/M′ have composition series, then it is easy to use them to get a composition series for M. If M has a composition series, then one can get a maximal chain of submodules of M with M′ as one of the submodules, as in the previous section. Part of such a chain is a composition series for M′, and the other part can be used to get a composition series for M/M′.

If (25.2.1) holds, then

(25.2.4) length(M) = length(M′) + length(M/M′),

as in Proposition 6.9 on p77 of [8]. This can be seen using the remarks in the preceding paragraph.

Proposition 6.10 on p78 of [8] says that for a vector space V over a field, the ascending descending chain conditions for linear subspaces of V are equivalent to each other, and to finite length, as well as finite dimension. Essentially the same arguments work for left and right vector spaces over a division ring. It is easy to see that finite dimension implies finite length, which implies both chain conditions, as before. If V does not have finite dimension, then it has an infinite sequence of linearly independent elements. One can use this to get infinite sequences of increasing or decreasing linear subspaces of V that are not eventually constant, as in [8].

Let M_{1}, …, M_{n} be finitely many all left or all right modules over A. If M_{l} has a composition series for each l, then one can check that

(25.2.5)  ⨁_{l=1}^{n} M_{l} has a composition series.

This can be verified directly, or obtained from the fact that (25.2.3) implies (25.2.1). This could also be obtained from the analogous statements for chain conditions. Using either of the first two types of argument, we get that

(25.2.6) length ( ⨁_{l=1}^{n} M_{l}) = ∑_{l=1}^{n} length(M_{l}).

Suppose that A has a composition series, as a left or right module over itself. If n is a positive integer, then the space A^{n} of n-tuples of elements of A has a composition series as a left or right module over A, as appropriate, as in the preceding paragraph. In fact,

(25.2.7) length(A^{n}) = n length(A),

as in (25.2.6). If

(25.2.8) M is a finitely-generated left or right module over A,

as appropriate, then M has a composition series. This follows from the fact that there is a module homomorphism from A^{n} onto M when M is generated by n elements.

Note that (25.2.1) implies (25.2.8), because the ascending chain condition is equivalent to the Noetherian property, as in Section 9.7. If M is not finitely generated as a left or right module over A, as appropriate, then there is an infinite sequence of strictly increasing finitely-generated submodules of M. This implies that there are finitely-generated submodules of M of arbitrarily large length.

25.3 A result about lengths

Let R be an integral domain with dimension one, as in Section 14.8. Also let K = K(R) be the corresponding field of fractions of R, and let M be a module over R. Suppose that (25.3.1) M is torsion-free over R,

so that if t ∈ R, v ∈ M, t, v ≠ 0, then t · v ≠ 0, as in Section 11.1.

Of course, S_{0} = R \ {0} (25.3.2)

is a multiplicatively closed set in R, and K is the same as S_{0}^{−1} R, as in Section 14.2. Let S_{0}^{−1} M be the module of fractions corresponding to M and S_{0} as in Section 12.2. This is a module over S_{0}^{−1} R, as before, which means that S_{0}^{−1} M is a vector space over K. Remember that

v ↦ v/1 (25.3.3)

is a homomorphism from M into S_{0}^{−1} M, as modules over R. This homomorphism is injective, because of (25.3.1), as before.

Under these conditions, the rank of M may be defined by

rank(M) = dim_{K}(S_{0}^{−1} M), (25.3.4)

the dimension of S_{0}^{−1} M as a vector space over K, as on p265 of [92]. If M is finitely generated as a module over R, then S_{0}^{−1} M is finitely generated as a module over K, so that rank(M) < ∞. (25.3.5)

Note that rank(M) > 0 when M ≠ {0}, because of the injectivity of (25.3.3).

If x ∈ R, then the ideal in R generated by x is often denoted (x), x R, or R x. Similarly, x M = {x · v : v ∈ M} (25.3.6)

is a submodule of M. Observe that

(25.3.7) R/(x R) has dimension zero

when x ≠ 0, in the sense of Section 14.8.

Suppose from now on in this section that

(25.3.8) R is Noetherian.

This implies that (25.3.9) R/(x R) is Noetherian

as a ring for every x ∈ R, as mentioned in Section 18.5. If x ≠ 0, then we get that (25.3.10) R/(x R) is Artinian

as a ring, because of (25.3.7), as in Section 18.9.

Equivalently, if x ≠ 0, then R/x R is Noetherian and Artinian as a module over itself, and thus as a module over R, as in Section 18.5. This means that

length(R/(x R)) < ∞, (25.3.11)

with R/(x R) considered as a module over itself or R, as in the previous section. Of course, the length of R/(x R) is the same when R/(x R) is considered as a module over itself or over R, because the submodules of R/(x R) are the same in both cases.

Note that R is not Artinian as a ring, because R has dimension 1, by hypothesis, as in Section 18.9. More precisely, Artin rings have dimension zero, as in Section 18.7. In fact, an integral domain is Artinian only when it is a field. This means that the length of R as a module over itself is infinite, as in the previous section.

Lemma 11.12 on p265 of [92] states that

(25.3.12) length(M/(x M)) ≤ rank(M) length(R/(x R)),

with equality when M is finitely generated as a module over R. It seems to be implicit here that x ≠ 0, and anyway the x = 0 case is not very interesting. Of course, if x is invertible in R, then R/(x R) = M/(x M) = {0}, and so we shall suppose that x is not invertible.

If one takes M = K, considered as a module over R, then M/(x M) = {0} when x ≠ 0, and the rank of M is one, because S_{0}^{−1} M = K. This shows that the inequality in (25.3.12) may be strict when M is not finitely generated as a module over R, as in [92].

25.4 Some initial steps

Let us continue with the same notation and hypotheses as in the previous section, and put r = rank(M). (25.4.1)

We may as well suppose that this is positive and finite. We also suppose at first that (25.4.2) M is finitely generated as a module over R,

as in [92].

Let us identify M with a submodule of S_{0}^{−1} M, as a module over R, using the homomorphism (25.3.3). We can find

m_{1}, …, m_{r} ∈ M (25.4.3)

such that

(25.4.4) m_{1}, …, m_{r} is a basis for S_{0}^{−1} M, as a vector space over K,

as in [92]. Indeed, we can start with any basis for S_{0}^{−1} M, as a vector space over K, and multiply the elements of the basis by suitable nonzero elements of R to get a basis consisting of elements of M.

Let R^{r} be the space of r-tuples of elements of R, considered as a module over R with respect to coordinatewise addition and scalar multiplication, as usual. Consider the mapping α from R^{r} into M defined by

α(y) = ∑_{l=1}^{r} y_{l} · m_{l}, (25.4.5)

as in [92]. This is a homomorphism from R^{r} into M, as modules over R.

Using α, we get a homomorphism S_{0}^{−1} α from S_{0}^{−1} T^{r} into S_{0}^{−1} M, as modules over S_{0}^{−1} R, as in Section 12.2. We may identify S_{0}^{−1} R^{r} with (S_{0}^{−1} R)^{r} = K^{r}, as a module over S_{0}^{−1} R = K, so that S_{0}^{−1} α corresponds to a linear mapping from K^{r} into S_{0}^{−1} M, as vector spaces over K. We may also indentify R^{r} with a submodule of K^{r}, as a module over R. Using these identifications, S_{0}^{−1} α is the same as α on R^{r}, and

(S_{0}^{−1}α)(y) = ∑_{l=1}^{r} y_{l} · m_{l} (25.4.6)

for each y ∈ K^{r}.

Note that

(25.4.7) S_{0}^{−1} α is an isomorphism from K^{r} onto S_{0}^{−1} M,

as vector spaces over K, because of (25.4.4). In particular,

(25.4.8) α is injective on R^{r}, as mentioned in [92]. If we identify R^{r} with a submodule of M, as a module over R, then we can put

N = M/R^{r} = M/(α(R^{r})), (25.4.9)

considered as a module over R, as in [92].

25.5 Some induced homomorphisms

Let us continue with the same notation and hypotheses as in the previous two sections. Of course, α(x R^{r}) = x α(R^{r}) ⊆ x M. (25.5.1)

Thus we get an induced homomorphism \overline{α} from R^{r}/(x R^{r}) into M/(x M), as modules over R, as on p266 of [92]. More precisely, let q_{R^{r}} be the natural quotient homomorphism from R^{r} onto R^{r}/(x R^{r}), and let q_{M} be the natural quotient homomorphism from M onto M/(x M). By definition, \overline{α} is the unique homomorphism from R^{r}/(x R^{r}) into M/(x M) such that

\overline{α} ∘ q_{R^{r}} = q_{M} ∘ α. (25.5.2)

Let q = q_{M,N} be the natural quotient mapping from M onto N, and note that q(x M) = x N. (25.5.3)

This leads to a homomorphism \overline{q} from M/(x M) onto N/(x N), as modules over R. This corresponds to a mapping implicitly indicated on p266 of [92]. More precisely, \overline{q} is the unique homomorphism from M/(x M) onto N/(x N) such that

\overline{q} ∘ q_{M} = q_{N} ∘ q. (25.5.4)

Observe that

\overline{q} ∘ \overline{α} ∘ q_{R^{r}} = \overline{q} ∘ q_{M} ∘ α = q_{N} ∘ q ∘ α = 0. (25.5.5)

This implies that \overline{q} ∘ \overline{α} = 0, (25.5.6)

so that \overline{α}(R^{r}/(x R^{r})) ⊆ ker \overline{q}. (25.5.7)

In fact, one can check that

ker \overline{q} = \overline{α}(R^{r}/(x R^{r})), (25.5.8)

as indicated on p266 of [92]. Indeed,

ker(\overline{q} ∘ q_{M}) = ker(q_{N} ∘ q) = q^{−1}(x N) = α(R^{r}) + (x M). (25.5.9)

Using this, we get that ker \overline{q} = q_{M}(α(R^{r})). (25.5.10)

602 CHAPTER 25. COMPOSITION SERIES AND CHAIN CONDITIONS Equivalently, ker \overline{q} = \overline{α}(q_{R^{r}}(R^{r})), (25.5.11) by (25.5.2). This is the same as (25.5.8), by definition of q_{R^{r}}.

Similarly, ker(\overline{α} ∘ q_{R^{r}}) = ker(q_{M} ∘ α) = α^{−1}(x M). (25.5.12) This means that ker \overline{α} = q_{R^{r}}(α^{−1}(x M)). (25.5.13) Note that x R^{r} ⊆ α^{−1}(x M), (25.5.14) by (25.5.1). Thus the right side of (25.5.13) may be identified with α^{−1}(x M)/(x R^{r}). (25.5.15) This is isomorphic to (α(R^{r}) ∩ (x M))/α(x R^{r}) = (α(R^{r}) ∩ (x M))/(x α(R^{r})), (25.5.16) as a module over R, because of (25.4.8).

We can identify M with a submodule of S_{0}^{−1} M, as a module over R, and multiplication by x^{−1} is defined on S_{0}^{−1} M, because it is a vector space over K. It follows that (25.5.16) is isomorphic to ((x^{−1} α(R^{r})) ∩ M)/α(R^{r}), (25.5.17) as a module over R.

Let ξ be the homomorphism from M into itself, as a module over R, corresponding to multiplication by x. Similarly, let \overline{ξ} be the homomorphism from N into itself, as a module over R, corresponding to multiplication by x. Thus \overline{ξ} ∘ q = q ∘ ξ. (25.5.18)

Note that (25.5.17) may be considered as a submodule of N. In fact, (25.5.17) is the same as the kernel of \overline{ξ}. This implies that (25.5.19) ker \overline{α} is isomorphic to ker \overline{ξ}, as modules over R. This corresponds to a description of the kernel of \overline{α} on p266 of [92].

25.6 Some properties of M and N

Let us continue with the same notation and hypotheses as in the previous three sections. Remember that m_{1}, …, m_{r} ∈ M are as in Section 25.4, and note that m_{1}, …, m_{r} ∈ α(R^{r}), (25.6.1)

25.6. SOME PROPERTIES OF M AND N 603

by construction. If w ∈ S_{0}^{−1} M, then it is easy to see that there is a t ∈ R such that t ≠ 0 and t · w ∈ α(R^{r}), (25.6.2)

because of (25.4.4). It follows that there is an f ∈ R such that f ≠ 0 and

f M ⊆ R^{r}, (25.6.3)

because M is finitely generated as a module over R, by hypothesis, as in (25.4.2). This implies that f N = {0}, (25.6.4)

as on p266 of [92].

Thus N may be considered as a module over R/(f R). Remember that R/(f R) has finite length as a module over itself, as in (25.3.11). Note that N is finitely generated as a module over R, because M is finitely generated, so that N is finitely generated as a module over R/(f R). It follows that

(25.6.5) N has finite length,

as a module over R/(f R), as in Section 25.2, and as mentioned on p266 of [92]. This is the same as saying that (25.6.5) holds when N is considered as a module over R.

Using (25.6.5), we get that ker \overline{ξ}, \overline{ξ}(N) = x N, and N/(x N) have finite length as modules over R, as in Section 25.2. In fact,

(25.6.6) length(N) = length(ker \overline{ξ}) + length(\overline{ξ}(N))

and (25.6.7) length N = length(x N) + length(N/(x N)),

as in (25.2.4). This implies that

(25.6.8) length(ker \overline{ξ}) = length(N/(x N)),

as on p266 of [92]. This means that

(25.6.9) length(ker \overline{α}) = length(N/(x N)),

because of (25.5.19).

Of course, x (M/(x M)) = x (N/(x N)) = {0}, (25.6.10)

so that M/(x M) and N/(x N) may be considered as modules over R/(x R). It is easy to see that M/(x M) and N/(x N) are finitely generated as modules over R/(x R), because M is finitely generated as a module over R, by hypothesis. Remember that R/(x R) has finite length as a module over itself or R, as in (25.3.11). This implies that

(25.6.11) M/(x M) and N/(x N) have finite length 604 CHAPTER 25. COMPOSITION SERIES AND CHAIN CONDITIONS as modules over R/(x R), as in Section 25.2. This means that they have finite length as modules over R, as before.

Observe that R^{r}/(x R^{r}) is isomorphic to (R/(x R))^{r}, which has finite length, as in Section 25.2. In fact, (25.6.12) length(R^{r}/(x R^{r})) = length((R/(x R))^{r}) = r length(R/(x R)), where the second step is as in (25.2.7).

We also have that the kernel and image of \overline{α} have finite length, with (25.6.13) length(R^{r}/(x R^{r})) = length(ker \overline{α}) + length(\overline{α}(R^{r}/(x R^{r}))), as in (25.2.4). This means that (25.6.14) r length(R/(x R)) = length(N/(x N)) + length(\overline{α}(R^{r}/(x R^{r}))), because of (25.6.9) and (25.6.12).

Remember that \overline{q} is a homomorphism from M/(x M) onto N/(x N) whose kernel is equal to the image of \overline{α}, as in Section 25.5. It follows that (25.6.15) length(M/(x M)) = length(\overline{α}(R^{r}/(x R^{r}))) + length(N/(x N)).

Combining (25.6.14) and (25.6.15), we obtain that (25.6.16) length(M/(x M)) = r length(R/(x R)). This implies that equality holds in (25.3.12), because of (25.4.1).

25.7 Not necessarily finitely generated modules

Let us continue with the same notation and hypotheses as in Section 25.3 and the beginning of Section 25.4, except that now we do not ask that M be finitely generated as a module over R, as in (25.4.2). Let M′ be a submodule of M, and suppose that (25.7.1) M′ is finitely generated as a module over R. If S_{0}^{−1} M′ is the module of fractions corresponding to M′ and S_{0}, then S_{0}^{−1} M′ corresponds to a linear subspace of S_{0}^{−1} M, as a vector space over K, as in Section 12.2. In particular, (25.7.2) rank(M′) = dim_{K}(S_{0}^{−1} M′) ≤ rank(M) = r.

Note that M′ is torsion-free over R, because M is, by hypothesis, as in (25.3.1). It follows that (25.7.3) length(M′/(x M′)) = rank(M′) length(R/(x R)), as in (25.3.12) in the finitely-generated case. This implies that

(25.7.4) length(M′/(x M′)) ≤ r length(R/(x R)),

because of (25.7.2).

Remember that q_{M} is the natural quotient mapping from M onto M/(x M), as in Section 25.5, and let q_{M′} be the natural quotient mapping from M′ onto x M′. Of course, x M′ ⊆ x M, so that the obvious inclusion mapping ι_{M′} from M′ into M induces a homomorphism \overline{ι}_{M′} from M′/(x M′) into M/(x M), as modules over R, with \overline{ι}_{M′} ∘ q_{M′} = q_{M} ∘ ι_{M′}. (25.7.5)

Observe that

(25.7.6) length(\overline{ι}_{M′}(M′/(x M′))) ≤ length(M′/(x M′)),

so that (25.7.7) length(\overline{ι}_{M′}(M′/(x M′))) ≤ r length(R/(x R)),

by (25.7.4). Equivalently,

(25.7.8) length(q_{M}(M′)) ≤ r length(R/(x R)),

because q_{M}(M′) = \overline{ι}_{M′}(M′/(x M′)).

It is easy to see that every finitely-generated submodule of M/(x M) may be expressed as q_{M}(M′) for some finitely-generated submodule M′ of M. If M/(x M) is finitely generated as a module over R, then (25.3.12) follows from (25.7.8).

If M/(x M) is not finitely generated as a module over R, then it contains finitely-generated submodules of arbitrarily large length, as in Section 25.2. This would contradict (25.7.8). This corresponds to some remarks on p267 of [92].

25.8 The Krull–Akizuki theorem

Let R be a Noetherian integral domain with dimension one, as in Section 14.8. Also let K = K(R) be the corresponding field of fractions, and let L be a finite extension of K. Remember that R may be identified with a subring of K, as in Section 14.2, and let S be a subring of L. If

R ⊆ S, (25.8.1)

then the Krull–Akizuki theorem states that

(25.8.2) S is Noetherian

and (25.8.3) the dimension of S is less than or equal to one,

as in Theorem 11.13 on p267 of [92]. In particular, the integral closure of R in L is Noetherian, as mentioned in Section 20.12.

To show this, let J be a nonzero ideal in S, and let b be a nonzero element of J. Note that b is algebraic over K, as in Section 19.9, so that

a_{n} b^{n} + ⋯ + a_{1} b + a_{0} = 0 (25.8.4)

for some positive integer n and a_{0}, a_{1}, …, a_{n} ∈ K, with a_{n} ≠ 0. We may as well suppose that a_{0} ≠ 0, because otherwise we can reduce to that case. We may also suppose that a_{0}, a_{1}, …, a_{n} ∈ R, (25.8.5)

by multiplying these coefficients by a suitable element of R. It follows that

a_{0} ∈ J ∩ R, (25.8.6)

as mentioned on p268 of [92], and where we have used the argument in the proof of Lemma 4.16 on p130 of [92].

Of course, J may be considered as a module over R, and a_{0} S is an ideal in S, which may be considered as a module over R as well. Note that

a_{0} S ⊆ J, (25.8.7)

by (25.8.6). We would like to show that

(25.8.8) J/(a_{0} S) has finite length,

as a module over R. In fact, we shall show that

(25.8.9) S/(a_{0} S) has finite length,

as a module over R. This will imply (25.8.8), as in Sections 25.1 and 25.2, because J/(a_{0} S) may be considered as a submodule of S/(a_{0} S), as a module over R.

Remember that S_{0} = R \ {0}, as in Section 25.3, and that S_{0}^{−1} S is the module of fractions corresponding to S, as a module over R, as in Section 12.2. Note that S may be considered as a submodule of L, as a module over R, and in particular S is torsion-free over R. The module of fractions S_{0}^{−1} L corresponding to L, as a module over R, may be identified with L, as a vector space over K. This means that S_{0}^{−1} S may be identified with a linear subspace of L, as a vector space over K. It follows that

(25.8.10) rank(S) = dim_{K}(S_{0}^{−1} S) ≤ dim_{K}L,

where the right side is finite, by hypothesis.

This implies (25.8.9), by the result in Section 25.3, with M = S and x = a_{0}. This works for any a_{0} ∈ R with a_{0} ≠ 0. Equivalently, this means that S/(a_{0} S) satisfies the ascending and descending chain conditions, as a module over R, as in Section 25.2. In particular, S/(a_{0} S) satisfies the ascending and descending chain conditions as a module over S. This is the same as saying that S/(a_{0} S) satisfies the ascending and descending chain conditions as a module over itself, so that it is Noetherian and Artinian as a ring.

It follows that (25.8.11) the dimension of S/(a_{0} S) is 0,

as in Section 18.7 and 18.9, at least if a_{0} is not invertible in S, so that S/(a_{0} S) ≠ {0}. This means that any proper prime ideal in S that contains a_{0} S is maximal. Thus any proper prime ideal in S that contains J is maximal when J ≠ S, because of (25.8.7). It is easy to get (25.8.3) from this.

Similarly, the ascending chain condition for ideals in S/(a_{0} S) implies the ascending chain condition for ideals in S that contain a_{0}. This implies the ascending chain condition for ideals in S that contain J. One can use this to get the ascending chain condition for all ideals in S, so that (25.8.2) holds.

Some examples of Nagata

26.1 Some commutative Noetherian rings

Let k be a commutative ring with a multiplicative identity element. Remember that if I is a proper prime ideal in k, then S_{I} = k \ I is multiplicatively closed in k, and put k_{I} = S_{I}^{−1} k, as in Section 12.4.

Suppose that k has the following two properties. First, (26.1.1) for every maximal proper ideal I in k, k_{I} is Noetherian. Second, (26.1.2) if x ∈ k and x ≠ 0, then there are only finitely many

maximal proper ideals in k that contain x.

Under these conditions, we would like to show that k is Noetherian, as in Exercise 9 on p85 of [8].

To do this, let L be an ideal in k, and let us show that L is finitely generated as an ideal in k. Of course, we may as well suppose that L ≠ 0. More precisely, we are going to choose a finitely-generated ideal L_{0} in k such that L_{0} ⊆ L, (26.1.3) and we are going to want to show that L_{0} = L. (26.1.4)

If I is a maximal proper ideal in k, then S_{I}^{−1} L_{0} may be considered as a submodule of S_{I}^{−1} L, as a module over k_{I}, as in Section 12.2. In order to get (26.1.4), it suffices to verify that S_{I}^{−1} L_{0} = S_{I}^{−1} L (26.1.5) for every maximal proper ideal I in k, as in Section 12.4.

26.1. SOME COMMUTATIVE NOETHERIAN RINGS 609

Similarly, S_{I}^{−1} L_{0}, S_{I}^{−1} L may be considered as submodules of k_{I}, as a module over itself. Note that (26.1.5) holds when S_{I}^{−1} L_{0} = k_{I}. (26.1.6)

Let x_{0} be a nonzero element of L. We are going to choose L_{0} so that x_{0} ∈ L_{0}. (26.1.7) This implies that (26.1.6) holds when x_{0} ∈ k \ I. (26.1.8)

There are only finitely many maximal proper ideals in k that contain x_{0}, as in (26.1.2). Let I_{1}, …, I_{r} (26.1.9) be the maximal proper ideals in k that contain L, and let I_{r+1}, …, I_{r+s} (26.1.10) be the maximal proper ideals in k that contain x_{0} and not L.

Let us choose x_{j} ∈ L \ I_{r+j} (26.1.11) for each j = 1, …, s. We are going to choose L_{0} so that x_{1}, …, x_{s} ∈ L_{0}. (26.1.12) This implies that (26.1.6) holds when I = I_{r+1}, …, I_{r+s}.

Observe that (26.1.13) S_{I_{i}}^{−1} L is finitely generated as an ideal in k_{I_{i}} for each i = 1, …, r, because k_{I_{i}} is Noetherian, as in (26.1.1). Using this, we can find finitely many elements x_{s+1}, …, x_{s+t} (26.1.14) of L such that for each i = 1, …, r, S_{I_{i}}^{−1} L is generated by the images of (26.1.14) under the natural mapping from k into k_{I_{i}}, as an ideal in k_{I_{i}}.

Let us now take L_{0} to be the ideal in k generated by x_{0}, x_{1}, …, x_{s}, x_{s+1}, …, x_{t}. (26.1.15) It is easy to see that S_{I_{i}}^{−1} L_{0} = S_{I_{i}} L (26.1.16) for each i = 1, …, r, by the remarks in the preceding paragraph. This implies that (26.1.5) holds for every maximal proper ideal I in k, so that (26.1.4) holds. as before.

26.2 Some sets of indeterminates

Let k be a commutative ring with a multiplicative identity element, and let V be a module over k. Also let I be a nonempty set, and let {T_{j}}_{j∈I} be a family of distinct commuting indeterminates indexed by I. If I_{0} is a nonempty subset of I, then the space V [{T_{j}}_{j∈I_{0}}] (26.2.1)

of formal polynomials in the T_{j}’s, j ∈ I_{0}, with coefficients in V may be defined as a module over k as in Section 24.1. More precisely, this may be considered as a module over k[{T_{j}}_{j∈I_{0}}], as before. This may be interpreted as being V when I_{0} = ∅.

Remember that a function α on I_{0} with values in the set Z_{+} ∪ {0} of nonnegative integers is said to be a multi-index associated to I_{0} if α(j) = 0 for all but finitely many j ∈ I_{0}, as in Section 24.1. The set Mult(I_{0}) of all multiindices associated to I_{0} is a commutative semigroup with respect to addition, as before. Every multi-index α associated to I_{0} can be extended to a multi-index associatied to I by putting α(j) = 0 when j ∈ I \ I_{0}. Using this,

(26.2.2) Mult(I_{0}) may be identified with a sub-semigroup of Mult(I).

Similarly,

(26.2.3) V [{T_{j}}_{j∈I_{0}}] may be identified with a submodule of V [{T_{j}}_{j∈I}],

as a module over k.

Let A be an algebra over k in the strict sense, so that A[{T_{j}}_{j∈I_{0}}] may be considered as an algebra in the strict sense over k as well, as in Section 24.1. This may be identified with a subalgebra of A[{T_{j}}_{j∈I}], as in the preceding paragraph.

In particular,

(26.2.4) k[{T_{j}}_{j∈I_{0}}] may be identified with a subalgebra of k[{T_{j}}_{j∈I}],

as an algebra over k. Using this, V [{T_{j}}_{j∈I}] may be considered as a module over k[{T_{j}}_{j∈I_{0}}], and (26.2.3) holds in this sense too.

26.3 Pairs of complementary subsets

Let us continue with the same notation and hypotheses as in the previous section, and suppose now that I_{1}, I_{2} are disjoint nonempty subsets of I such that

I_{1} ∪ I_{2} = I. (26.3.1)

Note that (26.3.2) Mult(I_{1}) × Mult(I_{2})

is a commutative semigroup with respect to coordinatewise addition, which is isomorphic in a simple way to Mult(I).

Consider the space (V [{T_{j}}_{j∈I_{1}}])[{T_{l}}_{l∈I_{2}}] (26.3.3)

of formal polynomials in the T_{l}’s, l ∈ I_{2}, with coefficients in V [{T_{j}}_{j∈I_{1}}]. There is a one-to-one correspondence between V [{T_{j}}_{j∈I}] and (26.3.3), in which a polynomial in the T_{j}’s, j ∈ I, is expressed as a polynomial in the T_{j}’s, j ∈ I_{2}, whose coefficients are polynomials in the T_{j}’s, j ∈ I_{1}. A simpler version of this was mentioned in Section 21.12.

More precisely, remember that V [{T_{j}}_{j∈I}] may be defined as the space c_{00}(Mult(I), V) of V -valued functions on Mult(I) with finite support, as in Section 24.1. Similarly, V [{T_{j}}_{j∈I_{1}}] may be defined as the space

c_{00}(Mult(I_{1}), V) (26.3.4)

of V -valued functions on Mult(I_{1}) with finite support. In the same way, (26.3.3) may be defined as the space

c_{00}(Mult(I_{2}), V [{T_{j}}_{j∈I_{1}}]) (26.3.5)

of functions on Mult(I_{2}) with values in V [{T_{j}}_{j∈I_{1}}] and finite support. This is the same as the space

(26.3.6) c_{00}(Mult(I_{2}), c_{00}(Mult(I_{1}), V))

of functions on Mult(I_{2}) with values in (26.3.4) an finite support. It is easy to see that this corresponds to c_{00}(Mult(I), V) in a simple way, using the isomorphism between Mult(I) and (26.3.2) mentioned earlier.

This defines an isomorphism between V [{T_{j}}_{j∈I}] and (26.3.3), as modules over k, as before. One can get an analogous isomorphism between V [{T_{j}}_{j∈I}] and (V [{T_{l}}_{l∈I_{2}}])[{T_{j}}_{j∈I_{1}}], (26.3.7)

as modules over k.

Similarly, we get an isomorphism between A[{T_{j}}_{j∈I}] and

(A[{T_{j}}_{j∈I_{1}}])[{T_{l}}_{l∈I_{2}}], (26.3.8)

as algebras over k in the strict sense. There is an analogous isomorphism between A[{T_{j}}_{j∈I}] and (A[{T_{l}}_{l∈I_{2}}])[{T_{j}}_{j∈I_{1}}], (26.3.9)

as algebras over k in the strict sense.

26.4 Some ideals and homomorphisms

Let us continue with the same notation and hypotheses as in the previous two sections. In particular, we can identify Mult(I_{2}) with a sub-semigroup of Mult(I), as before. If f(T) = ∑_{α∈Mult(I)} f_{α} T^{α} is a formal polynomial in the T_{j}’s, j ∈ I, with coefficients in V, then put

ϕ_{2}(f(T)) = ∑_{α∈Mult(I_{2})} f_{α} T^{α}, (26.4.1)

which is a formal polynomial in the T_{j}’s, j ∈ I_{2}, with coefficients in V. This defines a homomorphism ϕ_{2} = ϕ_{2,V} (26.4.2)

from V [{T_{j}}_{j∈I}] onto V [{T_{j}}_{j∈I_{2}}], as modules over k.

Similarly, we can define a mapping

ϕ_{2} = ϕ_{2,A} (26.4.3)

from A[{T_{j}}_{j∈I}] onto A[{T_{j}}_{j∈I_{2}}] in this way. One can check that

(26.4.4) ϕ_{2,A} is an algebra homomorphism.

This uses the fact that if α, β are multi-indices associated to I and

α + β = 0 on I_{1}, (26.4.5)

then α, β = 0 on I_{1}.

The kernel of ϕ_{2} consists of formal polynomials f(T) in the T_{j}’s, j ∈ I, such that f_{α} = 0 for every α ∈ Mult(I) that corresponds to a multi-index associated to I_{2}, which means that α = 0 on I_{1}. (26.4.6)

This is the same as saying that f_{α} ≠ 0 only when

α(j) ≠ 0 for some j ∈ I_{1}. (26.4.7)

This is a two-sided ideal in A[{T_{j}}_{j∈I}], as in Section 21.15.

Let us now take A = k, so that we get a homomorphism ϕ_{2} = ϕ_{2,k} from k[{T_{j}}_{j∈I}] onto k[{T_{j}}_{j∈I_{2}}], as algebras over k. This homomorphism takes T_{j} to itself when j ∈ I_{2}, and to 0 when j ∈ I_{1}.

Suppose that k is an integral domain, so that k[{T_{j}}_{j∈I_{2}}] is an integral domain, as in Section 24.5. This implies that the kernel of ϕ_{2} = ϕ_{2,k} is a proper prime ideal in k[{T_{j}}_{j∈I}].

26.5 Some more ideals and homomorphisms

Let k_{0} be a commutative ring with a multiplicative identity element, let V_{0} be a module over k_{0}, let I_{0} be a nonempty set, and let {T_{j}}_{j∈I_{0}} be a family of distinct commuting indeterminates indexed by I_{0}. If

f(T) = ∑_{α∈Mult(I_{0})} f_{α} T^{α} (26.5.1)

26.5. SOME MORE IDEALS AND HOMOMORPHISMS 613 is a formal polynomial in the T_{j}’s, j ∈ I_{0}, with coefficients in V_{0}, then put ϕ_{0}(f(T)) = f(0) = f_{0}, (26.5.2) where the second step is as in Section 24.2. This defines a homomorphism ϕ_{0} = ϕ_{0,V_{0}} (26.5.3) from V_{0}[{T_{j}}_{j∈I_{0}}] onto V_{0}, as modules over k_{0}. This may be interpreted as being the analogue of (26.4.1) with k = k_{0}, V = V_{0}, I = I_{1} = I_{0}, and I_{2} = ∅. If A_{0} is an algebra over k_{0} in the strict sense, then we get an algebra homomorphism ϕ_{0} = ϕ_{0,A_{0}} (26.5.4) from A_{0}[{T_{j}}_{j∈I_{0}}] onto A_{0}.

Let k be a commutative ring with a multiplicative identity element, let V be a module over k, let I be a nonempty set, and let {T_{j}}_{j∈I} be a family of distinct commuting indeterminates indexed by I, as in Section 26.2. Also let I_{1}, I_{2} be disjoint nonempty subsets of I such that I_{1} ∪ I_{2} = I, as in Section 26.3. Suppose for the moment that k_{0} = k and V_{0} = V [{T_{l}}_{l∈I_{2}}]. (26.5.5) Remember that V [{T_{j}}_{j∈I}] is isomorphic to V_{0}[{T_{j}}_{j∈I_{1}}], (26.5.6) as modules over k, in a simple way, as in Section 26.3. If I_{0} = I_{1}, then the homomorphism ϕ_{2,V} as in (26.4.2) corresponds to ϕ_{0,V_{0}} as in the preceding paragraph, using the isomorphism just mentioned.

Let k_{0} be any commutative ring with a mutliplicative identity element again, so that we get a homomorphism ϕ_{0} = ϕ_{0,k_{0}} from k_{0}[{T_{j}}_{j∈I_{0}}] onto k_{0}, as algebras over k_{0}, as before. If k_{0} is an integral domain, then the kernel of this homomorphism is a proper prime ideal in k_{0}[{T_{j}}_{j∈I_{0}}].

Suppose for the moment again that k_{0} = k[{T_{l}}_{l∈I_{2}}], (26.5.7) and remember that k[{T_{j}}_{j∈I}] is isomorphic to k_{0}[{T_{j}}_{j∈I_{1}}], (26.5.8) as algebras over k, in a simple way, as in Section 26.3. If I_{0} = I_{1}, then the homomorphism ϕ_{2,k} in the previous section corresponds to ϕ_{0,k_{0}} here, using the isomorphism just mentioned.

26.6 Ideals, homomorphisms, and fractions

Let k_{0} be an integral domain, let I_{0} be a nonempty set, and let {T_{j}}_{j∈I_{0}} be a family of distinct commuting indeterminates indexed by I_{0}. Thus k_{0}[{T_{j}}_{j∈I_{0}}] is an integral domain, as in Section 24.5. Similarly, if Q_{k_{0}} is the field of fractions of k, then Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] is an integral domain. Of course, k_{0} may be identified with a subring of Q_{k_{0}}, as in Section 14.2. This means that k_{0}[{T_{j}}_{j∈I_{0}}] may be identified with a subring of Q_{k_{0}}[{T_{j}}_{j∈I_{0}}].

The field of fractions of Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] is the field Q_{k_{0}}({T_{j}}_{j∈I_{0}}) of formal rational functions in the T_{j}’s, j ∈ I_{0}, with coefficients in Q_{k_{0}}, as in Section 24.5. We may identify Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] with a subring of Q_{k_{0}}({T_{j}}_{j∈I_{0}}), as before. Thus k_{0}[{T_{j}}_{j∈I_{0}}] may be identified with a subring of Q_{k_{0}}({T_{j}}_{j∈I_{0}}) too.

Note that every element of Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] may be expressed as the product of an element of k_{0}[{T_{j}}_{j∈I_{0}}] and the reciprocal of a nonzero element of k_{0}. This means that every element of Q_{k_{0}}({T_{j}}_{j∈I_{0}}) may be expressed as the quotient of elements of k_{0}[{T_{j}}_{j∈I_{0}}], where the denominator is not zero. It follows that Q_{k_{0}}({T_{j}}_{j∈I_{0}}) may be considered as the field of fractions of k_{0}[{T_{j}}_{j∈I_{0}}] as well.

Let ϕ_{0,k_{0}} be the homomorphism from k_{0}[{T_{j}}_{j∈I_{0}}] onto k_{0}, as algebras over k_{0}, defined as in (26.5.2), and remember that the kernel of ϕ_{k_{0}} is a proper prime ideal in k_{0}[{T_{j}}_{j∈I_{0}}]. This implies that

S_{k_{0}} = k_{0}[{T_{j}}_{j∈I_{0}}] \ (ker ϕ_{0,k_{0}}) (26.6.1)

is multiplicatively closed in k_{0}[{T_{j}}_{j∈I_{0}}], as in Section 12.4. The corresponding ring of fractions S_{k_{0}}^{−1} k_{0}[{T_{j}}_{j∈I_{0}}] (26.6.2)

may be considered as a subring of Q_{k_{0}}({T_{j}}_{j∈I_{0}}), as in Section 14.2.

Similary, let ϕ_{0,Q_{k_{0}}} be the homomorphism from Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] onto Q_{k_{0}}, as algebras over Q_{k_{0}}, defined as in (26.5.2), and note that the kernel of ϕ_{0,Q_{k_{0}}} is a maximal proper ideal in Q_{k_{0}}[{T_{j}}_{j∈I_{0}}]. In particular, the kernel of ϕ_{0,Q_{k_{0}}} is a proper prime ideal in Q_{k_{0}}[{T_{j}}_{j∈I_{0}}], so that

S_{Q_{k_{0}}} = Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] \ (ker ϕ_{0,Q_{k_{0}}} ) (26.6.3)

is multiplicatively closed in Q_{k_{0}}[{T_{j}}_{j∈I_{0}}]. The corresponding ring of fractions

S_{Q_{k_{0}}}^{−1} Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] (26.6.4)

may be considered as a subring of Q_{k_{0}}({T_{j}}_{j∈I_{0}}) as well.

We may consider k_{0} as a subalgebra of k_{0}[{T_{j}}_{j∈I_{0}}], as in Section 24.1, and the restriction of ϕ_{0,k_{0}} to k_{0} is the identity mapping. Thus k_{0} \ {0} is contained in (26.6.1), so that Q_{k_{0}} is contained in (26.6.2). In fact, one can check that (26.6.2) and (26.6.4) are the same.

If I_{0} has only finitely many element, then

(26.6.5) Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] is Noetherian as a ring, by Hilbert’s basis theorem. This implies that

(26.6.6) S_{Q_{k_{0}}}^{−1} Q_{k_{0}}[{T_{j}}_{j∈I_{0}}] is Noetherian,

as in Section 14.6. It follows that

(26.6.7) S_{k_{0}}^{−1} k_{0}[{T_{j}}_{j∈I_{0}}] is Noetherian

under these conditions.

26.7 These examples of Nagata

Let k be a field, and let X_{1}, X_{2}, X_{3}, … be an infinite sequence of distinct commuting indeterminates. Thus

A = k[{X_{j}}_{j∈Z_{+}}] = k[X_{1}, X_{2}, X_{3}, …] (26.7.1)

is a commutative associative algebra over k with a multiplicative identity element, as in Section 24.1. More precisely, this is an integral domain, as in Section 24.5.

Let m_{1}, m_{2}, m_{3}, … be a strictly increasing sequence of nonnegative integers. One may take m_{1} = 0, and anyway m_{i} ≥ 1 when i ≥ 2. We would also like to have that m_{i+1} − m_{i} → ∞ as i → ∞. (26.7.2)

Note that this holds when

m_{i+1} − m_{i} > m_{i} − m_{i−1} (26.7.3)

for all i > 1. Let I_{i} = I(X_{m_{i}+1}, …, X_{m_{i+1}}) (26.7.4)

be the ideal in A generated by X_{m_{i}+1}, …, X_{m_{i+1}} for each i ≥ 1.

Let i be a positive integer, and let A_{i} be the algebra of formal polynomials in X_{j} with j ≤ m_{i} or j > m_{i+1} and coefficients in k. It is easy to see that there is a homomorphism ϕ_{i} from A onto A_{i}, as algebras over k, that sends X_{j} to itself when j ≤ m_{i} and j > m_{i+1}, and sends X_{j} to 0 when m_{i} < j ≤ m_{i+1}. In fact, ker ϕ_{i} = I_{i}, (26.7.5)

by construction. This corresponds to some of the remarks in Section 26.4. Note that A_{i} is an integral domain, as in Section 24.5. This implies that

(26.7.6) I_{i} is a proper prime ideal in A,

as before.

It follows that S_{i} = A \ I_{i} (26.7.7) is a multiplicatively closed set in A, as in Section 12.4. This means that

S = ⋂_{i=1}^{∞} S_{i} = A \ ( ⋃_{i=1}^{∞} I_{i}) (26.7.8)

is multiplicatively closed in A as well.

Thus S^{−1} A may be defined as a commutative ring with a multiplicative identity element, as in Section 12.1. More precisely, S^{−1} A is an integral domain, as in Section 14.2.

We would like to show that

(26.7.9) S^{−1} A is Noetherian

and that (26.7.10) the dimension of S^{−1} A is + ∞,

in the sense of Section 14.8, as in Exercise 4 on p126 of [8]. We shall continue with this in the next four sections.

26.8 A helpful fact about ideals in A

Let us continue with the same notation and hypotheses as in the previous section. Let L be an ideal in A, and suppose that

L ⊆ ⋃_{i=1}^{∞} I_{i}. (26.8.1)

We would like to show that

L ⊆ ⋃_{i=1}^{n} I_{i} (26.8.2)

for some positive integer n.

Suppose for the sake of a contradiction that (26.8.2) does not hold for any n. In particular, this means that L ≠ {0}, and we let a(X) be a nonzero element of L. Let i_{1} be a positive integer such that a(X) is a formal polynomial in X_{1}, …, X_{m_{i_{1}+1}} with coefficients in k.

If (26.8.2) does not hold with n = i_{1}, then there is a b(X) ∈ L such that

b(X) ∉ ⋃_{i=1}^{i_{1}} I_{i}. (26.8.3)

We also have that b(X) ∈ I_{i_{2}} (26.8.4)

for some i_{2} > i_{1}, because of (26.8.1).

If α, β are multi-indices associated to Z_{+} such that the coefficients of X^{α}, X^{β} in a(X), b(X), respectively, are nonzero, then it is easy to see that

α ≠ β, (26.8.5)

26.9. IDEALS IN S^{−1} A 617 because of (26.8.4). This implies that the coefficients of X^{α} and X^{β} in a(X) + b(X) (26.8.6) are both nonzero.

One can use this to get that a(X) + b(X) ∉ I_{i} (26.8.7) for any i. More precisely, this also uses (26.8.3) when i ≤ i_{1}, and the way that i_{1} was chosen for i > i_{1}.

This contradicts (26.8.1), because a(X) + b(X) ∈ L. (26.8.8) Thus (26.8.2) holds for some n.

It follows that L ⊆ I_{i} (26.8.9) for some i, as in Section 17.3. This also uses (26.7.6).

26.9 Ideals in S^{−1} A

Let us continue with the same notation and hypotheses as in Section 26.7. Remember that A is an integral domain, as in Section 24.5. This implies that the usual homomorphism from A into S^{−1} A is injective, as in Section 12.1. Thus we may consider A as a subring of S^{−1} A.

If L_{1} is an ideal in A, then S^{−1} L_{1} is an ideal in S^{−1} A, and this is the same as the extension of L_{1} in S^{−1} A, as in Section 12.13. If L_{2} is an ideal in S^{−1} A, then L_{2} ∩ A (26.9.1) is an ideal in A, which is the contraction of L_{2} in A, as in Section 12.7. Remember that L_{2} = L_{2}^{ce} = S^{−1} (L_{2} ∩ A), (26.9.2) as in Section 12.13.

If L_{2} is a proper ideal in S^{−1} A, then L_{2} ∩ S = ∅. (26.9.3) This means that

L_{2} ∩ A ⊆ ⋃_{i=1}^{∞} I_{i}, (26.9.4)

by the definition (26.7.8) of S. It follows that L_{2} ∩ A ⊆ I_{i} (26.9.5)

618 CHAPTER 26. SOME EXAMPLES OF NAGATA

for some i, as in (26.8.9). This implies that L_{2} ⊆ S^{−1} I_{i}, (26.9.6) because of (26.9.2).

Note that S^{−1} I_{i} ≠ S^{−1} A (26.9.7) for each i. More precisely, I_{i} is disjoint from S, by construction, and this implies (26.9.7), as in Section 12.13.

If i_{1} ≠ i_{2}, then one can check that I_{i_{1}} ⊈ S^{−1} I_{i_{2}}. (26.9.8) In particular, this means that S^{−1} I_{i_{1}} ⊈ S^{−1} I_{i_{2}}. (26.9.9) It follows that (26.9.10) S^{−1} I_{i} is a maximal proper ideal in S^{−1} A for each i, and that (26.9.11) every maximal proper ideal in S^{−1} A is of this form.

26.10 Some Noetherian conditions

Let us continue with the same notation and hypotheses as in Section 26.7, and let i be a positive integer. Put I_{1,i} = {m_{i} + 1, …, m_{i+1}} and I_{2,i} = Z_{+} \ I_{1,i}. (26.10.1) Note that A = k[{X_{j}}_{j∈Z_{+}}] is isomorphic to (k[{X_{j}}_{j∈I_{1,i}}])[{X_{l}}_{l∈I_{2,i}}] (26.10.2) and (k[{X_{j}}_{l∈I_{2,i}}])[{X_{j}}_{j∈I_{1,i}}], (26.10.3) as algebras over k, as in Section 26.3.

The algebra A_{i} defined in Section 26.7 may be expressed as A_{i} = k[{X_{l}}_{l∈I_{2,i}}], (26.10.4) using this notation. The homomorphism ϕ_{i} from A onto A_{i} defined earlier corresponds to the homomorphism ϕ_{2,k} in Section 26.4 with I = Z_{+}, I_{1} = I_{1,i}, and I_{2} = I_{2,i}, as before.

Put k_{0,i} = A_{i} = k[{X_{l}}_{l∈I_{2,i}}], (26.10.5)

26.10. SOME NOETHERIAN CONDITIONS 619

so that k_{0,i} is an integral domain, as before. Thus k_{0,i}[{X_{j}}_{j∈I_{1,i}}] (26.10.6) is the same as (26.10.3), and isomorphic to A, as before. It is helpful to put I_{0,i} = I_{1,i}, to be more compatible with the notation in Sections 26.5 and 26.6, so that (26.10.6) is the same as k_{0,i}[{X_{j}}_{j∈I_{0,i}}]. (26.10.7) Let ϕ_{0,k_{0,i}} be the homomorphism from (26.10.7) onto k_{0,i} discussed in Section 26.5, that sends a formal polynomial to its constant term. This corresponds to the homomorphism ϕ_{i} from A onto A_{i} mentioned in the preceding paragraph.

Note that S_{k_{0,i}} = k_{0,i}[{X_{j}}_{j∈I_{0,i}}] \ (ker ϕ_{0,k_{0,i}}) (26.10.8) is multiplicatively closed in k_{0,i}[{X_{j}}_{j∈I_{0,i}}], as in Section 26.6. This leads to a corresponding ring of fractions S_{k_{0,i}}^{−1} k_{0,i}[{X_{j}}_{j∈I_{0,i}}], (26.10.9) as before. Of course, I_{0,i} = I_{1,i} has only finitely many elements, by construction, as in (26.10.1). It follows that (26.10.10) S_{k_{0,i}}^{−1} k_{0,i}[{X_{j}}_{j∈I_{0,i}}] is Noetherian, as before.

Observe that (26.10.8) corresponds to S_{i} = A \ I_{i}, in the notation of Section 26.7. Thus (26.10.9) corresponds to the ring of fractions S_{i}^{−1} A. (26.10.11) Similarly, (26.10.10) means that (26.10.12) S_{i}^{−1} A is Noetherian.

Let U_{i} be the image of S_{i} in S^{−1} A, under the natural homomorphism from A into S^{−1} A. This is a multiplicatively closed set in S^{−1} A, and we have that (26.10.13) S_{i}^{−1} A and U_{i}^{−1} (S^{−1} A) are isomorphic as rings, as in Section 14.1, with T = S_{i}. This uses the fact that S ⊆ S_{i}, by construction, so that S S_{i} = S_{i}. This implies that (26.10.14) U_{i}^{−1} (S^{−1} A) is Noetherian, because of (26.10.12).

Note that S^{−1} I_{i} is a proper prime ideal in S^{−1} A for each i, which could also be obtained as in Section 12.13. This implies that S^{−1} A \ S^{−1} I_{i} (26.10.15) is a multiplicatively closed set in S^{−1} A, as in Section 12.4. Consider the corresponding ring of fractions

(S^{−1} A \ S^{−1} I_{i})^{−1}(S^{−1} A) = (S^{−1} A)_{S^{−1} I_{i}}, (26.10.16)

as before.

Remember that S^{−1} A may be identified with a subring of the field Q_{A} of fractions of A, because A is an integral domain, as in Section 14.2. It is easy to see that U_{i} ⊆ S^{−1} A \ S^{−1} I_{i}, (26.10.17)

because S ⊆ S_{i} = A \ I_{i}, and I_{i} is a prime ideal in A. More precisely, one can check that S^{−1} I_{i} consists of the elements of S^{−1} A such that in any representation of the element as a quotient of an element of A and an element of S, the numerator is an element of I_{i}. Thus (26.10.15) consists of the elements of S^{−1} A that can be expressed as the quotient of an element of A \ I_{i} and an element of S. Equivalently, this means that (26.10.15) consists of products of elements of U_{i} and reciprocals of elements of S.

It follows that

U_{i}^{−1} (S^{−1} A) and (S^{−1} A \ S^{−1} I_{i})^{−1} (S^{−1} A) (26.10.18)

are isomorphic as rings,

and indeed they correspond to the same subring of Q_{A}. This shows that

(26.10.19) (S^{−1} A \ S^{−1} I_{i})^{−1} (S^{−1} A) is Noetherian,

as in (26.10.14).

26.11 Some more properties of S^{−1} A

We continue with the same notation and hypotheses as in Section 26.7. We would like to show that (26.11.1) S^{−1} A is Noetherian,

as mentioned earlier. To do this, we use the criterion discussed in Section 26.1, as indicated in Exercise 4 on p126 of [8]. The first condition mentioned in Section 26.1 follows from the characterization of the maximal proper ideals in S^{−1} A discussed in Section 26.9, and the fact that the corresponding rings of fractions of S^{−1} A are Noetherian, as in the previous section.

The second condition mentioned in Section 26.1 is that every nonzero element of S^{−1} A should be contained in only finitely many maximal proper ideals in S^{−1} A. This means that

(26.11.2) every nonzero element of S^{−1} A is contained in S^{−1} I_{i} for only finitely many i ≥ 1, 26.11. SOME MORE PROPERTIES OF S^{−1} A 621 as in Section 26.9. One can check that this is equivalent to saying that (26.11.3) every nonzero element of A is contained in I_{i}

for only finitely many i ≥ 1,

because S ⊆ S_{i} = A \ I_{i} and I_{i} is a prime ideal in A for each i. It is easy to see that this holds, by the definition of I_{i} in Section 26.7. It follows that (26.11.1) holds, as in Section 26.1.

We would also like to show that (26.11.4) the dimension of S^{−1} A is + ∞, in the sense of Section 14.8, as in Section 26.7. Let I_{i,l} = I(X_{m_{i}+1}, …, X_{m_{i}+l}) (26.11.5) be the ideal in A generated by X_{m_{i}+1}, …, X_{m_{i}+l} for each i, l ≥ 1. Clearly I_{i,l} ⊆ I_{i,l+1} (26.11.6) for each i, l, and I_{i,m_{i+1}−m_{i}} = I_{i} (26.11.7) for each i. We also have that (26.11.8) I_{i,l} is a proper prime ideal in A for each i, l, as in Sections 26.4 and 26.7. Note that I_{i,1} ≠ {0} (26.11.9) for each i, and that I_{i,l} ≠ I_{i,l+1} (26.11.10) for each i, l.

If 1 ≤ l ≤ m_{i+1} − m_{i}, (26.11.11) then I_{i,l} ⊆ I_{i}, (26.11.12) and thus I_{i,l} ∩ S = ∅. (26.11.13) This implies that (26.11.14) S^{−1} I_{i,l} is a proper prime ideal in S^{−1} A, because of (26.11.8), as in Section 12.13. Of course, S^{−1} I_{i,1} ≠ {0}, (26.11.15)

622 CHAPTER 26. SOME EXAMPLES OF NAGATA because of (26.11.9), and S^{−1} I_{i,l} ⊆ S^{−1} I_{i,l+1}, (26.11.16) by (26.11.6). If l + 1 ≤ m_{i+1} − m_{i}, so that I_{i,l+1} ∩ S = ∅, then S^{−1}I_{i,l} ≠ S^{−1} I_{i,l+1}, (26.11.17) as in Section 12.13 again.

Thus {0} ⊆ S^{−1} I_{i,1} ⊆ ⋯ ⊆ S^{−1} I_{i,m_{i+1}−m_{i}} (26.11.18) is a chain of proper prime ideals in S^{−1} A of length m_{i+1} − m_{i} (26.11.19) for each i, as in Section 14.8. This means that (26.11.20) the dimension of S^{−1} A is greater than or equal to m_{i+1} − m_{i} for each i. This implies that (26.11.4) holds when the numbers (26.11.19) are unbounded.

26.12 Some properties of Dedekind domains

Let k be a Dedekind domain, as in Section 15.11, and suppose that k is not a field. Also let L_{0} be an ideal in k, with L_{0} ≠ {0}. The first part of Exercise 7 on p99 of [8] states that (26.12.1) every ideal in k/L_{0} is a principal ideal. Equivalently, this means that (26.12.2) every ideal in k that contains L_{0} is generated by L_{0}

and another element of k.

The second part of Exercise 7 on p99 of [8] asks one to use this to obtain that (26.12.3) every ideal in k can be generated by two elements.

Let L be an ideal in k with L ≠ {0}, and suppose that x ∈ L, x ≠ 0. If I(x) is the ideal in k generated by x, then I(x) ≠ {0} (26.12.4) and I(x) ⊆ L. (26.12.5)

26.13. NONTRIVIAL IDEALS IN K 623

If y ∈ k, then let I(x, y) be the ideal in k generated by x and y. Consider the statement that there is a y ∈ L such that

L = I(x, y). (26.12.6)

This follows from (26.12.2) with

L_{0} = I(x). (26.12.7)

Conversely, let L_{0} be any nontrivial ideal in k again, and let x_{0} be a nonzero element of L_{0}. If we can show that the property in the preceding paragraph holds, then we get that

(26.12.8) every ideal in k that contains L_{0} is generated by x_{0} and another element of k.

This means that (26.12.2) holds. A proof of the property mentioned in the preceding paragraph will be discussed in the next section. The argument is analogous in some ways to the one in Section 26.1.

26.13 Nontrivial ideals in k

Let k be a Dedekind domain that is not a field again. If I is a proper prime ideal in k, then S_{I} = k \ I is multiplicatively closed in k, as in Section 12.4. If I ≠ {0} too, then S_{I}^{−1} k = k_{I} is a discrete valuation ring, as in Section 15.11. Remember that this leads to a discrete valuation v_{I} on the field Q_{k} of fractions of k, as in Section 18.12.

Let L be a nontrivial ideal in k, so that S_{I}^{−1} L is a nontrivial ideal in k_{I}. Remember that there is a unique nonnegative integer v_{I}(L) such that

S_{I}^{−1} L = {z ∈ k_{I} : v_{I}(z) ≥ v_{I}(L)}, (26.13.1)

as in Section 18.14. In fact,

v_{I}(L) = min {v_{I}(z) : z ∈ S_{I}^{−1} L} = min{v_{I}(z) : z ∈ L}, (26.13.2)

as before. We also have that

(26.13.3) L = {z ∈ k : v_{I}(z) ≥ v_{I}(L) for every nonzero proper

prime ideal I in k},

as in Section 18.15.

Let x be a nonzero element of L, and remember that

v_{I}(x) = 0 (26.13.4)

for all but finitely many nonzero proper prime ideals I in k, as in Section 18.12. Note that v_{I}(L) = 0 (26.13.5) when (26.13.4) holds.

If y ∈ k, then let I(x, y) be the ideal in k generated by x and y, as before. It is easy to see that

v_{I}(I(x, y)) = min{v_{I}(z) : z ∈ I(x, y)} = min(v_{I}(x), v_{I}(y)) (26.13.6)

for every nonzero proper prime ideal I in k. We would like to show that there is a y ∈ k such that I(x, y) = L, (26.13.7)

as in the previous section. Note that this would imply in particular that y ∈ L. It suffices to show that v_{I}(I(x, y)) = v_{I}(L) (26.13.8)

for every nonzero proper prime ideal I in k, as in Section 18.15 and (26.13.3).

Observe that (26.13.8) holds when (26.13.4) holds. Let I_{1}, …, I_{r} be the finitely many nonzero proper prime ideals in k such that (26.13.4) does not hold, without repetitions. We would like to find y ∈ k such that

v_{I_{m}}(y) = v_{I_{m}}(L) (26.13.9)

for each m = 1, …, r. In particular, this means that

v_{I_{m}}(y) ≤ v_{I_{m}}(x) (26.13.10)

for each m = 1, …, r, because x ∈ L. Under these conditions, we would get that v_{I_{m}}(I(x, y)) = min(v_{I_{m}}(x), v_{I_{m}}(y)) = v_{I_{m}}(L) (26.13.11)

for every m = 1, …, r.

Of course, there is a y_{m} ∈ L such that

v_{I_{m}}(y_{m}) = v_{I_{m}}(L) (26.13.12)

for each m = 1, …, r, as in (26.13.2). There is a y ∈ k such that

v_{I_{m}}(y − y_{m}) > v_{I_{m}}(L) (26.13.13)

for every m = 1, …, r, as in Section 19.6. One can use this to get (26.13.9).

It follows that (26.13.8) holds for every nonzero proper prime ideal I in k, because of (26.13.11).

Some polynomials and formal sums

27.1 Polynomials in one variable

Let k be a commutative ring with a multiplicative identity element, let A be an algebra over k in the strict sense, as in Section 20.13, and let T be an indeterminate. Also let

f(T) = ∑_{j=0}^{n} f_{j} T^{j} (27.1.1)

be a formal polynomial in T with coefficients in A, as in Sections 4.3 and 20.13. If f_{n} ≠ 0, (27.1.2)

then we say that f(T) has degree n, which may be expressed as deg f(T). If f(T) = 0, then the degree of f(T) may be interpreted as being −∞, as on p97 of [151], p117 of [177], and p107 of [195].

Let

g(T) = ∑_{l=1}^{p} g_{l} T^{l} (27.1.3)

be another formal polynomial in T with coefficients in A, and note that

(27.1.4) deg(f(T) g(T)) ≤ deg f(T) + deg g(T).

If f_{n} g_{p} ≠ 0, (27.1.5)

then (27.1.6) deg(f(T) g(T)) = deg f(T) + deg g(T).

Note that this holds automatically when either factor is 0, with suitable interpretations. If products of nonzero elements of A are always nonzero, then this holds for all polynomials in T with coefficients in A, as in (14) on p97 of [151]. We also have that

(27.1.7) deg(f(T) + g(T)) ≤ max(deg f(T), deg g(T)),

as in 15 on p98 of [151].

Suppose now that A is an associative algebra with a multiplicative identity element e_{A}. Suppose for the moment that A is an integral domain in the strict sense, as in Section 22.1, so that products of nonzero elements of A are nonzero. This implies that

(27.1.8) A[T] is an integral domain in the strict sense,

as on p97 of [151]. In this case, one can check that the only invertible elements of A[T] are the invertible elements of A, as on p98 of [151].

Let g(T) ∈ A(T) be as in (27.1.3), and suppose that

(27.1.9) g_{p} is invertible in A.

More precisely, we ask that A ≠ {0}, so that g_{p} ≠ 0, and deg g(T) = p. If f(T) ∈ A[T], then there are q_{1}(T), r_{1}(T) ∈ A[T] such that

f(T) = q_{1}(T) g(T) + r_{1}(T) (27.1.10)

and deg r_{1}(T) < deg g(T), (27.1.11)

as on p98 of [151]. We also have that q_{1}(T) and r_{1}(T) are uniquely determined by these properties, as on p99 of [151]. Similarly, there are unique q_{2}(T), r_{2}(T) in A[T] such that f(T) = g(T) q_{2}(T) + r_{2}(T) (27.1.12)

and deg r_{2}(T) < deg g(T). (27.1.13)

In particular, this works when g(T) ≠ 0 and A is a division ring. This case was mentioned previously in Section 9.12.

27.2 The factor theorem

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let f(T) ∈ A[T] be as in (27.1.1) again. If c ∈ A, then put

f_{L}(c) = ∑_{j=0}^{n} c^{j} f_{j} (27.2.1)

and

f_{R}(c) = ∑_{j=0}^{n} f_{j} c^{j}, (27.2.2)

where c^{j} is interpreted as being equal to e_{A} when j = 0, as usual. This corresponds to (18), (19) on p99 of [151]. Of course, (27.2.1) and (27.2.2) are the same when c commutes with f_{j} for each j ≥ 1.

Note that f(T) ↦ f_{L}(c) (27.2.3)

and f(T) ↦ f_{R}(c) (27.2.4)

are homomorphisms from A[T] into A, as modules over k. More precisely, (27.2.3) is a homomorphism from A[T] into A, as right modules over A, and (27.2.4) is a homomorphism from A[T] into A, as left modules over A. If A is commutative, then (27.2.1) and (27.2.2) are the same, and we get an algebra homomorphism from A[T] into A, as in Section 4.4. This also works when c is in the center of A, without asking A to be commutative, as before.

Let us take g(T) = T − c in the remarks in the previous section, as in [151]. If j is a positive integer, then

T^{j} − c^{j} = (T − c) ∑_{l=0}^{j−1} c^{j−l−1} T^{l} = ( ∑_{l=0}^{j−1} c^{j−l−1} T^{l}) (T − c), (27.2.5)

by standard arguments. One can use this to get that

f(T) − f_{R}(c) = q_{1}(T) (T − c), (27.2.6)

with

q_{1}(T) = ∑_{j=1}^{n} f_{j} ( ∑_{l=0}^{j−1} c^{j−l−1} T^{l}), (27.2.7)

as in [151]. Similarly,

f(T) − f_{L}(c) = (T − c) q_{2}(T), (27.2.8)

with

q_{2}(T) = ∑_{j=1}^{n} ( ∑_{l=0}^{j−1} c^{j−l−1} f_{j} T^{l}). (27.2.9)

If f_{R}(c) = 0, (27.2.10)

then (27.2.6) reduces to f(T) = q_{1}(T) (T − c). (27.2.11)

Similarly, if f_{L}(c) = 0, (27.2.12)

then (27.2.8) reduces to f(T) = (T − c) q_{2}(T). (27.2.13)

Conversely, if (27.2.11) or (27.2.13) holds for some q_{1}(T) or q_{2}(T) in A[T], then (27.2.10) or (27.2.12) holds, as appropriate. This follows from the uniqueness statements for (27.1.10) and (27.1.12) in the previous section. This corresponds to the factor theorem on p99 of [151].

Of course, one can also obtain (27.2.10), (27.2.12) from (27.2.11), (27.2.13), respectively, more directly. This is much simpler when A is commutative, or c is in the center of A.

Let V be a module over k, and suppose now that f(T) ∈ V [T] is as in (27.1.1). If V is a left module over A, then f_{L}(c) may be defined as an element of V as in (27.2.1). It is easy to see that (27.2.3) defines a homomorphism from V [T] into V, as modules over k. Similarly, if V is a right module over A, then f_{R}(c) may be defined as an element of V as in (27.2.2). In this case, (27.2.4) is a homomorphism from V [T] into V, as modules over k.

Suppose that V is a two-sided module over A, so that V is a left and right module over A, and the actions of A on V on the left and the right commute with each other. This implies that V [T] is a two-sided module over A, where A acts on elements of V [T] termwise on the left and on the right. Under these conditions, (27.2.3) is a homomorphism from V [T] into V, as right modules over A, and (27.2.4) is a homomorphism from V [T] into V, as left modules over A.

27.3 More on one-variable polynomials

Let B be a ring with a multiplicative identity element e = e_{B}, and let A be a subring of B that contains e. One is often interested in studying B relative to A, as mentioned on p92 of [151]. Note that B may be considered as a two-sided module over A. If A is contained in the center of B, so that every element of A commutes with every element of B, then A is commutative in particular, and B may be considered as an algebra over A.

Suppose that u ∈ B is in the centralizer of A in B, so that

a u = u a (27.3.1)

for every a ∈ A. Let T be an indeterminate, and let f(T) ∈ A[T] be as in (27.1.1). Put

f(u) = ∑_{j=0}^{n} f_{j} u^{j} = ∑_{j=0}^{n} u^{j} f_{j}, (27.3.2)

which is an element of B. One can check that

f(T) ↦ f(u) (27.3.3)

defines a ring homomorphism from A[T] into B. This is related to some of the remarks in Section 4.4.

Thus {f(u) : f(T) ∈ A[T]} (27.3.4) is a subring of B. It is easy to see that this ring contains A and u, and in fact it is the smallest subring of B that contains A and u. This subring may be expressed as A[u], as on p93 of [151].

If f(u) = 0 (27.3.5)

for some f(T) ∈ A[T], f(T) ≠ 0, then u is said to be algebraic relative to A, as on p94 of [151]. Otherwise, u is said to be transcendental relative to A, as on p93 of [151]. This means that (27.3.3) is a ring isomorphism from A[T] onto A[u], as in Theorem 5 on p97 of [151]. If B is commutative, then this is the same as saying that u is algebraically independent over A as a single element of B, as in Section 24.4, and as on p116 of [177].

27.4 More on several-variable polynomials

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, as in Section 20.13. Also let n be a positive integer, let T_{1}, …, T_{n} be n distinct commuting indeterminates, and remember that the space A[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in A may be considered as an algebra over k in the strict sense too, as in Sections 4.3 and 20.13. If n ≥ 2, then one can get an isomorphism from A[T_{1}, …, T_{n}] onto (A[T_{1}, …, T_{n−1}])[T_{n}], as algebras over k in the strict sense, as in Section 21.12.

Suppose for the moment that products of nonzero elements of A are nonzero. In this case,

(27.4.1) products of nonzero elements of A[T_{1}, …, T_{n}] are nonzero too.

This follows from the analogous statement for polynomials in over variable in Section 27.1, and the remarks in the preceding paragraph. If A is an integral domain in the strict sense, as in Section 22.1, then we get that

(27.4.2) A[T_{1}, …, T_{n}] is an integral domain in the strict sense,

as in (1) on p106 of [151].

Suppose for the moment again that A is an integral domain in the strict sense with a multiplicative identity element. One can check that

(27.4.3) the only invertible elements of A[T_{1}, …, T_{n}] are the invertible elements of A,

as in (2) on p107 of [151]. This can be obtained from the analogous statement for polynomials in one variable in Section 27.1 using the remarks at the beginning of the section, as before.

Now let B be a ring with a multiplicative identity element e = e_{B}, and let A be a subring of B that contains e again. Suppose that u_{1}, …, u_{n} ∈ B are n commuting elements of the centralizer of A in B, so that

u_{j} u_{l} = u_{l} u_{j} (27.4.4) for each j, l = 1, …, n, and a u_{j} = u_{j} a (27.4.5)

for every a ∈ A and j = 1, …, n. If α is a multi-index of length n, then we put u^{α} = u_{1}^{α_{1}} ⋯ u_{n}^{α_{n}}, as in Section 4.4. If

f(T) = f(T_{1}, …, T_{n}) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} T^{α} (27.4.6)

is an element of A[T_{1}, …, T_{n}], then put

f(u) = f(u_{1}, …, u_{n}) = ∑_{α∈(Z_{+}∪{0})^{n}} f_{α} u^{α}, (27.4.7)

which is an element of B. One can check that

f(T) ↦ f(u) (27.4.8)

is a ring homomorphism from A[T_{1}, …, T_{n}] into B, for basically the same reasons as in Section 4.4.

In particular, {f(u) : f(T) ∈ A[T_{1}, …, T_{n}]} (27.4.9)

is a subring of B. This is the smallest subring of B that contains A and u_{1}, …, u_{n}, as in the previous section, when n = 1. This subring may be expressed as A[u_{1}, …, u_{n}], as on p105 of [151].

We say that u_{1}, …, u_{n} are algebraically independent over A if

f(u_{1}, …, u_{n}) = 0 (27.4.10)

for some f(T) ∈ A[T_{1}, …, T_{n}] only when f(T) = 0, as on p105 of [151]. This means that (27.4.8) is a ring isomorphism from A[T_{1}, …, T_{n}] onto A[u_{1}, …, u_{n}]. If n = 1, then this is the same as saying that u_{1} is transcendental relative to A, as in the previous section. This is also the same as in Section 24.4 when B is commutative.

Note that the algebraic independence of u_{1}, …, u_{n} over A does not depend on the order in which these elements of B are listed, as mentioned in [151]. It is easy to see that their algebraic independence implies the algebraic independence of any nonempty subcollection of these elements of B. The lemma on p105 of [151] states that u_{1}, …, u_{n} are algebraically independent over A if and only if for each l = 1, …, n,

(27.4.11) u_{l} is transcendental over A[u_{1}, …, u_{l−1}].

27.5 Families of indeterminates

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, as in Section 20.13. Also let I be a nonempty set, and let {T_{j}}_{j∈I} be a family of distinct commuting indeterminates indexed by I. Remember that the space A[{T_{j}}_{j∈I}] of formal polynomials in the T_{j}’s with coefficients in A may be considered as an algebra over k in the strict sense, as in Section 24.1.

Suppose for the moment that products of nonzero elements of A are nonzero. Under these conditions,

(27.5.1) products of nonzero elements of A[{T_{j}}_{j∈I}] are nonzero as well.

This is the same as (27.4.1) when I has only finitely many elements, and otherwise one can reduce to that case, as in Section 24.5. If A is an integral domain in the strict sense, as in Section 22.1, then it follows that

(27.5.2) A[{T_{j}}_{j∈I}] is an integral domain in the strict sense.

Suppose for the moment again that A is an integral domain in the strict sense with a multiplicative identity element. One can verify that

(27.5.3) the only invertible elements of A[{T_{j}}_{j∈I}] are the invertible elements of A.

This is the same as (27.4.3) when I has only finitely many elements, and otherwise one can reduce to that case, as before.

Let B be a ring with a multiplicative identity element e = e_{B}, and suppose now that A is a subring of B that contains e. Also let u = {u_{j}}_{j∈I} be a family of commuting elements of the centralizer of A in B indexed by I, so that (27.4.4) holds for all j, l ∈ I, and (27.4.5) holds for all a ∈ A and j ∈ I. Remember that a function α on I with values in the set of nonnegative integers is said to be a multi-index associated to I when α(j) = 0 for all but finitely many j ∈ I, as in Section 24.1, and that the set of all multi-indices associated to I is denoted Mult(I). If α ∈ Mult(I), then

u^{α} = ∏_{j∈I} u_{j}^{α(j)} (27.5.4)

defines an element of B, as in Section 24.2, where u_{j}^{α(j)} is interpreted as being equal to e when α(j) = 0, so that this reduces to a finite product.

Let

f(T) = f({T_{j}}_{j∈I}) = ∑_{α∈Mult(I)} f_{α} T^{α} (27.5.5)

be an element of A[{T_{j}}_{j∈I}], as in Section 24.1. Under these conditions,

f(u) = f({u_{j}}_{j∈I}) = ∑_{α∈Mult(I)} f_{α} u^{α} (27.5.6)

defines an element of B, where all but finitely many terms in the sum are equal to 0. One can verify that f(T) ↦ f(u) (27.5.7) is a ring homomorphism from A[{T_{j}}_{j∈I}] into B, for essentially the same reasons as in Section 24.2. This is the same as in the previous section when I has only finitely many elements.

As before, {f(u) : f(T) ∈ A[{T_{j}}_{j∈I}]} (27.5.8)

is a subring of B, and the smallest subring of B that contains A and the u_{j}’s, j ∈ I. This subring may be expressed as A[{u_{j}}_{j∈I}], as usual.

Let us say that the u_{j}’s, j ∈ I, are algebraically independent over A if

f(u) = 0 (27.5.9)

for some f(T) ∈ A[{T_{j}}_{j∈I}] only when f(T) = 0. This is the same as in Section 24.4 when B is commutative, and as in the previous section when I has only finitely many elements. Equivalently, this means that (27.5.7) is a ring isomorphism from A[{T_{j}}_{j∈I}] onto A[{u_{j}}_{j∈I}], as before.

Of course, if I_{0} is a nonempty subset of I, and if the family of u_{j}’s, j ∈ I, is algebraically independent over A, then the family of u_{j}’s, j ∈ I_{0}, is algebraically independent over A. It is easy to see that the family of u_{j}’s, j ∈ I, is algebraically independent over A when the family of u_{j}’s, j ∈ I_{0}, is algebraically independent over A for each nonempty finite subset I_{0} of I. This is analogous to a remark in Section 24.5.

27.6 Some modules of formal sums

Let R be the set of real numbers, and let X be an indeterminate. If α is a real number, then X^{α} (27.6.1)

is formally considered as X to the power α. If β is another real number, then we consider X^{α} times X^{β} to be given by

X^{α} X^{β} = X^{α+β}. (27.6.2)

Let k be a commutative ring with a multiplicative identity element, let V be a module over k, and let E be a nonempty subset of R. We would like to consider formal sums of the form

f(X) = ∑_{α∈E} f_{α} X^{α}, (27.6.3)

where f_{α} ∈ V for every α ∈ E, and f_{α} = 0 for all but finitely many α ∈ E. The space of these formal sums may be denoted V [X^{E}]. This may be defined more precisely as the space of all V -valued functions on E with finite support. This is a module over k with respect to pointwise addition and scalar multiplication, which corresponds to termwise addition and scalar multiplication of formal sums as in (27.6.3).

Of course, if E = Z_{+} ∪ {0}, then V [X^{E}] is the same as the space V [X] of formal polynomials in X with coefficients in V, as in Section 4.3. If

0 ∈ E, (27.6.4)

then we may identify V with the submodule of V [X^{E}] consisting of f(X) as in (27.6.3) with f_{α} = 0 when α ≠ 0, as before.

Let E_{0} be another subset of R, with

E ⊆ E_{0}. (27.6.5)

Under these conditions, we can identify V [X^{E}] with a submodule of V [X^{E_{0}}], as a module over k, by identifying f(X) as in (27.6.3) with an element of V [X^{E_{0}}], by putting f_{α} = 0 when α ∈ E_{0} \ E.

Let E_{1}, E_{2} be nonempty subsets of R, so that

E_{1} + E_{2} = {α + β : α ∈ E_{1}, β ∈ E_{2}} (27.6.6)

is a nonempty subset of R as well. Also let W and Z be modules over k, and let b be a mapping from V × W into Z that is bilinear over k. Suppose that f(X) ∈ V [X^{E_{1}}] is as in (27.6.3), with E = E_{1}, and that

g(X) = ∑_{β∈E_{2}} g_{β} X^{β} (27.6.7)

is an element of W [X^{E_{2}}].

Let E_{3} be another subset of R, with

E_{1} + E_{2} ⊆ E_{3}. (27.6.8)

If γ ∈ E_{1} + E_{2}, then put

h_{γ} = ∑_{{α∈E_{1}, β∈E_{2} \atop α+β=γ}} b(f_{α}, g_{β}). (27.6.9)

This defines an element of Z, because all but finitely many terms in the sum are equal to 0. Let us interpret this as being equal to 0 when γ ∈ E_{3} \ (E_{1} + E_{2}), so that

h(X) = ∑_{γ∈E_{3}} h_{γ} X^{γ} (27.6.10)

is an element of Z[X^{E_{3}}].

Put b̂(f(X), g(X)) = h(X). (27.6.11)

This defines a mapping β̂ from V [X^{E_{1}}] × W [X^{E_{2}}] into Z[X^{E_{3}}] that is bilinear over k.

27.7 Some algebras of formal sums

Let E be a nonempty subset of R, and suppose that E is a sub-semigroup of R, as a commutative group with respect to addition, so that

E + E ⊆ E. (27.7.1)

Also let k be a commutative ring with a multiplicative identity element, let A be an algebra over k in the strict sense, as in Section 20.13, and let X be an indeterminate. Suppose that f(X) ∈ A[X^{E}] is as in (27.6.3), and that g(X) ∈ A[X^{E}] is as in (27.6.7), with E_{2} = E. If γ ∈ E, then put

h_{γ} = ∑_{{α,β∈E \atop α+β=γ}} f_{α} g_{β}, (27.7.2)

which is interpreted as being equal to 0 when γ ∉ E + E. Note that all but finitely many terms in the sum on the right are equal to 0, as before.

Thus h(X) defines an element of A[X^{E}] as in (27.6.10), with E_{3} = E. Put

f(X) g(X) = h(X), (27.7.3)

so that A[X^{E}] is an algebra over k in the strict sense. This is the same as the algebra A[X] of formal polynomials in X with coefficients in A when E = Z_{+} ∪ {0}, as in Sections 4.3 and 20.13. If 0 ∈ E, then A corresponds to a subalgebra of A[X^{E}], as in the previous section.

If A is commutative or associative, then it is easy to see that A[X^{E}] has the same property. If A has a multiplicative identity element e_{A}, and if 0 ∈ E, then e_{A} corresponds to the multiplicative identity element in A[X^{E}]. If A is associative with a multiplicative identity element and 0 ∈ E, then A[X^{E}] is essentially the same as the semigroup algebra of E over A, as in Section 4.9.

Suppose that the product of any two nonzero elements of A is nonzero. If f(X), g(X) are nonzero elements of A[X^{E}], then one can check that

f(X) g(X) ≠ 0. (27.7.4)

This is related to Exercise 2 on p118 of [151].

Let E_{0} be another sub-semigroup of R that contains E, so that A[X^{E}] may be identified with a submodule of A[X^{E_{0}}], as a module over k, as in the previous section. In this case,

(27.7.5) A[X^{E}] corresponds to a subalgebra of A[X^{E_{0}}].

Suppose that 0 ∈ E, and let E_{2} be a nonempty subset of R such that

E + E_{2} ⊆ E_{2}. (27.7.6)

Also V be a module over k, so that V [X^{E_{2}}] may be defined as a module over k as in the previous section. If f(X) ∈ k[X^{E}] is as in (27.6.3), and g(X) ∈ V [X^{E_{2}}] is as in (27.6.7), then put

h_{γ} = ∑_{{α∈E, β∈E_{2} \atop α+β=γ}} f_{α} · g_{β} (27.7.7)

for every γ ∈ E_{2}, which is interpreted as being equal to 0 when γ ∉ E + E_{2}. As usual, all but finitely many terms on the sum on the right are equal to 0, so that the sum defines an element of V. It follows that h(X) defines an element of V [X^{E_{2}}] as in (27.6.10), with E_{3} = E_{2}.

Of course, k[X^{E}] is a commutative associative algebra over k with a multiplicative identity element, as before. One can check that V [X^{E_{2}}] is a module over k[X^{E}], where the product of f(X) and g(X) is equal to h(X). Similarly, if A is an associative algebra over k with a multiplicative identity element, and V is a left or right module over A, then V [X^{E_{2}}] may be considered as a left or right module over A[X^{E}]. This corresponds to some remarks in Section 4.3 when E = E_{2} = Z_{+} ∪ {0}. This also corresponds to some remarks in Section 4.9, with E acting on V [X^{E_{2}}] by multiplication by powers of X with exponent in E.

Suppose that E_{2} is a nonempty subset of R that satisfies (27.7.6) and

E_{2} ⊆ E. (27.7.8)

If A is any algebra over k in the strict sense, then A[X^{E_{2}}] corresponds to a two-sided ideal in A[X^{E}].

27.8 Formal sums and algebraic independence

Let A be a ring with a multiplicative identity element e_{A}, and let E be a subsemigroup of R, as a commutative group with respect to addition, with 0 ∈ E. If X is an indeterminate, then A[X^{E}] may be considered as a ring that contains A as a subring, with e_{A} as the multiplicative identity element, as in the previous section. If β ∈ E, then X^{β} = e_{A} X^{β} may be considered as an element of A[X^{E}], with the coeffients of all of the other powers of X equal to 0. Note that X^{β} commutes with all other elements of A[X^{E}], so that it is an element of the center of A[X^{E}].

Let n be a positive integer, let T_{1}, …, T_{n} be n distinct commuting indeterminates, and let

F (T) = F (T_{1}, …, T_{n}) = ∑_{α∈(Z_{+}∪{0})^{n}} F_{α} T^{α} (27.8.1)

be a formal polynomial in T_{1}, …, T_{n} with coefficients in A. If α, β ∈ R^{n}, then put

α · β = ∑_{j=1}^{n} α_{j} β_{j}, (27.8.2)

636 CHAPTER 27. SOME POLYNOMIALS AND FORMAL SUMS

as usual. In particular, if β_{1}, …, β_{n} ∈ E, then α · β ∈ E (27.8.3) for every multi-index α. This means that

F (X^{β_{1}}, …, X^{β_{n}}) = ∑_{α∈(Z_{+}∪{0})^{n}} F_{α} X^{α·β} (27.8.4)

defines an element of A[X^{E}], as in Section 27.4.

Suppose for the moment that (27.8.5) β_{1}, …, β_{n} are linearly independent over Q. This means that if α ∈ Q^{n} and α · β = 0, (27.8.6) then α = 0. This is the same as saying that α ↦ α · β (27.8.7) is injective as a mapping from Q^{n} into R. In particular, this implies that (27.8.7) is injective as a mapping from (Z_{+} ∪ {0})^{n} into E. It follows that F (X^{β_{1}}, …, X^{β_{n}}) = 0 (27.8.8) in A[X^{E}] only when F (T) = 0. This implies that (27.8.9) X^{β_{1}}, …, X^{β_{n}} are algebraically independent over A in A[X^{E}], as in Section 27.4. If n = 1, then this is the same as saying that X^{β_{1}} is transcendental over A when β_{1} ≠ 0, as in Section 27.3.

Conversely, if (27.8.5) does not hold, then there is an α ∈ Q^{n} such that α ≠ 0 and (27.8.6) holds. More precisely, we may as well take α ∈ Z^{n}. This implies that there are multi-indices α′, α′′, not both equal to 0, such that α′ · β = α′′ · β. (27.8.10) If A ≠ {0}, so that e_{A} ≠ 0, then it is easy to see that there is an F (T) in A[T_{1}, …, T_{n}] such that F (T) ≠ 0 and (27.8.8) holds. Of course, this means that (27.8.9) does not hold.

Let I be a nonempty set, and let {β_{j}}_{j∈I} be a family of elements of E indexed by I. If (27.8.11) the β_{j}’s, j ∈ I, are linearly independent over Q, then (27.8.12) the X^{β_{j}}’s, j ∈ I, are algebraically independent over A in A[X^{E}]. This follows from the earlier remarks, corresponding to the case where I has only finitely many elements. Conversely, (27.8.12) implies (27.8.11) when A ≠ {0}, as in the preceding paragraph.

27.9 Another algebraic dependence property

Let A be a ring with a multiplicative identity element e_{A}, and let E be a subsemigroup of R, as a commutative group with respect to addition, with 0 ∈ E. Also let E_{0} be another sub-semigroup of R that contains E, and let X be an indeterminate, so that A[X^{E}] corresponds to a subring of A[X^{E_{0}}], as in Section 27.7. Note that E − E = {α − β : α, β ∈ E} (27.9.1)

is the smallest subgroup of R that contains E.

Let β_{0} ∈ E_{0} be given, and suppose for the moment that

l_{0} β_{0} ∈ E − E (27.9.2)

for some positive integer l_{0}. This means that

α′ + l_{0} β_{0} = α′′ (27.9.3)

for some α′, α′′ ∈ E. Equivalently, we have that

X^{α′} (X^{β_{0}})^{l_{0}} = X^{α′′}. (27.9.4)

It follows that (27.9.5) X^{β_{0}} is algebraic relative to A[X^{E}],

as an element of A[X^{E_{0}}], as in Section 27.3.

Conversely, suppose that (27.9.5) holds, and let us check that (27.9.2) holds for some positive integer l_{0}. Let T be an indeterminate, and let

F (T) = ∑_{j=0}^{n} F_{j} T^{j} (27.9.6)

be a formal polynomial in T with coefficients in A[X^{E}]. This means that

F_{j} = F_{j}(X) = ∑_{α∈E} F_{j,α} X^{α} (27.9.7)

for each j, where F_{j,α} ∈ A, and F_{j,α} = 0 for all but finitely many α ∈ E. Note that F (X^{β_{0}}) is defined as an element of A[X^{E_{0}}] by

F (X^{β_{0}}) = ∑_{j=0}^{n} F_{j}(X) X^{j β_{0}}, (27.9.8)

as in Section 27.3. Using (27.9.5), we get that

F (X^{β_{0}}) = 0 (27.9.9)

for some F (T) ∈ (A[X^{E}])[T], with F (T) ≠ 0.

We may as well suppose that F_{n}(X) ≠ 0, so that

F_{n,α′} ≠ 0 (27.9.10)

for some α′ ∈ E. It follows from this and (27.9.9) that there are j < n and α′′ ∈ E such that F_{j,α′′} ≠ 0 (27.9.11)

and α′ + n β_{0} = α′′ + j β_{0}. (27.9.12)

This implies (27.9.3), with l_{0} = n − j.

If l_{0} β_{0} ∈ E (27.9.13)

for some positive integer l_{0}, then

(X^{β_{0}})^{l_{0}} = X^{l_{0} β_{0}} ∈ A[X^{E}]. (27.9.14)

If A is commutative, then this implies that X^{β_{0}} is integral over A[X^{E}], as in Section 16.3. Conversely, suppose that (27.9.9) holds for some F (T) ∈ (A[X^{E}])(T), with F_{n}(X) = e_{A}. (27.9.15)

If A ≠ {0}, so that e_{A} ≠ 0, then (27.9.10) holds with α′ = 0. In this case, it is easy to see that (27.9.13) holds for some l_{0} ≥ 1, as in the preceding paragraph.

27.10 Some subrings of A[X^{E}]

Let I be a nonempty set, and let β = {β_{j}}_{j∈I} be a family of real numbers indexed by I. Of course, this is really the same as a real-valued function on I, although it is convenient to use this notation here. Also let α be a real-valued function on I with finite support, so that α(j) = 0 for all but finitely many j ∈ I. Thus

α · β = ∑_{j∈I} α(j) β_{j} (27.10.1)

may be defined as a real number, because all but finitely many terms in the sum are equal to 0.

Remember that Mult(I) is the set of all multi-indices associated to I, which are functions α on I with values in the set of nonnegative integers such that α(j) = 0 for all but finitely many j ∈ I, as in Section 24.1. Note that Mult(I) is a commutative semigroup with respect to addition, and that

α ↦ α · β (27.10.2)

is a homomorphism from Mult(I) into R, as a commutative semigroup with respect to addition. This homomorphism maps Mult(I) onto

E_{1}(β) = {α · β : α ∈ Mult(I)}, (27.10.3) which is the sub-semigroup of R generated by 0 and the β_{j}’s, j ∈ I.

The space c_{00}(I, Q) of all Q-valued functions on I with finite support is a linear subspace of the space c(I, Q) of all Q-valued functions on I, as a vector space over Q with respect to pointwise addition and scalar multiplication. We also have that (27.10.2) defines a linear mapping from C_{00}(I, Q) into R, as vector spaces over Q. This mapping is injective if and only if the β_{j}’s, j ∈ I, are linearly independent in R, as a vector space over Q. In this case, (27.10.2) is injective as a mapping from Mult(I) into R.

Conversely, if the β_{j}’s, j ∈ I, are linearly dependent in R, as a vector space over Q, then one can check that (27.10.2) is not injective as a mapping from Mult(I) into R. This was basically mentioned already in Section 27.8 when I has only finitely many elements, and it is easy to reduce to that case.

Let E be a sub-semigroup of R, as a commutative group with respect to addition, with 0 ∈ E, and suppose that

β_{j} ∈ E (27.10.4)

for each j ∈ I. This implies that

E_{1}(β) ⊆ E, (27.10.5)

so that E_{1}(β) may be considered as the sub-semigroup of E generated by 0 and the β_{j}’s, j ∈ I.

Let A be a ring with a multiplicative identity element e_{A}, and let X be an indeterminate. We may identify A[X^{E_{1}(β)}] with a subring of A[X^{E}], because of (27.10.5), as in Sections 27.6 and 27.7. Equivalently,

(27.10.6) A[X^{E_{1}(β)}] corresponds to the subring of A[X^{E}] generated by A and the X^{β_{j}}’s, j ∈ I.

Let {T_{j}}_{j∈I} be a family of distinct commuting indeterminates indexed by I, so that F (T) = F ({T_{j}}_{j∈I}) ↦ F ({X^{β_{j}}}_{j∈I}) (27.10.7)

defines a ring homomorphism from A[{T_{j}}_{j∈I}] into A(X^{E}), as in Section 27.5. More precisely, this homomorphism maps A[{T_{j}}_{j∈I}] onto A[X^{E_{1}(β)}], because of (27.10.6). If the β_{j}’s, j ∈ I, are linearly independent over Q, then this defines a ring isomorphism from A[{T_{j}}_{j∈I}] onto A[X^{E_{1}(β)}], as in Section 27.8.

27.11 Quotients of formal sums

Let k be a field, and let E be a sub-semigroup of R, as a commutative group with respect to addition, with 0 ∈ E. Also let X be an indeterminate, and note that k[X^{E}] is an integral domain, as in Section 27.7. Let

k(X^{E}) (27.11.1) be the field Q_{k[X^{E}]} of fractions of k[X^{E}], as in Sections 11.4 and 12.1. If E = Z_{+} ∪{0}, so that k[X^{E}] is the same as the algebra of formal polynomials in X with coefficients in k, then (27.11.1) is the same as the field of formal rational functions in X with coefficients in k, as in Section 21.12.

Let E_{0} be another subsemigroup of R that contains E, so that k[X^{E}] may be identified with a subalgebra of k[X^{E_{0}}], as an algebra over k, as in Section 27.7. This means that

(27.11.2) k(X^{E}) corresponds to a subfield of k(X^{E_{0}}).

Let n be a positive integer, and suppose for the moment that

(27.11.3) β_{1}, …, β_{n} ∈ E are linearly independent over Q.

This implies that

(27.11.4) X^{β_{1}}, …, X^{β_{n}} are algebraically independent over k

in k[X^{E}], as in Section 27.8. If T_{1}, …, T_{n} are n distinct commuting indeterminates, then F (T) = F (T_{1}, …, T_{n}) ↦ F (X^{β_{1}}, …, X^{β_{n}}) (27.11.5)

is an injective homomorphism from k[T_{1}, …, T_{n}] into k[X^{E}], as algebras over k, as in Sections 24.4 and 27.4. This leads to an injective homomorphism from the field k(T_{1}, …, T_{n}) of formal rational functions in T_{1}, …, T_{n} with coefficients in k into k(X^{E}), as in Section 24.5. In particular, if n = 1, then this works when β_{1} ≠ 0.

Let I be a nonempty set, and let β = {β_{j}}_{j∈I} be a family of elements of E indexed by I that is linearly independent over Q. This implies that the X^{β_{j}}’s, j ∈ I, are algebraically independent over A in A[X^{E}], as in Section 27.8. Let {T_{j}}_{j∈I} be a family of distinct commuting indeterminates indexed by I, so that

F (T) = F ({T_{j}}_{j∈I}) ↦ F ({X^{β_{j}}}_{j∈I}) (27.11.6)

defines an injective homomorphism from k[{T_{j}}_{j∈I}] into k[X^{E}], as algebras over k, as in Sections 24.4 and and 27.5. This leads to an injective homomorphism from the field k({T_{j}}_{j∈I}) of formal rational functions in the T_{j}’s, j ∈ I, with coefficients in k into k(X^{E}), as in Section 24.5.

More precisely, if E_{1}(β) is as in (27.10.3), then (27.11.6) defines an isomorphism from k[{T_{j}}_{j∈I}] onto k[X^{E_{1}(β)}], as algebras over k, as in the previous section. This leads to an isomorphism from k({T_{j}}_{j∈I}) onto k(X^{E_{1}(β)}), as a subfield of k(X^{E}).

Suppose now that {β_{j} : j ∈ I} (27.11.7)

is maximal among subsets of E that are linearly independent over Q. If γ ∈ E is not in (27.11.7), then γ may be expressed as a linear combination of the β_{j}’s, j ∈ I, with coefficients in Q. This implies that there is a positive integer l such that l γ ∈ E_{1}(β) − E_{1}(β). (27.11.8)

This means that α′ + l γ = α′′ (27.11.9)

for some α′, α′′ ∈ E_{1}(β). It follows that

(27.11.10) X^{γ} is algebraic relative to k[X^{E_{1}(β)}],

as an element of k[X^{E}], as in Section 27.9.

In particular, (27.11.11) X^{γ} is algebraic over k(X^{E_{1}(β)}),

as an element of k(X^{E}). If γ_{1}, …, γ_{n} are finitely many elements of E, then it follows that the subfield

(k(X^{E_{1}(β)}))(X^{γ_{1}}, …, X^{γ_{n}}) (27.11.12)

of k(X^{E}) generated by k(X^{E_{1}(β)}) and X^{γ_{1}}, …, X^{γ_{n}} is a finite extension of k(X^{E_{1}(β)}), as in Section 19.9. This implies that every element of (27.11.12) is algebraic over k(X^{E_{1}(β)}), as before. This means that

(27.11.13) k(X^{E}) is algebraic over k(X^{E_{1}(β)}),

because every element of k(X^{E}) is contained in a subfield of the form (27.11.12). This type of argument was also mentioned in Section 24.6.

Equivalently,

(27.11.14) every element of k(X^{E}) is algebraically dependent on {X^{β_{j}} : j ∈ I},

as in Section 24.5. This shows that

(27.11.15) {X^{β_{j}} : j ∈ I} is a transcendence basis for k(X^{E}) over k,

as in Section 24.6, because the X^{β_{J}} ’s, j ∈ I, are algebraically independent over k, as before.

A few related results

28.1 Some remarks about graded rings

Let A be a ring with multiplicative identity element e_{A}, and let I be a two-sided ideal in A. The quotient A/I may be considered as a ring, and as a two-sided module over A. Suppose that A is left or right Noetherian as a ring, and thus as a left or right module over itself, as appropriate, as in Section 9.7. This implies that A/I is Noetherian as a left or right module over A, as appropriate, as before. It follows that

(28.1.1) A/I is left or right Noetherian as a ring,

as appropriate, because the left and right ideals in A/I are the same as the submodules of A/I as a left or right module over A, as appropriate.

Suppose now that A is a graded ring, as in Section 9.14, and that A is commutative. Remember that A^{0} is a subring of A that contains e_{A}, so that A may be considered as an associative algebra over A^{0}. Proposition 10.7 on p106 of [8] states that (28.1.2) A is Noetherian as a ring

if and only if (28.1.3) A^{0} is Noetherian as a ring,

and (28.1.4) A is finitely generated as an algebra over A^{0}.

The “if” part follows from Hilbert’s basis theorem, as in Section 29.7.

To get the “only if” part, let A^{+} be the subset of A corresponding to the direct sum of A^{j}, j ≥ 1, and note that A^{+} is an ideal in A. Suppose that (28.1.2) holds, A is Noetherian as a ring, so that (28.1.3) follows from (28.1.1). We also get that

(28.1.5) A^{+} is finitely generated as a module over A, 28.2. FINITE GROUPS OF FIELD AUTOMORPHISMS 643

because A is Noetherian as a module over itself. More precisely, this implies that A^{+} is generated as a module over A by finitely many homogeneous elements x_{1}, …, x_{r}, (28.1.6) where x_{i} is homogeneous of degree l_{i} > 0 for each i = 1, …, r. This can be seen by taking the homogeneous components of any finite set of generators of A^{+}, as in Section 5.9.

Let A′ be the subalgebra of A, as an algebra over A^{0}, generated by e_{A} and (28.1.6). We would like to show that A = A′, (28.1.7) in order to get (28.1.4). It suffices to verify that A^{n} ⊆ A′ (28.1.8) for each nonnegative integer n. Of course, this holds when n = 0, by construction. Suppose that n ≥ 1, and that A^{j} ⊆ A′ (28.1.9) when j < n.

Let y ∈ A^{n} be given, so that y ∈ A^{+}, and thus

y = ∑_{i=1}^{r} a_{i} x_{i} (28.1.10)

for some a_{1}, …, a_{r} ∈ A. More precisely, we can take a_{i} ∈ A^{n−l_{i}} (28.1.11) for each i, with a_{i} = 0 when n < l_{i}. It follows that a_{i} ∈ A′ for each i, as in (28.1.9), so that y ∈ A′.

28.2 Finite groups of field automorphisms

Let K be a field, and let G be a finite group of automorphisms of K with exactly n elements. Note that (28.2.1) k = {u ∈ K : θ(u) = u for every θ ∈ G} is a subfield of K. Under these conditions, [K : k] ≤ n, (28.2.2) as in Theorem 16 on p471 of [24], and the lemma on p460 of [195]. This is also related to Theorem 2 on p194 of [177], and we shall say more about that in Section 28.4. This goes back to Artin, as in [24, 177, 195].

Suppose for the sake of a contradiction that there are n + 1 elements

u_{1}, …, u_{n+1} (28.2.3)

of K that are linearly independent over k. Consider the n homogeneous linear equations x_{1} θ(u_{1}) + ⋯ + x_{n+1} θ(u_{n+1}) = 0 (28.2.4)

in x_{1}, …, x_{n+1} ∈ K, with θ ∈ G. There is a nontrivial solution to these equations, by standard results in linear algebra. Let r be the smallest positive integer for which there is a nontrivial solution to these equations with x_{i} ≠ 0 for r i’s. We can rearrange the u_{i}’s, if necessary, to get a nontrivial solution of these equations with x_{1}, …, x_{r} ≠ 0, x_{r+1}, …, x_{n+1} = 0. (28.2.5)

We may as well take x_{1} = 1, because the equations are homogeneous. Observe that r ≥ 2, because otherwise u_{1} = 0. We also have that x_{i} ∉ k for some i ≤ r, because (28.2.3) are linearly independent over k, and we can take θ to be the identity mapping on K in (28.2.4). We may as well suppose that

x_{2} ∉ k, (28.2.6)

by rearranging the u_{i}’s some more, if necessary. This means that

θ_{0}(x_{2}) ≠ x_{2} (28.2.7)

for some θ_{0} ∈ G, by definition of k.

We can take θ_{0} of both sides of (28.2.4) to get that

θ_{0}(x_{1}) θ_{0}(θ(u_{1})) + ⋯ + θ_{0}(x_{n+1}) θ_{0}(θ(u_{n+1})) = 0. (28.2.8)

This implies that θ_{0}(x_{1}), …, θ_{0}(x_{n+1}) (28.2.9)

is another solution to the earlier system of equations, because G is a group. It follows that θ_{0}(x_{1}) − x_{1}, θ_{0}(x_{2}) − x_{2}, …, θ_{0}(x_{n+1}) − x_{n+1} (28.2.10)

is a solution to this system of equations as well. Of course, θ_{0}(x_{1})−x_{1} = 1−1 = 0, so that at most r − 1 of the terms in (28.2.10) are nonzero. This contradicts the minimality of r, because θ_{0}(x_{2}) − x_{2} ≠ 0, by (28.2.7).

Clearly G is a subgroup of the Galois group Gal(K/k) of K over k, by the definition of the Galois group, as in Section 19.15. Observe that

(28.2.11) k = {u ∈ K : θ(u) = u for every θ ∈ Gal(K/k)}

under these conditions, because k is contained in the right side by definition of the Galois group, and the right side is contained in the right side of (28.2.1).

This means that K is normal over k in the sense defined on p459 of [195], as mentioned in Section 19.15. This implies that

(28.2.12) K is normal and separable over k in the senses used here, so that

(28.2.13) K is a Galois extension of k,

as in Theorem 12 on p459 of [195], and as mentioned in Section 20.8. It follows that the number of elements of Gal(K/k) is equal to [K : k], as before.

This shows that G = Gal(K/k), (28.2.14)

because of (28.2.2). This corresponds to some remarks on p461 of [195].

28.3 A lemma about separable extensions

Let k be a field, and let E be an algebraic extension of k that is separable over k. Suppose that there is a positive integer n such that for each α ∈ E, the degree of α over k is less than or equal to n. Under these conditions, E is a finite extension of k, with [E : k] ≤ n, (28.3.1)

as in Lemma 1 on p194 of [177].

If α ∈ E, then let k(α) be the subfield of E generated by k and α, as usual. Thus [k(α) : k] ≤ n (28.3.2)

for every α ∈ E, by hypothesis. If E is a finite extension of k, then there is a primitive element of E over k, as in Section 21.10. In this case, (28.3.1) follows directly from (28.3.2).

Let α be an element of E such that

[k(α) : k] = m (28.3.3)

is maximal. We would like to show that

E = k(α). (28.3.4)

Suppose for the sake of a contradiction that E ≠ k(α), and let β be an element of E not contained in k(α). Thus the subfield k(α, β) of E generated by k, α, and β contains k(α) as a proper subfield.

Note that k(α, β) is separable over k, as in Section 20.4. We also have that k(α, β) is a finite extension of k, as in Section 19.9. This implies that there is an element γ of k(α, β) such that

k(α, β) = k(γ), (28.3.5)

as in Section 21.8. Of course,

[k(γ) : k] = [k(α, β) : k(α)] [k(α) : k], (28.3.6)

as in Section 19.9. This means that

[k(γ) : k] > m, (28.3.7)

because [k(α, β) : k(α)] > 1, contradicting the maximality of m.

28.4 Another approach to Artin’s result

Let K be a field, let G be a finite group of automorphisms of K with exactly n elements, and let k be the subfield of K consisting of elements of K that are fixed by every element of G, as in (28.2.1). Theorem 2 on p194 of [177] states that K is a finite Galois extension of k with Galois group G and

[K : k] = n. (28.4.1)

This was mentioned in Section 28.2, using another argument, and a result whose proof has not been included here.

Let α ∈ K be given, and let

σ_{1}, …, σ_{r} (28.4.2)

be a maximal list of elements of G such that

σ_{1}(α), …, σ_{r}(α) (28.4.3)

are distinct elements of K. If σ is any element of G, then it follows that σ(α) occurs in (28.4.3) exactly once. If τ ∈ G, then

τ(σ_{1}(α)), ⋯, τ(σ_{r}(α)) (28.4.4)

corresponds to a permutation of (28.4.3). Note that r ≤ n.

Of course, α is a root of the polynomial

f(X) = ∏_{l=1}^{r}(X − σ_{l}(α)) (28.4.5)

in the indeterminate X with coefficients in K. This polynomial may be expressed as

f(X) = ∑_{l=0}^{r} f_{l} X^{l}, (28.4.6)

with f_{l} ∈ K for each l. If τ ∈ G, then we put

f^{τ} (X) = ∑_{l=0}^{r} τ(f_{l}) X^{l}, (28.4.7)

as in Section 19.12. Observe that

f^{τ} (X) = ∏_{l=1}^{r}(X − τ(σ_{l}(α))) = f(X) (28.4.8)

in this case. This means that f_{l} ∈ k (28.4.9)

for each l.

By construction, f(X) is separable, as in Section 20.3. It follows that K is algebraic and separable over k, as in Sections 19.9 and 20.4. We also get that K is normal over k, as in Section 19.15, so that K is Galois over k, as in Section 20.8. The result in the previous section implies that K is a finite extension of k, with degree less than or equal to n. To obtain (28.4.1) and Gal(K/k) = G, one can use the fact that the number of elements of Gal(K/k) is equal to [K : k], as in Section 28.2.

Note that the argument used in this section is similar to one in Section 21.6, corresponding to part of the proof of Proposition 12 on p189 of [177]. The previous argument was only concerned with separability in positive characteristic, although the characteristic was not important for that part. The same argument could be used to get that K is normal over K_{1}, in the previous notation, in any characteristic, as in the argument discussed in this section.

28.5 General and special linear groups

Let A be a ring with a multiplicative identity element e_{A}. The group of invertible elements of A may be denoted G(A). (28.5.1)

Let n be a positive integer, and remember that the space M_{n}(A) of n × n matrices with entries in A is a ring with respect to entrywise addition and matrix multiplication, with multiplicative identity element equal to the identity matrix I_{n}. The group GL(n, A) = G(M_{n}(A)) (28.5.2)

of invertible elements of M_{n}(A) is known as the nth general linear group of matrices with entries in A.

Suppose now that A is commutative, so that the determinant of an element of M_{n}(A) may be defined as an element of A in the usual way. It is well known that an element of M_{n}(A) is invertible if and only if its determinant is invertible as an element of A, so that

GL(n, A) = {a ∈ M_{n}(A) : det a ∈ G(A)}. (28.5.3)

If k is a field, then we get that

GL(n, k) = {a ∈ M_{n}(k) : det a ≠ 0}. (28.5.4)

It is easy to see that the determinant defines a group homomorphism from GL(n, A) onto G(A). The kernel

SL(n, A) = {a ∈ M_{n}(A) : det a = e_{A}} (28.5.5)

of this homomorphism is the nth special linear group of matrices with entries in A.

If V is a vector space over k, then the group

GL(V) (28.5.6) of invertible linear mappings from V onto itself is called the general linear group of V. If V has positive finite dimension, then GL(V) consists of the linear mappings from V into itself with nonzero determinant. More precisely, the determinant defines a group homomorphism from GL(V) onto the multiplicative group of nonzero elements of k in this case. The kernel

SL(V) (28.5.7)

of this homomorphism is the special linear group of V.

Of course, the space k^{n} of n-tuples of elements of k may be considered as a vector space over k, with respect to coordinatewise addition and scalar multiplication. There is a standard isomorphism between the algebra of linear mappings from k^{n} into itself and M_{n}(k), which leads to a group isomorphism between GL(k^{n}) and GL(n, k) that takes SL(k^{n}) onto SL(n, k). Similarly, if V has dimension n, then one can use a basis for V to get an isomorphism between the algebra of linear mappings from V into itself and M_{n}(k), and thus a group isomorphism between GL(V) and GL(n, k) that takes SL(V) onto SL(n, k).

Let e_{1}, …, e_{n} be the standard basis vectors in k^{n}, so that the lth coordinate of e_{j} is equal to 1 when j = l, and to 0 otherwise. A linear mapping T from V into itself is invertible if and only if

T (e_{1}), …, T (e_{n}) (28.5.8)

is a basis for k^{n}, and indeed it suffcies to check that they are linearly independent in k^{n}, by standard results in linear algebra. This is the same as saying that T (e_{1}) ≠ 0, and that T (e_{l}) is not in the linear span of

T (e_{1}), …, T (e_{l−1}) (28.5.9)

when l ≥ 2. If k has only finitely many elements, then one can use this to get that the number of elements of GL(k^{n}) is equal to

∏_{l=1}^{n}((#k)^{n} − (#k)^{l−1}). (28.5.10)

It follows that the number of elements of SL(k^{n}) is equal to

(#k − 1)^{−1} ∏_{l=1}^{n}((#k)^{n} − (#k)^{l−1}). (28.5.11)

28.6 Projective spaces

Let k be a field, and let V be a vector space over k. The space

P (V) (28.6.1)

of one-dimensional linear subspaces of V is called the projective space associated to V. There is an obvious mapping from V \ {0} onto P (V), that sends v in V \ {0} to the line L(v) = L_{V} (v) in V with v ∈ L(v). Note that u, v ∈ V \ {0} satisfy L(u) = L(v) (28.6.2)

if and only if v = t u (28.6.3)

for some t ∈ k \ {0}. If k has only finitely many elements and V has positive finite dimension n, then V has (#k)^{n} elements, and P (V) has

\frac{(#k)^{n} − 1}{#k − 1} (28.6.4)

elements. This is related to parts of Execise 4 on p316 of [24], and Exercise 4 on p596 of [195].

Let W be another vector space over k, and let T be a one-to-one linear mapping from V into W. If v ∈ V \ {0}, then

T (L_{V} (v)) = L_{W} (T (v)), (28.6.5)

which leads to a one-to-one mapping T_{∗} from P (V) into P (W). If T also maps V onto W, then T_{∗} maps P (V) onto P (W), as mentioned on p595 of [195]. In this case, the induced mapping T_{∗} is called a projective isomorphism from P (V) onto P (W), as in [195].

If Z is another vector space over k, and R is a one-to-one linear mapping from W into Z, then R ∘ T is a one-to-one linear mapping from V into Z, and

(R ∘ T)_{∗} = R_{∗} ∘ T_{∗}, (28.6.6)

as mappings from P (V) into P (Z). If I_{V} is the identity mapping on V, then (I_{V})_{∗} is the identity mapping on P (V). Similarly, if T (V) = W, then

(T^{−1})∗ = (T_{∗})^{−1}, (28.6.7)

as mappings from P (W) onto P (V).

In particular, T ↦ T_{∗} (28.6.8)

defines a homomorphism from GL(V) into the group of one-to-one mappings from P (V) onto itself. If t ∈ k \ {0}, then (t I_{V})_{∗} is the identity mapping on P (V). In fact, the kernel of (28.6.8) consists exactly of the elements of GL(V) of the form t I_{V} , (28.6.9)

t ∈ k \ {0}, as in Theorem 33 on p314 of [24], and Theorem 17 on p595 of [195]. The quotient group GL(V)/{t I_{V} : t ∈ k \ {0}} (28.6.10)

may be called the projective general linear group of V, and denoted

P GL(V). (28.6.11)

This notation is used on p595 of [195] for the isomorphic group of induced mappings on P (V), called the projective group.

If n is a positive integer, then the n-dimensional projective group defined on p314 of [24] is the group of induced mappings on P (k^{n+1}). The quotient group

P GL(n, k) = GL(n, k)/{t I_{n} : t ∈ k \ {0}} (28.6.12)

may be called the nth projective general linear group corresponding to k, although this notation is used on p596 of [195] for the group of induced mappings on P (k^{n+1}). The standard isomorphism between GL(k^{n}) and GL(n, k) leads to an isomorphism between P GL(k^{n}) and P GL(n, k). If V has dimension n, then one can use a basis for V to get an isomorphism between GL(V) and GL(n, k), as in the previous section, and this leads to an isomorphism between P GL(V) and P GL(n, k). If k has only finitely many elements, then the number of elements of P GL(k^{n}) is equal to (28.5.11).

If v = (v_{1}, …, v_{n}) ∈ k^{n}, then

(v_{1}, …, v_{n}, 1) (28.6.13)

is a nonzero element of k^{n+1}, that determines an element of P (k^{n+1}). This defines a one-to-one mapping from k^{n} into P (k^{n+1}). Of course,

(v_{1}, …, v_{n}) ↦ (v_{1}, …, v_{n}, 0) (28.6.14)

is an injective linear mapping from k^{n} into k^{n+1}, and this leads to an injective mapping from P (k^{n}) into P (k^{n+1}), as before. Every element of P (k^{n+1}) corresponds to exactly one of these two types.

The n = 1 case will be discussed further in the next section.

28.7 Projective lines

Let k be a field, and let ∞ be a point not in k. It is convenient to put

a + ∞ = ∞ + a = ∞ and a/∞ = 0 (28.7.1)

for every a ∈ k, and a · ∞ = ∞ · a = ∞, a/0 = ∞ (28.7.2)

when a ∈ k \ {0}, as usual.

Let a, b, c, d ∈ k be given, and let T be the linear mapping from k^{2} into itself defined by T (v) = (a v_{1} + b v_{2}, c v_{1} + d v_{2}) (28.7.3)

for each v = (v_{1}, v_{2}) ∈ k^{2}. This corresponds to the matrix \binom{a b}{c d} in the usual way, so that det T = a d − b c. (28.7.4)

Suppose that a d − b c ≠ 0, (28.7.5) so that T is a one-to-one linear mapping from k^{2} onto itself. This leads to a oneto-one mapping T_{∗} from P (k^{2}) onto itself, as in the previous section. Remember that T ↦ T_{∗} is a homomorphism from GL(k^{2}) into the group of one-to-one mappings from P (k^{2}) onto itself, and that the kernel of this homomorphism consists of nonzero multiples of the identity mapping on k^{2}.

If x ∈ k, then T ((x, 1)) = (a x + b, c x + d). (28.7.6)

We would like to put

ϕ(x) = \frac{a x + b}{c x + d}. (28.7.7)

If the denominator is equal to 0, then the numerator is not equal to 0, because (28.7.6) is nonzero, and the right side of (28.7.7) is interpreted as being equal to ∞, as before. If x ≠ 0, then

ϕ(x) = \frac{a + b/x}{c + d/x}, (28.7.8)

and we interpret this as meaning that

ϕ(∞) = a/c. (28.7.9)

This defines an element of k ∪ {∞}, because at least one of a, c is nonzero, by (28.7.5).

We can identify k ∪ {∞} with P (k^{2}), where x ∈ k corresponds to the onedimensional subspace of k^{2} that contains (x, 1), as in the previous section, and ∞ corresponds to k × {0}. Using this identification, ϕ corresponds to T_{∗}. In particular, ϕ is a one-to-one mapping from k ∪ {∞} onto itself.

28.8 Some formal compositions

Let k be a commutative ring with a multiplicative identity element 1 = 1_{k}, let X be an indeterminate, and let A be an associative algebra over k with a multiplicative identity element e_{A}. If f(X) is a formal polynomial in X with coefficients in k and α ∈ A, then f(α) ∈ A may be defined as in Section 4.4, and this defines a homomorphism from k[X] into A, as algebras over k. It is easy to see that this is the unique algebra homomorphism from k[X] into A that sends 1_{k} to e_{A} and X to α.

In particular, we can take A = k[X], so that if a(X) ∈ k[X], then

(f ∘ a)(X) = f(a(X)) (28.8.1)

may be defined as an element of k[X] as before. This may be considered as a formal composition of a(X) and f(X), as formal polynomials in X with coefficients in k. If α ∈ A, then one can check that

(f ∘ a)(α) = f(a(α)). (28.8.2)

652 CHAPTER 28. A FEW RELATED RESULTS

More precisely, it is easy to see that f(X) ↦ (f ∘ a)(α) (28.8.3) is a homomorphism from k[X] into A, as algebras over k, that sends 1_{k} to e_{A} and X to a(α). It follows that this mapping is the same as f(X) ↦ f(a(α)), as in the preceding paragraph.

If b(X) ∈ k[X], then (28.8.2) implies that (f ∘ a)(b(X)) = f((a ∘ b)(X)). (28.8.4) This means that ((f ∘ a) ∘ b)(X) = (f ∘ (a ∘ b))(X). (28.8.5) This may be used to define f(a(b(X))) (28.8.6) as an element of k[X]. If k is an integral domain, and the degrees of f(X) and a(X) are positive, then it is easy to see that (28.8.7) deg f(a(X)) = (deg f(X)) (deg a(X)). Of course, if f(X) or a(X) is a constant polynomial, then f(a(X)) is a constant polynomial as well.

Suppose now that a(X), b(X) ∈ k[X] are of the form a(X) = a_{1} X + a_{0}, b(X) = b_{1} X + b_{0} (28.8.8) with a_{1}, a_{0}, b_{1}, b_{0} ∈ k. Observe that a(b(X)) = a_{1} b_{1} X + a_{1} b_{0} + a_{0} (28.8.9) and b(a(X)) = a_{1} b_{1} X + b_{1} a_{0} + b_{0}. (28.8.10) If a_{1} is invertible in k, then we can take b_{1} = a_{1}^{−1}, b_{0} = −a_{1}^{−1} a_{0} (28.8.11) to get that a(b(X)) = b(a(X)) = X. (28.8.12) This means that f(X) ↦ f(a(X)) (28.8.13) is an automorphism of k[X], as an algebra over k, with inverse f(X) ↦ f(b(X)). (28.8.14) This corresponds to Exercise 7 on p140 of [177].

Every algebra homomorphism from k[X] into itself that sends 1_{k} to itself is of the form (28.8.13) for a unique a(X) ∈ k[X], as before, and in particular every algebra automorphism of k[X] is of this form. If k is an integral domain, then it is easy to see that (28.8.13) is injective if and only if

deg a(X) ≥ 1. (28.8.15)

If (28.8.13) is an automorphism of k[X], then one can check that

deg a(X) = 1, (28.8.16)

so that a(X) is as in (28.8.8), with a_{1} ≠ 0. More precisely, one can verify that

(28.8.17) a_{1} is invertible in k

under these conditions. This corresponds to Exercise 8 on p140 of [177].

28.9 Some irreducible formal polynomials

Let k be a field, let Y, Z be distinct commuting indeterminates, and let f(Z) and g(Z) be nonzero formal polynomials in Z with coefficients in k and no common factors of positive degree. Of course,

f(Z) − Y g(Z) (28.9.1)

may be considered as a formal polynomial in Y with coefficients in the algebra k[Z] of formal polynomials in Z with coefficients in k. Remember that k[Z] is an integral domain, as mentioned in Section 21.12. Using this, it is easy to see that (28.9.1) is not invertible in the algebra (k[Z])[Y] of formal polynomials in Y with coefficients in k[Z].

Suppose for the sake of a contradiction that (28.9.1) is reducible in (k[Z])[Y], so that it can be expressed as the product of two noninvertible elements of (k[Z])[Y]. Observe that one of these elements of (k[Z])[Y] has degree 0 in Y and the other has degree 1 in Y, because (28.9.1) has degree 1 in Y. The element of (k[Z])[Y] of degree 0 in Y corresponds to an element of k[Z] of positive degree in Z, because it is not invertible in k[Z]. This element of k[Z] is a common factor of f(Z) and g(Z), which is not permitted, by hypothesis. This shows that

(28.9.2) f(Z) − Y g(Z) is irreducible in (k[Z])[Y].

Remember that (k[Z])[Y] may be identified with the algebra k[Z, Y] of formal polynomials in Z, Y with coefficients in k, as in Section 21.12. This may also be identified with the algebra (k[Y])[Z] of formal polynomials in Z with coefficients in k[Y]. Thus (28.9.2) is the same as saying that

(28.9.3) f(Z) − Y g(Z) is irreducible in (k[Y])[Z].

It is well known that k[Y] is a principal ideal domain, and thus a unique factorization domain, as mentioned in Section 22.12. Remember that the field k(Y) of formal rational functions in Y with coefficients in k is defined to be the field of fractions of k[Y], as in Section 21.12. Using (28.9.3), we get that

(28.9.4) f(Z) − Y g(Z) is irreducible in (k(Y))[Z],

the algebra of formal polynomials in Z with coefficients in k(Y), as in Section 22.12 again. This is related to some remarks on p157 of [153], and we shall say more about that in the next section.

28.10 Some formal rational functions

Let k be a field, let X be an indeterminate, and remember that k(X) is the field of formal rational functions in X with coefficients in k, as in Section 21.12. Also let η(X) be an element of k(X) that is not in k. Thus

η(X) = f(X)/g(X) (28.10.1)

for some nonzero formal polynomials f(X) and g(X) in X with coefficients in k, and we may suppose that f(X) and g(X) have no common factors of positive degree. Put (28.10.2) n = max(deg f(X), deg g(X)) > 0,

which may be called the degree of η(X), as in Exercise 1 on p229 of [177]. This means that f(X) and g(X) may be expressed as

f(X) = ∑_{l=0}^{n} f_{l} X^{l} (28.10.3)

and

g(X) = ∑_{l=0}^{n} g_{l} X^{l}, (28.10.4)

respectively, where f_{n} or g_{n} is nonzero.

Of course, f(X) − η(X) g(X) = 0, (28.10.5)

so that

∑_{l=0}^{n}(f_{l} − η(X) g_{l}) X^{l} = 0. (28.10.6)

Note that f_{n} − η(X) g_{n} ≠ 0, (28.10.7)

because η(X) is not in k, by hypothesis. Let k(η(X)) be the subfield of k(X) generated by k and η(X). It follows that X is the root of a polynomial equation of degree n with coefficients in k(η(X)), as on p157 of [153]. This means that X is algebraic over k(η(X)), so that k(X) is a finite extension of k(η(X)), as in Section 19.9.

We can use this to get that

(28.10.8) η(X) is transcendental over k,

as in [153]. Indeed, if η(X) were algebraic over k, then X would be algebraic over k.

Let Y, Z be two more distinct commuting indeterminates that are distinct from X and commute with X. Note that k(η(X)) is isomorphic to the field k(Y) of formal rational functions in Y with coefficients in k, in such a way that η(X) corresponds to Y. Let

f(Z) = ∑_{l=0}^{n} f_{l} Z^{l}, g(Z) = ∑_{l=0}^{n} g_{l} Z^{l} (28.10.9)

be the formal polynomials in Z with the same coeficients in k as f(X), g(X). Using (28.9.4), we get that

(28.10.10) f(Z) − η(X) g(Z) is irreducible in (k(η(X)))[Z],

as on p157 of [153].

Note that f(Z) − η(X) g(Z) has degree n as a formal polynomial in Z with coefficients in k(η(X)). It follows that X is algebraic of degree n over k(η(X)), so that [k(X) : k(η(X))] = n, (28.10.11)

as in Theorem 7 on p158 of [153]. This corresponds to part (a) of Exercise 1 on p229 of [177] as well.

28.11 Some more formal compositions

Let k be a field, and let X be an indeterminate. Also let K be a field that contains k as a subfield, so that K may be considered as a commutative associative algebra over k in particular. If f(X) is a formal polynomial in X with coefficients in k and α ∈ K, then f(α) may be defined as an element of K as in Section 4.4. Observe that

S_{α} = {f(X) ∈ k(X) : f(α) ≠ 0} (28.11.1)

is a multiplicative closed subset of k[X], as in Section 12.1, that contains k\{0}. The corresponding ring of fractions S_{α}^{−1} k[X] may be identified with the subring of the field k(X) of formal rational functions in X with coefficients in k consisting of quotients of elements of k[X] and elements of S_{α}, as in Section 14.2. More precisely, this is a subalgebra of k(X), as an algebra over k. If f(X) is an element of this subalgebra, and if f(X) is expressed as a quotient of formal polynomials in X with coefficients in k with no common factors of positive degree, then it is easy to see that the denominator is in S_{α}.

If f(X) ∈ S_{α}^{−1} k[X], (28.11.2)

then f(α) may be defined as an element of K, extending the previous definition for formal polynomials. More precisely, there is a unique ring homomorphism from S_{α}^{−1} k[X] into K extending the previous definition of f(α) for f(X) in k[X], as in Section 12.12. Equivalently, this is the unique homomorphism from S_{α}^{−1} k[X] into K that is equal to the identity mapping on k and sends X to α. Of course, if S_{α} = k[X] \ {0}, (28.11.3)

then S_{α}^{−1} k[X] = k(X). In this case, we get a unique embedding of k(X) into K that is the identity mapping on k and sends X to α.

If f(X) is a formal polynomial in X with coefficients in k and a(X) ∈ k(X), then (f ∘ a)(X) = f(a(X)) (28.11.4)

may be defined as an element of k(X). If a(X) ∈ k[X], then this is the same as in Section 28.8. Suppose that

a(X) ∈ k(X) is not in k, (28.11.5)

so that (28.11.4) is nonzero when f(X) ≠ 0, as in (28.10.8). This implies that (28.11.4) may be defined as an element of k(X) for every f(X) ∈ k(X), as in the preceding paragraph. This is the unique embedding of k(X) into itself that is the identity mapping on k and sends X to a(X), as before.

If a(X) ∈ S_{α}^{−1} k[X], (28.11.6)

then a(α) may be defined as an element of K as before. If f(X) ∈ k[X], then

(f ∘ a)(X) = f(a(X)) ∈ S_{α}^{−1} k[X], (28.11.7)

because S_{α}^{−1} k[X] is a subalgebra of k(X), as an algebra over k. This means that (f ∘ a)(α) may be defined as an element of K as before, and one can verify that (f ∘ a)(α) = f(a(α)). (28.11.8)

Indeed, f(X) ↦ (f ∘ a)(α) (28.11.9)

is a homomorphism from k[X] into K that is the identity mapping on k and sends X to a(α). This implies that this mapping is the same as

f(X) ↦ f(a(α)), (28.11.10)

as in Section 28.8.

Of course, S_{a(α)} may be defined as in (28.11.1). If f(X) ∈ S_{a(α)}, then

(f ∘ a)(α) = f(a(α)) ≠ 0, (28.11.11) which implies that 1/(f ∘ a)(X) ∈ S_{α}^{−1} k[X]. (28.11.12)

If f(X) ∈ S_{a(α)}^{−1} k[X], (28.11.13)

then one can use the previous statement to check that (28.11.7) holds. We also get that (28.11.8) holds under these conditions, where both sides of the equation are defined as elements of K, because of the previous version for formal polynomials. Note that this works when a(X) is an element of k, so that a(X) is the same as a(α), and (28.11.4) may be defined as being equal to f(a(α)).

28.12 More on these formal compositions

Let us continue with the same notation and hypotheses as in the previous section. In particular, we suppose that (28.11.5) holds, so that (28.11.4) is defined as an element of k(X) for every f(X) ∈ k(X). If f(X) is not in k, then is is easy to see that (28.11.4) is not in k.

Let b(X) be another element of k(X) not in k, so that

(a ∘ b)(X) = a(b(X)) (28.12.1)

defines an element of k(X) not in k, as in the preceding paragraph. Similarly, if f(X) ∈ k(X), then

(f ∘ (a ∘ b))(X) = f((a ∘ b)(X)) (28.12.2)

and ((f ∘ a) ∘ b)(X) = (f ∘ a)(b(X)) (28.12.3)

are defined as elements of k(X). These define embeddings of k(X) into itself equal to the identity mapping on k, and sending X to (28.12.1). It follows that these two embeddings are the same, so that

(f ∘ (a ∘ b))(X) = ((f ∘ a) ∘ b)(X) (28.12.4)

for every f(X) ∈ k(X). Alternatively, one can verify that

f((a ∘ b)(X)) = (f ∘ a)(b(X)) (28.12.5)

using some of the remarks in the previous section, with K = k(X) and α = b(X).

The embedding f(X) ↦ (f ∘ a)(X) = f(a(X)) (28.12.6)

of k(X) into itself maps k(X) onto the subfield k(a(X)) generated by k and a(X). Note that every embedding of k(X) into itself that is the identity mapping on k is of this form for a unique a(X) as in (28.11.5). This embedding maps k(X) onto itself exactly when

(28.12.7) a(X) has degree 1 as a formal rational function in X with coefficients in k, as in Section 28.10. Thus every automorphism on k(X) that is the identity mapping on k is of this type. This corresponds to some semarks on p158 of [153], Exercise 9 on p141 of [177], and part (b) of Exercise 1 on p229 of [177].

The condition (28.12.7) means that

a(X) = \frac{a_{1,1} X + a_{1,2}}{a_{2,1} X + a_{2,2}} , (28.12.8)

where a_{1,1}, a_{1,2}, a_{2,1}, a_{2,2} ∈ k, a_{1,1} ≠ 0 or a_{2,1} ≠ 0, and where

a_{1,1} X + a_{1,2} (28.12.9)

and a_{2,1}X + a_{2,2} (28.12.10)

have no common factor of positive degree. This third part is the same as saying that neither of (28.12.9) and (28.12.10) is a constant multiple of the other. One can check that this happens if and only if

det \binom{a_{1,1} a_{1,2}}{a_{2,1} a_{2,2}} = a_{1,1} a_{2,2} − a_{1,2} a_{2,1} ≠ 0, (28.12.11)

as in [153, 177]. Of course, if t ∈ k and t ≠ 0, then

a(X) = \frac{t a_{1,1} X + t a_{1,2}}{t a_{2,1} X + t a_{2,2}} . (28.12.12)

One can verify that these are the only ways to express the same element of k(X) of degree 1.

Let

b(X) = \frac{b_{1,1} X + b_{1,2}}{b_{2,1} X + b_{2,2}} (28.12.13)

be another element of k(X) of degree 1, and let

c(X) = (a ∘ b)(X) = a(b(X)) (28.12.14)

be as in (28.12.1). It is well known and not difficult to see that

c(X) = \frac{c_{1,1} X + c_{1,2}}{c_{2,1} X + c_{2,2}} , (28.12.15)

with

\binom{c_{1,1} c_{1,2}}{c_{2,1} c_{2,2}} = \binom{a_{1,1} a_{1,2}}{a_{2,1} a_{2,2}} \binom{b_{1,1} b_{1,2}}{b_{2,1} b_{2,2}}. (28.12.16)

Formal series and other matters

29.1 Some formal series

Let k be a commutative ring with a multiplicative identity element, let V be a module over k, and let T be an indeterminate. Consider the space V ((T)) of formal series of the form

f(T) = ∑_{j=j_{0}}^{∞} f_{j} T^{j}, (29.1.1)

where j_{0} ∈ Z and f_{j} ∈ V for each j ≥ j_{0}. More precisely, we consider f_{j} to be defined as an element of V for every integer j, with f_{j} = 0 when j < j_{0}. This type of formal series is discussed on p27 of [52], and on p285 of [195], with coefficients in a field. A formal series of this type is called an extended formal power series in [195].

The space c(Z, V) of all V -valued functions may be considered as a module over k, with respect to pointwise addition and scalar multiplication of functions, as in Section 1.2. We may define V ((T)) as the subset of c(Z, V) whose elements are equal to 0 at all but finitely many negative integers, and which may be expressed as in (29.1.1). This is a submodule of c(Z, V), so that V ((T)) may be considered as a module over k as well, with respect to termwise addition and scalar multiplication of these formal series. An element of V ((T)) be also be expressed as

f(T) = ∑_{j>>−∞} f_{j} T^{j}, (29.1.2)

to indicate that f_{j} = 0 for all but finitely many j < 0, as on p27 of [52].

Let A be an algebra over k in the strict sense, as in Section 20.13, so that A((T)) may be defined as a module over k, as before. Let f((T)) ∈ A((T)) be as in (29.1.1), and let

g(T) = ∑_{l>>−∞} g_{l} T^{l} (29.1.3)

be another element of A((T)). We would like to put

h_{n} = ∑_{j+l=n} f_{j} g_{l} (29.1.4)

for each n ∈ Z, where more precisely the sum is taken over all j, l ∈ Z such that j + l = n. It is easy to see that all but finitely many terms in the sum are equal to 0, because f_{j} = 0 for all but finitely many j < 0, and g_{l} = 0 for all but finitely many l < 0. Thus the sum is defined as an element of A for every n ∈ Z, and one can check that

(29.1.5) h_{n} = 0 for all but finitely many n < 0.

It follows that

h(T) = ∑_{n>>−∞} h_{n} T^{n} (29.1.6)

defines an element of A((T)), and we put

f(T) g(T) = h(T). (29.1.7)

It is easy to see that this definition of multiplication on A((T)) is bilinear over k, so that A((T)) is an algebra in the strict sense over k.

Remember that A[[T]] is the space of formal power series in T with coefficients in A, as in Sections 4.3 and 21.13. This may be identified with the subset of A((T)) consisting of formal series for which the coefficient of T^{j} is equal to 0 for every j < 0. Using this identification, A[[T]] corresponds to a subalgebra of A((T)). Similarly, we can identify A with the subset of A((T)) consisting of formal series for which the coefficient of T^{j} is equal to 0 when j ≠ 0, which is compatible with the usual identification of A with a subalgebra of A[[T]].

If multiplication on A is associative, then one can check that multiplication on A((T)) is associative. Similarly, if multiplication on A is commutative, then multiplication on A((T)) is commutative too. If A has a multiplicative identity element e_{A}, then this is the multiplicative identity element in A((T)) as well.

29.2 More on these formal series

Let us continue with the same notation and hypotheses as in the previous section. Thus k((T)) is a commutative associative algebra over k. If V is a module over k again, then the product of an element of k((T)) and an element of V ((T)) may be defined in the same way as before, so that V ((T)) is a module over k((T)). Note that V [[T]] may be considered as a submodule of V ((T)), as a module over k[[T]].

Suppose that A is an associative algebra over k with a multiplicative identity element e_{A}. If V is a left or right module over A, then the product of an element of V ((T)) and an element of A((T)) on the left or right, as appropriate, may be defined in the same way as before, so that V ((T)) is a left or right module over A((T)), as appropriate. We also have that V [[T]] may be considered as a submodule of V ((T)), as a left or right module over A[[T]], as appropriate. If V is a two-sided module over A, then V ((T)) is a two-sided module over A((T)).

Of course, invertible elements of A[[T]] are invertible in A((T)). Remember that an element of A[[T]] is invertible in A[[T]] when the coefficient of T^{j} is invertible in A for j = 0, as in Section 4.5. If f(T) ∈ A((T)) is as in (29.1.1), and if f_{j_{0}} is invertible as an element of A, then it follows that

(29.2.1) f(T) is invertible in A((T)).

More precisely, f(T) T^{−j_{0}} is invertible as an element of A[[T]] in this case, and

f(T)^{−1} = (f(t) T^{−j_{0}})^{−1} T^{−j_{0}}. (29.2.2)

If A is a division algebra, then we get that

(29.2.3) A((T)) is a division algebra

too.

Suppose that V is a module over k, and that f(T) ∈ V ((T)) is as in (29.1.1). The formal derivative of f(T) may be defined as an element of V ((T)) by

f′(T) = ∂f(T) = ∑_{j=j_{0}}^{∞} j · f_{j} T^{j−1} = ∑_{j=j_{0}−1}^{∞}(j + 1) · f_{j+1} T^{j}. (29.2.4)

This is the same as in Section 4.6 when f(T) ∈ V [[T]]. Note that this defines a homomorphism from V ((T)) into itself, as a module over k.

If A is an algebra in the strict sense over k and f(T), g(T) ∈ A((T)), then one can check that

∂(f(T) g(T)) = (∂f(T)) g(T) + f(T) (∂g(T)), (29.2.5)

so that ∂ defines a derivation on A((T)), as in Section 21.13. There is an anlogous statement for the product of an element of k((T)) and V ((T)). If A is an associative algebra over k with a multiplicative identity element, and V is a left or right module over A, then there is also an analogous statement for the product of an element of V ((T)) and an element of A((T)) on the left or right, as appropriate.

29.3 Some derivations on formal series

Let k be a commutative ring with a multiplicative identity element, let T be an indeterminate, and let B be an algebra over k in the strict sense. Also let A be a subalgebra of B, and let δ_{A} be a derivation from A into B. If f(T) ∈ A((T)) is as in (29.1.1), then put

δ̂_{A}(f(T)) = ∑_{j=j_{0}}^{∞} δ_{A}(f_{j}) T^{j}, (29.3.1)

which is an element of B((T)). One can check that

(29.3.2)  δ̂_{A} is a derivation from A((T)) into B((T)).

Of course, δ̂_{A} = δ_{A} on A, (29.3.3)

by construction. This is analogous to some of the remarks in Section 23.10, with n = 1. In particular, this is the same as before when f(T) ∈ A[[T]].

Suppose now that δ_{A} is a derivation from A into B((T)), and that

δ_{A}(A) ⊆ T^{r} B[[T]] (29.3.4)

for some r ∈ Z. Under these conditions, one can define (29.3.1) as an element of B((T)), where the coefficient of T^{l} is equal to 0 unless

l ≥ j_{0} + r. (29.3.5)

This is because the coefficient of T^{l} in δ_{A}(f_{j}) T^{j} is equal to 0 unless j ≥ j_{0} and j + r ≤ l. One can check that (29.3.2) and (29.3.3) hold, as before. If δ_{A} is a derivation from A into B[[T]], then (29.3.4) holds with r = 0, and this definition of δ̂_{A} is the same as in Section 23.10 on A[[T]].

Now let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V_{0} be a module over k that is a two-sided module over A, so that V_{0}((T)) is a two-sided module over A((T)), as mentioned in the previous section. If δ_{A} is a derivation from A into V_{0} and f(T) ∈ A((T)) is as in (29.1.1), then δ̂_{A}(f(T)) ∈ V_{0}((T)) (29.3.6)

may be defined as in (29.3.1), and satisfies (29.3.3). One can verify that

(29.3.7)  δ̂_{A} is a derivation from A((T)) into V_{0}((T)),

as before. This is anlaogous to some remarks in Section 23.10 again, with n = 1, and in particular this is the same as before on A[[T]].

Suppose that δ_{A} is a derivation from A into V_{0}((T)), and that

δ_{A}(A) ⊆ T^{r} V_{0}[[T]] (29.3.8)

for some r ∈ Z. If f(T) ∈ A((T)), then one can define (29.3.1) as an element of V ((T)), in the same way as before. Clearly (29.3.3) holds in this case, and one can verify that (29.3.7) holds too. If δ_{A} is a derivation from A into V_{0}[[T]], then (29.3.8) holds with r = 0, and this definition of δ̂_{A} is the same as in Section 23.10 on A[[T]].

29.4. MORE ON DERIVATIONS ON A((T)) 663

29.4 More on derivations on A((T))

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let T be an indeterminate. Also let V be a two-sided module over A((T)), and let v be an invariant element of V, as in Section 23.13. Under these conditions, L_{v}(f(T)) = (∂f(T)) · v = v · (∂f(T)) (29.4.1) is a derivation from A((T)) into V, as in Section 23.14.

Let V_{0} be a module over k that is a two-sided module over A, so that V_{0}((T)) may be considered as a two-sided module over A((T)), as in Section 29.2. If u(T) ∈ V_{0}((T)) and a(T) ∈ A((T)) has coefficients in the centralizer C_{A}^{V_{0}}(V_{0}) of V_{0} in A, as in Section 23.13, then a(T) · u(T) = u(T) · a(T). (29.4.2) This means that (29.4.3) a(T) is in the centralizer of V_{0}((T)) in A((T)).

Let δ be a derivation from A((T)) into V_{0}((T)). Observe that (29.4.4) v(T) = δ(e_{A} T) is an invariant element of V_{0}((T)), as in Section 23.13, because e_{A} T is an element of the center of A((T)), and of the centralizer of V_{0}((T)) in A((T)). This implies that L_{v(T)}(f(T)) = (∂f(T)) · v(T) = v(T) · (∂f(T)) (29.4.5) is a derivation from A((T)) into V_{0}((T)), as before.

Suppose that δ(A[[T]]) ⊆ T^{−r_{1}} V_{0}[[T]] (29.4.6) for some nonnegative integer r_{1}. In particular, this means that δ(e_{a} T) ∈ T^{−r_{1}} V_{0}[[T]]. (29.4.7) If a(T) ∈ A[[T]] and l_{0} ∈ Z, then one can check that δ(a(T) T^{l_{0}}) ∈ T^{l_{0}−1−2 r_{1}} V_{0}[[T]]. (29.4.8)

Let δ_{A} be the restriction of δ to A, which is a derivation from A into V_{0}((T)), as a two-sided module over A. Note that δ_{A} satisfies (29.3.8) with r = −r_{1}, because of (29.4.6). Thus we get an extension δ̂_{A} of δ_{A} to a derivation from A((T)) into V_{0}((T)), as in the previous section.

We would like to check that δ = δ̂_{A} + L_{v(T)} (29.4.9) on A((T)) under these conditions. This equation holds on A and at e_{A} T, by construction. It follows that this equation holds at e_{A} T^{l} for every integer l, because both sides are derivations from A into V_{0}((T)).

To get that (29.4.9) holds on A((T)), one can use an approximation argument. This also uses (29.4.8), and analogous properties of δ̂_{A} and L_{v(T)}. This is analogous to some of the remarks in Section 23.15, with n = 1.

29.5 Some derivations on A[[T]]

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let T be an indeterminate. Also let V be a two-sided module over A((T)), which may be considered as a two-sided module over A[[T]] in particular. If v ∈ V is an invariant element of V as a two-sided module over A[[T]], as in Section 23.13, then it is easy to see that

(29.5.1) v is an invariant element of V, as a two-sided module over A((T)).

Let V_{0} be a module over k that is a two-sided module over A, so that V_{0}((T)) may be considered as a two-sided module over A((T)), as in Section 29.2. In particular, V_{0}((T)) may be considered as a two-sided module over A[[T]]. If δ is a derivation from A[[T]] into A((T)), as a two-sided module over A[[T]], then

(29.5.2) v(T) = δ(e_{A} T) is an invariant element of V_{0}((T)),

as a two-sided module over A[[T]], as in Section 23.13, because e_{A} T is an element of the center of A[[T]], and of the centralizer of V_{0}((T)) in A[[T]]. This implies that (29.5.2) holds with V_{0}((T)) considered as a two-sided module over A((T)), as in the preceding paragraph.

Suppose that (29.4.6) holds for some nonnegative integer r_{1}. This implies that (29.4.8) holds for every nonnegative integer r, as before.

Let δ_{A} be the restriction of δ to A, which is a derivation from A into V_{0}((T)), as a two-sided module over A. We can get an extension δ̂_{A} of δ_{A} to a derivation from A((T)) into V_{0}((T)) as in Section 29.3, for the same reason as in the previous section.

Using v(T), we get a derivation L_{v(T)} from A((T)) into V_{0}((T)) as in (29.4.5). One can check that δ = δ̂_{A} + L_{v(T)} (29.5.3)

on A[[T]], in basically the same way as in the previous section. Note that this gives an extension of δ to a derivation from A((T)) into V_{0}[[T]]. One can also get such an extension more directly, as we shall see in a moment.

Let V be any two sided module over A((T)), and let δ_{0} be a derivation from A((T)) into V. If δ_{0} = 0 on A[[T]], (29.5.4)

then it is easy to see that δ_{0} = 0 on A((T)), (29.5.5) using the remarks in Section 23.3. It follows that a derivation from A[[T]] into V has at most one extension to a derivation from A((T)) into V.

Suppose now that

(29.5.6) e_{A} T is an element of the centralizer of V in A((T)),

as in Section 23.13. Of course, this condition is satisfied by V_{0}((T)). Let δ be a derivation from A[[T]] into V, as a two-sided module over A[[T]]. Note that

(29.5.7) v = δ(e_{A} T) is an invariant element of V,

as in Section 23.13, because e_{A}T is also an element of the center of A[[T]].

Under these conditions, there is an extension of δ to a derivation from A((T)) into V. In fact, any such extension satisfies

(29.5.8) δ(e_{A} T^{−1}) = −(e_{A} T)^{−1} · δ(e_{A} T) · (e_{A} T)^{−1}
= −(e_{A} T)^{−2} · δ(e_{A} T) = −δ(e_{A} T) · (e_{A} T)^{−2},

where the first step is as in as in Section 23.3. Similarly, any such extension satisfies

δ(e_{A} T^{−l}) = −l (e_{A} T)^{−l−1} · δ(e_{A} T) = −l δ(e_{A} T) · (e_{A} T)^{−1−1} (29.5.9)

for every positive integer l. One can define an extension of δ to A((T)) directly with these properties, and check that it is a derivation.

29.6 Some remarks about modules

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V be a module over k that is a two-sided module over A, as in Section 23.3. Remember that v ∈ V is said to be an invariant element of V if

y · v = v · y (29.6.1)

for every y ∈ A, as in Section 23.13. Note that

(29.6.2) the collection of invariant elements of V is a submodule of V, as a module over k.

Of course, A may be considered as a two-sided module over itself. In this case, the invariant elements of A are the same as the elements of the center of A.

Let U, W be modules over k that are left modules over A, and consider the space Hom_{k}(U, W) of homomorphisms from U into W, as modules over k. Remember that Hom_{k}(U, W) is a module over k that may be considered as both a left and right module over A, as in Sections 1.7 and 1.8. It is easy to see that these actions of A on Hom_{k}(U, W) on the left and right commute with each other, so that Hom_{k}(U, W) may be considered as a two-sided module over A. In this case, the invariant elements of Hom_{k}(U, W) are the same as the homomorphisms from U into W, as left modules over A. There are analogous statements for right modules over A, as usual.

Suppose now that A is commutative. If V is a module over k that is a onesided module over A, then V may be considered as a two-sided module over A, where the actions of A on V on the left and the right are the same. This means that every element of V is an invariant element of V, as a two-sided module over A, by construction.

Let T be an indeterminate, and consider the algebra k[T] of formal polynomials in T with coefficients in k, as in Section 4.3. If V is a module over k, then Hom_{k}(V, V) is an associative algebra over k with a multiplicative identity element, which is the identity mapping on V. If ϕ is a homomorphism from V into itself, as a module over k, and f(T) ∈ k[T], then f(ϕ) may be defined as a homomorphism from V into itself, as in Section 4.4. Using this, V may be considered as a left module over k[T], where

f(T) · v = (f(ϕ))(v) (29.6.3)

for every f(T) ∈ k[T] and v ∈ V.

Similarly, if ψ is a homomorphism from V into itself, a a module over k, then V may be considered as a right module over k[T], where

v · g(T) = (g(ψ))(v) (29.6.4)

for every g(T) ∈ k[T] and v ∈ V. If

ϕ ∘ ψ = ϕ ∘ ϕ (29.6.5)

on V, then f(ϕ) ∘ g(ψ) = g(ψ) ∘ f(ϕ) (29.6.6) on V for every f(T), g(T) ∈ k[T]. This means that these two actions of k[T] on V on the left and on the right commute with each other, so that V may be considered as a two-sided module over k[T] in this way.

Under these conditions, v ∈ V is an invariant element of V exactly when

(f(ϕ))(v) = (f(ψ))(v) (29.6.7)

for every f(T) ∈ k[T]. This is the same as saying that ϕ^{n}(v) = ψ^{n}(v) for every positive integer n. If n ≥ 2, then

ϕ^{n} − ψ^{n} = ( ∑_{l=1}^{n−1} ϕ^{l} ∘ ψ^{n−1−l}) ∘ (ϕ − ψ), (29.6.8)

because of (29.6.5). This implies that

ker(ϕ − ψ) ⊆ ker(ϕ^{n} − ψ^{n}). (29.6.9)

This means that the set of invariant elements of V is equal to

ker(ϕ − ψ). (29.6.10)

29.7 Hilbert’s basis theorem

Let A be a ring with a multiplicative identity element e_{A}, and let X be an indeterminate, so that the corresponding ring A[X] of formal polynomials in X with coefficients in A may be defined as in Section 4.3. If

(29.7.1) A is left Noetherian,

as in Section 9.7, then Hilbert’s basis theorem states that

(29.7.2) A[X] is left Noetherian,

as on p171 of [151]. This corresponds to Theorem 7.5 on p81 of [8] and Theorem 1 on p144 of [177] when A is commutative. Of course, there is an analogous statement when A is right Noetherian.

Let L be a left ideal in A[X], and for each nonnegative integer n, let I_{n}(L) be the set of b ∈ A for which there is an element of L of the form

∑_{j=0}^{n} b_{j} X^{j}, (29.7.3)

with b = b_{n}. Observe that

(29.7.4) I_{n}(L) is a left ideal in A.

It is easy to see that I_{n}(L) ⊆ I_{n+1}(L), (29.7.5)

by multiplying (29.7.3) by X.

Because of (29.7.1), there is a nonnegative integer N such that

I_{n}(L) = I_{N}(L) (29.7.6)

when n ≥ N. We also have that I_{n}(L) is finitely generated as a left ideal in A for each n. Let b_{n,1}, …, b_{n,m_{n}} (29.7.7)

be finitely many elements of I_{n}(L) that generate I_{n}(L) as a left ideal in A for each n = 0, 1, 2, …, N. This means that for each n = 0, 1, 2, …, N and l = 1, …, m_{n} there is an f_{n,l}(X) ∈ L of the form

(29.7.8) f_{n,l}(X) = b_{n,l} X^{n} + lower-order terms.

We would like to show that L is generated by these polynomials, as a left ideal in A[X].

Let (29.7.9) g(X) = c_{r} X^{r} + lower-order terms

be an element of L for some r ≥ 0. If r ≤ N, then c_{r} can be expressed as

c_{r} = a_{r,1} b_{r,1} + ⋯ + a_{r,m_{r}} b_{r,m_{r}} (29.7.10) for some a_{r,1}, …, a_{r,m_{r}} ∈ A. This implies that

g(X) − ∑_{l=1}^{m_{r}} a_{r,l} f_{r,l}(X) (29.7.11)

is an element of L with degree strictly less than r, as a formal polynomial in X. This means that (29.7.11) is equal to 0 when r = 0. Similarly, if r > N, then

c_{r} = a_{r,1} b_{N,1} + ⋯ + a_{r,m_{N}} b_{N,m_{N}} (29.7.12)

for some a_{r,1}, …, a_{r,m_{N}} ∈ A. It follows that

g(X) − ∑_{l=1}^{m_{N}} a_{r,l} X^{r−N} f_{N,l}(X) (29.7.13)

is an element of L with degree strictly less than r. Thus we can repeat the process to get that g(X) is an element of the left ideal in A[X] by the polynomials f_{n,l}(X).

Let X_{1}, …, X_{m} be finitely many distinct commuting indeterminates, so that the ring A[X_{1}, …, X_{m}] of formal polynomials in X_{1}, …, X_{m} with coefficients in A may be defined as in Section 4.3 again. If (29.7.1) holds, then

(29.7.14) A[X_{1}, …, X_{m}] is left Noetherian,

as in Corollary 1 on p172 of [151]. This corresponds to Corollary 7.6 on p81 of [8] when A is commutative. This uses the fact that A[X_{1}, …, X_{m}] is isomorphic to (A[X_{1}, …, X_{m−1}])[X_{m}] when m ≥ 2, as in Section 21.12. There is an analogous statement when A is right Noetherian, as before.

In particular, if A is a division ring, then

(29.7.15) A[X_{1}, …, X_{m}] is both left and right Noetherian,

as in Corollary 2 on p172 of [151]. If A is a principal ideal domain, then we also get that A[X_{1}, …, X_{m}] is Noetherian.

Suppose now that A is commutative, and that B is a finitely-generated associative algebra over A with a multiplicative identity element. This implies that there is a homomorphism from A[X_{1}, …, X_{m}] onto B, as algebras over A, for some m. If A is Noetherian, so that A[X_{1}, …, X_{m}] is Noetherian, then

(29.7.16) B is Noetherian

as a ring as well. This corresponds to Corollary 7.7 on p81 of [8], and to the corollary on p145 of [177]. More precisely, B is Noetherian as a module over A[X_{1}, …, X_{m}], as in Section 9.7, and thus as a module over itself.

29.8 Noetherian conditions and power series

Let A be a ring with a multiplicative identity element e_{A} again, and let X be an indeterminate, so that the corresponding ring A[[X]] of formal power series in X with coefficients in A may be defined as in Section 4.3. If A is left Noetherian, then (29.8.1) A[[X]] is left Noetherian,

as in Exercise 2 on p172 of [151]. This is mentioned in a remark on p81 of [8] when A is commutative, and it corresponds to Theorem 2 on p147 in this case. Essentially the same argument works without asking that A be commutative. This is also analogous to the argument in the previous section, except that we consider terms of lowest degree in formal power series in X, as mentioned in [8, 177].

Let L be a left ideal in A[[X]], and for each nonnegative integer n, let K_{n}(L) be the set of b ∈ A for which there is an element of L of the form

∑_{j=n}^{∞} b_{j} X^{j}, (29.8.2)

with b = b_{n}. It is easy to see that

(29.8.3) K_{n}(L) is a left ideal in A

and K_{n}(L) ⊆ K_{n+1}(L), (29.8.4)

as before. If A is left Noetherian, then there is a nonnegative integer N such that K_{n}(L) = K_{N}(L) (29.8.5)

when n ≥ N. We also have that K_{n}(L) is finitely generated as a left module over A for each n.

Let b_{n,1}, …, b_{n,m_{n}} (29.8.6)

be finitely many elements of K_{n}(L) that generate K_{n}(L) as a left module over A for each n = 0, 1, 2, …, N. Thus for each n = 0, 1, 2, …, N and l = 1, …, m_{n} there is an f_{n,l}(X) ∈ L of the form

(29.8.7) f_{n,l}(X) = b_{n,l} X^{n} + higher-order terms.

We would like to show that L is generated by these formal power series, as a left ideal in A[[X]].

Let (29.8.8) g(X) = c_{r} X^{r} + higher-order terms

be an element of L for some r ≥ 0. Suppose for the moment that r ≤ N. As before, c_{r} may be expressed as

c_{r} = a_{r,1} b_{r,1} + ⋯ + a_{r,m_{r}} b_{r,m_{r}} (29.8.9) for some a_{r,1}, …, a_{r,m_{r}} ∈ A. Under these conditions,

g(X) − ∑_{l=1}^{m_{r}} a_{r,l} f_{r,l}(X) (29.8.10)

is an element of L for which the coefficient of X^{j} is equal to 0 for j = 0, 1, 2, …, r. Repeating the process, we can reduce to the case where r > N.

If r > N, then c_{r} = a_{r,1} b_{N,1} + ⋯ + a_{r,m_{N}} b_{N,m_{N}} (29.8.11)

for some a_{r,1}, …, a_{r,m_{N}} ∈ A. This implies that the coefficient of X^{j} in

g(X) − ∑_{l=1}^{m_{N}} a_{r,l} X^{r−N} f_{N,l}(X) (29.8.12)

is equal to 0 for j = 0, 1, 2, …, r. We can repeat the process to get

a_{t,1}, …, a_{t,m_{N}} ∈ A (29.8.13)

for each t ≥ r, with

g(X) = ∑_{l=1}^{m_{N}} ( ∑_{t=r}^{∞} a_{t,r} X^{t−N}) f_{N,l}(X), (29.8.14)

as in [177].

Let X_{1}, …, X_{m} be finitely many distinct commuting indeterminates, so that the ring A[[X_{1}, …, X_{m}]] of formal power series in X_{1}, …, X_{m} with coefficients in A may be defined as in Section 4.3. If A is left Noetherian, then

(29.8.15) A[[X_{1}, …, X_{m}]] is left Noetherian,

as in the corollary on p147 of [177]. This also corresponds to Corollary 10.27 on p113 of [8] when A is commutative.

More precisely, if m ≥ 2, then one can check that A[[X_{1}, …, X_{m}]] is isomorphic to (A[[X_{1}, …, X_{m−1}]])[[X_{m}]]. (29.8.16)

This is analogous to the corresponding statement for formal polynomials mentioned in Section 21.12. There are analogous statements when A is right Noetherian, as usual. In particular, one can take A to be a division ring, or a principal ideal domain, as before.

29.9 Some remarks about Noetherian rings

Let A be a ring with a multiplicative identity element e_{A}, and let B, C be subrings of A with C ⊆ B (29.9.1)

29.9. SOME REMARKS ABOUT NOETHERIAN RINGS 671 and e_{A} ∈ C. (29.9.2) Thus A may be considered as a left and right module over B, and A, B may be considered as left and right modules over C. Suppose from now on in this section that (29.9.3) C is left Noetherian as a ring. Of course, there are analogous statements when C is right Noetherian as a ring.

Suppose that (29.9.4) B is finitely generated, as a left module over C. This implies that (29.9.5) B is Noetherian, as a left module over C, as in Section 9.7. One can use this to get that (29.9.6) B is left Noetherian as a ring, because every left ideal in B is a submodule of B, as a left module over C.

Similarly, if V is a left module over B, then V may be considered as a left module over C. If (29.9.7) V is finitely generated as a left module over C, then (29.9.8) V is Noetherian, as a left module over C. It follows that (29.9.9) V is Noetherian, as a left module over B, because every submodule of V, as a left module over B, is a submodule of V, as a left module over C.

In particular, if (29.9.10) A is finitely generated as a left module over C, then (29.9.11) A is Noetherian, as a left module over C. This implies that (29.9.12) A is Noetherian, as a left module over B, as before.

29.10 Sums of rings

Let I be a nonemtpy set, and let A_{j} be a ring for each j ∈ I. The Cartesian product

∏_{j∈I} A_{j} (29.10.1)

is also a ring, with respect to coordinatewise addition and multiplication. The direct sum

⨁_{j∈I} A_{j} (29.10.2)

consists of the elements of (29.10.1) for which all but finitely many coordinates are equal to 0, and is a subring of (29.10.1). More precisely, (29.10.2) is an ideal in (29.10.1). Of course, (29.10.1) and (29.10.2) are the same when I has only finitely many elements.

Suppose that A_{j} has a multiplicative identity element e_{A_{j}} for each j ∈ I, and let e be the element of (29.10.1) whose jth coordinate is equal to e_{A_{j}} for each j ∈ I. This is the multiplicative identity element of (29.10.1).

Let n be a positive integer, and suppose now that A_{j} is a ring for each j = 1, …, n, with multiplicative identity element e_{A_{j}}. Thus

∏_{j=1}^{n} A_{j} (29.10.3)

is a ring, with multiplicative identity element

e = (e_{A_{1}}, …, e_{A_{n}}), (29.10.4)

as before. If I_{j} is a left ideal in A_{j} for each j = 1, …, n, then

∏_{j=1}^{n} I_{j} (29.10.5)

is a left ideal in (29.10.3). Conversely, one can check that every left ideal in (29.10.3) is of this form. There are analogous statements for right ideals and two-sided ideals.

Using this, one can verify that

(29.10.6)  ∏_{j=1}^{n} A_{j} is left Noetherian

as a ring if and only if (29.10.7) A_{j} is left Noetherian

as a ring for each j. There is an analogous statement for the right Noetherian property, as usual.

If 1 ≤ l ≤ n, then put

Ã_{l} = {a ∈ ∏_{j=1}^{n} A_{j} : a_{j} = 0 when j ≠ l}, (29.10.8)

which is a two-sided ideal in (29.10.3). Of course, Ã_{l} is isomorphic to A_{l} as a ring in an obvious way. We may consider Ã_{l} as a left and right module over (29.10.3), and (29.10.3) is isomorphic to the direct sum of Ã_{l}, l = 1, …, n, as a left or right module over (29.10.3). This implies that

(29.10.9) Ã_{l} is projective as a left or right module over ∏_{j=1}^{n} A_{j}

for each l = 1, …, n, as in Section 2.7.

Note that a submodule of Ã_{l}, as a left or right module over (29.10.3), corresponds exactly to a left or right ideal in A_{l}. One can check that A_{l} is Noetherian as a left or right module over (29.10.3), if and only if A_{l} is left or right Noetherian as a ring, as appropriate.

If 1 ≤ l_{1} ≠ l_{2} ≤ n, then one can verify that a tensor product of Ã_{l_{1}}, as a right module over (29.10.3), and Ã_{l_{2}}, as a left module over (29.10.3), is equal to {0}. Similarly, the only homomorphism from Ã_{l_{1}} into Ã_{l_{2}}, as both left or right modules over (29.10.3), is 0.

Let A be a ring, and suppose now that A_{j} = A for each j = 1, …, n. Note that a ↦ (a, …, a) (29.10.10)

is an injective ring homomorphism from A into (29.10.3). Let

Â_{n} = {(a, …, a) : a ∈ A} (29.10.11)

be the image of A under this homomorphism, which is a subring of (29.10.5). We may consider (29.10.3) as a left or right module over Â_{n}, which corresponds to the direct sum of n copies of A, as a left or right module over itself, as appropriate.

29.11 Some more spaces of matrices

Let k be a commutative ring with a multiplicative identity element, and let A be a module over k. Also let m, n be positive integers, and let M_{m,n}(A) be the space of m × n matrices with entries in A. This is a module over k, with respect to entrywise addition and scalar multiplication of matrices. If m = n, then this may be denoted M_{n}(A), as before.

Suppose that A is an algebra in the strict sense over k, as in Section 20.13. If n_{1} is another positive integer, a ∈ M_{m,n}(A), and b ∈ M_{n,n_{1}}(A), then a b may be defined as an element of M_{m,n_{1}}(A) by

(a b)_{j,r} = ∑_{l=1}^{n} a_{j,l} b_{l,r} (29.11.1)

for j = 1, …, m and r = 1, …, n_{1}, as usual. Of course,

(a, b) ↦ a b (29.11.2)

is bilinear over k as a mapping from M_{m,n}(A) × M_{n,n_{1}}(A) into M_{m,n_{1}}(A). In particular, M_{n}(A) is an algebra over k in the strict sense with respect to matrix multiplication.

Suppose from now on in this section that A has a multiplicative identity element e_{A}. The corresponding identity matrices I_{m}, I_{n} in M_{m}(A), M_{n}(A), respectively, may be defined in the usual way, as in Section 4.7. In this case, it is easy to see that I_{m} a = a I_{n} = a (29.11.3)

for every a ∈ M_{m,n}(A).

Suppose also for the rest of this section that A is associative. If n_{2} is another positive integer, and c ∈ M_{n_{1},n_{2}}(A), then

(a b) c = a (b c), (29.11.4)

as elements of M_{m,n_{2}}(A). In particular, this means that M_{n}(A) is an associative algebra over k.

Using (29.11.3) and (29.11.4), we get that M_{m,n}(A) is a left module over M_{m}(A), using matrix multiplication. Similarly, M_{m,n}(A) is a right module over M_{n}(A).

Note that we can identify M_{m,n_{1}+n_{2}}(A) with the direct sum of M_{m,n_{1}}(A) and M_{m,n_{2}}(A), as modules over k, and left modules over M_{m}(A). Of course, M_{m,m}(A) = M_{m}(A) is a free left module of rank one over itself. One can use this to get that

(29.11.5) M_{m,n}(A) is projective as a left module over M_{m}(A)

for each n, as in Section 2.7.

Similarly, if m_{1} and m_{2} are any two positive integers, then we can identify M_{m_{1}+m_{1},n}(A) with the direct sum of M_{m_{1},n}(A) and M_{m_{1},n}(A), as modules over k, and as right modules over M_{n}(A). We also have that M_{n,n}(A) = M_{n}(A) is a free right module of rank one over itself. One can use this to get that

(29.11.6) M_{m,n}(A) is projective as a right module over M_{n}(A)

for each m.

Let A^{m} be the space of m-tuples of elements of A, as in Section 4.7. We can identify A^{m} with M_{m,1}(A), where the jth component of an element of A^{m} corresponds to the (j, 1) entry of an element of M_{m,1}(A). The action of an 29.12. MORE ON M_{N}(A) 675

element of M_{m}(A) on M_{m,1}(A) by multiplication on the left corresponds exactly to the action of this element of M_{m}(A) on A^{m} using multiplication on the left as in Section 4.7, so that M_{m,1}(A) corresponds to A^{m} in this way as left modules over M_{m}(A). In particular,

(29.11.7) A^{m} is projective as a left module over M_{m}(A),

as in (29.11.5).

Similarly, we can identify A^{n} with M_{1,n}(A), where the lth component of an element of A^{n} corresponds to the (1, l) entry of an element of M_{n,1}(A). If α ∈ M_{n}(A), then the action of α on M_{1,n}(A) by multiplication on the right corresponds to the mapping T_{α^{t}}^{R} from A^{n} into itself defined in Section 4.7, where α^{t} is the transpose of α, as before. If we consider A^{n} as a right module over M_{n}(A) in this way, then

(29.11.8) A^{n} is projective as a right module over M_{n}(A),

as in (29.11.6).

29.12 More on M_{n}(A)

Let k be a commutative ring with a multiplicative identity element 1_{k}, and let A_{1}, A_{2} be algebras over k in the strict sense, as in Section 20.13 again. Thus A_{1} and A_{2} are modules over k in particular, and we let A = A_{1} ⨂_{k} A_{2} be a tensor product of A_{1} and A_{2}, as modules over k. One can define multiplication on A in such a way that

(a_{1} ⊗ a_{2}) (b_{1} ⊗ b_{2}) = (a_{1} b_{1}) ⊗ (a_{2} b_{2}) (29.12.1)

for every a_{1}, b_{1} ∈ A_{1} and a_{2}, b_{2} ∈ A_{2}, as in Section 4.1. This makes A an algebra over k in the strict sense, as before. If A_{1} and A_{2} have multiplicative identity elements e_{1} = e_{A_{1}} and e_{2} = e_{A_{2}}, then e_{1} ⊗ e_{2} is the multiplicative identity element in A, as before.

Let n be a positive integer, and let us now take

A_{1} = M_{n}(k), (29.12.2)

the algebra of n × n matrices with entries in k. In this case, we can take

A = M_{n}(A_{2}), (29.12.3)

as a module over k. Of course, M_{n}(k) is a free as a module over k of rank n^{2}, so that one can take A = M_{n}(k) ⨂_{k}A_{2} (29.12.4)

to be a direct sum of n^{2} copies of A_{2}, as a module over k, as in Section 1.3.

More precisely, if a_{1} ∈ M_{n}(k) and a_{2} ∈ A_{2}, then a_{1} ⊗ a_{2} ∈ M_{n}(A_{2}) may be defined by taking its entries to be a_{2} times the corresponding entries of a_{1}. This defines a mapping from M_{n}(k) × A_{2} into M_{n}(A_{2}) that is bilinear over k, and this is the mapping used in considering M_{n}(A_{2}) as a tensor product of M_{n}(k) and A_{2} over k. One can check that matrix multiplication on M_{n}(A_{2}) satisfies (29.12.1), using this mapping from M_{n}(k) × A_{2} into M_{n}(A_{2}).

If A_{2} has a multiplicative identity element e_{2} = e_{A_{2}}, then

I_{M_{n}(A_{2})} = I_{M_{n}(k)} ⊗ e_{2}, (29.12.5)

where I_{M_{n}(k)}, I_{M_{n}(A_{2})} are the identity matrices in M_{n}(k), M_{n}(A_{2}), respectively, as before.

Let V be a module over k, and let V^{n} be the space of n-tuples of elements of V, as usual, considered as a module over k with respect to coordinatewise addition and scalar multiplication. Also let A be an associative algebra over k with a multiplicative identity element e_{A}, and suppose for the moment that V is a left module over A. If a ∈ M_{n}(A) and v ∈ V^{n}, then a · v may be defined as an element of V^{n} by

(a · v)_{j} = ∑_{l=1}^{n} a_{j,l} · v_{l} (29.12.6)

for each j = 1, …, n. One can check that this makes V^{n} a left module over M_{n}(A). If V = A, considered as a left module over itself, then this is the same as the action of M_{n}(A) on A^{n} using multiplication on the left in Section 4.7.

Suppose now that V is a right module over A. If a ∈ M_{n}(A) and v ∈ V^{n}, then v · a may be defined as an element of V^{n} by

(v · a)_{l} = ∑_{j=1}^{n} v_{j} · a_{j,l} (29.12.7)

for each l = 1, …, n. One can verify that this makes V^{n} a right module over M_{n}(A). If V = A, considered as a right module over itself, then this corresponds to the mapping T_{a^{t}}^{R} from A^{n} into itself defined in Section 4.7, where a^{t} is the transpose of a, as usual.

In particular, A^{n} may be considered as a left and right module over M_{n}(A). However, if n ≥ 2, then the actions of elements of M_{n}(A) on A^{n} from the left and the right do not normally commute with each other.

Of course, A^{n} may also be considered as a left and right module over A, using coordinatewise multiplication on the left and right, respectively. The action of M_{n}(A) on A^{n} on the left commutes with this action of A on A^{n} on the right, and similarly the action of M_{n}(A) on A^{n} on the right commutes with this action of A on A^{n} on the left.

29.13 More on M_{m,n}(A)

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let V be a module over k. If m, n are positive integers, then M_{m,n}(A), M_{m,n}(V)

29.13. MORE ON M_{M,N}(A) 677

may be considered as modules over k with respect to entrywise addition and scalar multiplication, as in Section 29.11. Suppose for the moment that V is a left module over A. If n_{1} is another positive integer, b ∈ M_{m,n}(A), and v ∈ M_{n,n_{1}}(V), then b · v may be defined as an element of M_{m,n_{1}}(V) by

(b · v)_{j,r} = ∑_{l=1}^{n} b_{j,l} · c_{l,r} (29.13.1)

for each j = 1, …, m and r = 1, …, n_{1}. If V = A, considered as a left module over itself, then this is the same as the usual multiplication of matrices, as in Section 29.11.

Note that (b, v) ↦ b · v (29.13.2)

is bilinear over k as a mapping from M_{m,n}(A) × M_{n,n_{1}}(V) into M_{m,n_{1}}(V). If m_{0} is another positive integer and a ∈ M_{m_{0},m}(A), then a b ∈ M_{m_{0},n}(A), and one can check that (a b) · v = a · (b · v), (29.13.3)

as elements of M_{m_{0},n_{1}}(V). If I_{n} is the identity matrix in M_{n}(A), as usual, then it is easy to see that I_{n} · v = v. (29.13.4)

Thus M_{n,n_{1}}(V) is a left module over M_{n}(A). If we identify V^{n} with M_{n,1}(V) in the usual way, then this action of M_{n}(A) on M_{n,1}(V) on the left corresponds to the action of M_{n}(A) on V^{n} on the left defined in the previous section.

Suppose now that V is a right module over A. If v ∈ M_{m_{0},m}(V) and b is an element of M_{m,n}(A), then v · b may be defined as an element of M_{m_{0},n}(V) by

(v · b)_{j,r} = ∑_{l=1}^{n} v_{j,l} · b_{l,r} (29.13.5)

for each j = 1, …, m_{0} and r = 1, …, n. This is the same as the usual multiplication of matrices when V = A, considered as a right module over itself. As before, (v, b) ↦ v · b (29.13.6)

is bilinear over k as a mapping from M_{m_{0},m}(V) × M_{m,n}(A) into M_{m_{0},n}(V). It is easy to see that v · I_{m} = v, (29.13.7)

where I_{m} is the identity matrix in M_{m}(A).

If c ∈ M_{n,n_{1}}(A), then b c ∈ M_{m,n_{1}}(A), and one can check that

v · (b c) = (v · b) · c, (29.13.8)

as elements of M_{m_{0}}(V). In particular, M_{m_{0},m}(A) is a right module over M_{m}(A). If we identify V^{m} with M_{1,m}(V) in the usual way, then this action of M_{m}(A) on M_{1,m}(V) on the right corresponds to the action of M_{m}(A) on V^{m} on the right defined in the previous section.

Suppose that V is both a left and right module over A, where the actions of A on V on the left and right commute with each other. Of course, V = A has this property, where A acts on itself on the left and right using multiplication in A. If a ∈ M_{m_{0},m}(A), v ∈ M_{m,n}(V), and b ∈ M_{n,n_{1}}(A), then a · v ∈ M_{m_{0},n}(V), v · b ∈ M_{m,n_{1}}(V), and one can check that

(a · v) · b = a · (v · b), (29.13.9)

as elements of M_{m_{0},n_{1}}(V).

29.14 Some tensor products using matrices

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let n be a positive integer, and remember that M_{n}(A) is the algebra of n × n matrices with entries in A, with respect to matrix multiplication. If m_{1}, m_{2} are positive integers, then we let M_{m_{1},m_{2}}(A) be the space of m_{1} × m_{2} matrices with entries in A, as in Section 29.11. This is a module over k with respect to entrywise addition and scalar multiplication, as before.

Remember that M_{1,n}(A) is a right module over M_{n}(A) with respect to matrix multiplication, and that M_{n,1}(A) is a left module over M_{n}(A), as in Section 29.11. Consider the mapping μ from M_{1,n}(A) × M_{n,1}(A) into M_{1,1}(A) defined by matrix multiplication, so that

μ(x, y) = x y (29.14.1)

for every x ∈ M_{1,n}(A) and y ∈ M_{n,1}(A). This mapping is bilinear over k, and satisfies μ(x a, y) = (x a) y = x (a y) = μ(x, a y) (29.14.2)

for every x ∈ M_{1,n}(A), a ∈ M_{n}(A), and y ∈ M_{n,1}(A). We would like to check that M_{1,1}(A) satisfies the requirements of a tensor product of M_{1,n}(A) and M_{n,1}(A) over A with respect to μ, as in Section 1.9.

Let W be a module over k, and let b be a mapping from M_{1,n}(A) × M_{n,1}(A) into W that is bilinear over k and satisfies

b(x a, y) = b(x, a y) (29.14.3)

for every a ∈ M_{1,n}(A), a ∈ M_{n}(A), and y ∈ M_{n,1}(A). We would like to show that there is a unique homomorphism c from M_{1,1}(A) into W, as modules over k, such that b(x, y) = c(μ(x, y)) (29.14.4)

for every x ∈ M_{n,1}(A) and y ∈ M_{n,1}(A), as in Section 1.9.

Let l_{1}, l_{2} be positive integers less than or equal to n. Suppose for the moment that x_{1,l} = 0 when l ≠ l_{1} (29.14.5)

and y_{l,1} = 0 when l ≠ l_{2}. (29.14.6)

Under these conditions, it is easy to see that there is an a ∈ M_{n}(A) such that

x a = x, a y = 0. (29.14.7)

This implies that b(x, y) = b(x a, y) = b(x, a y) = 0. (29.14.8)

If r is a positive integer less than or equal to n, then define u(r) ∈ M_{1,n}(A) and v(r) ∈ M_{n,1}(A) by

(29.14.9)  (u(r))_{1,l} = (v(r))_{l,1} = e_{A} when l = r
= 0 when l ≠ r.

Of course, M_{1,n}(A) and M_{n,1}(A) may also be considered as both left and right modules over A, with respect to entrywise multiplication by elements of A on the left or right, respectively. Using this notation, we may express any x ∈ M_{1,n}(A) and y ∈ M_{n,1}(A) as

x = ∑_{r=1}^{n} u(r) · x_{1,r}, y = ∑_{r=1}^{n} y_{r,1} · v(r), (29.14.10)

respectively. It follows that

b(x, y) = ∑_{r=1}^{n} b(u(r) · x_{1,r}, y_{r,1} · v(r)), (29.14.11)

by the remarks in the preceding paragraph.

If w ∈ A, then put c_{r}(w) = b(u(r) · w, v(r)) (29.14.12)

for each r = 1, …, n. This defines a homomorphism from A into W, as modules over k. It is easy to see that

b(u(r) · z, z′ · v(r)) = b(u(r) · (z z′), v(r)) = c_{r}(z z′) (29.14.13)

for all z, z′ ∈ A, because of (29.14.3). Combining this with (29.14.11), we get that

b(x, y) = ∑_{r=1}^{n} c_{r}(x_{1,r} y_{r,1}) (29.14.14)

for every x ∈ M_{1,n}(A) and y ∈ M_{n,1}(A).

One can verify that the c_{r}’s are all the same, using (29.14.3). This means that

b(x, y) = c_{1}( ∑_{r=1}^{n} x_{1,r} y_{r,1}) (29.14.15)

for every x ∈ M_{1,n}(A) and y ∈ M_{n,1}(A). This is essentially the same as (29.14.4), because x y is the 1 × 1 matrix whose entry is ∑_{r=1}^{n} x_{1,r} y_{r,1}. It is easy to see that c is uniquely determined by (29.14.4).

Note that the action of A on M_{1,n}(A) on the left commutes with the action of M_{n}(A) on M_{1,n}(A) on the right, and similarly that the action of A on M_{n,1}(A) on the right commutes with the action of M_{n}(A) on M_{n,1}(A) on the left. This means that a tensor product of M_{1,n}(A) and M_{n,1}(A) over M_{n}(A) should also be a module over A on the left and the right, as in Section 1.10. One can check that this corresponds to the usual actions of A on M_{1,1}(A) on the left and on the right.

29.15 Some spaces of homomorphisms

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let n be a positive integer again. As before, M_{n,1}(A) is a left module over M_{n}(A), and so we may consider the space

Hom_{M_{n}(A)}(M_{n,1}(A), M_{n,1}(A)) (29.15.1)

for homomorphisms ϕ from M_{n,1}(A) into itself, as a left module over M_{n}(A), as in Section 1.7. This means that ϕ is a homomorphism from M_{n,1}(A) into itself, as a module over k, with ϕ(a y) = a ϕ(y) (29.15.2)

for every a ∈ M_{n}(A) and y ∈ M_{n,1}(A).

We may also consider M_{n,1}(A) as a right module over A, with respect to entrywise multiplication by elements of A on the right, and this action of elements of A on M_{n,1}(A) on the right commutes with the action of any element of M_{n}(A) on M_{n,1}(A) on the left. If z ∈ A, then

y ↦ y · z (29.15.3)

defines a homomorphism from M_{n,1}(A) into itself, as a left module over M_{n}(A). Note that z is uniquely determined by this homomorphism. We would like to verify that every homomorphism ϕ from M_{n,1}(A) into itself, as a left module over M_{n}(A), is of this form.

Let l_{2} be a positive integer less than or equal to n, and suppose for the moment that y ∈ M_{n,1}(A) satisfies (29.14.6). Suppose also that a ∈ M_{n}(A) satisfies

(29.15.4)  a_{j,l} = e_{A} when j = l = l_{2}
= 0 when j ≠ l_{2}.

This implies that a y = y, and that

(ϕ(y))_{j,1} = (ϕ(a y))_{j,1} = (a ϕ(y))_{j,1} = 0 (29.15.5)

when j ≠ l_{2}.

If y is any element of M_{n,1}(A) and r is a positive integer less than or equal to n, then let y(r) be the element of M_{n,1}(A) defined by

(29.15.6)  (y(r))_{l,1} = y_{r,1} when l = r
= 0  when l ≠ r.

Thus

y = ∑_{r=1}^{n} y(r), (29.15.7)

so that

ϕ(y) = ∑_{r=1}^{n} ϕ(y(r)). (29.15.8)

This implies that (ϕ(y))_{r,1} = ϕ(y(r))_{r,1} (29.15.9)

for each r = 1, …, n, by the remarks in the preceding paragraph.

We may consider M_{n,1}(A) as a left module over A, with respect to entrywise multiplication by elements of A on the left. If w ∈ A, then let ŵ ∈ M_{n}(A) be the diagonal matrix whose diagonal entries are all equal to w. The action of w on M_{n,1}(A) on the left, as a left module over A, is the same as multiplication by ŵ on the left. This means that

ϕ(w · y) = ϕ(ŵ y) = ŵ ϕ(y) = w · ϕ(y) (29.15.10)

for every y ∈ M_{n,1}(A).

Let v(r) ∈ M_{n,1}(A) be as in (29.14.9) for each r = 1, …, n. If y ∈ M_{n,1}(A) and y(r) is as in (29.15.6), then

y(r) = y_{r,1} · v(r) (29.15.11)

for each r = 1, …, n. This implies that

ϕ(y) = ∑_{r=1}^{n} y_{r,1} · ϕ(v(r)), (29.15.12)

by (29.15.9) and (29.15.10). We also have that (ϕ(v(r)))_{j,1} = 0 when j ≠ r, as before.

One can check that (ϕ(v(r)))_{r,1} is the same for each r = 1, …, n, using (29.15.2). If z is this element of A, then

ϕ(v(r)) = v(r) · z (29.15.13) for each r = 1, …, n. It follows that

ϕ(y) = ∑_{r=1}^{∞} y_{r,1} · (v(r) · z) = y · z (29.15.14)

for every y ∈ M_{n,1}(A). Similarly, homomorphisms from M_{1,n}(A) into itself, as a right module over M_{n}(A), correspond exactly to the actions of elements of A on M_{1,n}(A) by entrywise multiplication on the left.

Note that (29.15.1) may be considered as a left and right module over A, because M_{n,1}(A) is a right module over A, and the actions of elements of A on M_{n,1}(A) on the right commute with the actions of elements of M_{n}(A) on N_{n,1}(A) on the left, as in Section 1.8. The action of an element of A on (29.15.1) on the right is defined pointwise on the values of a homomorphism from M_{n,1}(A) into itself, as a left module over M_{n}(A). The action of an element of A on (29.15.1) on the left is defined by acting on elements of M_{n,1}(A) on the right, and then composing that with a homomorphism from M_{n,1}(A) into itself. Using the previous characterization of elements of (29.15.1) in terms of the action of A on M_{n,1}(A) on the right, these actions of A on (29.15.1) on the right and left correspond to the usual actions of A on itself on the right and left. There are analogous statements for the space of homomorphisms from M_{1,n}(A) into itself, as a right module over M_{n}(A).

Part VIII

Some semisimplicity and other topics Some vector spaces and topologies

30.1 Some basic topological spaces

Let I be a nonempty set, and let Y_{j} be a nonempty set for each j ∈ I. If Y_{j} is equipped with the discrete topology for each j ∈ I, then we get a corresponding product topology on the Cartesian product

∏_{j∈I} Y_{j}. (30.1.1)

If j_{1}, …, j_{n} are finitely many elements of I and

y_{j_{l}} ∈ Y_{j_{l}} (30.1.2)

for each l = 1, …, n, then the set of points in (30.1.1) whose j_{l}th coordinate is equal to y_{j_{l}} for each l = 1, …, n is an open set with respect to this topology, and the collection of these open sets is a basis for this topology. Note that these basic open sets are closed sets with respect to this topology as well. If I has only finitely many elements, then this topology on (30.1.1) is the same as the discrete topology.

If X and Y are nonempty sets, then the space of mappings from X into Y is sometimes denoted Y^{X}. This is the same as the Cartesian product (30.1.1), with Y_{j} = Y for each j, and I = X. We may use the notation c(X, Y) for this set, as in Section 1.2. The topology defined on c(X, Y) as in the preceding paragraph may be described as the topology of pointwise convergence, with respect to the discrete topology on Y. If x_{1}, …, x_{n} are finitely many elements of X and

y_{1}, …, y_{n} ∈ Y, (30.1.3)

then {f ∈ c(X, Y) : f(x_{l}) = y_{l} for l = 1, …, n} (30.1.4) is both open and closed with respect to this topology, and the collection of these basic open sets is a base for this topology on c(X, Y), as before.

Let Z be another nonempty set, so that we may also consider the sets c(X, Z) and c(Y, Z) of mappings from X and Y, respectively, into Z. The topologies of pointwise convergence may be defined on these spaces using the discrete topology on Z in the same way as in the previous paragrph. Using these topologies, we get a corresponding product topology on

c(Y, Z) × c(X, Y). (30.1.5)

Of course, composition of mappings defines a mapping from this product space into c(X, Z). Let us check that this mapping is continuous.

Let n be a positive integer, and let

x_{1} …, x_{n} ∈ X (30.1.6)

be given. Also let f_{0}, f be mappings from X into Y, and let g_{0}, g be mappings from Y into Z. If f(x_{l}) = f_{0}(x_{l}) (30.1.7)

and g(f_{0}(x_{l})) = g_{0}(f_{0}(x_{l})) (30.1.8)

for each l = 1, …, n, then

g(f(x_{l})) = g_{0}(f_{0}(x_{l})) (30.1.9)

for l = 1, …, n. This shows that composition of mappings is continuous at (f_{0}, g_{0}), as a mapping from (30.1.5) into c(X, Z).

In particular, we can take X = Y = Z. Note that the space c(X, X) of mappings from X into itself is a semigroup with respect to composition of mappings, with the identity mapping on X as the multiplicative identity element. The continuity of composition of mappings as multiplication on c(X, X) is related to a remark on p249 of [152], and we shall say more about this soon.

Part IX

Rings, modules, satellites, and augmentations Some more rings and modules

31.1 Some local rings

Let A be a division ring, with multiplicative identity element e_{A} ≠ 0, and let T be an indeterminate. Thus the corresponding ring A[[T]] of formal power series in T with coefficients in A may be defined as in Section 4.3. Consider

B_{1} = {f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ A[[T]] : f_{1} = 0}, (31.1.1)

as in Exercise 10 on p160 of [50]. It is easy to see that

(31.1.2) B_{1} is a subring of A[[T]].

Note that e_{A} ∈ B_{1}.

Put

I_{1} = {f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ B_{1} : f_{0} = 0} = (A[[T]]) T^{2}. (31.1.3)

This is a two-sided ideal in A[[T]] that is contained in B_{1}, and thus a two-sided ideal in B_{1} in particular. Of course,

B_{1} \ I_{1} = {f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ A[[T]] : f_{0} ≠ 0, f_{1} = 0}. (31.1.4)

Remember that an element of A[[T]] is invertible in A[[T]] when the constant term is invertible in A, as in Section 4.5. In this case, it suffices to ask that the constant term be nonzero, because A is a division ring, by hypothesis. If

f(T) ∈ B_{1} \ I_{1}, (31.1.5) then it follows that f(T) is invertible in A[[T]]. In fact, one can check that

f(T)^{−1} ∈ B_{1}, (31.1.6)

using the description of the multiplicative inverse in Section 4.5.

Note that the elements of I_{1} have neither a left or right inverse in A[[T]], as in Section 4.14. In particular, this means that they do not have a left or right inverse in B_{1}. It follows that

(31.1.7) B_{1} is a local ring,

as in Section 4.13. This corresponds to the first part of Exercise 10 on p160 of [50].

Remember that f(T) ↦ f_{0} (31.1.8)

defines a ring homomorphism from A[[T]] onto A, as in Section 4.3. Of course, this homomorphism also maps B_{1} on A. The kernel of the restriction of this homomorphism to B_{1} is equal to I_{1}. This leads to a ring isomorphism from B_{1}/I_{1} onto A.

31.2 Some remarks about a(T) B_{1}

Let us continue with the same notation and hypotheses as in the previous section. Let a(T) = ∑_{m=0}^{∞} a_{m} T^{m} be a nonzero element of A[[T]], and let m_{0} be the smallest nonnegative integer such that

a_{m_{0}} ≠ 0. (31.2.1)

Thus a(T) = a_{m_{0}} T^{m_{0}} (e_{A} − c(T)), (31.2.2)

where c(T) = ∑_{r=0}^{∞} c_{r} T^{r} ∈ A[[T]], with

c_{0} = 0 and c_{1} = −a_{m_{0}}^{−1} a_{m_{0}+1}. (31.2.3)

It is easy to see that

(e_{A} − c(T))^{−1} − e_{A} − c_{1} T ∈ (A[[T]]) T^{2}, (31.2.4)

using the description of (e_{A} − c(T))^{−1} in Section 4.5.

Consider

(e_{A} − c(T)) B_{1} = {(e_{A} − c(T)) f(T) : f(T) ∈ B_{1}} (31.2.5)

which is a subset of A[[T]]. This is contained in

{g(T) = ∑_{l=0}^{∞} g_{l} T^{l} ∈ A[[T]] : g_{1} = −c_{1} g_{0}}. (31.2.6)

31.3. SOME RIGHT MODULES OVER B_{1} 689

If g(T) is an element of this set, then one can check that

(e_{A} − c(T))^{−1} g(T) ∈ B_{1}, (31.2.7)

using (31.2.4). This implies that (31.2.6) is contained in (31.2.5). Thus (31.2.5) is the same as (31.2.6).

If follows that

a(T) B_{1} = {h(T) = ∑_{n=0}^{∞} h_{n} T^{n} ∈ A[[T]] : h_{n} = 0 when n < m_{0},

h_{m_{0}+1} = −c_{1} h_{m_{0}}}. (31.2.8)

Another part of Exercise 10 on p160 of [50] says that I_{1} is not free as a right module over B_{1}. Using (31.2.8), we get that I_{1} is not generated by a single element, as a right module over B_{1}. Any set of generators of I_{1}, as a right module over B_{1}, should include at least one element for which the coefficient of T^{2} is not 0. Another generator is needed to get arbitrary coefficients of T^{3} of elements of I_{1}. However, one can verify that I_{1} cannot be freely generated in this way, as a right module over B_{1}.

31.3 Some right modules over B_{1}

We continue with the same notation and hypotheses as in the previous two sections. Of course, A[[T]] may be considered as a right module over B_{1}, and B_{1} and I_{1} may be considered as submodules of A[[T]], as a right module over B_{1}. If m_{0} is a nonnegative integer, then

(A[[T]]) T^{m_{0}} (31.3.1)

may be considered as a submodule of A[[T]], as a right module over B_{1}, and (31.2.8) is a submodule of A[[T]], as a right module over B_{1}. Note that c_{1} ∈ A may be arbitrary in (31.2.8).

Let V be a submodule of A[[T]], as a right module over B_{1}, and suppose that V ≠ {0}. Let m_{0}(V) be the largest nonnegative integer such that

V ⊆ (A[[T]]) T^{m_{0}(V)}. (31.3.2)

Equivalently, m_{0}(V) is the smallest nonnegative integer such that V contains an element of (A[[T]]) T^{m_{0}(V)} for which the coefficient of T^{m_{0}(V)} is nonzero. This implies that V contains a submodule of the form (31.2.8) with m_{0} = m_{0}(V) for some c_{1} ∈ A. If V is not equal to this module, then one can check that

V = (A[[T]]) T^{m_{0}(V)}. (31.3.3)

It follows that

(31.3.4) A[[T]] is Noetherian as a right module over B_{1}, as in Section 9.7. More precisely, every submodule of A[[T]], as a right module over B_{1}, is generated by one or two elements. One could also look at the Noetherian property in terms of the ascending chain condition. Thus I_{1} is Noetherian as a right module over B_{1}, and B_{1} is right Noetherian as a ring.

Another part of Exercise 10 on p160 of [50] states that

(31.3.5) I_{1} is not projective, as a right module over B_{1}.

More precisely, if I_{1} were projective as a right module over B_{1}, then one could get that I_{1} is free as a right module over B_{1}, as in Section 13.9. To see this, let I_{1} · I_{1} be the subset of I_{1} consisting of sums of products of pairs of elements of I_{1} = (A[[T]]) T^{2}, as in Section 13.4. In this case,

I_{1} · I_{1} = (A[[T]]) T^{4}. (31.3.6)

This is a submodule of I_{1}, as a right module over B_{1}, as in Section 13.7.

The quotient I_{1}/(I_{1} · I_{1}) (31.3.7)

may be defined in the usual way as a right module over B_{1}. In fact, this may be considered as a right module over B_{1}/I_{1}, as in Section 13.7. This corresponds to considering (31.3.7) as a right module over A here. As a right module over A, (31.3.7) is freely generated by the images of T^{2} and T^{3} in the quotient.

Note that I_{1} is generated by T^{2} and T^{3}, as a right module over B_{1}. If I_{1} were projective as a right module over B_{1}, then I_{1} should be freely generated by T^{2} and T^{3}, as a right module over B_{1}, as in Section 13.9. However, I_{1} is not freely generated by T^{2} and T^{3}, as a right module over B_{1}, and so I_{1} is not projective as a right module over B_{1}.

Let us consider the Cartesian product B_{1} × B_{1} as a right module over B_{1}, which is the same as the direct sum of two copies of B_{1}, as a right module over itself. As in Exercise 10 on p160 of [50],

ϕ((b_{1}(T), b_{2}(T))) = b_{1}(T) T^{2} − b_{2}(T) T^{3} (31.3.8)

defines a homomorphism from B_{1} × B_{1} into I_{1}, as right modules over B_{1}. In fact, ϕ(B_{1} × B_{1}) = I_{1}. (31.3.9)

We also have that ψ(b(T)) = (b(T) T, b(T)) (31.3.10)

defines an injective homomorphism from I_{1} into B_{1} × B_{1}, as right modules over B_{1}. Note that ϕ ∘ ψ = 0 (31.3.11)

on I_{1}.

One can check that ker ϕ = ψ(I_{1}), (31.3.12)

as mentioned in [50]. One can use this to get a projective resolution of I_{1} as a right module over B_{1}, as in Section 10.2.

31.4 Some related Notherian conditions

Let A be a ring with a multiplicative identity element e_{A}, let T be an indeterminate, and consider the subring B_{1} of A[[T]] as in (31.1.1). Also let A[[T^{2}]] be the subring of A[[T]] consisting of formal power series in T^{2} with coefficients in A, and note that A[[T^{2}]] ⊆ B_{1}. (31.4.1) Thus B_{1} may be considered as a left and right module over A[[T^{2}]].

It is easy to see that

(31.4.2) B_{1} is finitely generated, as a left or right module over A[[T^{2}]].

Indeed, B_{1} is generated by e_{A} and T^{3}, as a left or right module over A[[T^{2}]]. More precisely, B_{1} corresponds to the direct sum of two copies of A[[T^{2}]], as a left or right module over itself.

Let X be another indeterminate, and note that A[[T^{2}]] is isomorphic to A[[X]] as a ring, by sending a formal power series in X to the corresponding formal power series in T^{2}. If A is left or right Noetherian as a ring, then A[[X]] has the same property, as in Section 29.8. This means that A[[T^{2}]] is left or right Noetherian as a ring, as appropriate. It follows that

(31.4.3) B_{1} is Noetherian as a left or right module over A[[T^{2}]],

as appropriate, as in Section 9.7.

Of course, every left or right ideal in B_{1} is a submodule of B_{1}, as a left or right module over A[[T^{2}]], as appropriate. One can use this to get that

(31.4.4) B_{1} is left or right Noetherian as a ring,

as appropriate, when A is left or right Noetherian as a ring, as in Section 29.9.

Similarly, A[[T]] corresponds to the direct sum of two copies of A[[T^{2}]], as a left or right module over A[[T^{2}]]. If A is left or right Noetherian as a ring, then A[[T]] is Noetherian as a left or right module over A[[T^{2}]], as appropriate, as before. One can use this to get that

(31.4.5) A[[T]] is Noetherian as a left or right module over B_{1},

as appropriate, under these conditions, as in Section 29.9.

31.5 Some more local rings

Let A be a division ring again, with multiplicative identity element e_{A} ≠ 0, and let T_{1}, T_{2} be distinct commuting indeterminates. Consider the corresponding ring A[[T_{1}, T_{2}]] of formal power series in T_{1}, T_{2} with coefficients in A, as in Section 4.3. Put

(31.5.1)  B_{2} = {f(T) = ∑_{α∈(Z_{+}∪{0})^{2}} f_{α} T^{α} ∈ A[[T_{1}, T_{2}]] :

f_{α} = 0 when α_{1} + α_{2} is odd},

as in Exercise 11 on p160 of [50]. Observe that

(31.5.2) B_{2} is a subring of A[[T_{1}, T_{2}]],

with e_{A} ∈ B_{2}.

It is easy to see that

I_{2} = {f(T) = ∑_{α∈(Z_{+}∪{0})^{2}} f_{α} T^{α} ∈ B_{2} : f_{0} = 0} (31.5.3)

is a two-sided ideal in B_{2}. Remember that f(T) ↦ f_{0} is a ring homomorphism from A[[T_{1}, T_{2}]] onto A, as in Section 4.3. Equivalently, I_{2} is the intersection of the kernel of this homomorphism with B_{2}, which is the same as the kernel of the restriction of this homomorphism to B_{2}. This homomorphism also maps B_{2} onto A, which leads to a ring isomorphism from B_{2}/I_{2} onto A.

An element of A[[T_{1}, T_{2}]] is invertible in A[[T_{1}, T_{2}]] when its constant term is invertible in A, as in Section 4.5, which means that the constant term is nonzero in this case, because A is a division ring. If

f(T) ∈ B_{2} \ I_{2}, (31.5.4)

then f(T) is invertible in A[[T_{1}, T_{2}]], and one can check that

f(T)^{−1} ∈ B_{2}, (31.5.5)

using the description of f(T)^{−1} in Section 4.5.

The elements of I_{2} do not have one-sided inverses in A[[T_{1}, T_{2}]], as in Section 4.14, and in particular they do not have local inverses in B_{2}. This implies that

(31.5.6) B_{2} is a local ring,

as in Section 4.13. This corresponds to the first part of Exercise 11 on p160 of [50].

Of course, A[[T_{1}, T_{2}]] may be considered as a right module over B_{2}, and B_{2}, I_{2} may be considered as submodules of A[[T_{1}, T_{2}]], as a right module over B_{2}. Consider

(31.5.7)  W_{2} = {f(T) = ∑_{α∈(Z_{+}∪{0})^{2}} f_{α} T^{α} ∈ A[[T_{1}, T_{2}]] :

f_{α} = 0 when α_{1} + α_{2} is even}.

It is easy to see that this is also a submodule of A[[T_{1}, T_{2}]], as a right module over B_{2}, as in Exercise 11 on p160 of [50].

31.6. MORE ON W_{2} 693

31.6 More on W_{2}

Let us continue with the same notation and hypotheses as in the previous section. Observe that

(31.6.1) W_{2} is generated by T_{1}, T_{2}, as a right module over B_{2}.

Let W_{2} · I_{2} be the subset of W_{2} consisting of finite sums of products of elements of W_{2} and I_{2}, as in Section 13.4. This is a submodule of W_{2}, as a right module over B_{2}, as in Section 13.7. One can check that

(31.6.2)  W_{2} · I_{2} = {f(T) = ∑_{α∈(Z_{+}∪{0})^{2}} f_{α} T^{α} ∈ W_{2} :

f_{α} = 0 when α_{1} + α_{2} = 1}.

The quotient W_{2}/(W_{2} · I_{2}) (31.6.3)

may be defined as a right module over B_{2} in the usual way. This may also be considered as a right module over B_{2}/I_{2}, as in Section 13.7. This means that (31.6.3) may be considered as a right module over A here.

One can check that (31.6.3) is freely generated by the images of T_{1} and T_{2} in the quotient, as a right module over A, using (31.6.2). If W_{2} were projective as a right module over B_{2}, then W_{2} should be freely generated by T_{1} and T_{2}, as a right module over B_{2}, as in Section 13.9. However, it is easy to see that W_{2} is not freely generated by T_{1} and T_{2}, as a right module over B_{2}. This implies that (31.6.4) W_{2} is not projective, as a right module over B_{2}.

This is another part of Exercise 11 on p160 of [50].

Let us consider B_{2} × B_{2} as a right module over B_{2}, which is the same as the direct sum of two copies of B_{2}, as a right module over itself. Observe that

ϕ((b_{1}(T), b_{2}(T))) = b_{1}(T) T_{2} − b_{2}(T) T_{1} (31.6.5)

defines a homomorphism from B_{1} × B_{2} into W_{2}, as right modules over B_{2}, as in Exercise 11 on p160 of [50]. We also have that

ϕ(B_{2} × B_{2}) = W_{2}, (31.6.6)

as in (31.6.1). Clearly

ψ(a(T)) = (a(T) T_{1}, a(T) T_{2}) (31.6.7)

defines an injective homomorphism from W_{2} into B_{1} ×B_{2}, as right modules over B_{2}. It is easy to see that ϕ ∘ ψ = 0 on W_{2}. In fact, one can verify that

ker ϕ = ψ(W_{2}), (31.6.8)

as mentioned in Exercise 11 on p160 of [50]. This can be used to get a projective resolution of W_{2} as a right module over B_{2}, as in Section 10.2.

31.7 Some more Noetherian conditions

Let A be a ring with a multiplicative identity element e_{A}, let T_{1}, T_{2} be distinct commuting indeterminates, and consider the subring B_{2} of A[[T_{1}, T_{2}]] as in (31.5.1). Also let A[[T_{1}^{2}, T_{2}^{2}]] be the subring of A[[T_{1}, T_{2}]] consisting of formal power series in T_{1}^{2} and T_{2}^{2} with coefficients in A, and observe that

A[[T_{1}^{2}, T_{2}^{2}]] ⊆ B_{2}. (31.7.1)

It follows that B_{2} may be considered as a left and right module over A[[T_{1}^{2}, T_{2}^{2}]]. We also have that

(31.7.2) B_{2} is finitely generated, as a left or right module over A[[T_{1}, T_{2}^{2}]].

One can use e_{A} and T_{1} T_{2} as the generators of B_{1}, as a left or right module over A[[T_{1}^{2}, T_{2}^{2}]]. In fact, B_{2} corresponds to the direct sum of two copies of A[[T_{1}^{2}, T_{2}^{2}]], as a left or right module over itself.

Let X_{1}, X_{2} be another pair of distinct comuting indeterminates, and note that a[[T_{1}^{2}, T_{2}^{2}]] is isomorphic to A[[X_{1}, X_{2}]] as a ring, by sending a formal power series in X_{1}, X_{2} to the correspoinding formal power series in T_{1}^{2}, T_{2}^{2}. If A is left or right Noetherian as a ring, then A[[X_{1}, X_{2}]] has the same property, as in Section 29.8, so that A[[T_{1}^{2}, T_{2}^{2}]] has the same property as well. This implies that (31.7.3) B_{2} is Noetherian as a left or right module over A[[T_{1}^{2}, T_{2}^{2}]],

as appropriate, as in Section 9.7.

Any left or right ideal in B_{2} may be considered as a submodule of B_{2}, as a left or right module over A[[T_{1}^{2}, T_{2}^{2}]], as appropriate. One can use this to get that (31.7.4) B_{2} is left or right Noetherian as a ring,

as appropriate, when A is left or right Noetherian as a ring, as in Section 29.9. In particular, (31.7.5) B_{2} is left and right Noetherian as a ring

when A is a division ring.

Observe that A[[T_{1}, T_{2}]] corresponds to the direct sum of four copies of A[[T_{1}^{2}, T_{2}^{2}]], as a left or right module over A[[T_{1}^{2}, T_{2}^{2}]]. This uses the submodules of A[[T_{1}, T_{2}]] generated by

e_{A}, T_{1}, T_{2}, and T_{1} T_{2}. (31.7.6)

If A is left or right Noetherian as a ring, then it follows that A[[T_{1}, T_{2}]] is Noetherian as a left or right module over A[[T_{1}^{2}, T_{2}^{2}]], as appropriate. One can use this to get that

(31.7.7) A[[T_{1}, T_{2}]] is Noetherian as a left or right module over B_{2},

as appropriate, as in Section 29.9.

Let W_{2} be as in (31.5.7), so that W_{2} is a submodule of A[[T_{1}, T_{2}]], as a left and right module over B_{2}. In particular, W_{2} is a submodule of A[[T_{1}, T_{2}]], as a left and right module over A[[T_{1}^{2}, T_{2}^{2}]]. In fact, W_{2} corresponds to the direct sum of two copies of A[[T_{1}^{2}, T_{2}^{2}]], as a left or right module over A[[T_{1}^{2}, T_{2}^{2}]]. This uses the submodules of W_{2} generated by T_{1}, T_{2}, as a module over A[[T_{1}^{2}, T_{2}^{2}]]. Note that A[[T_{1}, T_{2}]] corresponds to the direct sum of B_{2} and W_{2}, as a left or right module over B_{2}, and thus as a left or right module over A[[T_{1}^{2}, T_{2}^{2}]].

If A is left or right Noetherian as a ring, then W_{2} is Noetherian as a left or right module over A[[T_{1}^{2}, T_{2}^{2}]], as appropriate. This implies that

(31.7.8) W_{2} is Noetherian as a left or right module over B_{2},

as appropriate, as in Section 29.9. Of course, this could also be obtained from (31.7.7), as in Section 9.7.

31.8 Another family of local rings

Let A be a division ring, with multiplicative identity element e_{A} ≠ 0, and let T_{1}, T_{2} be distinct commuting indeterminates again. Consider the corresponding rings A[[T_{1}]] and A[[T_{2}]] of formal power series in T_{1} and T_{2}, respectively, with coefficients in A, as in Section 4.3. The Cartesian product

(A[[T_{1}]]) × (A[[T_{2}]]) (31.8.1)

of A[[T_{1}]] and A[[T_{2}]] is also a ring, with respect to coordinatewise addition and scalar multiplication, which is the same as the direct sum of these two rings. Note that (e_{A}, e_{A}) is the multiplicative identity element in (31.8.1).

Consider

B_{3} = {(f(T_{1}), g(T_{2})) ∈ (A[[T_{1}]]) × (A[[T_{2}]]) : f(0) = g(0)}, (31.8.2)

as in Exercise 12 on p160 of [50], where f(0), g(0) are the same as the constant terms in f(T_{1}), g(T_{2}). It is easy to see that

(31.8.3) B_{3} is a subring of (A[[T_{1}]]) × (A[[T_{2}]]).

Put I_{3} = {(f(T), g(T)) ∈ B_{3} : f(0) = g(0) = 0}. (31.8.4)

One can check that this is a two-sided ideal in B_{3}. In fact, this is a two-sided ideal in (31.8.1).

Of course, the Cartesian product A × A of A with itself is a ring with respect to coordinatewise addition and multiplication too, and (e_{A}, e_{A}) is the multiplicative identity element in A × A. Clearly

(f(T_{1}), g(T_{2})) ↦ (f(0), g(0)) (31.8.5)

defines a ring homomorphism from (31.8.1) onto A × A, whose kernel is equal to I_{3}.

Note that {(a, a) : A ∈ A} (31.8.6)

is a subring of A × A, and that

a ↦ (a, a) (31.8.7)

is a ring isomorphism from A onto (31.8.6). The restriction of (31.8.5) to B_{3} corresponds to a ring homomorphism

(f(T_{1}), g(T_{2})) ↦ f(0) = g(0) (31.8.8)

from B_{3} onto A in this way. The kernel of this homomorphism is equal to I_{3} as well, so that B_{3}/I_{3} is isomorphic to A as a ring.

It is easy to see that an element (f(T_{1}), g(T_{2})) of (31.8.1) has a multiplicative inverse in (31.8.1) if and only if f(T_{1}) and g(T_{2}) have multiplicative inverses in A[[T_{1}]] and A[[T_{2}]], respectively, with

(f(T_{1}), g(T_{2}))^{−1} = (f(T_{1})^{−1}, g(T_{2})^{−1}). (31.8.9)

This happens exactly when f(0) and g(0) are invertble in A, as in Section 4.5, which is the same as saying that f(0), g(0) ≠ 0 in this case, because A is a division ring. Under these conditions, the constant terms in f(T_{1})^{−1} and g(T_{2})^{−1} are the same as the multiplicative inverses of the constant terms in f(T_{1}), g(T_{2}), respectively. If (f(T_{1}), g(T_{2})) ∈ B_{3} and

f(0) = g(0) ≠ 0, (31.8.10)

then it follows that (f(T_{1}), g(T_{2}))^{−1} ∈ B_{3}. (31.8.11)

Similarly, an element (f(T_{1}), g(T_{2})) of (31.8.1) has a one-sided inverse in (31.8.1) if and only if f(T_{1}) and g(T_{2}) have the same type of one-sided inverses in A[[T_{1}]] and A[[T_{2}]], respectively. In this case, f(0) and g(0) have the same type of one-sided inverses in A, as in Section 4.14, so that f(0), g(0) ≠ 0. Thus the elements of I_{3} do not have one-sided inverses in (31.8.1), and in particular they do not have one-sided inverses in B_{3}. This means that

(31.8.12) B_{3} is a local ring,

as in Section 4.13. This corresponds to the first part of Exercise 12 on p160 of [50].

Observe that (f(T_{1}), g(T_{2})) ↦ f(T_{1}) (31.8.13)

and (f(T_{1}), g(T_{2})) ↦ g(T_{2}) (31.8.14)

define ring homomorphisms from (31.8.1) onto A[[T_{1}]] and A[[T_{2}]], respectively. The restrictions of these homomorphisms to B_{3} map B_{3} onto A[[T_{1}]] and A[[T_{2}]], respectively, as well.

31.9 Some right modules over B_{3}

Let us continue with the same notation and hypotheses as in the previous section. We may consider (31.8.1) as a right module over B_{3}, and B_{3}, I_{3} are submodules of (31.8.1), as a right module over B_{3}.

Put V_{3} = {(f(T_{1}) T_{1}, 0) : f(T_{1}) ∈ A[[T_{1}]]} (31.9.1)

and W_{3} = {(0, g(T_{2}) T_{2}) : g(T_{2}) ∈ A[[T_{2}]]}. (31.9.2)

It is easy to see that these are submodules of (31.8.1), as a right module over B_{3}, as on p161 of [50]. In fact, V_{3} is the same as the kernel of the restriction of (31.8.14) to B_{3}, and W_{3} is the same as the kernel of the restriction of (31.8.13) to B_{3}.

Observe that V_{3} is generated, as a right module over B_{3}, by any element (f(T_{1}) T_{1}, 0) with f(0) ≠ 0. Any collection of generators of V_{3}, as a right module over B_{3}, has to contain an element of this type. However, one can check that V_{3} is not freely generated as a right module over B_{3}. Similarly, W_{3} is not freely generated as a right module over B_{3}. This is another part of Exercise 12 on p160 of [50].

Let V_{3} · I_{3} be the subset of V_{3} consisting of finite sums of propducts of elements of V_{3} and I_{3}, as in Section 13.4, and similarly for W_{3} · I_{3}. These are submodules of V_{3} and W_{3}, respectively, as right modules over B_{3}, as in Section 13.7. One can verify that

V_{3} · I_{3} = {(f(T_{1}) T_{1}^{2}, 0) : f(T_{1}) ∈ A[[T_{1}]]} (31.9.3)

and W_{3} · I_{3} = {(0, g(T_{2}) T_{2}^{2}) : g(T_{2}) ∈ A[[T_{2}]]}. (31.9.4)

The quotients V_{3}/(V_{3} · I_{3}) (31.9.5)

and W_{3}/(W_{3} · I_{3}) (31.9.6)

may be defined as right modules over B_{3}, as usual. These may also be considered as right modules over B_{3}/I_{3}, as in Section 13.7. This means that (31.9.5) and (31.9.6) may be considered as right modules over A.

It is easy to see that (31.9.5) is freely generated by the image of (T_{2}, 0) in the quotient, as a right module over A, because of (31.9.3). Similarly, (31.9.6) is freely generated by the image of (0, T_{2}) in the quotient, as a right module over A, because of (31.9.4). If V_{3} were projective as a right module over B_{3}, then V_{3} should be freely generated by (T_{1}, 0), as a right module over B_{3}, as in Section 9.10. Similarly, if W_{3} were projective as a right module over B_{3}, then it should be freely generated by (0, T_{2}), as a right module over B_{3}. It follows that

(31.9.7) V_{3} and W_{3} are not projective, as right modules over B_{3}, 698 CHAPTER 31. SOME MORE RINGS AND MODULES because V_{3} and W_{3} are not freely generated in this way, as on p161 of [50].

Observe that ϕ((f(T_{1}), g(T_{2}))) = (0, g(T_{2}) T_{2}) (31.9.8) defines a homomorphism from B_{3} onto W_{3}, as right modules over B_{3}, as on p161 of [50]. Similarly, ϕ′((f(T_{1}), g(T_{2}))) = (f(T_{1}) T_{1}, 0) (31.9.9) defines a homomorphism from B_{3} onto V_{3}, as right modules over B_{3}.

One can check that ker ϕ = V_{3} (31.9.10) and ker ϕ′ = W_{3}, (31.9.11) as mentioned on p161 of [50]. One can use ϕ and ϕ′ to get projective resolutions of V_{3} and W_{3} are right modules over B_{3}, as in Section 10.2.

31.10 Noetherian conditions related to B_{3}

Let A be a ring with a multiplicative identity element e_{A}, let T_{1}, T_{2} be distinct commuting indeterminates, and let B_{3} be the subring of the Cartesian product (31.8.1) defined in (31.8.2). Let X be another indeterminate, and define C_{3} ⊆ (A[[T_{1}]]) × (A[[T_{2}]]) (31.10.1) by C_{3} = {(f(T_{1}), f(T_{2})) : f(X) ∈ A[[X]]}. (31.10.2) More precisely, if f(X) ∈ A[[X]], then f(T_{1}), f(T_{2}) are the formal power series in T_{1}, T_{2} with the same coefficients as f(X), so that (f(T_{1}), f(T_{2})) is an element of the Cartesian product (31.8.1). It is easy to see that C_{3} is a subring of (31.8.1), and that f(X) ↦ (f(T_{1}), f(T_{2})) (31.10.3) is a ring isomorphism from A[[X]] onto C_{3}. Note that C_{3} ⊆ B_{3}. (31.10.4)

Thus B_{3} may be considered as a left and right module over C_{3}. One can check that (31.10.5) B_{3} is finiteley generated, as a left or right module over C_{3}. More precisely, B_{3} is generated by (e_{A}, e_{A}) and (T_{1}, 0) (31.10.6) as a left or right module over C_{3}. If we consider B_{3} as a left or right module over A[[X]], then B_{3} corresponds to the direct sum of two copies of A[[X]], as a left or right module over itself, as appropriate.

If A is left or right Noetherian as a ring, then A[[X]] has the same property, as in Section 29.8, which is the same as saying that C_{3} has the same property. This implies that

(31.10.7) B_{3} is Noetherian as a left or right module over C_{3},

as appropriate, as in Section 9.7.

A left or right ideal in B_{3} may be considered as a submodule of B_{3}, as a left or right module over C_{3}, as appropriate. One can use this to get that

(31.10.8) B_{3} is left or right Noetherian as a ring

when A is left or right Noetherian as a ring, respectively, as in Section 29.9.

We may consider the Cartesion product (31.8.1) as a left and right module over C_{3}, because of (31.10.1). This corresponds to the direct sum of two copies of A[[X]], as a left and right module over itself. If A is left or right Noetherian as a ring, then (31.8.1) is left or right Noetherian as a module over C_{3}, as appropriate. One can use this to get that

(31.10.9) (31.8.1) is Noetherian as a left or right module over B_{3},

as appropriate, as in Section 29.9.

Let V_{3} and W_{3} be as in (31.9.1) and (31.9.2), respectively, which are twosided ideals in (31.8.1). In particular, V_{3} and W_{3} may be considered as submodules of (31.8.1), as a left or right module over B_{3} or C_{3}. As left and right modules over C_{3}, V_{3} and W_{3} correspond to A[[X]] as a left and right module over itself, respectively.

If A is left or right Noetherian as a ring, then V_{3}, W_{3} are Noetherian as left or right modules over C_{3}, as appropriate. This implies that

(31.10.10) V_{3}, W_{3} are Noetherian as left or right modules over B_{3},

as appropriate, as in Section 29.9.

31.11 Direct systems of sets

Let (I, ⪯) be a nonempty pre-directed set, and let E_{j} be a set for each j ∈ I. Suppose that for every j, l ∈ I with j ⪯ l we have a mapping θ_{j,l} from E_{j} into E_{l}. We ask that θ_{j,j} be the identity mapping on E_{j} for each j ∈ I, and that for every j, l, r ∈ I with j ⪯ l ⪯ r, we have that

θ_{l,r} ∘ θ_{j,l} = θ_{j,r}. (31.11.1)

In this case, the family of sets E_{j} and mappings θ_{j,l} is said to form a direct or inductive system over (I, ⪯), as usual. Note that if j, l ∈ I, j ⪯ l, and l ⪯ j, then θ_{j,l} and θ_{l,j} are inverses of each other.

Let Ẽ_{j} be a nonempty set that contains E_{j} for each j ∈ I. If x, y ∈ ∏_{j∈I} Ẽ_{j}, then put x ∼ y when there is an l ∈ I such that

x_{j} = y_{j} (31.11.2)

for each j ∈ I with l ⪯ j. One can check that this defines an equivalence relation on ∏_{j∈I} Ẽ_{j}. This permits us to define to corresponding quotient set

( ∏_{j∈I} Ẽ_{j})/ ∼ (31.11.3)

as the set of equivalence classes in ∏_{j∈I} A_{j} with respect to ∼.

If l ∈ I and w_{l} ∈ E_{l}, then there are elements x of ∏_{j∈I} Ẽ_{j} such that

x_{j} = θ_{l,j}(w_{l}) (31.11.4)

for every j ∈ I with l ⪯ j. Any two such elements of ∏_{j∈I} Ẽ_{j} are clearly equivalent with respect to ∼. This defines a mapping θ_{l} from E_{l} into the quotient set (31.11.3). If r ∈ I and l ⪯ r, then it is easy to see that

θ_{l} = θ_{r} ∘ θ_{l,r}. (31.11.5)

Consider

lim_{⟶} E_{j} = ⋃_{l∈I} θ_{l}(E_{l}), (31.11.6)

which is a subset of the quotient set (31.11.3). This is the direct or inductive limit of the direct system of E_{j}’s, j ∈ I. More precisely, the direct limit consists of (31.11.6) together with the mappings θ_{l} from E_{l} into this set for each l ∈ I. If l, r ∈ I and l ⪯ r, then θ_{l}(E_{l}) ⊆ θ_{r}(E_{r}), (31.11.7)

by (31.11.5).

Let l_{1}, l_{2} ∈ I, w_{1,l_{1}} ∈ E_{l_{1}}, and w_{2,l_{2}} ∈ E_{l_{2}} be given. If

θ_{l_{1}}(w_{1,l_{1}}) = θ_{l_{2}}(w_{2,l_{2}}), (31.11.8)

then one can check that there is an r ∈ I such that

l_{1}, l_{2} ⪯ r (31.11.9)

and θ_{l_{1},r}(w_{1,l_{1}}) = θ_{l_{2},r}(w_{2,l_{2}}). (31.11.10)

Conversely, if there is an r ∈ I such that (31.11.9) and (31.11.10) hold, then

θ_{l_{1},t}(w_{1,l_{1}}) = θ_{l_{2},t}(w_{2,l_{2}}) (31.11.11)

for every t ∈ I such that r ⪯ t, by (31.11.1). Under these conditions, (31.11.8) holds, by construction.

Let C be another set, and let γ_{l} be a mapping from E_{l} into C for each l ∈ I. Suppose that for every l, r ∈ I with l ⪯ r, we have that

γ_{l} = γ_{r} ∘ θ_{l,r}. (31.11.12)

Under these conditions, one can check that there is a unique mapping γ from (31.11.6) into C such that γ_{l} = γ ∘ θ_{l} (31.11.13)

for every l ∈ I. More precisely, the main point is to verify that γ is well defined in this way. This determines the direct limit uniquely, up to suitable equivalence.

Let Ẽ be a set, and suppose now that E_{j} is a subset of Ẽ for every j ∈ I. If j, l ∈ I and j ⪯ l, then we ask that

E_{j} ⊆ E_{l}, (31.11.14)

and we let θ_{j,l} be the obvious inclusion mapping from E_{j} into E_{l}. These mappings satisfy the conditions in the definition of a direct system of sets, and the direct limit is equivalent to

⋃_{j∈I} E_{j}. (31.11.15)

Of course, this uses the obvious inclusion mapping from E_{l} into (31.11.15) for each l ∈ I.

31.12 Some limits of free modules

Let us continue with the same notation and hypotheses as at the beginning of the previous section, and put

E = lim_{⟶} E_{j}. (31.12.1)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a nonzero multiplicative identity element e_{A}. If j ∈ I, then let F_{E_{j}} be the free left or right module over A generated by E_{j}. This may be defined precisely as the space c_{00}(E_{j}, A) of A-valued functions on E_{j} with finite support, where each element of E_{j} is identified with the A-valued function on E_{j} equal to e_{A} at that element, and to 0 otherwise. Similarly, let F_{E} be the free left or right module over A generated by E, which may be defined precisely as c_{00}(E, A).

If j, l ∈ I and j ⪯ l, then there is a unique homomorphism Θ_{j,l} from F_{E_{j}} into F_{E_{l}}, as modules over A, corresponding to θ_{j,l}. This mapping is the identity mapping on F_{E_{j}} when j = l, and it is easy to see that

Θ_{l,r} ∘ Θ_{j,l} = Θ_{j,r} (31.12.2)

for all j, l, r ∈ I with j ⪯ l ⪯ r, by (31.11.1). In particular, if j, l ∈ I, j ⪯ l, and l ⪯ j, then Θ_{j,l} and Θ_{l,j} are inverses of each other. Thus the F_{E_{j}}’s, j ∈ I, together with the homomorphisms Θ_{j,l}, form a direct system of modules over (I, ⪯).

Similarly, if l ∈ I, then there is a unique homomorphism Θ_{l} from F_{E_{l}} into F_{E}, as modules over A, corresponding to θ_{l}. If r ∈ I and l ⪯ r, then

Θ_{l} = Θ_{r} ∘ Θ_{l,r}, (31.12.3)

because of (31.11.5).

Let Z be another left or right module over A, as appropriate, and let ζ_{l} be a homomorphism from F_{E_{l}} into Z, as modules over A, for each l ∈ I. Note that ζ_{l} is determined by its values on on the elements of E_{l}, as generators of F_{E_{l}}, as a module over A. Suppose that for each l, r ∈ I with l ⪯ r, we have that

ζ_{l} = ζ_{r} ∘ Θ_{l,r}. (31.12.4)

If l ∈ I, then let γ_{l} be the restriction of ζ_{l} to E_{l}, considered as a subset of F_{E_{l}}, as before. It follows from (31.12.4) that (31.11.12) holds for every l, r ∈ I with l ⪯ r. This implies that there is a unique mapping γ from E into Z such that (31.11.13) holds for every l ∈ I, as in the previous section.

This leads to a unique homomorphism ζ from F_{E} into Z, as modules over A, that corresponds to γ on E. If l ∈ I, then

ζ_{l} = ζ ∘ Θ_{l}, (31.12.5)

by (31.11.13).

This shows that the direct limit of the F_{E_{j}}’s, j ∈ I, may be identified with F_{E}, using the homomorphisms Θ_{l}, l ∈ I, as in Section 3.2.

31.13 More on systems of modules

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let (I, ⪯) be a nonempty pre-directed set, and let V_{j} be a left module over A for every j ∈ I. Suppose that for every j, l ∈ I with j ⪯ l we have a homomorphism ν_{j,l} from V_{j} into V_{l}, as left modules over A, that satisfies the conditions mentioned in Section 3.2. This means that ν_{j,j} is the identity mapping on V_{j} for every j ∈ I, and that if j, l, r ∈ I satisfy j ⪯ l ⪯ r, then ν_{l,r} ∘ ν_{j,l} = ν_{j,r}. The direct limit V = lim_{⟶} V_{j} of the V_{j}’s can be defined as a left module over A, as before.

Of course, ∏_{j∈I} V_{j} may be considered as a left module over A, with respect to coordinatewise addition and left multiplication by elements of A. Let W be the set of w ∈ ∏_{j∈I} V_{j} for which there is an l ∈ I such that

w_{j} = 0 (31.13.1)

for each j ∈ I with l ⪯ j. It is easy to see that W is a submodule of ∏_{j∈I} V_{j}, as a left module over A. Thus the quotient

( ∏_{j∈I} V_{j})/W (31.13.2)

is defined as a left module over A.

If l ∈ I and v_{l} ∈ V_{l}, then there are elements y of ∏_{j∈I} V_{j} such that

y_{j} = ν_{l,j}(v_{l}) (31.13.3)

for each j ∈ I with l ⪯ j. The difference between any two such elements of ∏_{j∈I} V_{j} is an element of W. This defines a mapping η_{l} from V_{l} into the quotient (31.13.2). It is easy to see that η_{l} is a homomorphism from V_{l} into (31.13.2), as modules over A. If r ∈ I and l ⪯ r, then

η_{l} = η_{r} ∘ ν_{l,r}. (31.13.4)

Remember that the direct limit V is equipped with a homomorphism ν_{l} from V_{l} into V, as modules over A, for each l ∈ I, as in Section 3.2. It follows from the remarks in the preceding paragraph that there is a unique homomorphism η from V into the quotient (31.13.2), as modules over A, such that

η_{l} = η ∘ ν_{l} (31.13.5)

for every l ∈ I, as in Section 3.2. Remember that

V = ⋃_{l∈I} ν_{l}(V_{l}), (31.13.6)

as in Section 3.2. This implies that

η(V) = ⋃_{l∈I} η(ν_{l}(V_{l})) = ⋃_{l∈I} η_{l}(V_{l}). (31.13.7)

Suppose that η(v) = 0 for some v ∈ V. As in (31.13.6), v = ν_{l}(v_{l}) for some l ∈ I and v_{l} ∈ V_{l}. Thus we get that

η_{l}(v_{l}) = η(ν_{l}(v_{l})) = η(v) = 0. (31.13.8)

This means that if y is an element of ∏_{j∈I} V_{j} that satisfies (31.13.3), then y ∈ W. It follows that there is an r ∈ I such that y_{j} = 0 for each j ∈ I with r ⪯ j, by definition of W. If j ∈ I and l, r ⪯ j, then we get that

ν_{l,j}(v_{l}) = y_{j} = 0. (31.13.9)

In this case, we obtain that

v = ν_{l}(v_{l}) = ν_{j}(ν_{l,j}(v_{l})) = 0, (31.13.10)

where the second step is as in Section 3.2. This shows that η is injective, so that the definition of the direct limit of the V_{j}’s as modules in Section 3.2 is compatible with the definition of the direct limit of the V_{j}’s as sets in Section 31.11.

31.14 Projective resolutions of direct limits

Let us continue with the same notation and hypotheses as at the beginning of the previous section. Under these conditions, Lemma 9.5* on p100 of [50] says that one can find a projective resolution X_{j} of V_{j} for each j ∈ I, such that the family of X_{j}’s, j ∈ I, form a direct system, whose limit X = lim_{⟶} X_{j} is a projective resolution of V. Remember that direct systems of graded modules and complexes were discussed in Section 8.15. Of course, there are analogous statements for right modules.

If j ∈ I, then let X_{0,j} be the free left module F_{V_{j}} over A generated by V_{j}, as in Section 31.12. Similarly, let X_{0} be the free left module F_{V} over A generated by V. We may consider the V_{j}’s, j ∈ I, as a direct system of sets, so that the X_{0,j}’s, j ∈ I, form a direct system of left modules over A, as before. In fact, we may identify V with the direct limit of the V_{j}’s, j ∈ I, as sets, as in the previous section. This permits us to identify X_{0} with the direct limit of the X_{0,j}’s, as modules over A, as in Section 31.12.

If j ∈ I, then there is a natural homomorphism ϕ_{j} from X_{0,j} onto V_{j}, as modules over A. This homomorphism sends an element of V_{j}, considered as an element of X_{0,j}, to itself in V_{j}. The family of ϕ_{j}’s, j ∈ I, defines a homomorphism between the direct systems of X_{0,j}’s and V_{j}’s, as in Section 3.4. Similarly, there is a homomorphism ϕ from X_{0} onto V, as modules over A, which sends an element of V, considered as an element of X_{0}, to itself in V. This corresponds to the direct limit lim_{⟶} ϕ_{j} of the ϕ_{j}’s, as before.

Let R_{j} be the kernel of ϕ_{j} for each j ∈ I, and let R be the kernel of ϕ. It is easy to see that the family of R_{j}’s, j ∈ I, is a direct system of left modules over A. More precisely, if j, l ∈ I and j ⪯ l, then the homomorphism from X_{0,j} into X_{0,l} obtained from ν_{j,l} as in Section 31.12 maps R_{j} into R_{l}, and satisfies the conditions in the definition of a direct system of modules. The family of inclusion mappings from the R_{j}’s into the X_{0,j}’s, j ∈ I, defines a homomorphism between the direct systems of R_{j}’s and X_{0,j}’s. The direct limit of these inclusion mappings defines an isomorphism from lim_{⟶} R_{j} onto R, as modules over A, as in Section 3.4.

Thus R can be identified with the direct limit of the R_{j}’s, j ∈ I. We can repeat the process to get the desired projective resolutions, as in Section 10.2.

Some satellites

32.1 Some basic comparisons

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, V_{1}, P, and P_{1} be all left or all right modules over A, let β, β_{1} be homomorsphisms from P, P_{1} onto V, V_{1}, respectively, and let g be a homomorphism from V into V_{1}.

Thus g ∘β is a homomorphism from P into V_{1}, as modules over A. We would like to consider some related mappings and their properties, as in the discussion starting on p33 of [50].

Suppose that P is projective, so that there is a homomorphism f from P into P_{1}, as modules over A, such that

β_{1} ∘ f = g ∘ β. (32.1.1)

Note that f(ker β) ⊆ ker β_{1} (32.1.2)

in this case. This means that the restriction of f to ker β defines a homomorphism from ker β into ker β_{1}, as modules over A.

If f̃ is another homomorphism from P into P_{1}, then

β_{1} ∘ f̃ = g ∘ β (32.1.3)

if and only if (f − f̃)(P) ⊆ ker β_{1}. (32.1.4)

This is the same as saying that f − f̃ may be considered as the composition of a homomorphism from P into ker β_{1} with the obvious inclusion mapping from ker β_{1} into P_{1}.

If t ∈ k, then t g is a homomorphism from V into V_{1} as well, as modules over A. Of course, t f is a homomorphism from P into P_{1}, whose composition with β_{1} is equal to the composition of β with t g.

Let ĝ be another homomorphism from V into V_{1}, as modules over A, so that ĝ ∘ β is a homomorphism from P into V_{1}. As before, there is a homomorphism f̂ from P into P_{1}, as modules over A, such that

β_{1} ∘ f̂ = ĝ ∘ β, (32.1.5)

because P is projective. It follows that

β_{1} ∘ (f + f̂) = (g + ĝ) ∘ β. (32.1.6)

Let V_{2} and P_{2} be left or right modules over A, as appropriate, let β_{2} be a homomorphism from P_{2} onto V_{2}, and let g_{1} be a homomorphism from V_{1} into V_{2}. Note that g_{1} ∘ β_{1} is a homomorphism from P_{1} into V_{2}, as modules over A. Suppose now that P_{1} is projective as well, so that there is a homomorphism f_{1} from P_{1} into P_{2}, as modules over A, such that

β_{2} ∘ f_{1} = g_{1} ∘ β_{1}. (32.1.7)

Of course, g_{1} ∘g is a homomorphism from V into V_{2}, and g_{1} ∘g ∘β is a homomorphism from P into V_{2}. Under these conditions, f_{1} ∘ f is a homomorphism from P into P_{2}, with

β_{2} ∘ f_{1} ∘ f = g_{1} ∘ β_{1} ∘ f = g_{1} ∘ g ∘ β. (32.1.8)

Suppose that V_{2} = V, P_{2} = P, β_{2} = β, and g_{1} ∘ g is the identity mapping on V. In this case, f_{1} ∘ f is a homomorphism from P into itself, with

β ∘ f_{1} ∘ f = β, (32.1.9)

by (32.1.8). Equivalently, this means that f_{1} ∘ f minus the identity mapping on P maps P into the kernel of β.

32.2 Comparisons and tensor products

Let us continue with the same notations and hypotheses as in the previous section. Put M = ker β, M_{1} = ker β_{1}, (32.2.1)

which are submodules of P, P_{1}, respectively, as modules over A. Let α, α_{1} be the obvious inclusion mappings from M, M_{1} into P, P_{1}, respectively. Remember that f maps M into M_{1}, as in (32.1.2), and let f′ be the restriction of f to M, considered as a homomorphism from M into M_{1}, as modules over A. Thus

α_{1} ∘ f′ = f ∘ α, (32.2.2)

as homomorphisms from M into P_{1}, by construction.

Suppose now that V, V_{1}, P, P_{1}, and so on are right modules over A, and let W be a left module over A. Let M ⨂_{A} W, P ⨂_{A} W, M_{1} ⨂_{A} W, and P_{1} ⨂_{A} W be tensor products of M, P, M_{1}, and P_{1} with W over A, respectively. Consider the homomorphisms

(32.2.3) α^{W} , α_{1}^{W} from M ⨂_{A}W, M_{1} ⨂_{A}W into P ⨂_{A}W, P_{1} ⨂_{A}W,

respectively, as modules over k, corresponding to α, α_{1} and the identity mapping on W, as in Section 1.9. Similarly, we get homomorphisms

(32.2.4) f′_{W} , f_{W} from M ⨂_{A}W, P ⨂_{A}W into M_{1} ⨂_{A}W, P_{1} ⨂_{A}W,

respectively, as modules over k, corresponding to f′, f and the identity mapping on W. Note that α_{1}^{W} ∘ f′_{W} = f_{W} ∘ α^{W} , (32.2.5)

as homomorphisms from M ⨂_{A} W into P_{1} ⨂_{A} W, because of (32.2.2), as in (2) on p34 of [50].

Using (32.2.5), we get that

f′_{W} (ker α^{W}) ⊆ ker α_{1}^{W} . (32.2.6)

Consider the homomorphism

(32.2.7) θ_{1}(g) from ker α^{W} into ker α_{1}^{W} ,

as modules over k, corresponding to the restriction of f′_{W} to ker α^{W}, as on p34 of [50].

Let us check that

(32.2.8) θ_{1}(g) does not depend on the choice of f

from the previous section, as in Proposition 1.1 on p34 of [50]. Let f̃ be another homomorphism from P into P_{1}, as modules over A, such that (32.1.3) holds. Thus f − f̃ maps P into the kernel of β_{1}, as in (32.1.4). Let h be the homomorphism from P into ker β_{1}, as modules over A, that is equal to f − f̃. This means that f − f̃ = α_{1} ∘ h, (32.2.9)

as homomorphisms from P into P_{1}.

Let f̃′ be the restriction of f̃ to M, considered as a homomorphism from M into M_{1}. Note that f′ − f̃′ = h ∘ α. (32.2.10)

Let f̃′_{W} be the homomorphism from M ⨂_{A} W into M_{1} ⨂_{A} W corresponding to f̃′ and the identity mapping on W, and let h_{W} be the homomorphism from P ⨂_{A} W into M_{1} ⨂_{A} W that corresponds to h and the identity mapping on W. Thus f′_{W} − f̃′_{W} = h_{W} ∘ α^{W} , (32.2.11) as homomorphisms from M ⨂_{A} W into M_{1} ⨂_{A} W, because of (32.2.10). This implies that f′_{W} = f̃′_{W} on ker α^{W} , (32.2.12)

as desired.

If t ∈ k, then t g is a homomorphism from V into V_{1}, and t f is a homomorphism from P into P_{1} that is related to t g in the usual way. It is easy to see that θ_{1}(t g) = t θ_{1}(g), (32.2.13)

as homomorphisms from ker α^{W} into ker α_{1}^{W}.

Let ĝ be another homomorphism from V into V_{1}, as in the previous section, and let f̂ be a homomorphism from P into P_{1} such that (32.1.5) holds. Under these conditions, g + ĝ is a homomorphism from V into V_{1}, and f + f̂ is a homomorphism from P into P_{1} that satisfies (32.1.6). This means that homomorphisms

(32.2.14) θ_{1}(ĝ) and θ_{1}(g + ĝ) from ker α^{W} into ker α_{1}^{W}

may be defined using f̂ and f + f̂, respectively, as before. It follows that

θ_{1}(g + ĝ) = θ_{1}(g) + θ_{1}(ĝ), (32.2.15)

as in Proposition 1.1 on p34 of [50].

Suppose for the moment that V_{1} = V, P_{1} = P, β_{1} = β, and g is the identity mapping I_{V} on V. In this case, we can take f to be the identity mapping I_{P} on P, so that f′ is the identity mapping I_{M} on M = M_{1}. This means that f′_{W} is the identity mapping on M ⨂_{A} W = M_{1} ⨂_{A} W. Note that α = α_{1}, so that α_{1}^{W} = α^{W}, and thus ker α_{1}^{W} = ker α^{W}. Under these conditions, we get that

(32.2.16) θ_{1}(g) is the identity mapping on ker α^{W} .

Let V_{1}, P_{1}, β_{1}, and g be as before, and suppose now that P_{1} is projective as a module over A. Also let V_{2}, P_{2}, β_{2}, g_{1}, and f_{1} be as in the previous section. Put M_{2} = ker β_{2}, (32.2.17)

which is a submodule of P_{2}, and let α_{2} be the obvious inclusion mapping from M_{2} into P_{2}. Let M_{2} ⨂_{A} W and P_{2} ⨂_{A} W be tensor products of M_{2} and P_{2} with W over A, and consider the homomorphism

(32.2.18) α_{2}^{W} from M_{2} ⨂_{A}W into P_{2} ⨂_{A}W

corresponding to α_{2} and the identity mapping on W. We can define a homomorphism (32.2.19) θ_{1}(g_{1}) from ker α_{1}^{W} into ker α_{2}^{W} ,

as modules over k, using f_{1}, as before. Similarly, we can define a homomorphism

(32.2.20) θ_{1}(g_{1} ∘ g) from ker α^{W} into ker α_{2}^{W} , as modules over k, using f_{1} ∘ f, because of (32.1.8). One can check that

θ_{1}(g_{1} ∘ g) = θ_{1}(g_{1}) ∘ θ_{1}(g), (32.2.21)

using the analogous property for induced mappings between tensor products. This is another part of Proposition 1.1 on p34 of [50].

Suppose that V_{2} = V, P_{2} = P, and β_{2} = β, so that M_{2} = M, α_{2} = α, α_{2}^{W} = α^{W}, and thus ker α_{2}^{W} = ker α^{W}. If g_{1} ∘ g is the identity mapping on V, then (32.2.22) θ_{1}(g_{1} ∘ g) is the identity mapping on ker α^{W} ,

as in (32.2.16). Similarly, if g ∘ g_{1} is the identity mapping on V_{1}, then

(32.2.23) θ_{1}(g ∘ g_{1}) is the identity mapping on ker α_{1}^{W} .

If g and g_{1} are inverses of each other, then it follows that

(32.2.24) θ_{1}(g) and θ_{1}(g_{1}) are inverses of each other,

because of (32.2.21).

Of course, there are analogous statements when we start with left modules over A as in the previous section, and consider their tensor products with a right module W over A.

32.3 Left satellites of tensor products

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V be a right module over A, let W be a left module over A, and let

T (V) = V ⨂_{A}W (32.3.1)

be a tensor product of V and W over A.

Suppose that P is a projective right module over A, and that β is a homomorphism from P onto V, as modules over A. Put M = ker β, and let α be the obvious inclusion mapping from M into P, as in the previous section. Let M ⨂_{A} W, P ⨂_{A} W be tensor products of M, P with W again, respectively, and consider the homomorphism

(32.3.2) α^{W} from M ⨂_{A}W into P ⨂_{A}W,

as modules over k, corresponding to α and the identity mapping on W, as before. Put S_{1}T (V) = ker α^{W} , (32.3.3)

which is a submodule of M ⨂_{A} W, and a module over k in particular. This corresponds to (5) on p35 of [50].

Let P_{1} be another projective right module over A, and let β_{1} be a homomorphism from P_{1} onto V, as modules over A. Put M_{1} = ker β_{1}, and let α_{1} be the obvious inclusion mapping from M_{1} into P_{1}, as before. Let M_{1} ⨂_{A} W, P_{1} ⨂_{A} W be tensor products of M_{1}, P_{1} with W, respectively, and consider the homomorphism

(32.3.4) α_{1}^{W} from M_{1} ⨂_{A}W into P_{1} ⨂_{A}W,

as modules over k, corresponding to α_{1} and the identity mapping on W. This is the same as in the previous section, with V = V_{1}.

If we take g to be the identity mapping on V, then we get a homomorphism θ_{1}(g) from ker α^{W} into ker α_{1}^{W}, as modules over k, as in (32.2.8). Of course, this depends on P, P_{1} and β, β_{1}.

Let us take V_{2} = V, P_{2} = P, and β_{2} = β, so that we have M_{2} = M, α_{2} = α, and α_{2}^{W} = α^{W}, as in the previous section. If we take g_{1} to be the identity mapping on V_{1} = V, then we get a homomorphism θ_{1}(g_{1}) from ker α_{1}^{W} into ker α_{2}^{W} = ker α^{W}, as modules over k, as in (32.2.19). This depends on P_{1}, P_{2} = P and β_{1}, β_{2} = β, as before.

Because g_{1} ∘ g is the identity mapping on V, and g ∘ g_{1} is the identity mapping on V_{1} = V, we get that θ_{1}(g) and θ_{1}(g_{1}) are inverses of each other, as in (32.2.24). This shows that (32.3.3) is unique up to natural isomorphisms, as on p35f of [50]. This is called the left satellite of (32.3.1), as on p36 of [50].

If V is projective as a right module over A, the we can take P = V and β to be the identity mapping on V, so that M = {0}, α = 0, α^{W} = 0, and

S_{1}T (V) = {0}. (32.3.5)

This corresponds to part of Proposition 1.3 on p37 of [50].

If W is projective as a left module over A, then α^{W} is injective, because α is injective, as mentioned near the end of Section 2.7. This implies that (32.3.5) holds in this case too. This corresponds to part of Proposition 1.2 on p36 of [50].

Of course, there are analogous statements when V is a left module over A, W is a right module over A, and T (V) = W ⨂_{A} V is a tensor product of W and V, as modules over A.

32.4 Comparisons and Hom_{A}(·, W)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. As in Section 32.1, we let V, V_{1}, P, and P_{1} be all left or all right modules over A, we let β, β_{1} be homomorphisms from P, P_{1} into V, V_{1}, respectively, and we let g be a homomorphism from V into V_{1}. We also let M, M_{1} be the kernels of β, β_{1}, respectively, and let α, α_{1} be the obvious inclusion mappings from M, M_{1} into P, P_{1}, respectively.

32.4. COMPARISONS AND HOM_{A}(·, W) 711

Suppose that P is projective again, so that there is a homomorphism f from P into P_{1} such that β_{1} ∘ f = g ∘ β. Remember that

f(M) ⊆ M_{1}, (32.4.1)

and let f′ be the restriction of f to M, considered as a homomorphism from M into M_{1}.

Let W be another left or right module over A, depending on whether the other modules are left or right modules. Consider the homomorphisms

(32.4.2) Hom(α, I_{W}) from Hom_{A}(P, W) into Hom_{A}(M, W)

and (32.4.3) Hom(α_{1}, I_{W}) from Hom_{A}(P_{1}, W) into Hom_{A}(M_{1}, W)

associated to α, α_{1} and the identity mapping I_{W} on W as in Section 6.3. Similarly, we get homomorphisms

(32.4.4) Hom(f′, I_{W}) from Hom_{A}(M_{1}, W) into Hom_{A}(M, W)

and (32.4.5) Hom(f, I_{W}) from Hom_{A}(P_{1}, W) into Hom_{A}(P, W).

It is easy to see that

(32.4.6) Hom(f′, I_{W}) ∘ Hom(α_{1}, I_{W}) = Hom(α, I_{W}) ∘ Hom(f, I_{W}),

as homomorphisms from Hom_{A}(P_{1}, W) into Hom_{A}(M, W), because α_{1} ∘ f′ = f ∘ α. This corresponds to a remark on p34 of [50].

Remember that the cokernel of Hom(α, I_{W}) is the quotient of Hom_{A}(M, W) by the image of Hom_{A}(P, W) under Hom(α, I_{W}), as a module over k. Similarly, the cokernel of Hom(α_{1}, I_{W}) is the quotient of Hom_{A}(M_{1}, W) by the image of Hom_{A}(P_{1}, W) under Hom(α_{1}, I_{W}). It follows from (32.4.6) that Hom(f′, I_{W}) induces a homomorphism

(32.4.7) θ^{1}(g) from the cokernel of Hom(α_{1}, I_{W})

into the cokernel of Hom(α, I_{W}),

as modules over k, as on p34 of [50].

We would like to verify that

(32.4.8) θ^{1}(g) does not depend on the choice of f,

as in Proposition 1.1 on p34 of [50]. If f̃ is another homomorphism from P into P_{1} with β_{1} ∘ f̃ = g ∘ β, then f − f̃ maps P into the kernel of β_{1}, as before. If h is the homomorphism from P into M_{1} = ker β_{1} defined by f − f̃, then f − f̃ = α_{1} ∘ h, as homomorphisms from P into P_{1}. Let f̃′ be the restriction of f̃ to M, as a homomorphism into M_{1}, so that f′ − f̃′ = h ∘ α, as before.

Using f̃′, we get a homomorphism

(32.4.9) Hom(f̃′, I_{W}) from Hom_{A}(M_{1}, W) into Hom_{A}(M, W).

Observe that

Hom(f′, I_{W}) − Hom(f̃′, I_{W}) = Hom(f′ − f̃′, I_{W}) = Hom(h ∘ α, I_{W})
(32.4.10)  = Hom(α, I_{W}) ∘ Hom(h, I_{W}),

where Hom(h, I_{W}) is a homomorphism from Hom_{A}(M_{1}, W) into Hom_{A}(P, W), as modules over k. In particular,

(32.4.11) Hom(f′ − f̃′, I_{W}) maps Hom_{A}(M_{1}, W) into the image of Hom_{A}(P, W) under Hom(α, W).

This implies that Hom(f′, I_{W}) and Hom(f̃′, I_{W}) induce the same homomorphisms into the cokernel of Hom(α, I_{W}), as desired.

If t ∈ k, then t g is a homomorphism from V into V_{1}, and t f is a homomorphism from P into P_{1} that is related to t g in the same way as before. Using this, we get that θ^{1}(t g) = t θ^{1}(g). (32.4.12)

Let ĝ be another homomorphism from V into V_{1}, and let f̂ be a homomorphism from P into P_{1} such that β_{1}∘f̂ = ĝ∘β. Thus g+ĝ is a homomorphism from V into V_{1}, f +f̂is a homomorphism from P into P_{1}, and β_{1}∘(f +f̂) = (g+ĝ)∘β. It follows that homomorphisms θ^{1}(ĝ) and θ^{1}(g + ĝ) may be defined using f̂ and f + f̂, respectively, as before. This implies that

θ^{1}(g + ĝ) = θ^{1}(g) + θ^{1}(ĝ), (32.4.13)

as in Proposition 1.1 on p34 of [50].

If V_{1} = V, P_{1} = P, β_{1} = β, and g is the identity mapping I_{V} on V, then we can take f to be the identity mapping I_{P} on P, so that f′ is the identity mapping I_{M} on M = M_{1}. Under these conditions,

Hom(f′, I_{W}) = Hom(I_{M}, I_{W}) (32.4.14)

is the identity mapping on Hom_{A}(M, W) = Hom_{A}(M_{1}, W). We also have that α = α_{1}, so that Hom(α, I_{W}) = Hom(α_{1}, I_{W}), and thus their cokernels are the same. This means that

(32.4.15) θ^{1}(g) is the identity mapping on the cokernel of Hom(α, I_{W})

in this case.

Let V_{1}, P_{1}, β_{1}, and g be as before, and let V_{2}, P_{2} be left or right modules over A, depending on whether the other modules are left or right modules over A. Also let β_{2} be a homomorphism from P_{2} onto V_{2}, and let g_{1} be a homomorphism from V_{1} into V_{2}, as modules over A. Put M_{2} = ker β_{2} again, and let α_{2} be the 32.5. THE RIGHT SATELLITE OF HOM_{A}(·, W) 713 obvious inclusion mapping from M_{2} into P_{2}. Suppose that P_{1} is projective as a module over A, so that there is a homomorphism f_{1} from P_{1} into P_{2}, as modules over A, such that β_{2} ∘ f_{1} = g_{1} ∘ β_{1}, as before.

As usual, we get a homomorphism (32.4.16) Hom(α_{2}, I_{W}) from Hom_{A}(P_{2}, W) into Hom_{A}(M_{2}, W), as modules over k. We can define a homomorphism (32.4.17) θ^{1}(g_{1}) from the cokernel of Hom(α_{2}, I_{W})

into the cokernel of Hom(α_{1}, I_{W}),

as modules over k, using f_{1}, as before. Similarly, we can define a homomorphism (32.4.18) θ^{1}(g_{1} ∘ g) from the cokernel of Hom(α_{2}, I_{W})

into the cokernel of Hom(α, I_{W}),

as modules over k, using f_{1} ∘ f. One can verify that θ^{1}(g_{1} ∘ g) = θ^{1}(g) ∘ θ^{1}(g_{1}), (32.4.19) as in Proposition 1.1 on p34 of [50] again.

Suppose that V_{2} = V, P_{2} = P, and β_{2} = β, so that M_{2} = M, α_{2} = α, Hom(α_{2}, I_{W}) = Hom(α, I_{W}), and the cokernels of the latter homomorphisms are the same. If g_{1} ∘ g is the identity mapping on V, then (32.4.20) θ^{1}(g_{1} ∘ g) is the identity mapping on the cokernel of Hom(α, I_{W}), as in (32.4.15). Similarly, if g ∘ g_{1} is the identity mapping on V_{1}, then (32.4.21) θ^{1}(g ∘ g_{1}) is the identity mapping on the cokernel of Hom(α_{1}, I_{W}). If g and g_{1} are inverse of each other, then we get that (32.4.22) θ^{1}(g) and θ^{1}(g_{1}) are inverses of each other, using (32.4.19).

32.5 The right satellite of Hom_{A}(·, W)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative agebra over k with a multiplicative identity element e_{A}. Also let V, W be both left or both right modules over A, and put T (V) = Hom_{A}(V, W). (32.5.1)

Suppose that P is a projective left or right module over A, as appropriate, and that β is a homomorphism from P onto V, as modules over A. Put M = ker β, and let α be the obvious inclusion mapping from M into P, as before. Using α, we get a homomorphism Hom(α, I_{W}) as in (32.4.2). Under these conditions, we put

(32.5.2) S^{1}T (V) = the cokernel of Hom(α, I_{W})
= Hom_{A}(M, W)/Hom(α, I_{W})(Hom_{A}(P, W)),

which is a module over k. This corresponds to (6a) on p36 of [50].

Let P_{1} be another projective left or right module over A, as appropriate, and let β_{1} be a homomorphism from P_{1} onto V, as modules over A. This corresponds to taking V_{1} = V in the previous section. Put M_{1} = ker β_{1} again, and let α_{1} be the obvious inclusion mapping from M_{1} into P_{1}. We can use α_{1} to get a homomorphism Hom(α_{1}, I_{W}) as in (32.4.3).

If we take g to be the identity mapping on V, then we get a homomorphism θ^{1}(g) from the cokernel of Hom(α_{1}, I_{W}) into the cokernel of Hom(α, I_{W}), as modules over k, as in (32.4.7). Note that this depends on P, P_{1}, β, and β_{1}.

Let us take V_{2} = V, P_{2} = P, and β_{2} = β, so that M_{2} = M, α_{2} = α, Hom(α_{2}, I_{W}) = Hom(α, I_{W}), and the cokernels of the latter homomorphisms are the same, as in the previous section. We can take g_{1} to be the identity mapping on V_{1} = V, to get a homomorphism θ^{1}(g_{1}) from the cokernel of Hom(α, I_{W}) into the cokernel of Hom(α_{1}, I_{W}), as in (32.4.17). Of course, this depends on P_{1}, P_{2} = P, β_{1}, and β_{2} = β, as before.

In particular, g_{1} ∘ g is the identity mapping on V, and g ∘ g_{1} is the identity mapping on V_{1} = V, so that θ^{1}(g) and θ^{1}(g_{1}) are inverses of each other, as in (32.4.22). This means that (32.5.2) is unique up to natural isomorphisms, as in [50]. This is the right satellite of (32.5.1), as on p36 of [50].

If V is projective as a module over A, then we can take P = V and β to be the identity mapping on V, so that M = {0}, α = 0, Hom(α, I_{W}) = 0, and

S^{1}T (V) = {0}. (32.5.3)

This corresponds to another part of Proposition 1.3 on p37 of [50].

If W is injective as a module over A, then Hom(α, I_{W}) maps Hom_{A}(P, W) onto Hom_{A}(M, W). This means that (32.5.3) holds in this case as well, as in Proposition 1.2 on p36 of [50]. This also corresponds to the sufficiency of injectivity in Corollary 2.2a on p111 of [50].

32.6 Necessity of injectivity

Let us continue with the same notation and hypotheses as in the previous section. Note that (32.5.3) holds if and only if

(32.6.1) Hom(α, I_{W})(Hom_{A}(P, W)) = Hom_{A}(M, W),

by (32.5.2). More precisely, we have seen that this does not depend on the choice of projective module P over A, or homomorphism β from P onto V.

Let P be any projective left or right module over A, as appropriate, and let M be a submodule of P. Suppose that V is the quotient module

V = P/M, (32.6.2)

and β is the natural quotient mapping from P onto V. If (32.5.3) holds, then (32.6.1) holds, as in the preceding paragraph.

Supposee that

(32.6.3) (32.5.3) holds for every left or right module V over A,

as appropriate. This implies that (32.6.1) holds for every projective left or right module P over A, as appropriate, and every submodule M of P, as before.

Under these conditions, we get that

(32.6.4) W is injective, as a module over A,

as in Section 9.2. This corresponds to the necessity of injectivity in Corollary 2.2a on p111 of [50].

32.7 Some more comparisons

Let k be a commutative ring with a mutliplicative identity element again, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let W, W_{1}, Q, and Q_{1} be all left or all right modules over A, let β, β_{1} be injective homomorphisms from W, W_{1} into Q, Q_{1}, respectively, and let g be a homomorphism from W_{1} into W.

Note that g ∘ β is a homomorphism from W_{1} into Q, as modules over A. Let us consider some related mappings and their properties, as mentioned beginning on p34 of [50].

Suppose that Q is injective, as a module over A. This implies that there is a homomorphism f from Q_{1} into Q, as modules over A, such that

f ∘ β_{1} = β ∘ g. (32.7.1)

In particular, f(β_{1}(W_{1})) = β(g(W_{1})) ⊆ β(W). (32.7.2)

Put N = Q/β(W), N_{1} = Q_{1}/β_{1}(W_{1}), (32.7.3)

which are modules over A, and let α, α_{1} be the natural quotient mappings from Q, Q_{1} onto N, N_{1}, respectively. It follows from (32.7.2) that f induces a homomorphism (32.7.4) f′ from N_{1} into N,

as modules over A, such that

f′ ∘ α_{1} = α ∘ f. (32.7.5)

If f̃ is another homomorphism from Q_{1} into Q, then

f̃∘ β_{1} = β ∘ g (32.7.6)

if and only if β_{1}(W_{1}) ⊆ ker(f − f̃). (32.7.7) This holds exactly when there is a homomorphism h from N_{1} into Q such that

f − f̃ = h ∘ α_{1}. (32.7.8)

Let f̃′ be the homomorphism from N_{1} into N, as modules over A, induced by f̃, so that f̃′ ∘ α_{1} = α ∘ f̃, (32.7.9) as in (32.7.5). Using (32.7.5), (32.7.8), and (32.7.9), we get that

(f′ − f̃′) ∘ α_{1} = α ∘ (f − f̃) = α ∘ h ∘ α_{1}. (32.7.10)

This implies that f′ − f̃′ = α ∘ h, (32.7.11) as homomorphisms from N_{1} into N, because α_{1} maps Q_{1} onto N_{1}.

If t ∈ k, then t g is a homomorphism from W_{1} into W too, as modules over k. In this case, t f is a homomorphism from Q_{1} into Q, and the composition of β_{1} with t f is equal to the composition of t g with β.

Let ĝ be another homomorphism from W_{1} into W, as modules over A, so that β ∘ ĝ is a homomorphism from W_{1} in Q. Because Q is injective, there is a homomorphism f̂ from Q_{1} into Q, as modules over A, such that

f̂∘ β_{1} = β ∘ ĝ, (32.7.12)

as before. Thus (f + f̂) ∘ β_{1} = β ∘ (g + ĝ). (32.7.13)

Let W_{2} and Q_{2} be left or right modules over A, as appropriate, let β_{2} be an injective homomorphism from W_{2} into Q_{2}, and let g_{1} be a homomorphism from W_{2} into W_{1}. Thus β_{1} ∘ g_{1} is a homomorphism from W_{2} into Q_{1}, as modules over A. If Q_{1} is injective as a module over A, then there is a homomorphism f_{1} from Q_{2} into Q_{1}, as modules over A, such that

f_{1} ∘ β_{2} = β_{1} ∘ g_{1}. (32.7.14)

Note that g ∘ g_{1} is a homomorphism from W_{2} into W, and that β ∘ g ∘ g_{1} is a homomorphism from W_{2} into Q. Similarly, f ∘ f_{1} is a homomorphism from Q_{2} into Q, with f ∘ f_{1} ∘ β_{2} = f ∘ β_{1} ∘ g_{1} = β ∘ g ∘ g_{1}. (32.7.15)

Suppose now that W_{2} = W, Q_{2} = Q, β_{2} = β, and g ∘ g_{1} is the identity mapping on W. Under these conditions, f ∘ f_{1} is a homomorphism from Q into itself, with f ∘ f_{1} ∘ β = β, (32.7.16) as in (32.7.15). This is the same as saying that β(W) is contained in the kernel of f ∘ f_{1} minus the identity mapping on Q.

32.8. COMPARISONS AND HOM_{A}(V, ·) 717

32.8 Comparisons and Hom_{A}(V, ·)

Let us continue with the same notation and hypotheses as in the previous section, and let V be a left or right module over A, depending on whether the other modules are left or right modules. Consider the homomorphisms

(32.8.1) Hom(I_{V} , α) from Hom_{A}(V, Q) into Hom_{A}(V, N)

and (32.8.2) Hom(I_{V} , α_{1}) from Hom_{A}(V, Q_{1}) into Hom_{A}(V, N_{1})

associated to the identity mapping I_{V} on V and α, α_{1} as in Section 6.3. Similarly, we have homomorphisms

(32.8.3) Hom(I_{V} , f) from Hom_{A}(V, Q_{1}) into Hom_{A}(V, Q)

and (32.8.4) Hom(I_{V} , f′) from Hom_{A}(V, N_{1}) into Hom_{A}(V, N).

Observe that

(32.8.5) Hom(I_{V} , f′) ∘ Hom(I_{V} , α_{1}) = Hom(I_{V} , α) ∘ Hom(I_{V} , f),

as homomorphisms from Hom_{A}(V, Q_{1}) into Hom_{A}(V, N), because of (32.7.5), as in (2a) on p35 of [50].

Remember that the cokernel of Hom(I_{V} , α_{1}) is the quotient of Hom_{A}(V, N_{1}) by the image of Hom_{A}(V, Q_{1}) under Hom(I_{V} , α_{1}), and similarly that the cokernel of Hom(I_{V} , α) is the quotient of Hom_{A}(V, N) by the image of Hom_{A}(V, Q) under Hom(I_{V} , α). Using (32.8.5), we get that Hom(I_{V} , f′) induces a homomorphism

(32.8.6) θ^{1}(g) from the cokernel of Hom(I_{V} , α_{1})

into the cokernel of Hom(I_{V} , α),

as modules over k, as on p35 of [50]. We would like to check that

(32.8.7) θ^{1}(g) does not depend on the choice of f,

as in Proposition 1.1a on p35 of [50].

Let f̃ be another homomorphism from Q_{1} into Q that satisfies (32.7.6), so that (32.7.7) holds. Remember that f̃ induces a homomorphism f̃′ from N_{1} into N as in (32.7.9). We have seen that there is a homomorphism h from N_{1} into Q, as modules over A, such that (32.7.11) holds.

We can use f̃′ to get a homomorphism

(32.8.8) Hom(I_{V} , f̃′) from Hom_{A}(V, N_{1}) into Hom_{A}(V, N).

We also have that

Hom(I_{V} , f′) − Hom(I_{V} , f̃′) = Hom(I_{V} , f′ − f̃′) = Hom(I_{V} , α ∘ h)
(32.8.9)  = Hom(I_{V} , α) ∘ Hom(I_{V} , h),

using (32.7.11) in the second step. This implies that

(32.8.10) Hom(I_{V} , f′ − f̃′) maps Hom_{A}(V, N_{1}) into the image of Hom_{A}(V, Q) under Hom(I_{V} , α),

because Hom(I_{V} , h) maps Hom_{A}(V, N_{1}) into Hom_{A}(V, Q). Using this, we get that Hom(I_{V} , f′) and Hom(I_{V} , f̃′) induce the same homomorphisms into the cokernel of Hom(I_{V} , α), as desired.

If t ∈ k, then t g is a homomorphism from W_{1} into W, t f is a homomorphism from Q_{1} into Q, and they are related as in (32.7.1). One can verify that

θ^{1}(t g) = t θ^{1}(g). (32.8.11)

Let ĝ be another homomorphism from W_{1} into W, and let f̂ be a homomorphism from Q_{1} into Q such that (32.7.12) holds. This means that g + ĝ is a homomorphism from W_{1} into W, f + f̂ is a homomorphism from Q_{1} into Q, and that (32.7.13) holds. Thus θ^{1}(ĝ) and θ^{1}(g + ĝ) may be defined using f̂ and f + f̂, respectively, as before. It follows that

θ^{1}(g + ĝ) = θ^{1}(g) + θ^{1}(ĝ), (32.8.12)

as in Proposition 1.1a pn p35 of [50].

If W_{1} = W, Q_{1} = Q, β_{1} = β, and g is the identity mapping I_{W} on W, then we can take f to be the identity mapping I_{Q} on Q, so that f′ is the identity mapping I_{N} on N. In this case,

Hom(I_{V} , f′) = Hom(I_{V} , I_{N}) (32.8.13)

is the identity mapping on Hom_{A}(V, N_{1}) = Hom_{A}(V, N). Note that α_{1} = α, so that Hom(I_{V} , α_{1}) = Hom(I_{V} , α), and their cokernels are the same. Thus

(32.8.14) θ^{1}(g) is the identity mapping on the cokernel of Hom(I_{V} , α_{1})

under these conditions.

Let W_{1}, Q_{1}, β_{1}, and g be as before, and let W_{2}, Q_{2}, β_{2}, and g_{1} be as in the previous section. Put N_{2} = Q_{2}/β_{2}(W_{2}), (32.8.15)

and let α_{2} be the natural quotient mapping from Q_{2} onto N_{2}. Suppose that Q_{1} is injective as a module over A, so that there is a homomorphism f_{1} from Q_{2} into Q_{1}, as modules over A, such that (32.7.14) holds.

Using α_{2}, we get a homomorphism

(32.8.16) Hom(I_{V} , α_{2}) from Hom_{A}(V, Q_{2}) into Hom_{A}(V, N_{2}),

as modules over k. We can use f_{1} to define a homomorphism

(32.8.17) θ^{1}(g_{1}) from the cokernel of Hom(I_{V} , α_{2})

into the cokernel of Hom(I_{V} , α_{1}), 32.9. THE RIGHT SATELLITE OF HOM_{A}(V, ·) 719

as modules over k, as before. Similarly, we can use f ∘ f_{1} to define a homomorphism

(32.8.18) θ^{1}(g ∘ g_{1}) from the cokernel of Hom(I_{V} , α_{2})

into the cokernel of Hom(I_{V} , α),

as modules over k. One can check that

θ^{1}(g ∘ g_{1}) = θ^{1}(g_{1}) ∘ θ^{1}(g), (32.8.19)

as in Proposition 1.1a on p35 of [50].

Suppose now that W_{2} = W, Q_{2} = Q, and β_{2} = β, so that N_{2} = N, α_{2} = α, Hom(I_{V} , α_{2}) = Hom(I_{V} , α), and the cokernels of the latter homomorphisms are the same. If g ∘ g_{1} is the identity mapping on W, then

(32.8.20) θ^{1}(g ∘ g_{1}) is the identity mapping on the cokernel of Hom(I_{V} , α),

as in (32.8.14). Simiarly, if g_{1} ∘ g is the identity mapping on W_{1}, then

(32.8.21) θ^{1}(g_{1} ∘ g) is the identity mapping on the cokernel of Hom(I_{V} , α_{1}).

If g and g_{1} are inverses of each other, then it follows that

(32.8.22) θ^{1}(g) and θ^{1}(g_{1}) are inverses of each other,

because of (32.8.19).

32.9 The right satellite of Hom_{A}(V, ·)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, W be both left or both right modules over A, and put

T (W) = Hom_{A}(V, W). (32.9.1)

Suppose that Q is an injective left or right module over A, depending on whether V, W are left or right modules over A, and let β be an injective homomorphism from W into Q, as modules over A. Put N = Q/β(W), and let α be the natural quotient mapping from Q onto N, as before. We can use α to get a homomorphism Hom(I_{V} , α) as in (32.8.1). Put

(32.9.2) S^{1}T (W) = the cokernel of Hom(I_{V} , α)
= Hom_{A}(V, N)/Hom(I_{V} , α)(Hom_{A}(V, Q)),

which is a module over k. This corresponds to (6) on p35 of [50].

Let Q_{1} be another injective left or right module over A, as appropriate, and let β_{1} be an injective homomorphism from W into Q_{1}. This corresponds to taking W_{1} = W in the previous two sections. Put N_{1} = Q_{1}/β(W), and let α_{1} be the natural quotient mapping from Q_{1} onto N_{1}, as before. Using α_{1}, we get a homomorphism Hom(I_{V} , α_{1}) as in (32.8.2).

If we take g to be the identity mapping on W_{1} = W, then we get a homomorphism θ^{1}(g) from the cokernel of Hom(I_{V} , α_{1}) into the cokernel of Hom(I_{V} , α), as modules over k, as in (32.8.6). This depends on Q, Q_{1}, β, and β_{1}.

Let us take W_{2} = W, Q_{2} = Q, and β_{2} = β, so that N_{2} = N, α_{2} = α, Hom(I_{V} , α_{2}) = Hom(I_{V} , α), and the cokernels of the latter homomorphisms are the same, as in the previous section. Let us also take g_{1} to be the identity mapping on W_{2} = W, as a mapping into W_{1} = W, to get a homomorphism θ^{1}(g_{1}) from the cokernel of Hom(I_{v}, α) into the cokernel of Hom(I_{V} , α_{1}), as in (32.8.17). This depends on Q_{1}, Q_{2} = Q, β_{1}, and β_{2} = β, as before.

Of course, g∘g_{1} is the identity mapping on W_{2} = W, and g_{1}∘g is the identity mapping on W_{1} = W. Thus θ^{1}(g) and θ^{1}(g_{1}) are inverses of each other, as in (32.8.22). This shows that (32.9.2) is unique up to natural isomorphisms, as on p36 of [50]. This is the right satellite of (32.9.1), as in [50].

If W is injective as a module over A, then we can take Q = W and β to be the identity mapping on W, so that N = {0}, α = 0, Hom(I_{V} , α) = 0, and

S^{1}T (W) = {0}. (32.9.3)

This corresponds to part of Proposition 1.3 on p37 of [50].

If V is projective as a module over A, then Hom(I_{V} , α) maps Hom_{A}(V, Q) onto Hom_{A}(V, N). It follows that (32.9.3) holds in this case too, as in Proposition 1.2 on p36 of [50]. This corresponds to the sufficiency of projectivity in Corollary 2.2 on p110 of [50] as well.

32.10 Necessity of projectivity

We continue with the same notation and hypotheses as in the previous section. Of course, (32.9.3) holds if and only if

(32.10.1) Hom(I_{V} , α)(Hom_{A}(V, Q)) = Hom_{A}(V, N),

by (32.9.2). This does not depend on the choice of injective module Q over A or injective homomorphism β from W into Q, as before.

Suppose now that Q is any injective left or right module over A, as appropriate, and that α is a homomorphism from Q onto another module N over A. Put W = ker α, (32.10.2)

and let β be the obvious inclusion mapping from W into Q. If (32.9.3) holds, then it follows that (32.10.1) holds.

Suppose that

(32.10.3) (32.9.3) holds for every left or right module W over A, as appropriate. In this case, (32.10.1) holds for every injective left or right module Q over A, as appropriate, and every homomorphism α from Q onto another module N over A, as in the preceding paragraph.

This implies that

(32.10.4) V is projective, as a module over A,

as in Section 9.1. This corresponds to the necessity of projectivity in Corollary 2.2 on p110 of [50].

32.11 A type of double satellite

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, P_{V} be right modules over A, and let W, P_{W} be left modules over A.

Suppose that P_{V}, P_{W} are projective as modules over A, and that

(32.11.1) β_{V} , β_{W} are homomorphisms from P_{V} , P_{W} onto V, W,

respectively, as modules over A. Put

M_{V} = ker β_{V} , M_{W} = ker β_{W} , (32.11.2)

which are submodules of P_{V}, P_{W}, respectively, as modules over A. Consider the obvious inclusion mappings

(32.11.3) α_{V} , α_{W} from M_{V} , M_{W} into P_{V} , P_{W} ,

respectively.

Let M_{V} ⨂_{A} M_{W}, P_{V} ⨂_{A} M_{W}, and M_{V} ⨂_{A} P_{W} be tensor products of the indicated modules over A. We can use α_{V} and the identity mapping on M_{W} to get a homomorphism

(32.11.4) \overline{α}_{V} from M_{V} ⨂_{A}M_{W} into P_{V} ⨂_{A}M_{W} ,

as modules over k. Similarly, we can use the identity mapping on M_{V} and α_{W} to get a homomorphism

(32.11.5) \overline{α}_{W} from M_{V} ⨂_{A}M_{W} into M_{V} ⨂_{A}P_{W} ,

as modules over k. Using \overline{α}_{V} and \overline{α}_{W}, we get a homomorphism

(32.11.6) from M_{V} ⨂_{A}M_{W} into (P_{V} ⨂_{A}M_{W} ) ⨁ (M_{V} ⨂_{A}P_{W} ),

as modules over k. Of course, the kernel of this homomorphism is equal to

(ker \overline{α}_{V}) ∩ (ker \overline{α}_{W}), (32.11.7)

722 CHAPTER 32. SOME SATELLITES which is a submodule of M_{V} ⨂_{A} M_{W}.

This corresponds to taking the left satellite of T (V, W) = V ⨂_{A}W (32.11.8) in the first variable, and then in the second variable, as in (6) on p50 of [50]. This also corresponds to taking the left satellite of (32.11.8) in the second variable, and then the first variable, as in (7) on p50 of [50].

32.12 Another type of double satellite

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let V, P_{V}, W, and Q_{W} be all left or all right modules over A.

Suppose that P_{V} is projective as a module over A, and that (32.12.1) β_{V} is a homomorphism from P_{V} onto V, as modules over A. Similarly, suppose that Q_{W} is injective as a module over A, and that (32.12.2) β_{W} is an injective homomorphism from W into Q_{W} , as modules over A. Put M_{V} = ker β_{V} , N_{W} = Q_{W} /β_{W} (W), (32.12.3) which are modules over A. Thus we get the obvious inclusion mapping (32.12.4) α_{V} from M_{V} into P_{V} , and the natural quotient mapping (32.12.5) α_{W} from Q_{W} onto N_{W} .

We can use α_{V} and the identity mapping I_{N_{W}} on N_{W} to get a homomorphism (32.12.6) Hom(α_{V} , I_{N_{W}}) from Hom_{A}(P_{V} , N_{W}) into Hom_{A}(M_{V} , N_{W}), as modules over k, as in Section 6.3. Similarly, we can use the identity mapping I_{M_{V}} on M_{V} and α_{W} to get a homomorphism (32.12.7) Hom(I_{M_{V}} , α_{W}) from Hom_{A}(M_{V} , Q_{W}) into Hom_{A}(M_{V} , N_{W}), as modules over k. Using these two homomorphisms, we get a homomorphism (32.12.8) from Hom_{A}(P_{V} , N_{W}) ⨁ Hom_{A}(M_{V} , Q_{W}) into Hom_{A}(M_{V} , N_{V}), as modules over k. The image of this homomorphism is the same as the submodule

Hom(α_{V} , I_{N_{W}})(Hom_{A}(P_{V} , N_{W})) (32.12.9)

+Hom(I_{M_{V}} , α_{W})(Hom_{A}(M_{V} , Q_{W}))

of Hom_{A}(M_{V} , N_{W}) generated by the images of (32.12.6) and (32.12.7). Thus the cokernel of the homomorphism as in (32.12.8) is the same as the quotient of Hom_{A}(M_{V} , N_{W}) by the submodule (32.12.9).

This corresponds to taking the right satellite of

T (V, W) = Hom_{A}(V, W) (32.12.10)

in the first variable, and then in the second variable, as in (6a) on p50 of [50], adjusted for the fact that (32.12.10) is contravariant in the first variable, as in [50]. This also corresponds to taking the right satellite of (32.12.10) in the second variable, and then in the first variable, as in (7a) on p50 of [50], suitably adjusted for this case, as before.

32.13 Left and right ideals

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let I_{L}, I_{R} be left and right ideals in A, respectively. In particular, I_{L}, I_{R} are submodules of A, as a module over k, so that their interesection is a submodule of A as a module over k as well.

Let I_{R} I_{L} be the subset of A consisting of finite sums of products of elements of I_{R} and I_{L}. This is another submodule of A, as a module over k, with

I_{R} I_{L} ⊆ I_{L} ∩ I_{R}. (32.13.1)

Of course, I_{L} and I_{R} may be considered as left and right modules over A, respectively. Let I_{R} ⨂_{A} I_{L} be a tensor product of I_{R} and I_{L} over A. There is an obvious homomorphism

(32.13.2) from I_{R} ⨂_{A}I_{L} into A,

as modules over k. This homomorphism is obtained from the restriction of multiplication on A to I_{R} × I_{L}, as a mapping into A. The image of the homomorphism as in (32.13.2) is the same as I_{R} I_{L}.

The quotients A/I_{L} and A/I_{R} may be defined as left and right modules over A in the usual way. Let I_{R} ⨂_{A}(A/I_{L}) be a tensor product of I_{R} and A/I_{L} over A. There is an obvious homomorphism

(32.13.3) from I_{R} ⨂_{A}(A/I_{L}) into A/I_{L},

as modules over k. This homomorphism is obtained from the restriction of the action of A on A/I_{L} on the left to I_{R} × (A/I_{L}), as a mapping into A/I_{L}.

There is a natural homomorphism

(32.13.4) from I_{R} ⨂_{A}A onto I_{R} ⨂_{A}(A/I_{L}),

as modules over k. This homomorphism is obtained from the identity mapping on I_{R}, and the natural quotient mapping from A onto A/I_{L}. This may be identified with a homomorphism

(32.13.5) from I_{R} onto I_{R} ⨂_{A}(A/I_{L}),

as modules over k, using the usual identification of I_{R} ⨂_{A} A with I_{R}.

The composition of the homomorphism as in (32.13.5) with the homomorphism as in (32.13.3) is a homomorphism

(32.13.6) from I_{R} into A/I_{L},

as modules over k. This homomorphism is the same as the restriction to I_{R} of the natural quotient mapping from A onto A/I_{L}. Thus the kernel of this homomorphism is I_{L} ∩ I_{R}.

It follows that the kernel of the homomorphism as in (32.13.3) is equal to the image of I_{L} ∩ I_{R} under the homomorphism as in (32.13.5). This uses the fact that the homomorphism as in (32.13.5) is surjective.

There is a natural homomorphism

(32.13.7) from I_{R} ⨂_{A}I_{L} into I_{R} ⨂_{A}A,

as modules over k. This homomorphism is obtained from the identity mapping on I_{R}, and the obvious inclusion mapping from I_{L} into A. This homomorphism maps I_{R} ⨂_{A} I_{L} onto the kernel of the homomorphism as in (32.13.4), as discussed in Section 2.5.

The homomorphism as in (32.13.7) corresponds to a homomorphism

(32.13.8) from I_{R} ⨂_{A}I_{L} into I_{R},

as modules over k, because of the usual identification of I_{R} ⨂_{A} A with I_{R}. This homomorphism is the same as the one obtained from the restriction of multiplication on A to I_{R} × I_{L}, as in (32.13.2), considered as a homomorphism into I_{R}. The image of this homomorphism is the same as I_{R} I_{L}, as before.

This means that the kernel of the homomorphism as in (32.13.5) is equal to I_{R} I_{L}. It follows that the image of I_{L} ∩ I_{R} under the homomorphism as in (32.13.5) is isomorphic to the quotient

(I_{L} ∩ I_{R})/(I_{R} I_{L}), (32.13.9)

as modules over k. This implies that the kernel of the homomorphism as in (32.13.3) is isomorphic to (32.13.9), as modules over k.

This corresponds to a satellite, as in Section 32.3. This is basically the same as the first part of Exercise 19 on p126 of [50], considered in terms of satellites.

32.14 More on left, right ideals

Let us continue with the same notation and hypotheses as in the previous section. There is a natural homomorphism

(32.14.1) from I_{R} ⨂_{A}I_{L} into A ⨂_{A}I_{L},

as modules over k. This homomorphism is obtained from the obvious inclusion mapping from I_{R} into A, and the identity mapping on I_{L}. This corresponds to a homomorphism (32.14.2) from I_{R} ⨂_{A}I_{L} into I_{L},

as modules over k, because of the usual identification of A ⨂_{A} I_{L} with I_{L}.

Of course, these homormophisms are analogous to those as in (32.13.7) and (32.13.8). As before, the homomorphism as in (32.14.2) is the same as the one obtained from the restriction of multiplication on A to I_{R} × I_{L}, as in (32.13.2), considered as a homomorphism into I_{L}. This may also be considered as a homomorphism (32.14.3) from I_{R} ⨂_{A}I_{L} onto I_{R} I_{L},

as modules over k.

Consider

(32.14.4) the kernel of the homomorphism as in (32.14.3),

which is a submodule of I_{R} ⨂_{A} I_{L}, as a module over k. This corresponds to a double satellite, as in Section 32.11. This is basically the same as the second part of Exercise 19 on p126 of [50], considered in terms of satellites.

Augmented rings

33.1 Left augmentations

Let A be a ring with a multiplicative identity element e_{A}. A left augmentation of A consists of a left module Q over A and a homomorphism ε from A onto Q, as left modules over A. Equivalently, A is said to be a left augmented ring when it is equipped with a left augmentation, as on p143 of [50]. In this case, Q is called the augmentation module, and ε is called the augmentation homomorphism (or epimorphism), as in [50]. The kernel I = I_{ε} of ε is called the augmentation ideal, as in [50], and is a left ideal in A.

Under these conditions, if V is a right module over A, then we put

T (V) = T^{L}(V) = V ⨂_{A}Q, (33.1.1)

which is a commutative group. Suppose for the moment that k is a commutative ring with a multiplicative identity element, and that A is an associative algebra over k. This means that modules over A may be considered as modules over k, and (33.1.1) is defined as a module over k too.

Of course, (v, b) ↦ v · b (33.1.2)

defines a mapping from V × A into V that is bilinear over Z. If a, b ∈ A and v ∈ V, then the values of this mapping at (v·a, b) and (v, a b) are the same. This mapping can be used to show that V satisfies the requirements of V ⨂_{A} A, as a commutative group, and in fact as a right module over A, as in Section 1.10.

We may also restrict (33.1.2) to V × I, to get a mapping into V that is bilinear over Z. This leads to a homomorphism

(33.1.3) from V ⨂_{A}I into V,

as commutative groups. This can be identified with the homomorphism

(33.1.4) from V ⨂_{A}I into V ⨂_{A}A, as commutative groups, that corresponds to the identity mapping on V and the obvious inclusion mapping from I into V, as in Section 1.9.

By hypothesis,

0 ⟶ I ⟶ A ⟶^{ε} Q ⟶ 0 (33.1.5)

is an exact sequence of left modules over A, using the obvious inclusion mapping from I into A. This leads to an exact sequence

V ⨂_{A}I ⟶ V ⨂_{A}A ⟶ V ⨂_{A}Q ⟶ 0, (33.1.6)

as commutative groups, as in Section 2.5. More precisely, this uses the mappings obtained from those in (33.1.5), and the identity mapping on V.

This leads to an isomorphism between (33.1.1) and

(33.1.7) the cokernel of the homomorphism as in (33.1.4),

which is to say the quotient of V ⨂_{A} A by the image of V ⨂_{A} I under the homomorphism as in (33.1.4). This corresponds to (2) on p144 of [50]. This is also mentioned in Exercise 2 on p31 of [8], in the case of commutative rings. This is essentially the same as in Section 13.15 as well.

If W is a left module over A, then put

U(W) = U_{L}(W) = Hom_{A}(Q, W), (33.1.8)

which is a commutative group. As before, if A is an algebra over a commutative ring k with a multiplicative identity element, then modules over A may be considered as modules over k, and (33.1.8) is defined as a module over k.

Of course, A may be considered as a left module over itself, so that the space Hom_{A}(A, W) of homomorphisms from A into W, as left modules over A, is defined as a commutative group. There is a natural isomorphism between Hom_{A}(A, W) and W, as commutative groups, as in Section 1.7, and in fact as left modules over A, as in Section 1.8. There is an obvious homomorphism

(33.1.9) from Hom_{A}(A, W) into Hom_{A}(I, W),

which sends a homomorphism from A into W to its restriction to I. This may be identified with a homomorphism

(33.1.10) from W into Hom_{A}(I, W),

using the natural isomorphism between Hom_{A}(A, W) and W mentioned before.

Homomorphisms from Q into W, as left modules over A, correspond to homomorphisms from A into W, as left modules over A, whose kernels contain I. Thus (33.1.8) is isomorphic in a natural way to

(33.1.11) the kernel of the homomorphism as in (33.1.9).

This corresponds to (2a) on p144 of [50].

33.2 Right augmentations, and related matters

Let A be a ring with a multiplicative identity element e_{A} again. A right augmentation of A consists of a right module Q over A and a homomorphism ε from A onto Q, as right modules over A. If A is equipped with a right augmentation, then A is said to be a right augmented ring. As before, Q is called the augmentation module, and ε is called the augmentation homomorphism (or epimorphism). The kernel I = I_{ε} is called the augmentation ideal, and is a right ideal in A.

In this case, if V is a left module over A, then we put

T (V) = T^{R}(V) = Q ⨂_{A}V, (33.2.1)

which is a commutative group. If W is a right module over A, then we put

U(W) = U_{R}(W) = Hom_{A}(Q, W), (33.2.2)

which is a commutative group. There are analogues of the statements in the previous section for right augmentations, as on p145 of [50].

Suppose now that I is a two-sided ideal in A. Thus

Q = A/I (33.2.3)

is a ring, which may be considered as both a left and right module over A. This means that A may be considered as both a left and right augmented ring, as on p145 of [50].

If V is a right module over A, then (33.1.1) may be considered as a right module over A too, as in Section 1.10. Similarly, if V is a left module over A, then (33.2.1) may considered as a left module over A.

If W is a left module over A, then (33.1.8) may be considered as a left module over A, as in Section 1.8. Similarly, if W is a right module over A, then (33.2.2) may be considered as a right module over A.

Let V be a right module over A again. Consider the subset

V · I (33.2.4)

of V consisting of finite sum of elements of V of the form v · b, where v ∈ V and b ∈ I. This is a subgroup of V, as a commutative group with respect to addition, which is the same as the image of V ⨂_{A} I under the homomorphism as in (33.1.3). Thus the quotient

V/(V · I) (33.2.5)

can be identified with (33.1.7), which is isomorphic to (33.1.1), as a commutative group, as before. Note that this works when I is any left ideal in A.

More precisely, (33.2.4) is a submodule of V here, as a right module over A, because I is also a right ideal in A. This means that the quotient (33.2.5) is a right module over A as well. Under these conditions, the homomorphisms as in (33.1.3) and (33.1.4) are homomorphisms between right modules over A. Similarly, the isomorphism between (33.1.1) and (33.1.7) mentioned in the previous section is an isomorphism between right modules over A.

Remember that the homomorphism from V ⨂_{A} A onto V ⨂_{A} Q in (33.1.6) is obtained from ε and the identity mapping on V. This is also a homomorphism between right modules over A in this case. It follows that (33.2.5) is isomorphic to (33.1.1), as right modules over A. This corresponds to (1) on p154 of [50].

If we take V = Q, considered as a right module over A, then (33.2.4) is equal to {0}. Of course, this means that (33.2.5) is equal to Q, so that (33.1.1) is isomorphic to Q, as a right module over A. This corresponds to (8) on p146 of [50].

33.3 Augmentations and homology

Let A be a ring with a multiplicative identity element e_{A}. Suppose that A is equipped with a left augmentation, with augmentation module Q, homomorphism ε, and ideal I = I_{ε}, as in Section 33.1. If V is a right module over A, then we put T (V) = T^{L}(V) = V ⨂_{A} Q, as before.

If n is a nonnegative integer, then the nth homology group T_{n}(V) of A as a left augmented ring with coefficients in V is defined as a commutative group on p143 of [50]. Another characterization of T_{n}(V), up to isomorphism, is given in Theorem 1.1 on p145 of [50]. In particular,

T_{0}(V) = T (V). (33.3.1)

If V is projective as a right module over A, then

(33.3.2) T_{n}(V) = {0} for every n ≥ 1.

As mentioned on p143 of [50], V may have additional actions that commute with the action of A on the right, which lead to suitable actions on T_{n}(V). Suppose for instance that k is a commutative ring with a multiplicative identity element, and that A is an associative algebra over k. This means that modules over A may be considered as modules over k, and T (V) is defined as a module over k. Similarly, T_{n}(V) is a module over k for each n ≥ 0, as in [50].

Remember that V satisfies the requirements of V ⨂_{A} A, as a commutative group, and in fact as a right module over A, as in Section 1.10. This uses the mapping (v, b) ↦ v · b from V × A into V. We can use the restriction of this mapping to V × I, to get a mapping into V that is bilinear over Z, and satisfies the usual compatibility condition with respect to the actions of A on V on the right and on I on the left.

This leads to a homomorphism

(33.3.3) from V ⨂_{A}I into V,

as commutative groups, as in Section 33.1. This can be identified with the homomorphism (33.3.4) from V ⨂_{A}I into V ⨂_{A}A, as commutative groups, that corresponds to the identity mapping on V and the obvious inclusion mappng from I into A, as in Section 1.9. It can be shown that

(33.3.5) T_{1}(V) is isomorphic to the kernel of the homomorphism as in (33.3.3),

as a commutative group, as in (3) on p144 of [50].

Suppose that X = X(V) is a projective resolution of V, as a right module over A, as in Section 10.2. We may consider X ⨂_{A} Q as a graded module over Z, with

(X ⨂_{A}Q)^{j} = X^{j} ⨂_{A}Q (33.3.6)

for each j ∈ Z, as in Section 7.5. More precisely, X ⨂_{A} Q may be considered as a complex over Z, as before. The homology of this complex can be used to obtain T_{n}(V), as on p143 of [50]. Namely,

T_{n}(V) = H(X ⨂_{A}Q)_{n} = H(X ⨂_{A}Q)^{−n} (33.3.7)

for each n ≥ 0.

Alternatively, suppose that X = X(Q) is a projective resolution of Q, as a left module over A. As before, V ⨂_{A} X may be considered as a graded module over Z, with

(V ⨂_{A}X)^{j} = V ⨂_{A}X^{j} (33.3.8)

for each j ∈ Z. In fact, V ⨂_{A} X may be considered as a complex over Z, as in Section 7.5 again. The homology of this complex can be used to obtain T_{n}(V), with T_{n}(V) = H(V ⨂_{A}X)_{n} = H(V ⨂_{A}X)^{−n} (33.3.9)

for each n ≥ 0, as on p144 of [50].

Remember that A is automatically free as a left module over itself, and thus projective. This means that the augmentation homomorphism ε may be considered as a homomorphism from a projective left module over A onto Q. This can be used as the initial step in the construction of a projective resolution of Q, as in Section 10.2, and mentioned on p144 of [50].

Now let X = X(V) be a projective resolution of V, as a right module over A, and let Y = Y (Q) be a projective resolution of Q, as a left module over A, as in Section 10.2 again. Under these conditions, X ⨂_{A} Y can be defined as a double complex over Z, as in Section 6.2. In particular, X ⨂_{A} Y is doubly-graded as a module over Z, as in Section 6.1. One can obtain a single grading on X ⨂_{A} Y from the double grading in the usual way, as in Section 5.13. One can also combine the two differentiation operators on X ⨂_{A} Y, as a double complex, to get a single differentiation operator, so that X ⨂_{A} Y becomes a single complex, as in Section 5.14.

The homology of X ⨂_{A} Y as a single complex can be used to get T_{n}(V), with T_{n}(V) = H(X ⨂_{A}Y )_{n} = H(X ⨂_{A}Y )^{−n} (33.3.10)

for each n ≥ 0. This uses one of the equivalent definitions of T_{n}(V) indicated on p143 of [50], which is based on a definition on p107 of [50]. The equivalence of the two definitions of T_{n}(V) on p143 of [50], as well as the descriptions in (33.3.7) and (33.3.9), are mentioned on p107 of [50] too.

The definition on p107 of [50] mentioned in the preceding paragraph is based on another definition on p84 of [50], which is related to the discussion in Section 10.11. The equivalence with (33.3.7) and (33.3.9) is related to the discussion in Section 10.12. The other equivalent definition of T_{n}(V) on p143 of [50] is based on “satellites”, which are also mentioned in Section 10.12.

Of course, there are analogous notions and statements for right augmented rings, as mentioned on p145 of [50].

33.4 Two-sided ideals and homology

Let us continue with the same notation and hypotheses as in the previous section. Suppose now that I is a two-sided ideal in A, so that Q = A/I is a ring. In this case, T_{n}(V) may be considered as a right module over A for every n ≥ 0, because Q is a right module over A, as on p145 of [50].

We may also take V = Q in the previous section, because Q is a right module over A. Remember that T_{0}(Q) = T (Q) was disccused in Section 33.2.

Of course, if q ∈ Q and a ∈ I, then

q · a = 0 (33.4.1)

in Q, by construction. This implies that the homomorphism

(33.4.2) from Q ⨂_{A}I into Q

as in (33.3.3), with V = Q, is equal to 0. This means that

(33.4.3) T_{1}(Q) is isomorphic to Q ⨂_{A}I,

as in (33.3.5). This corresponds to part of (9) on p146 of [50].

In this case,

0 ⟶ I ⟶ A ⟶^{ε} Q ⟶ 0 (33.4.4)

may be considered as an exact sequence of left and right modules over A, using the obvious inclusion mapping from I into A. This leads to an exact sequence

I ⨂_{A}I ⟶ A ⨂_{A}I ⟶ Q ⨂_{A}I ⟶ 0, (33.4.5)

as in Section 2.5. This uses the mappings obtained from those in (33.4.5), and the identity mapping on I.

As usual, I satisfies the requirements of A ⨂_{A} I, using the map (a, b) ↦ a b from A × I into I. Thus the mapping in the first step in (33.4.5) corresponds to the homomorphism (33.4.6) from I ⨂_{A}I into I

with a ⊗ b ↦ a b (33.4.7)

for every a, b ∈ I. The image I^{2} = I I (33.4.8)

of the homomorphism as in (33.4.6) consists of finite sums of products of the form a b, with a, b ∈ I. This is a submodule of I, as both a left and right module over A.

Using the exactness of (33.4.5), we get an isomorphism

(33.4.9) from I/I^{2} onto Q ⨂_{A}I,

as both left and right modules over A. This corresponds to another part of (9) on p146 of [50].

It can be shown that

(33.4.10) T_{2}(Q) is isomorphic to the kernel of the homomorphism as in (33.4.6),

as in (10) on p146 of [50].

33.5 Augmentations and cohomology

Let A be a ring with a multiplicative identity element e_{A}. Suppose that A is a left augmented ring, with augmentation module Q, homomorphism ε, and ideal I = I_{ε}, as in Section 33.1. If W is a left module over A, then we put U(W) = U_{L}(W) = Hom_{A}(Q, W), as before.

If n is a nonnegative integer, then the nth cohomology group U^{n}(W) of A as a left augmented ring with coefficients in W is defined as a commutative group on p143 of [50]. Theorem 1.1a on p145 of [50] gives another characterization of U^{n}(W), up to isomorphism. In particular,

U^{0}(W) = U(W). (33.5.1)

If W is injective as a left module over A, then

(33.5.2) U^{n}(W) = {0} for every n ≥ 1.

As on p143 of [50], W may have additional actions that commute with the action of A on the left, and these lead to suitable actions on U^{n}(W). In particular, if k is a commutative ring with a multiplicative identity element, and A is an associative algebra over k, then modules over A may be considered as modules over k, and U(W) is defined as a module over k. In this case, U^{n}(W) is a module over k for every n ≥ 0, as in [50].

Remember that the space Hom_{A}(A, W) of homomorphisms from A into W, as left modules over A, is isomorphic to W in a natural way, as commutative groups, and as left modules over A, as in Section 33.1. Using this isomorphism, the obvious homomorphism

(33.5.3) from Hom_{A}(A, W) into Hom_{A}(I, W),

which sends a homomorphism from A into W to its restriction to I, may be identified with a homomorphism

(33.5.4) from W into Hom_{A}(I, W),

as before. It can be shown that

(33.5.5) U^{1}(W) is isomorphic to the cokernel of the homomorphism as in (33.5.3),

as a commutative group, as in (3a) on p144 of [50].

Suppose that Y = Y (W) is an injective resolution of W, as a left module over A, as in Section 10.7. In particular, Y is a complex over A, so that Hom_{A}^{gr}(Q, Y) may be defined as a complex over Z, as in Section 7.12. Under these conditions,

U^{n}(W) = H(Hom_{A}^{gr}(Q, Y))^{n} (33.5.6)

for each n ≥ 0, as on p143 of [50].

Alternatively, suppose that X = X(Q) is a projective resolution of Q, as a left module over A, as in Section 10.2. Thus X is a complex over A in particular, and Hom_{A}^{gr}(X, W) may be defined as a complex over Z, as in Section 8.4. In this case, U^{n}(W) = H(Hom_{A}^{gr}(X, W))^{n} (33.5.7)

for each n ≥ 0, as on p144 of [50].

Now let X = X(Q) be a projective resolution of Q, as a left module over A, and let Y = Y (W) be an injective resolution of W as a left module over A, as in Sections 10.2 and 10.7. Under these conditions, Hom_{A}^{gr}(X, Y) can be defined as a double complex over Z, as in Section 6.5. This uses the definition of Hom_{A}^{gr}(X, Y) as a doubly-graded module over Z in Section 6.3. As usual, one can get a single grading on Hom_{A}^{gr}(V, W) from the double grading as in Section 5.13. The two differentiation operators on Hom_{A}^{gr}(X, Y), as a double complex, can be combined to get a single differentiation operator, so that Hom_{A}^{gr}(X, Y) becomes a single complex, as in Section 5.14.

The homology of Hom_{A}^{gr}(X, Y) as a single complex can be used to obtain U^{n}(W), with U^{n}(W) = H(Hom_{A}^{gr}(X, Y))^{n} (33.5.8)

for each n ≥ 0. This uses one of the equivalent definitions of U^{n}(W) mentioned on p143 of [50], which uses another definition on p107 of [50]. The equivalence of the two definitions of U^{n}(W) on p143 of [50] is mentioned on p107 of [50], as well as the equivalence with the descriptions in (33.5.6) and (33.5.7).

The definition on p107 of [50] mentioned in the preceding paragraph uses another definition on p83 of [50], which is related to the discussion in Section 10.13. The equivalence with (33.5.6) and (33.5.7) is related to the discussions in Sections 10.14 and 10.15. The equivalence with the descriptions in terms of “satellites” is mentioned in those sections as well.

There are analogous notions and statements for right augmented rings, as mentioned on p145 of [50].

If I is a two-sided ideal in A, then Q may be considered as a right module over A too, as in Section 33.2. In this case, U^{n}(W) may be considered as a left module over A for every n ≥ 0, as on p145 of [50].

33.6 Dual numbers and augmentations

Let A be a ring with a multiplicative identity element e_{A}, and let A[d] be the corresponding ring of dual number, as in Section 5.4. Also let ε be the mapping from A[d] onto A defined by

ε(a_{1} + a_{2} d) = a_{1} (33.6.1)

for every a_{1}, a_{2} ∈ A. This is a ring homomorphism, which is the same as in Section 9.14, with A[d] considered as a graded ring.

Thus A may be considered as a left and right module over A[d], using ε. Equivalently, the actions of d on A on the left and right are equal to 0.

We may consider A[d] as a left and right augmented ring, with augmentation homomorphism ε, and augmentation module A. The augmentation ideal is A d, which may be considered as a left and right module over A[d], because it is a two-sided ideal in A[d].

Observe that a ↦ a d (33.6.2)

defines an isomorphism (33.6.3) from A onto A d,

as left and right modules over A[d]. More precisely, the actions of d on A d on the left and on the right are equal to 0.

It is easy to see that a_{1} + a_{2} d ↦ a_{1} d (33.6.4)

is a homomorphism (33.6.5) from A[d] onto A d,

as left and right modules over A[d]. This is the same as the composition of ε with the isomorphism as in (33.6.2).

Consider the graded left and right module X over A[d] defined by

(33.6.6)  X^{j} = {0} when j > 0
= A[d] when j ≤ 0.

Let d_{X}^{j} be the mapping from X^{j} into X^{j+1} which is given by (33.6.4) when j < 0, and which is of course equal to 0 when j ≥ 0. Note that

d_{X}^{j+1} ∘ d_{X}^{j} = 0 (33.6.7)

for every j, by construction. Let d_{X} be the homomorphism from X into itself of degree 1 whose jth component is equal to d_{X}^{j} for each j. It follows that (X, d_{X}) is a graded left and right module over A[d] with differentiation that is a complex, as in Section 5.10.

We may consider A as a graded left and right module over A[d] with differentiation that is a complex by taking A^{0} = A, A^{j} = {0} when j ≠ 0, and differentiation operator d_{A} = 0, as before. Let ε_{X} be the homomorphism from X into A of degree 0 defined by taking ε_{X}^{0} to be the map from X^{0} = A[d] into A^{0} = A given by ε, and with ε_{X}^{j} = 0 when j ≠ 0. Observe that

ε_{X}^{0} ∘ d_{X}^{−1} = 0, (33.6.8)

by construction. This implies that ε_{X} is a homomorphism from X into A, as modules with differentiation, as in Section 10.1. More precisely, ε_{X} is a map from X into A, as complexes over A[d].

This makes X a left complex over A, as a left or right module over A[d], with augmentation ε_{X}, as in Section 10.1. Clearly X is projective over A[d], because A[d] is projective as a left or right module over itself. It is easy to see that X is acyclic as a left complex over A, using the remarks in Section 10.1. Thus X is a projective resolution of A, as a left or right module over A[d], as in Section 10.2. This corresponds to some remarks on p147 of [50].

33.7 Dual numbers and homology

Let us continue with the same notation and hypotheses as in the previous section. Also let V be a right module over A[d]. Equivalently, this means that V is a right module over A with differentiation, where the action of d on V on the right is the same as the differentiation operator d_{V}, as in Section 5.4.

Let us take Y = V ⨂_{A[d]} X, which may be considered as a graded right module over A[d]. More precisely,

(33.7.1)  Y^{j} = V ⨂_{A[d]}X^{j} = {0}  when j > 0
= V ⨂_{A[d]}A[d] when j ≤ 0,

by (33.6.6). We can identify Y^{j} with V when j ≤ 0, as a right module over A[d], in the usual way.

In fact, Y is a right module over A[d] with differentiation, where the differentiation operator d_{Y} is obtained from d_{X} and the identity mapping on V in the usual way, as in Section 7.5. More precisely, Y is a complex, and d_{Y}^{j} is obtained from d_{X}^{j} and the identity mapping on V in the usual way for each j. Of course, d_{Y}^{j} = 0 when j ≥ 0.

If j < 0, then Y^{j} and Y^{j+1} can be identified with V, as right modules over A[d], as before. Using this identification, one can check that d_{Y}^{j} corresponds exactly to d_{V}. This uses the fact that d_{X}^{j} corresponds to the mapping from A[d] into itself given by (33.6.4), as in the previous section.

If n ≥ 1, then we get that

T_{n}(V) = H(Y)_{n} = H(Y)^{−n} = H(V), (33.7.2)

as in (33.3.9). Here H(V) is the usual homology of V as a module with differentation, as in Section 5.1.

We also have that

(33.7.3) T (V) = T_{0}(V) = V ⨂_{A[d]}A is isomorphic to Z′(V),

where Z′(V) is as in Section 5.1, for V as a module with differentiation. This can be obtained from the description of T (V) in Section 33.2, when the augmentation ideal I is a two-sided ideal. Here I = A d, and it is easy to see that V · I = V · (A d) = V · d = d_{V} (V). (33.7.4)

One could obtain (33.7.3) from the fact that T_{0}(V) = H(Y)^{0}, as in Section 33.3, as well.

These remarks correspond to some of those on p147 of [50].

33.8 Dual numbers and cohomology

Let us continue with the same notation and hypotheses as in Section 33.6 again. Now let W be a left module over A[d]. Thus W may be considered as a left module over A with differentiation, where the differentiation operator d_{W} corresponds to the action of d on W on the left, as in Section 5.4.

Here we take Y = Hom_{A[d]}^{gr}(X, W), which may be considered initially as a graded module over Z, as in Section 8.4. In fact, Y is a graded left module over A[d], because X is a graded left and right module over A[d]. Using (33.6.6), we get that

Y^{j} = Hom_{A[d]}(X^{−j}, W) = Hom_{A[d]}(A[d], W) when j ≥ 0
(33.8.1)  = {0}  when j < 0.

We can identify Y^{j} with W when j ≥ 0, as a left module over A[d], as in Section 1.8.

Let ϕ ∈ Y be given, so that ϕ^{j} ∈ Hom_{A[d]}(X^{−j}, W) for each j ∈ Z. Under these conditions, d_{Y} (ϕ) ∈ Y is defined by putting

d_{Y} (ϕ)^{j+1} = ϕ^{j} ∘ d_{X}^{−j−1} (33.8.2) for each j, as in Section 8.4. Remember that Y is a complex with respect to d_{Y}, as before. Equivalently, d_{Y}^{j} is defined on Y^{j} by composition with d_{X}^{−j−1} for each j. If j < 0, then −j − 1 ≥ 0, so that d_{X}^{−j−1} = 0, and thus d_{Y}^{j} = 0.

If j ≥ 0, then Y^{j} and Y^{j+1} can be identified with W, as left modules over A[d], as before. In this case, one can check that d_{Y}^{j} corresponds to d_{W} with respect to this identification. More precisely, this uses the fact that d_{X}^{−j−1} corresponds to (33.6.4) as a mapping from A[d] into itself, by construction.

If n ≥ 1, then we obtain that

U^{n}(W) = H(Y)^{n} = H(W), (33.8.3)

as in (33.5.7). This is the usual homology of W as a module with differentiation, as in Section 5.1.

We also get that

(33.8.4) U(W) = U^{0}(W) = Hom_{A[d]}(A, W) is isomorphic to Z(W),

where Z(W) is as in Section 5.1. Indeed, Hom_{A[d]}(A, W) is isomorphic in a natural way to the submodule of Hom_{A[d]}(A[d], W) consisting of homomorphisms from A[d] into W, as left modules over A[d], that are equal to 0 on the augmentation ideal A d in A[d]. If we identify Hom_{A[d]}(A[d], W) with W in the usual way, then this submodule corresponds to Z(W). One could obtain (33.8.4) from the fact that U^{0}(W) = H(Y)^{0}, as in Section 33.5, too.

These remarks correspond to some more of those on p147 of [50].

33.9 Augmentations and homomorphisms

Let A, B be rings with multiplicative identity elements e_{A}, e_{B}, respectively. Suppose that A, B are both left augmented rings, with augmentation modules Q_{A}, Q_{B}, augmentation homomorphisms ε_{A}, ε_{B}, and augmentation ideals I_{A}, I_{B}, respectively. Of course, one could consider right augmentations as well.

Suppose that ϕ is a homomorphism from A into B as rings, with ϕ(e_{A}) = e_{B}. If ϕ(I_{A}) ⊆ I_{B}, (33.9.1)

then ϕ is said to be a map or homomorphism of augmented rings, as on p149 of [50].

In this case, there is a unique mapping ψ from Q_{A} into Q_{B} such that

ψ ∘ ε_{A} = ε_{B} ∘ ϕ. (33.9.2)

If a ∈ A and x ∈ Q_{A}, then it follows that

ψ(a · x) = ϕ(a) · ψ(x). (33.9.3)

Note that Q_{B} may be considered as a left module over A, where the action of a ∈ A on Q_{B} on the left is given by the action of ϕ(a) ∈ B on Q_{B} on the left, as in Section 2.9. Using this, ψ may be considered as a homomorphism from Q_{A} into Q_{B}, as left modules over A, as on p149 of [50].

Similarly, B may be considered as a right module over itself, and thus a right module over A, using ϕ, as in Section 2.9. Let

(ϕ)Q_{A} = B ⨂_{A}Q_{A} (33.9.4)

be a tensor product of B and Q_{A} as modules over A. This may be considered as a left module over B, which is the covariant ϕ-extension of Q_{A}, as in Section 2.9.

Consider the mapping from B × Q_{A} into Q_{B} defined by

(b, x) ↦ b · ψ(x) (33.9.5)

for every b ∈ B and x ∈ Q_{A}. This mapping is bilinear over Z, and compatible with the actions of A on B on the right and on Q_{A} on the left, by (33.9.3). It follows that there is a unique homomorphism

(33.9.6) g from (33.9.4) into Q_{B},

as commutative groups with respect to addition, such that

g(b ⊗ x) = b · ψ(x) (33.9.7)

for every b ∈ B and x ∈ Q_{A}, as on p149f of [50]. More precisely, g is a homomorphism as in (33.9.6), as left modules over B.

Let X_{A} be a projective resolution of Q_{A}, as a left module over A, and let X_{B} be a projective resolution of Q_{B}, as a left module over B. Also let

(ϕ)X_{A} = B ⨂_{A}X_{A} (33.9.8)

be a tensor product of B and X_{A}, as modules over A, which is the covariant ϕ-extension of X_{A}. This may be considered as a complex over Z, as in Section 7.5. More precisely, this may be considered as a left module over B, as before, and in fact as a complex over B. Note that

(_{(ϕ)}X_{A})^{j} = B ⨂_{A}X_{A}^{j} =_{(ϕ)} X_{A}^{j} (33.9.9)

is the covariant ϕ-extension of X_{A}^{j}, as a left module over A, for each integer j. One can check that

(33.9.10) (33.9.8) is a left complex over (33.9.4),

as left modules over B, in a natural way. The augmentation of (33.9.8) as a left complex can be obtained from the augmentation of X_{A}, as a left complex over Q_{A}, and the identity mapping on B. Observe that

(33.9.11) (33.9.9) is projective as a left module over B for each j, because X_{A}^{j} is projective as a left module over A for each j, by hypothesis, as in Section 2.10. This means that

(33.9.12) (33.9.8) is projective as a left complex over (33.9.4),

as left modules over B, as in Section 10.1. This is mentioned at the top of p150 of [50].

It follows that there is a map

(33.9.13) G from (33.9.8) into X_{B} over g,

as in Section 10.3. This map is unique up to homotopy, as in Section 10.4.

Let V be a right module over B, so that V may be considered as a right module over A using ϕ, as in Section 2.9. If we consider V as a right module over A, then a tensor product V ⨂_{A} X_{A} of V and X_{A} over A may be considered as a complex over Z, as in Section 7.5. The homology of this complex may be used to obtain the homology of A as a left augmented ring with coefficients in V, as in Section 33.3.

Remember that V satisfies the requirements of a tensor product V ⨂_{B} B. This permits us to identify V ⨂_{A} X_{A} with a tensor product of the form

(V ⨂_{B}B) ⨂_{A}X_{A}. (33.9.14)

There is a natural isomorphism between a tensor product of this form and a tensor product of the form

V ⨂_{B}(B ⨂_{A}X_{A}), (33.9.15)

as in Section 1.12. This was also mentioned in connection with covariant ϕextensions in Section 2.10. More precisely, one can check that this is an isomorphism between these tensor products as complexes.

A tensor product V ⨂_{B} X_{B} of V and X_{B} over B may be considered as a complex over Z as well, as in Section 7.5 again. The homology of this complex may be used to obtain the homology of B as a left augmented ring with coefficients in V, as in Section 33.3. Using G and the identity mapping on V, we get a map (33.9.16) from (33.9.15) into V ⨂_{B}X_{B},

as complexes, as in Section 7.6. Note that this map is uniquely determined up to homotopy by g. We may consider this as a map

(33.9.17) from V ⨂_{A}X_{A} into V ⨂_{B}X_{B},

as complexes, using the identification and isomorphism mentioned in the preceding paragraph.

The map as in (33.9.17) leads to a homomorphism

(33.9.18) from H(V ⨂_{A}X_{A}) into H(V ⨂_{B}X_{B}) of degree 0, as graded modules over Z, as in Section 5.11. This homomorphism is uniquely determined by g, because induced maps on homology are preserved by homotopies. This corresponds to (2) on p150 of [50]. This determines a homomorphism

(33.9.19) from the homology of A into the homology of B, as left augmented rings, with coefficients in V,

as in (1) on p149 of [50].

Similarly, let Z be a left module over B, which may be considered as a left module over A too, as in Section 2.9. Thus Hom_{A}^{gr}(X_{A}, Z) may be defined as a complex over Z, as in Section 8.4. The homology of this complex can be used to get the cohomology of A as a left augmented ring with coefficients in Z, as in Section 33.5.

We also have that Hom_{B}^{gr}(X_{B}, Z) and

Hom_{B}^{gr}(_{(ϕ)}X_{A}, Z) = Hom_{B}^{gr}(B ⨂_{A}X_{A}) (33.9.20)

are defined as complexes over Z, as in Section 8.4. Remember that G is in particular a map from (33.9.8) into X_{B} as complexes over B, as in Section 10.3. This implies that G induces a map

(33.9.21) from Hom_{B}^{gr}(X_{B}, Z) into (33.9.20),

as complexes over Z, as in Section 8.5. This map is uniquely determined up to homotopy by g.

There is a natural isomorphism

(33.9.22) between (33.9.20) and Hom_{A}^{gr}(X_{A}, Z),

as in Section 2.10. More precisely, one can reduce to the previous statements for ordinary spaces of homomorphisms between modules without gradings, to deal with graded modules here. One can also check that this is an isomorphism between complexes over Z. Using this, the map in (33.9.21) may be considered as a map (33.9.23) from Hom_{B}^{gr}(X_{B}, Z) into Hom_{A}^{gr}(X_{A}, Z),

as complexes over Z.

This leads to a homomorphism

(33.9.24) from H(Hom_{B}^{gr}(X_{B}, Z)) into H(Hom_{A}^{gr}(X_{A}, Z))

of degree 0, as graded modules over Z, as in Section 5.11 again. This homomorphism is uniquely determined by g, and corresponds to (2a) on p150 of [50]. This determines a homomorphism

(33.9.25) from the cohomology of B into the cohomology of A, as left augmented rings, with coefficients in Z,

as in (1a) on p149 of [50].

33.10 The Mapping Theorem

Let us continue with the same notation and hypotheses as in the previous section. One may be interested in having the homomorphisms as in (33.9.19) and (33.9.25) be isomorphisms. This is addressed in Theorem 3.1 on p150 of [50], which is called the Mapping Theorem. Of course, this is equivalent to the homomorphisms as in (33.9.18) and (33.9.24) being isomorphisms.

Suppose for the moment that V = B, considered as a right module over B. Thus V ⨂_{A}X_{A} = B ⨂_{A}X_{A} =_{(ϕ)} X_{A}, (33.10.1)

and X_{B} satisfies the requirements of V ⨂ X_{B} = B ⨂_{B} X_{B}. Suppose also that the homomorphism as in (33.9.18) is an isomorphism in this case, so that we have an isomorphism

(33.10.2) from H(B ⨂_{A}X_{A}) onto H(X_{B})

of degree 0, as graded modules over Z. This determines an isomorphism

(33.10.3) from H(B ⨂_{A}X_{A})_{n} onto H(X_{B})_{n}

for each n ≥ 0.

Remember that H(B ⨂_{A} X_{A})_{n} is isomorphic to

(33.10.4) the nth homology group of A, as a left augmented ring with coefficients in B, as a right module over A using ϕ,

for each n ≥ 0, as in Section 33.3.

The condition that we have an isomorphism as in (33.10.3) when n = 0 means that

(33.10.5) the homomorphism g in (33.9.6) is an isomorphism.

This is the same as (i) on p150 of [50].

Of course, H(X_{B})_{n} = {0} (33.10.6)

when n > 0, because X_{B} is acyclic as a left complex over Q_{B}, by hypothesis, as in Section 10.1. Thus the condition that we have an isomorphism as in (33.10.3) when n > 0 means that H(B ⨂_{A}X_{A})_{n} = {0} (33.10.7)

when n > 0. This is the same as saying that

(33.10.8) (33.10.4) is equal to {0}

when n > 0, as in (ii) on p150 of [50].

Suppose now that (33.10.5) holds, and that (33.10.8) holds for every n > 0. If X_{A} is any projective resolution of Q_{A}, as a left module over A, then (33.10.8) implies that (33.10.7) holds for every n > 0. We can use g to consider B ⨂_{A} X_{A} as a left complex over Q_{B}, instead of B ⨂_{A} Q_{A}, as before. Our hypotheses imply B ⨂_{A} X_{A} is acyclic as a left complex over Q_{B}, as in Section 10.1. It follows that (33.10.9) B ⨂_{A}X_{A} is a projective resolution of Q_{B},

as a left module over B, because of (33.9.11), as in Theorem 3.1 on p150 of [50]. This permits us to take

X_{B} = B ⨂_{A}X_{A} (33.10.10)

in this case. Under these conditions, the map G as in (33.9.13) is the identity mapping on (33.10.10). This implies that the homomorphisms as in (33.9.18) and (33.9.24) are isomorphisms, so that the homomorphisms as in (33.9.19) and (33.9.25) are isomorphisms, as in Theorem 3.1 on p150 of [50].

33.11 Some interesting conditions

Let A be a left augmented ring, with multiplicative identity element e_{A}, augmentation module Q, augmentation module Q, augmentation homomorphism ε, and augmentation ideal I, as in Section 33.1. Also let n be a positive integer, and let x_{1}, …, x_{n} be n commuting elements of A. Suppose that

(33.11.1) I is generated by x_{1}, …, x_{n}, as a left ideal in A.

Put I_{0} = {0}, and for each l = 1, …, n, let

(33.11.2) I_{l} be the left ideal in A generated by x_{1}, …, x_{l}.

Thus I_{n} = I, by (33.11.1). Consider the following condition:

(33.11.3) if a ∈ A, 1 ≤ l ≤ n, and a x_{l} ∈ I_{l−1}, then a ∈ I_{l−1}.

This is the condition (i) in Theorem 4.2 on p150 of [50].

Let us mention a couple of classes of examples where this holds, as on p151 of [50]. Let A_{0} be a ring with a multiplicative identity element, and let x_{1}, …, x_{n} be n distinct commuting indeterminates. The corresponding ring A_{0}[x_{1}, …, x_{n}] of formal polynomials in x_{1}, …, x_{n} with coefficients in A_{0}, as in Section 4.3, may be considered as a left augmented ring, where the augmentation homomorphism sends an element of A_{0}[x_{1}, …, x_{n}] to its constant term in A_{0}. The augmentation ideal consists of elements of A_{0}[x_{1}, …, x_{n}] with constant term equal to 0, and it is easy to see that this is generated by x_{1}, …, x_{n}, as a left ideal in A_{0}[x_{1}, …, x_{n}]. One can check that (33.11.3) holds in this case.

Similarly, the ring A_{0}[[x_{1}, …, x_{n}]] of formal power series in x_{1}, …, x_{n}, as in Section 4.3, may be considered as a left augmented ring, where the augmentation homomorphism sends an element of A_{0}[[x_{1}, …, x_{n}]] to its constant term. As before, the augmentation ideal consists of elements of A_{0}[[x_{1}, …, x_{n}]] with constant term equal to 0, and one can check that this is generated by x_{1}, …, x_{n}, as a left ideal in A_{0}[[x_{1}, …, x_{n}]]. One can verify that (33.11.3) holds in this case too.

Let us return now to the same notation and hypotheses as at the beginning of the section. Let T_{1}, …, T_{n} be n distinct commuting indeterminates, so that the ring Z[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in Z may be defined as in Section 4.3 again. As on p151 of [50],

(33.11.4) A may be considered as a right module over Z[T_{1}, …, T_{n}],

where for each a ∈ A and l = 1, …, n,

a · T_{l} = a x_{l}. (33.11.5)

Put J_{0} = {0}, and for each l = 1, …, n, let

(33.11.6) J_{l} be the ideal in Z[T_{1}, …, T_{n}] generated by T_{1}, …, T_{l}.

Observe that I_{l} = A · J_{l} (33.11.7)

for each l = 0, 1, …, n, as on p151 of [50].

Similarly, let V be any right module over Z[T_{1}, …, T_{n}]. Note that

V · J_{l} (33.11.8)

is a submodule of V, as a module over Z[T_{1}, …, T_{n}], for each l = 0, 1, …, n. Of course, if 0 ≤ l_{1} ≤ l_{2} ≤ n, then

J_{l_{1}} ⊆ J_{l_{2}}. (33.11.9)

This implies that V · J_{l_{1}} ⊆ V · J_{l_{2}}. (33.11.10)

Consider the following condition:

(33.11.11) if v ∈ V, 1 ≤ l ≤ n, and v · T_{l} ∈ V · J_{l−1}, then v ∈ V · J_{l−1}.

This is the same as the condition (i’) in Proposition 4.3 on p151 of [50]. If V = A, considered as a right module over Z[T_{1}, …, T_{n}], then this is the same as (33.11.3).

If 1 ≤ l ≤ n, then the quotient modules V/(V · J_{l−1}) and (V · J_{l})/(V · J_{l−1}) are defined as modules over Z[T_{1}, …, T_{n}]. The mapping

v ↦ v · T_{l} (33.11.12)

from V into V · J_{l} induces a homomorphism

(33.11.13) from V/(V · J_{l−1}) into (V · J_{l})/(V · J_{l−1}),

as modules over Z[T_{1}, …, T_{n}]. It is easy to see that this homomorphism is surjective. The condition (33.11.11) says that this induced homomorphism is injective, and thus an isomorphism, as mentioned at the top of p152 of [50].

33.12 A related left complex

Let n be a positive integer, and let T_{1}, …, T_{n} be n distinct commuting indeterminates, so that the ring B = Z[T_{1}, …, T_{n}] (33.12.1)

of formal polynomials in T_{1}, …, T_{n} with coefficients in Z may be defined as in Section 4.3, as before. Also let J_{l} be the ideal in Z[T_{1}, …, T_{n}] defined in the previous section for each l = 0, 1, …, n, and let V be a right module over Z[T_{1}, …, T_{n}].

Let y_{1}, …, y_{n} be another collection of n distinct indeterminates, and let

E_{B}(y_{1}, …, y_{n}) (33.12.2)

be the corresponding exterior ring in y_{1}, …, y_{n} with coefficients in B, as in Section 4.15. Similarly, we get a right module

E_{V} (y_{1}, …, y_{n}) (33.12.3)

over E_{B}(y_{1}, …, y_{n}), as before. More precisely, E_{B}(y_{1}, …, y_{n}) may be considered as a graded ring, as in Section 9.14, and E_{V} (y_{1}, …, y_{n}) may be considered as a graded module over E_{B}(y_{1}, …, y_{n}), as a graded ring, as in Section 9.15. Note that this does not use a grading on B.

Let us take X = E_{V} (y_{1}, …, y_{n}), (33.12.4)

as a right module over E_{B}(y_{1}, …, y_{n}), and thus a right module over B in particular. We may consider X as a graded module over B in the usual sense, without a grading on B, with

X_{j} = (E_{V} (y_{1}, …, y_{n}))^{j} (33.12.5)

for each j ∈ Z, as on p151 of [50]. Remember that

X_{j} = X^{−j} (33.12.6)

for each j ∈ Z, as in Section 5.9.

Let 1 ≤ r ≤ n be given, and let us define a homomorphism d_{r} from X_{r} into X_{r−1}, as modules over B, as follows. Let I = {j_{1}, …, j_{r}} be a subset of {1, …, n} with r elements, where j_{1} < j_{2} < ⋯ < j_{r}. Thus y_{I} = y_{j_{1}} ∧ ⋯ ∧ y_{j_{r}}, as in (4.15.2). If v ∈ V, then we put

d_{r}(v y_{I}) = ∑_{l=1}^{r}(−1)^{l+1} (v · T_{j_{l}}) y_{(I\{j_{l}})}, (33.12.7)

as on p151 of [50]. Of course,

y_{(I\{j_{l}})} = y_{j_{1}} ∧ ⋯ ∧ ŷ_{j_{l}} ∧ ⋯ ∧ y_{j_{r}}, (33.12.8) where ŷ_{j_{l}} indicates that y_{j_{l}} should be omitted from the right side.

It is easy to see that this defines a homomorphism from X_{r} into X_{r−1}, as modules over B, because B is commutative. If r ≤ 0 or r > n, then we take d_{r} = 0. One can check that d_{r−1} ∘ d_{r} = 0 (33.12.9)

for every r ∈ Z, as on p151 of [50].

This makes X a complex, as a module over B, where the restriction of the differentiation operator d on X to X_{r} is equal to d_{r} for each r ∈ Z. Note that X is negative as a graded module over B, because X^{j} = X_{−j} = {0} when j > 0.

Remember that the quotient module V/(V · J_{n}) is defined as a module over B, as in the previous section. We may consider this module as a complex in the usual way, where the jth submodule is the same module when j = 0 and is {0} otherwise, and where the differentiation operator is equal to 0.

In fact,

(33.12.10) we may consider X as a left complex over V/(V · J_{n}),

as on p151 of [50]. To do this, we should choose a map ε_{X} from X into V/(V ·J_{n}), as complexes, as in Section 10.1. Of course, ε_{X} is determined by its restriction ε_{X,0} to X_{0} in this case, as before. We can identify X_{0} with V in an obvious way, as a module over B. Using this identification, we take ε_{X,0} to be the natural quotient mapping from V onto V/(V · J_{n}), as in [50].

It is easy to see that ε_{X,0} ∘ d_{1} = 0, (33.12.11)

by construction, as in [50]. This implies that ε_{X} is a homomorphism from X into V/(V · J_{n}), as modules with differentiation, as in Section 10.1.

If V satisfies the condition (33.11.11), then Proposition 3.4 on p151 of [50] states that (33.12.12) X is acyclic as a left complex over V/(V · J_{n}),

as in Section 10.1. To show this, we shall consider some more left complexes, as in [50].

33.13 Some more left complexes

Let us continue with the same notation and hypotheses as in the previous section. If 1 ≤ l ≤ n, then E_{B}(y_{1}, …, y_{l}) (33.13.1)

can be identified with a subring of E_{B}(y_{1}, …, y_{n}) in an obvious way. If l = 0, then we interpret (33.13.1) as being B. Note that (33.13.1) is homogeneous as a subgroup of E_{B}(y_{1}, …, y_{n}), as a graded commutative group with respect to addition, and that the grading induced on (33.13.1) by the one on E_{B}(y_{1}, …, y_{n}) is the same as the grading defined on (33.13.1) as in Section 9.14. In particular, E_{V} (y_{1}, …, y_{n}) may be considered as a graded right module over (33.13.1), as a graded ring, as in Section 9.15.

Similarly, E_{V} (y_{1}, …, y_{l}) (33.13.2)

may be considered as a submodule of E_{V} (y_{1}, …, y_{n}), as a module over (33.13.1), when 1 ≤ l ≤ n. If l = 0, then we interpret (33.13.2) as being V. As before, (33.13.2) is homogeneous as a subgroup of E_{V} (y_{1}, …, y_{n}), as a graded commutative group with respect to addition, and the grading induced on (33.13.2) by the one on E_{V} (y_{1}, …, y_{n}) is the same as the grading defined on (33.13.2) as in Section 9.15.

If 1 ≤ l ≤ n, then we take

X^{(l)} = E_{V} (y_{1}, …, y_{l}), (33.13.3)

as a right module over (33.13.1), and thus a right module over B in particular, as on p152 of [50]. This is the same as (33.12.4) when l = n. As before, we may consider X^{(l)} as a graded module over B in the usual sense, without a grading on B, with X_{j}^{(l)} = (E_{V} (y_{1}, …, y_{l}))^{j} (33.13.4)

for each j ∈ Z. If l = 0, then we interpret (33.13.3) as being V, as in the preceding paragraph, and we interpret (33.13.4) as being V when j = 0, and {0} otherwise.

Note that X^{(l)} may be considered as a homogeneous submodule of X for each l, as in Section 5.9. Similarly, X^{(l−1)} may be considered as a homogeneous submodule of X^{(l)} when l ≥ 1.

The differentiation operator d defined on X in the previous section maps X^{(l)} into itself for each l, so that X^{(l)} may be considered as a submodule of X, as a module with differentiation. It follows that X^{(l)} is a complex with respect to the restriction of d to X^{(l)} for each l. The restriction of d_{r} to X_{r}^{(l)} is equal to 0 when r > l. In particular, the restriction of d to X^{(0)} is equal to 0.

The quotient module V/(V · J_{l}) is defined as a module over B for each l = 0, 1, …, n, as in Section 33.11. As usual, we may consider this module as a complex, where the jth submodule is the same module when j = 0 and is {0} otherwise, and with differentiation operator equal to 0. If l = 0, then this is the same as X^{(0)}.

We would like to choose a map ε_{X^{(l)}} from X^{(l)} into V/(V ·J_{l}), as complexes, for each l = 0, 1, …, n, as in Section 10.1. As before, ε_{X^{(l)}} is determined by its restriction ε_{X^{(l)},0} to X_{0}^{(l)} in this case, and we can identify X_{0}^{(l)} with V, as a module over B. We take ε_{X^{(l)},0} to be the natural quotient mapping from V onto V/(V · J_{l}), as on p152 of [50].

One can check that

(33.13.5) the composition of the restriction of d_{1} to X_{1}^{(l)} with ε_{X^{(l)},0} is equal to 0.

This means that ε_{X^{(l)}} is a homomorphism from X^{(l)} into V/(V ·J_{l}), as modules with differentiation, as in Section 10.1. Thus

(33.13.6) we may consider X^{(l)} as a left complex over V/(V · J_{l}), as on p152 of [50].

Suppose that V satisfies (33.11.11). We would like to show that

(33.13.7) X^{(l)} is acyclic as a left complex over V/(V · J_{l})

for each l = 0, 1, …, n, as on p152 of [50]. This is the same as (33.12.12) when l = n, because X = X^{(n)}. It is easy to see that (33.13.7) holds when l = 0, because J_{0} = {0}, as in Section 33.11, and X^{(0)} = V.

If 0 ≤ l ≤ n, then put

(33.13.8)  Y_{j}^{(l)} = X_{j}^{(l)} when j ≠ 0
= V · J_{l} when j = 0,

as on p152 of [50]. This defines Y^{(l)} as a graded module over B, which is in fact a homogeneous submodule of X^{(l)}, as in Section 5.9. More precisely, Y^{(l)} is a submodule of X^{(l)}, as a module with differentiation, because

d_{1}(X_{1}^{(l)}) ⊆ V · J_{l}. (33.13.9)

In particular, Y^{(l)} is a complex, with respect to the restriction of d to Y^{(l)}. One can verify that (33.13.7) holds if and only if

H(Y^{(l)}) = {0}, (33.13.10)

as on p152 of [50]. This uses some remarks about acyclicity in Section 10.1, and the fact that ε_{X^{(l)},0} maps X_{0}^{(l)} onto V/(V · J_{l}) with kernel equal to V · J_{l}.

In order to show (33.13.7), we use induction on l, as in [50]. The base case l = 0 has already been mentioned. Thus we suppose that l ≥ 1, and that the analogue of (33.13.7) for l − 1 holds. This means that

H(Y^{(l−1)}) = {0}, (33.13.11)

as in the preceding paragraph.

Note that Y^{(l−1)} is a homogeneous submodule of Y^{(l)}, as a graded module over B. We also have that Y^{(l−1)} is a submodule of Y^{(l)}, as a module with differentiation. Thus the quotient Y^{(l)}/Y^{(l−1)} can be defined as a complex in a natural way.

Using the obvious inclusion mapping from Y^{(l−1)} into Y^{(l)}, and the natural quotient mapping from Y^{(l)} onto Y^{(l)}/Y^{(l−1)}, we get a sequence of induced mappings H(Y^{(l−1)}) ⟶ H(Y^{(l)}) ⟶ H(Y^{(l)}/Y^{(l−1)}). (33.13.12)

This sequence is exact, as in some remarks near the beginning of Section 5.3. If we can show that H(Y^{(l)}/Y^{(l−1)}) = {0}, (33.13.13)

then it will follow that (33.13.10) holds, because of (33.13.11), as on p152 of

Of course,

(33.13.14) (Y^{(l)}/Y^{(l−1)})_{j} = X_{j}^{(l)}/X_{j}^{(l−1)}  when j ≠ 0
= (V · J_{l})/(V · J_{l−1}) when j = 0,

by construction. Multiplication by y_{l} on the right defines a homomorphism

(33.13.15) from X_{j−1}^{(l−1)} into X_{j}^{(l)},

as modules over B, for each j. We can compose this with the natural quotient mapping from X_{j}^{(l)} onto X_{j}^{(l)}/X_{j}^{(l−1)} to get a homomorphism

(33.13.16) from X_{j−1}^{(l−1)} into X_{j}^{(l)}/X_{j}^{(l−1)},

as modules over B, for each j. One can check that

the homomorphism as in (33.13.16)

(33.13.17)

is an isomorphism when j ≠ 0.

Remember that we also have an isomorphism

(33.13.18) from V/(V · J_{l−1}) onto (Y^{(l)}/Y^{(l−1)})_{0} = (V · J_{l})/(V · J_{l−1}),

as modules over B. This is the homomorphism as in (33.11.13), which is an isomorphism because of the hypothesis (33.11.11). This homomorphism was induced by the homomorphism from V into V · J_{l} defined by v ↦ v · T_{l}.

The acyclicity of X^{(l−1)} as a left complex over V/(V · J_{l−1}) is equivalent to the exactness of the sequence

0 ⟶ X_{l−1}^{(l−1)} ⟶^{d_{l−1}} X_{l−2}^{(l−1)} ⟶^{d_{l−2}} ⋯ ⟶^{d_{1}} X_{0}^{(l−1)} ⟶^{ε_{X^{(l−1)},0}} V/(V · J_{l−1}) ⟶ 0,

(33.13.19)

as in Section 10.1. More precisely, this uses the fact that X_{j}^{(l−1)} = {0} when j ≥ l, by construction. The condition (33.13.13) is equivalent to the exactness of the sequence

0 ⟶ X_{l}^{(l)}/X_{l}^{(l−1)} ⟶ X_{l−1}^{(l)}/X_{l−1}^{(l−1)} ⟶ ⋯ (33.13.20)

⟶ X_{1}^{(l)}/X_{1}^{(l−1)} ⟶ (V · J_{l})/(V · J_{l−1}) ⟶ 0,

where the homomorphisms are given by differentiation on Y^{(l)}/Y^{(l−1)}. This uses the fact that (Y^{(l)}/Y^{(l−1)})_{j} = {0} when j > l or j < 0.

We have just seen that we have isomomorphisms between the modules in these two sequences. One can verify that these isomorphisms are compatible with the homomorphisms in these two sequences, as on p152 of [50].

This means that the exactness of (33.13.20) follows from the exactness of (33.13.19), as on p152 of [50]. Thus (33.13.13) follows from our induction hypothesis. This implies that (33.13.10) holds, as before.

This is the same as saying that (33.13.7) holds, as before. Thus (33.13.7) holds for every l = 0, 1, …, n, as desired.

Remember that (33.12.12) is the same as (33.13.7) with l = n. This shows that (33.12.12) holds when V satisfies (33.11.11), as mentioned in the previous section.

33.14 Some nice projective resolutions

Let us now see how the results mentioned in the previous two sections can be used to deal with augmented rings as in Section 33.11, as on p152 of [50]. Let A be a left augmented ring again, with multiplicative identity element e_{A}, augmentation module Q, augmentation homomorphism ε, and augmentation ideal I. Also let n be a positive integer, let x_{1}, …, x_{n} be n commuting elements of A, and suppose that I is generated by x_{1}, …, x_{n}, as a left ideal in A, as before.

Let T_{1}, …, T_{n} be n distinct commuting indeterminates, and let us consider A as a right module over the ring Z[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in Z, as before. Thus a·T_{l} = a x_{l} for each a ∈ A and l = 1, …, n. Remember that I = A · J_{n}, (33.14.1)

where J_{n} is the ideal in Z[T_{1}, …, T_{n}] generated by T_{1}, …, T_{n}.

Let us take V = A, (33.14.2)

as a right module over Z[T_{1}, …, T_{n}]. Note that I is a submodule of V, as a module over Z[T_{1}, …, T_{n}]. This means that

V/I = V/(V · J_{n}) (33.14.3)

may be considered as a module over Z[T_{1}, …, T_{n}]. However, the action of T_{l} on (33.14.3) on the right is equal to 0 for each l = 1, …, n, by construction.

Let y_{1}, …, y_{n} be another collection of n distinct indeterminates, as in Section 33.12. Let us take

X = E_{V} (y_{1}, …, y_{n}) = E_{A}(y_{1}, …, y_{n}), (33.14.4)

as before. More precisely, this may be considered as a right module over B = Z[T_{1}, …, T_{n}], because V is a right module over B, as in Section 4.15. Similarly, X may be considered as a left module over A, because A is a left module over itself.

Put X_{j} = (E_{V} (y_{1}, …, y_{n}))^{j} = (E_{A}(y_{1}, …, y_{n}))^{j} (33.14.5)

for each j ∈ Z, as in Section 33.12. This is a submodule of X, as both a right module over B, and a left module over A. This defines a grading on X, as a right module over B, and a left module over A.

We can define a homomorphism d_{r} from X_{r} into X_{r−1}, as right modules over B, for each r ∈ Z, as in Section 33.12. It is easy to see that d_{r} is a homomorphism from X_{r} into X_{r−1} as left modules over A as well. This means that X is a complex, as a right module over B, and as a left module over A.

Remember that X is a left complex over V/(V · J_{n}), as right modules over B, with the augmentation map ε_{X} defined in Section 33.12. More precisely, the restriction ε_{X,0} of ε_{X} to X_{0} corresponds exactly to the natural quotient mapping from V onto V/(V · J_{n}). Of course,

Q = A/I = V/(V · J_{n}), (33.14.6)

as commutative groups with respect to addition, and this may be considered as both a left module over A, and a right module over B. Using this, it is easy to see that (33.14.7) we may consider X as a left complex over Q,

as left modules over A.

If I ⊆ {1, …, n}, then let y_{I} be as in (4.15.2). If 0 ≤ j ≤ n, then (E_{A}(y_{1}, …, y_{n}))^{j} is freely generated as a left module over A by y_{I}, where I has exactly j elements. Of course, X_{j} = {0} when j < 0 or j > n. It follows that X_{j} is projective as a left module over A for each j ∈ Z, so that X is projective as a left complex of modules over A, as in Section 10.1.

Suppose now that (33.11.3) holds, which means that (33.11.11) holds in this case. This implies that X is acyclic as a left complex over V/(V · J_{n}), as right modules over B, as in (33.12.12). Using this, we get that

(33.14.8) X is acyclic as a left complex over Q, as left modules over A.

It follows that

(33.14.9) X is a projective resolution of Q, as a left module over A,

as in Section 10.2. This corresponds to some remarks on p152 of [50], and to part of Theorem 4.2 on p150 of [50].

33.15 Using these projective resolutions

Let us continue with the same notation and hypotheses as in the previous section. Let W be a right module over A, and let

W ⨂_{A}X = W ⨂_{A}E_{A}(y_{1}, …, y_{n}) (33.15.1)

be a tensor product of W and X = E_{A}(y_{1}, …, y_{n}) over A. This is a graded module over Z, with

(W ⨂_{A}X)_{j} = W ⨂_{A}X_{j} = W ⨂_{A}(E_{A}(y_{1}, …, y_{n}))^{j} (33.15.2) for each j ∈ Z. It is easy to see that

W ⨂_{A}X = E_{W} (y_{1}, …, y_{n}), (33.15.3)

where E_{W} (y_{1}, …, y_{n}) is as in Section 4.15. Similarly,

(W ⨂_{A}X)_{j} = (E_{W} (y_{1}, …, y_{n}))^{j} (33.15.4)

for each j, where (E_{W} (y_{1}, …, y_{n}))^{j} is as in Section 9.15.

More precisely, W ⨂_{A} X may be considered as a complex over Z, as in Section 7.5. Remember that the differentiation operator d_{W ⨂_{A} X} on W ⨂_{A} X is obtained from the differentiation operator d = d_{X} on X and the identity mapping on W in the usual way. If r ∈ Z, then the corresponding homomorphism

(33.15.5) d_{W ⨂_{A}X,r} from (W ⨂_{A}X)_{r} into (W ⨂_{A}X)_{r−1}

is obtained from the homomorphism d_{r} from X_{r} into X_{r−1} and the identity mapping on W in the same way. In particular, this is equal to 0 when d_{r} = 0, which happens when r ≤ 0 and when r > n.

If 1 ≤ r ≤ n, then (33.15.5) can be described as follows. Let I = {j_{1}, …, j_{r}} be a subset of {1, …, n} with r elements, where j_{1} < j_{2} < ⋯ < j_{r}, so that y_{I} = y_{j_{1}} ∧ ⋯ ∧ y_{j_{r}}, as before. If w ∈ W, then

d_{W ⨂_{A}X,r}(w y_{I}) = ∑_{l=1}^{r}(−1)^{l+1}(w · x_{j_{l}}) y_{(I\{j_{l}})}, (33.15.6)

as in (2) on p152 of [50]. Remember that the homology of this complex may be used to obtain the homology of A, as a left augmented ring, with coefficients in W, as in Section 33.3.

This type of complex was first found by J. L. Koszul, in connection with cohomology theory of Lie groups, as mentioned on p153 of [50].

Similarly, let Z be a left module over A, so that

Hom_{A}^{gr}(X, Z) = Hom_{A}^{gr}(E_{A}(y_{1}, …, y_{n}), Z) (33.15.7)

may be defined as a graded module over Z as in Section 8.4. More precisely,

(33.15.8) (Hom_{A}^{gr}(X, Z))^{j} = Hom_{A}(X_{j}, Z) = Hom_{A}((E_{A}(y_{1}, …, y_{n}))^{j}, Z)

for each j ∈ Z. Of course, this reduces to {0} when j < 0, and when j > n. If 0 ≤ j ≤ n, then the elements of (33.15.8) may be indentified with Z-valued functions f(I) defined for subsets I of {1, …, n} such that the number |I| of elements of I is equal to j.

In fact, (33.15.7) may be considered as a complex over Z, as in Section 8.4. Let δ = d_{Hom_{A}^{gr}(X,Z)} (33.15.9) be the corresponding differentiation operator on (33.15.7), and let δ_{j} be its restriction to (33.15.8) for each j. This is the mapping from (33.15.8) into

(33.15.10) (Hom_{A}^{gr}(X, Z))^{j+1} = Hom_{A}(X_{j+1}, Z)
= Hom_{A}((E_{A}(y_{1}, …, y_{n}))^{j+1}, Z)

defined by composing d_{j+1} with a homomorphism from X_{j} into Z, as modules over A, to get a homomorphism from X_{j+1} into Z, as modules over A. Note that δ_{j} = 0 when j < 0 or j ≥ n.

Suppose that 0 ≤ j < n, and that f is a Z-valued function on the set of all subsets of {1, …, n} with exactly j elements, which is identified with an element of (33.15.8), as before. Let δ_{j}(f) be the Z-valued function on the set of all subsets of {1, …, n} with exactly j + 1 elements that corresponds to δ_{j} of the element of (33.15.8) corresponding to f. If I = {m_{1}, …, m_{j+1}} is a subset of {1, …, n} with j + 1 elements, where m_{1} < m_{2} < ⋯ < m_{j+1}, then

(δ_{j}(f))(I) = ∑_{l=1}^{j+1}(−1)^{l+1} (x_{l} · f(I \ {m_{l}})). (33.15.11)

This corresponds to (2a) on p153 of [50]. The homology of this complex may be used to obtain the cohomology of A, as a left augmented ring, with coefficients in Z, as in Section 33.5.

Some helpful facts

34.1 Projectivity and tensor products

Let k be a commutative ring with a multiplicative identity element, and let A, B be associative algebras over k with multiplicative identity elements e_{A}, e_{B}, respectively. Also let V, W be modules over k, and suppose that

(34.1.1) V is a right module over A, and W is a left module over A.

Thus we can let V ⨂_{A} W be a tensor product of V and W over A, which is a module over k.

Suppose for the moment that

(34.1.2) W is a right module over B,

where the actions of A and B on W commute with each other. Thus V ⨂_{A} W may be considered as a right module over B too, as in Section 1.10. If

(34.1.3) V is projective as a right module over A

and (34.1.4) W is projective as a right module over B,

then (34.1.5) V ⨂_{A}W is projective as a right module over B,

as in Proposition 5.3 on p28 of [50].

If V is free as a right module over A, then V ⨂_{A} W corresponds to a direct sum of copies of W, as a right module over B. This means that V ⨂_{A} W is projective as a right module over B, because W is projective.

If V is projective as a right module over A, then there is a free right module over A that is isomorphic to the direct sum of V and another right module U over A, as in Section 2.7. This implies that the direct sum of V ⨂_{A} W and a tensor product U ⨂_{A} W of U and W over A is projective as a right module over B, as in the preceding paragraph. It follows that V ⨂_{A} W is projective as a right module over B, as in Section 2.7.

Alternatively, let Y and Z be right modules over B, and let ψ be a homomorphism from Y onto Z, as modules over B. Consider the homomorphism

(34.1.6) Ψ from Hom_{B}(W, Y) into Hom_{B}(W, Z)

defined by composing a homomorphism from W into Y, as modules over B, with ψ to get a homomorphism from W into Z, as modules over B. Note that

(34.1.7) Ψ maps Hom_{B}(W, Y) onto Hom_{B}(W, Z),

because W is projective as a module over B, by hypothesis. More precisely, Hom_{B}(W, Y) and Hom_{B}(W, Z) may be considered as right modules over A, because W is a left module over A, and the actions of A and B on W commute, as in Section 1.8. It is easy to see that Ψ is a homomorphism as in (34.1.6), as right modules over A.

Using Ψ, we get a homomorphism

(34.1.8) Ψ̂ from Hom_{A}(V, Hom_{B}(W, Y)) into Hom_{A}(V, Hom_{B}(W, Z)),

as modules over k. As before, Ψ̂ is defined by composing a homomorphism from V into Hom_{B}(W, Y), as modules over A, with Ψ to get a homomorphism from V into Hom_{B}(W, Z), as modules over A. We also have that

(34.1.9) Ψ̂ maps Hom_{A}(V, Hom_{B}(W, Y)) onto Hom_{A}(V, Hom_{B}(W, Z)),

because V is projective as a right module over A, by hypothesis.

Consider the homomorphism

(34.1.10) Ψ̃ from Hom_{B}((V ⨂_{A}W ), Y ) into Hom_{B}((V ⨂_{A}W ), Z),

as modules over k, defined by composing a homomorphism from V ⨂_{A} W into Y, as modules over B, with ψ to get a homomorphism from V ⨂_{A} W into Z, as modules over B.

There is a natural isomorphism

(34.1.11) from Hom_{B}((V ⨂_{A}W ), Z) onto Hom_{A}(V, Hom_{B}(W, Z)),

as modules over k, as in Section 1.13. Of course, there is an analogous isomorphism with Z replaced by Y. One can check that Ψ̃ corresponds to Ψ̂ with respect to these isomorphisms. Thus the surjectivity of Ψ̃ follows from the surjectivity of Ψ̂. This implies (34.1.5).

Suppose now that (34.1.12) V is a left module over B,

where the actions of A and B on V commute with each other. This means that V ⨂_{A} W may be considered as a left module over B as well, as in Section 1.10. If (34.1.13) V is projective as a left module over B 34.2. INJECTIVITY OF HOM_{A}(V, Z) 755

and (34.1.14) W is projective as a left module over A,

then (34.1.15) V ⨂_{A}W is projective as a left module over B,

as on p28 of [50].

One can start with the case where W is free as a left module over A, and then reduce to that case using direct sums, as before. Alternatively, if Z is a left module over B, then there is a natural isomorphism

(34.1.16) from Hom_{B}((V ⨂_{A}W ), Z) onto Hom_{A}(W, Hom_{B}(V, Z)),

as modules over k, as in Section 1.13 again. This can be used to obtain (34.1.15), as before.

34.2 Injectivity of Hom_{A}(V, Z)

Let k be a commutative ring with a multiplicative identity element, and let A, B be associative algebras over k with multiplicative identity elements e_{A}, e_{B}, respectively. Also let V, Z be modules over k, and suppose that

(34.2.1) V and Z are both left or both right modules over A.

This means that Hom_{A}(V, Z) is defined as a module over k.

Suppose too that

(34.2.2) Z is a left or right module over B,

where the actions of A and B on Z commute. This implies that Hom_{A}(V, Z) may be considered as a left or right module over B, as appropriate, as well, as in Section 1.8. If (34.2.3) V is projective as a module over A,

and (34.2.4) Z is injective as a module over B,

then (34.2.5) Hom_{A}(V, Z) is injective as a module over B,

as in Exercise 5 on p32 of [50].

Suppose for the moment that V is free as a module over A, and thus isomorphic to a direct sum of copies of A. This implies that Hom_{A}(V, Z) is isomorphic to a direct product of copies of Hom_{A}(A, Z), as in Section 1.7. This means that Hom_{A}(V, Z) is isomorphic to a direct product of copies of Z, as modules over B. It follows that (34.2.5) holds, because Z is injective as a module over B, as in Section 2.8.

If (34.2.3) holds, then there is a left or right module U over A, as appropriate, such that the direct sum of U and V is free as a module over A, as in Section 2.7. In this case, Hom_{A} (U ⨁ V, Z) is injective as a module over B, as in the preceding paragraph. Of course, Hom_{A} (U ⨁ V, Z) is isomorphic to the direct sum of Hom_{A}(U, Z) and Hom_{A}(V, Z), as modules over B. It follows that (34.2.5) holds, as in Section 2.8 again.

Alternatively, let W be a left or right module over B, as appropriate, and let W_{0} be a submodule of W. There is an obvious homomorphism

(34.2.6) from Hom_{B}(W, Hom_{A}(V, Z)) into Hom_{B}(W_{0}, Hom_{A}(V, Z)),

as modules over k, which sends a homomorphism from W into Hom_{A}(V, Z) to its restriction to W_{0}. We would like to verify that this homomorphism is surjective, under the conditions considered here.

Note that Hom_{B}(W, Z) is a module over k which may be considered as a left or right module over A, as appropriate, as in Section 1.8 again. There is a natural isomorphism

(34.2.7) from Hom_{A}(V, Hom_{B}(W, Z)) onto Hom_{B}(W, Hom_{A}(V, Z)),

as modules over k, as in Section 1.14. Similarly, there is a natural isomorphism

(34.2.8) from Hom_{A}(V, Hom_{B}(W_{0}, Z)) onto Hom_{B}(W_{0}, Hom_{A}(V, Z)),

as modules over k. Using these isomorphisms, the homomorphism as in (34.2.6) corresponds to a homomorphism

(34.2.9) from Hom_{A}(V, Hom_{B}(W, Z)) into Hom_{A}(V, Hom_{B}(W_{0}, Z)),

as modules over k.

More precisely, there is an obvious homomorphism

(34.2.10) from Hom_{B}(W, Z) into Hom_{B}(W_{0}, Z),

as modules over k or A, which sends a homomorphism from W into Z to its restriction to W_{0}. One can check that the homomorphism as in (34.2.9) sends a homomorphism from V into Hom_{B}(W, Z) to its composition with the homomorphism as in (34.2.10).

The homomorphism as in (34.2.10) is surjective, because of (34.2.4). This implies that the homomorphism as in (34.2.9) is surjective, because of (34.2.3). This means that the homomorphism as in (34.2.6) is surjective, as desired.

34.3 Another injectivity property

Let k be a commutative ring with a multiplicative identity element, and let A, B be associative algebras over k with multiplicative identity elements e_{A}, e_{B}, respectively. Also let V, Z be modules over k, and suppose that

(34.3.1) V and Z are both left or both right modules over B, so that Hom_{B}(V, Z) is defined as a module over k.

In this section, we ask that

(34.3.2) V be a left or right module over A

too, where the actions of A and B on V commute. This means that Hom_{B}(V, Z) may be considered as a right or left module over A, respectively, as in Section 1.8. If (34.3.3) V is projective as a module over A and (34.3.4) Z is injective as a module over B, then (34.3.5) Hom_{B}(V, Z) is injective as a module over A, as in Proposition 1.4 on p107 of [50].

Suppose for the sake of simplicity that V is a right module over A, so that Hom_{B}(V, Z) is a left module over A. Let W be a left module over A, and let W_{0} be a submodule of W. There is an obvious homomorphism

(34.3.6) from Hom_{A}(W, Hom_{B}(V, Z)) into Hom_{A}(W_{0}, Hom_{B}(V, Z)),

as modules over k, which sends a homomorphism from W into Hom_{B}(V, Z), as modules over A, to its restriction to W_{0}. We would like to check that this homomorphism is a surjection.

Let V ⨂_{A} W, V ⨂_{A} W_{0} be tensor products of V with W, W_{0}, respectively, over A. These modules over k may also be considered as left or right modules over B, as appropriate, as in Section 1.10. There is a natural isomorphism

(34.3.7) from Hom_{A}(W, Hom_{B}(V, Z)) onto Hom_{B}((V ⨂_{A}W ), Z),

as modules over k, as in Section 1.13. Similarly, there is a natural isomorphism

(34.3.8) from Hom_{A}(W_{0}, Hom_{B}(V, Z)) onto Hom_{B}((V ⨂_{A}W_{0}), Z),

as modules over k. Using these isomomorphisms, the homomorphism as in (34.3.6) corresponds to a homomorphism

(34.3.9) from Hom_{B}((V ⨂_{A}W ), Z) into Hom_{B}((V ⨂_{A}W_{0}), Z),

as modules over k.

There is a natural homomorphism

(34.3.10) from V ⨂_{A}W_{0} into V ⨂_{A}W,

as modules over B, corresponding to the identity mapping on V, and the obvious inclusion mapping from W_{0} into W. One can verify that the homomorphism as in (34.3.9) sends a homomorphism from V ⨂_{A} W to Z, as modules over B, to its composition with the homomorphism as in (34.3.10). The homomorphism as in (34.3.10) is injective, because of (34.3.3), as mentioned near the end of Section 2.7. This implies that the homomorphism as in (34.3.9) is surjective, because of (34.3.4). It follows that the homomorphism as in (34.3.6) is surjective, as desired.

34.4 An associativity isomorphism

Let k be a commutative ring with a multiplicative identity element, and let A, B, and C be associative algebras over k, with multiplicative identity elements e_{A}, e_{B}, and e_{C}, respectively. Also let A ⨂_{k} B and B ⨂_{k} C be tensor products of A, B and B, C, respectively, over k. These may be considered as associative algebras over k with multiplicative identity elements too, as in Section 4.1.

Let V, W, and Z be modules over k. Suppose that

(34.4.1) V is a right module over A and B, where the actions of A and B on V commute with each other,

and that

(34.4.2) W is a left module over A and a right module over C, where the actions of A and C on W commute with each other.

Let V ⨂_{A} W be a tensor product of V and W over A which is a module over k. Remember that V ⨂_{A} W may be considered as a right module over each of B and C, as in Section 1.10. It is easy to see that the actions of B and C on V ⨂_{A} W commute with each other.

Similarly, suppose that

(34.4.3) Z is a left module over B and C, where the actions of B and C on Z commute with each other.

Let W ⨂_{C} Z be a tensor product of W and Z over C, which is a module over k. As before, W ⨂_{C} Z may be considered as a left module over A and B, where the actions of A and B on W ⨂_{C} Z commute with each other.

Note that V may be considered as a right module over A ⨂_{k} B, and Z may be considered as a left module over B ⨂_{k} C, as in Section 4.2. Similarly, V ⨂_{A} W may be considered as a right module over B ⨂_{k} C, and W ⨂_{C} Z may be considered as a left module over A ⨂_{k} B. Let

(V ⨂_{A}W ) ⨂_{B ⨂_{k}C}Z (34.4.4)

be a tensor product of V ⨂_{A} W and Z over B ⨂_{k} C, and let

V ⨂_{A ⨂_{k}B}(W ⨂_{C}Z) (34.4.5)

be a tensor product of V and W ⨂_{C} Z over A ⨂_{k} B.

Under these conditions, there is a unique homomorphism from (34.4.4) into (34.4.5), as modules over k, which sends

(v ⊗_{A} w) ⊗_{B ⨂_{k}C} z (34.4.6) to v ⊗_{A ⨂_{k}B} (w ⊗_{C} z) (34.4.7)

for every v ∈ V, w ∈ W, and z ∈ Z, as in Proposition 2.1 on p165 of [50]. More precisely, this homomorphism is an isomorphism, as in [50]. A simpler version of this was discussed in Section 1.12.

In particular, this homomorphism can be obtained directly using the definition of a tensor product, as before. One can start by fixing z ∈ Z, and considering the mapping from V ×W into (34.4.5) that sends (v, w) ∈ V ×W to (34.4.7). This leads to a unique homomorphism from V ⨂_{A} W into (34.4.5), as modules over k, that sends v ⊗_{A} w to (34.4.7) for every v ∈ V and w ∈ W. This may be considered as a mapping from (V ⨂_{A} W ) × Z into (34.4.5), which is bilinear over k and has other nice properties. Using this, we get a homomorphism from (34.4.4) into (34.4.5), as modules over k.

Similarly, there is a unique homomorphism from (34.4.5) into (34.4.4) that sends (34.4.7) to (34.4.6) for every v ∈ V, w ∈ W, and z ∈ Z. This homomorphism is the inverse of the previous one, so that these homomorphisms are isomorphisms.

34.5 Some more isomorphisms

Let k, A, B, and C be as in the previous section, as well as A ⨂_{k} B and B ⨂_{k} C. Also let V, W, and Z be modules over k again, and suppose that (34.4.1) and (34.4.2) hold. As before, we let V ⨂_{A} W be a tensor product of V and W over A, which is a module over k that may be considered as a right module over B and C, and where the actions of B and C on V ⨂_{A} W commute with each other.

Suppose now that

(34.5.1) Z is a right module over B and C, where the actions of B and C on Z commute with each other.

Thus Hom_{C}(W, Z) is defined as a module over k. This may be considered as a right module over A and B in this case, as in Section 1.8. It is easy to see that the actions of A and B on Hom_{C}(W, Z) commute with each other.

Remember that V may be considered as a right module over A ⨂_{k} B, and that V ⨂_{A} W may be considered as a right module over B ⨂_{k} C, as in the previous section. Similarly, Hom_{C}(W, Z) may be considered as a right module over A ⨂_{k} B, and Z may be considered as a right module over B ⨂_{k} C, as in Section 4.2. Thus Hom_{A ⨂_{k}B}(V, Hom_{C}(W, Z)) (34.5.2)

and Hom_{B ⨂_{k}C}((V ⨂_{A}W ), Z) (34.5.3)

are defined as modules over k.

Proposition 2.2 on p165 of [50] states that there is a unique homomorphism from (34.5.2) into (34.5.3), as modules over k, with the following property. If ϕ is an element of (34.5.2), then this homomorphism sends ϕ to an element of (34.5.3) with v ⊗_{A} w ↦ (ϕ(v))(w) (34.5.4)

for every v ∈ V and w ∈ W. More precisely, if v ∈ V, then ϕ(v) ∈ Hom_{C}(W, Z), so that (ϕ(v))(w) ∈ Z for each w ∈ W.

If ϕ is an element of (34.5.2), then

(v, w) ↦ (ϕ(v))(w) (34.5.5)

defines a mapping from V × W into Z that is bilinear over k. This leads to a unique homomorphism from V ⨂_{A} W into Z, as modules over k, that satisfies (34.5.4) for every v ∈ V and w ∈ W. One can check that this homomorphism is an element of (34.5.3). This defines a mapping from (34.5.2) into (34.5.3), which is easily seen to be linear over k.

Conversely, any element of (34.5.3) can be used to get a mapping from V ×W into Z that is bilinear over k and has other relevant properties. This can be used to get an element of (34.5.2). This defines a homomorphism from (34.5.3) into (34.5.2), as modules over k, which is the inverse of the previous homomorphism from (34.5.2) into (34.5.3). Thus we get an isomorphism between these modules over k, as in [50]. A simpler version of this was discussed in Section 1.13.

There is an analogous statement with actions on the left and right exchanged, and using a tensor product W ⨂_{A} V, as mentioned on p165 of [50]. The corresponding simplified version was discussed in Section 1.13.

34.6 Another projectivity property

Let us continue with the same notation and hypotheses as in the previous section. Suppose that

(34.6.1) V is projective as a right module over A ⨂_{k}B,

and that (34.6.2) W is projective as a right module over C.

Under these conditions,

(34.6.3) V ⨂_{A}W is projective as a right module over B ⨂_{k}C,

as in Proposition 2.3 on p165 of [50].

To see this, let Y and Z be modules over k that are right modules over B ⨂_{k} C. Equivalently, this means that (34.5.1) holds, and similarly

(34.6.4) Y is a right module over B and C, where the actions of B and C on Y commute with each other.

Let ψ be a homomorphism from Y onto Z, as right modules over B ⨂_{k} C. Consider the homomorphism

Ψ̃ from Hom_{B ⨂_{k}C}((V ⨂_{A}W ), Y ) into Hom_{B ⨂_{k}C}((V ⨂_{A}W ), Z), (34.6.5) as modules over k, defined by composing a homomorphism from V ⨂_{A} W into Y, as right modules over B ⨂_{k} C, with ψ to get a homomorphism from V ⨂_{A} W into Z, as right modules over B ⨂_{k} C. We would like to show that Ψ̃ is surjective.

As in the previous section, V and Hom_{C}(W, Z) may be considered as right modules over A ⨂_{k} B, and (34.5.2) is isomorphic to (34.5.3), as modules over k, in a natural way. Similarly, Hom_{C}(W, Y) may be considered as a right module over A ⨂_{k} B, and Hom_{A ⨂_{k}B}(V, Hom_{C}(W, Y)) (34.6.6)

is isomorphic to Hom_{B ⨂_{k}C}((V ⨂_{A}W ), Y ), (34.6.7)

as modules over k, in a natural way.

Consider the homomorphism

(34.6.8) Ψ from Hom_{C}(W, Y) into Hom_{C}(W, Z)

that sends a homomorphism from W into Y, as modules over C, to its composition with ψ. The composition is a homomorphism from W into Z, as modules over C, because ψ is a homomorphism from Y onto Z, as modules over C in particular. Observe that

(34.6.9) Ψ maps Hom_{C}(W, Y) onto Hom_{C}(W, Z),

because of (34.6.2). One can check that Ψ is a homomorphism as in (34.6.8), as right modules over A ⨂_{k} B. Equivalently, Ψ is a homomorphism as in (34.6.8), as modules over each of A and B.

This leads to a homomorphism

(34.6.10)  Ψ̂ from Hom_{A ⨂_{k}B}(V, Hom_{C}(W, Y))
into Hom_{A ⨂_{k}B}(V, Hom_{C}(W, Z)),

as modules over k, which sends a homomorphism from V into Hom_{C}(W, Y), as right modules over A ⨂_{k} B, to its composition with Ψ. In fact,

(34.6.11)  Ψ̂ maps Hom_{A ⨂_{k}B}(V, Hom_{C}(W, Y))
onto Hom_{A ⨂_{k}B}(V, Hom_{C}(W, Z)),

because of (34.6.1) and (34.6.9).

One can verify that Ψ̃ corresponds to Ψ̂ with respect to the isomorphisms between (34.5.2) and (34.5.3), and between (34.6.6) and (34.6.7). Thus the surjectivity of Ψ̃ follows from (34.6.11), as desired.

Of course, a simpler version of this was discussed in Section 34.1.

34.7 A couple of corollaries

Let k be a commutative ring with a multiplicative identity element, and let A, B, and C be associative algebras over k, with multiplicative identity elements e_{A}, e_{B}, and e_{C}, respectively. Also let A ⨂_{k} B and B ⨂_{k} C be tensor products of A, B and B, C over k again, considered as associative algebras with multiplicative identity elements, as in Section 4.1.

Suppose that V is a module over k that is a right module over A and B, where the actions of A and B on V commute with each other. Equivalently, this means that V is a right module over A ⨂_{k} B, as in Section 4.2. If

(34.7.1) V is projective as a right module over A ⨂_{k}B

and (34.7.2) A is projective as a module over k,

then (34.7.3) V is projective as a right module over B.

This follows from the remarks in the previous section, with C = k, and W = A, considered as a left module over A and a right module over C. In this case, B satisfies the requirements of a tensor product of B and C over k, and V satisfies the requirements of a tensor product of V and W over A. More precisely, it is easy to see that multiplication on B is the same as multiplication on B ⨂_{k} C. Similarly, the given actions of B and k on V are the same as the actions of B and C on V ⨂_{A} W considered previously.

This corresponds to Corollary 2.4 on p166 of [50], which is stated in terms of the opposite algebra A^{op} of A. Of course, A^{op} is the same as A as a module over k, so that (34.7.2) is the same as for A^{op}.

Let C be any associative algebra over k with a multiplicative identity element e_{C} again. Suppose now that V, W are modules over k, with V a right module over B, and W a right module over C. Let V ⨂_{k} W be a tensor product of V and W over k, which may be considered as a right module over B and C, as in Section 1.10. One can check that the actions of B and C on V ⨂_{k} W commute with each other, as before.

If (34.7.3) holds, and

(34.7.4) W is projective as a right module over C,

then (34.7.5) V ⨂_{k}W is projective as a right module over B ⨂_{k}C.

This follows from the remarks in the previous section, with A = k. More precisely, B satisfies the requirements of A ⨂_{k} B, and multiplication on B is the same as multiplication on A ⨂_{k} B. Thus (34.6.1) reduces to (34.7.3). This corresponds to Corollary 2.5 on p166 of [50].

34.8 Some alternate approaches

We would like to consider some alternative arguments for the projectivity conditions discussed in the previous two sections, which are analogous to the first argument in Section 34.1. Let k be a commutative ring with a multiplicative identity element again, and let A, B, and C be associative algebras over k, with multiplicative identity elements e_{A}, e_{B}, and e_{C}, respectively. Also let A ⨂_{k} B and B ⨂_{k} C be tensor products of of A, B and B, C, respectively, considered as associative algebras over k with multiplicative identity elements, as before.

Suppose that (34.8.1) A is projective as a module over k.

This implies that

(34.8.2) A ⨂_{k}B is projective as a right module over B,

as in Section 34.1. Let V be a module over k that is a right module over A and B, where the actions of A and B on V commute, so that V may be considered as a right module over A ⨂_{k} B. If V is free as a right module over A ⨂_{k} B, then (34.8.3) V is projective as a right module over B.

Indeed, V corresponds to the direct sum of copies of A ⨂_{k} B, and thus a direct sum of projective right modules over B.

Suppose now that

(34.8.4) V is projective as a right module over A ⨂_{k}B.

This implies that there is a free right module over A ⨂_{k} B that is isomorphic to the direct sum of V and another right module U over A ⨂_{k} B, as in Section 2.7. This means that the direct sum of V and U is projective as a right module over B, as in the preceding paragraph. It follows that (34.8.3) holds, as in Section 2.7 again.

Let V, W be modules over k, with V a right module over B, and W a right module over C, and let V ⨂_{k} W be a tensor product of V and W over k. As in the previous section, V ⨂_{k} W may be considered as a right module over B and C, where the actions of B and C on V ⨂_{k} W commute, so that V ⨂_{k} W may be considered as a right module over B ⨂_{k} C. If V and W are free as right modules over B and C, respectively, then V ⨂_{k} W is free as a right module over B ⨂_{k} C.

Suppose that (34.8.4) holds, and that

(34.8.5) W is projective as a right module over C.

Thus there are right modules U, Z over B, C, respectively, such that V ⨁ U and W ⨁ Z are free as right modules over B and C, respectively, as in Section 2.7. This implies that a tensor product

(V ⨁ U) ⨂_{k}(W ⨁ Z) (34.8.6) is free as a right module over B ⨂_{k} C, as in the preceding paragraph. This tensor product is isomorphic to the direct sum of V ⨂_{k} W and tensor products U ⨂_{k} W, V ⨂_{k} Z, and U ⨂_{k} Z, as a module over k, right modules over B and C, and hence over B ⨂_{k} C. This means that

(34.8.7) V ⨂_{k}W is projective as a right module over B ⨂_{k}C,

as in Section 2.7.

Let W be a module over k that is a left module over A and a right module over C, where the actions of A and C on W commute with each other. Let us consider A ⨂_{k} B as a right module over A for the moment, and let

(A ⨂_{k}B) ⨂_{A}W (34.8.8)

be a tensor product of A ⨂_{k} B and W over A. This may be considered as a right module over B and C, where the actions of B and C commute, and thus as a right module over B ⨂_{k} C. We would like to check that

(34.8.9) (34.8.8) is projective as a right module over B ⨂_{k}C.

Although this may be considered as a particular case of (34.6.3), we would like to use this to give another argument for (34.6.3).

Remember that A ⨂_{k} B is isomorphic to B ⨂_{k} A in a natural way, as in Section 1.4. Thus (34.8.8) is isomorphic to

(B ⨂_{k}A) ⨂_{A}W (34.8.10)

in a natural way. This is isomorphic to a tensor product of the form

B ⨂_{k}(A ⨂_{A}W ) (34.8.11)

in a natural way, as in Section 1.12. This reduces to a tensor product of the form B ⨂_{k} W, because W satisfies the requirements of A ⨂_{A} W. This means that (34.8.9) follows from (34.8.7), which is a bit simpler in this case.

Suppose now that V is a module over k that is a right module over A and B, where the actions of A and B on V commute, so that V may be considered as a right module over A ⨂_{k} B. If

(34.8.12) V is projective as a right module over A ⨂_{k}B,

and W is as in the previous two paragraphs, then

(34.8.13) V ⨂_{A}W is projective as a right module over B ⨂_{k}C,

as in Section 34.6. Alternatively, if V is free as a right module over A ⨂_{k} B, then V ⨂_{A} W corresponds to the direct sum of modules of the form (34.8.8), and is thus projective as a right module over B ⨂_{k} C. Otherwise, there is a free right module over A ⨂_{k} B that is isomorphic to the direct sum of V and another right module U over A ⨂_{k} B, as in Section 2.7. This implies that the direct sum of V ⨂_{A} W and a tensor product U ⨂_{A} W is projective as a right module over B ⨂_{k} C, so that (34.8.13) holds, as in Section 2.7 again.

34.9 Another result about injectivity

Let k be a commutative ring with a multiplicative identity element, and let A, B, and C be associative algebras over k, with multiplicative identity elements e_{A}, e_{B}, and e_{C}, respectively. As before, we let A ⨂_{k} B and B ⨂_{k} C be tensor products of A, B and B, C over k, respectively, considered as associative algebras over k with multiplicative identity elements, as in Section 4.1.

Let W and Z be modules over k. Suppose that

(34.9.1) W is a left module over A and a right module over C, where the actions of A and C on W commute with each other,

and that

(34.9.2) Z is a right module over B and C, where the actions of B and C on Z commute with each other.

In particular, Hom_{C}(W, Z) is defined as a module over k, and may be considered as a right module over A and B, as in Section 1.8. One can check that the actions of A and B on Hom_{C}(W, Z) commute with each other, as before. Remember that Z may be considered as a right module over B ⨂_{k} C, and that Hom_{C}(W, Z) may be considered as a right module over A ⨂_{k} B, as in Section 4.2.

Suppose that

(34.9.3) W is projective as a left module over A,

and that (34.9.4) Z is injective as a right module over B ⨂_{k}C.

Under these conditions,

(34.9.5) Hom_{C}(W, Z) is injective as a right module over A ⨂_{k}B,

as in Proposition 2.3a on p166 of [50]. Simpler versions of this were discussed in Sections 34.2 and 34.3.

Let V be a module over k that is a right module over A ⨂_{k} B. Thus V may be considered as a right module over A and B, where the actions of A and B on V commute with each other. Let V ⨂_{A} W be a tensor product of V and W over A, which is a module over k that may be considered as a right module over B and C. As before, the actions of B and C on V ⨂_{A} W commute with each other, so that V ⨂_{A} W may be considered as a right module over B ⨂_{k} C. Remember that there is a natural isomorphism

(34.9.6) from Hom_{A ⨂_{k}B}(V, Hom_{C}(W, Z))

onto Hom_{B ⨂_{k}C}((V ⨂_{A}W ), Z),

as modules over k, as in Section 34.5.

Let V_{0} be a submodule of V as a right module over A ⨂_{k} B, and thus as a module over A and B. Also let V_{0} ⨂_{A} W be a tensor product of V_{0} and W over A, which is another module over k that may be considered as a right module over B and C. The actions of B and C on V_{0} ⨂_{A} W commute with each other, so that V ⨂_{A} W may be considered as a right module over B ⨂_{k} C. There is a natural isomorphism

(34.9.7) from Hom_{A ⨂_{k}B}(V_{0}, Hom_{C}(W, Z))

onto Hom_{B ⨂_{k}C}((V_{0} ⨂_{A}W ), Z),

as modules over k, as in Section 34.5 again.

There is a natural homomorphism

(34.9.8) from Hom_{A ⨂_{k}B}(V, Hom_{C}(W, Z))

into Hom_{A ⨂_{k}B}(V_{0}, Hom_{C}(W, Z)),

as modules over k, that sends a homomorphism from V into Hom_{C}(W, Z), as right modules over A ⨂_{k} B, to its restriction to V_{0}. We would like to show that this homomorphism is surjective.

Using the isomorphisms as in (34.9.6) and (34.9.7), the homomorphism as in (34.9.8) corresponds to a homomorphism

from Hom_{B ⨂_{k}C}((V ⨂_{A}W ), Z) (34.9.9)

into Hom_{B ⨂_{k}C}((V_{0} ⨂_{A}W ), Z),

as modules over k. It suffices to show that this homomorphism is surjective. There is a natural homomorphism

(34.9.10) from V_{0} ⨂_{A}W into V ⨂_{A}W,

as modules over k, corresponding to the obvious inclusion mapping of V_{0} into V and the identity mapping on W. It is easy to see that this is a homomorphism as in (34.9.10), as modules over B and C. This homomorphism is injective, because of (34.9.3), as discussed near the end of Section 2.7.

One can check that the homomorphism as in (34.9.9) sends a homomorphism from V ⨂_{A} W into Z, as right modules over B ⨂_{k} C, to its composition with the homomorphism as in (34.9.10). It follows that the homomorphism as in (34.9.9) is surjective, because of (34.9.4), and the injectivity of the homomorphism as in (34.9.10).

34.10 Two corollaries about injectivity

Let k, A, B, C, A ⨂_{k} B, and B ⨂_{k} C be as in the previous section. Also let Z be a module over k that is a right module over B and C, where the actions of B and C commute. Thus Z may be considered as a right module over B ⨂_{k} C, as in Section 4.2. If (34.9.4) holds, and

(34.10.1) C is projective as a module over k,

then (34.10.2) Z is injective as a module over B.

This follows from the remarks in the previous section, with A = k, and W = C, considered as a left module over A and a right module over C.

More precisely, this uses the fact that B satisfies the requirements of a tensor product of A and B over k in this case, where multiplication on B is the same as muldiplication on A ⨂_{k} B. This also uses the usual identification of Hom_{C}(C, Z) with Z. Note that Z is the same as Hom_{C}(C, Z) as modules over k, and right modules over B, with respect to this identification. This corresponds to Corollary 2.4a on p166 of [50].

Let A be any associative algebra over k with a multiplicative identity element e_{A} again, and let W, Z be modules over k, with W a left module over A, and Z a right module over V. If (34.9.3) and (34.10.2) hold, then

(34.10.3) Hom_{k}(W, Z) is injective as a right module over A ⨂_{k}B,

as in Corollary 2.5a on p166 of [50]. This follows from the remarks in the previous section, with C = k. In this case, B satisfies the requirements of a tensor product of B and C over k, and multiplication on B is the same as multiplication on B ⨂_{k} C. This means that (34.9.4) redcues to (34.10.2).

34.11 Augmentations and tensor products

Let k be a commutative ring with a multiplicative identity element, and let A, C be associative algebras over k, with multiplicative identity elements e_{A}, e_{C}, respectively. Suppose that A has a left or right augmentation, with augmentation module Q_{A} and augmentation homomorphism ε_{A}. More precisely, we take Q_{A} to be a module over k that is a left or right module over A, as appropriate, and ε_{A} to be a homomorphism from A onto Q_{A}, as modules over k, and left or right modules over A, as appropriate.

Let C ⨂_{k} A be a tensor product of C and A over k, considered as an associative algebra over k, as in Section 4.1. Also let

Q_{C ⨂_{k}A} = C ⨂_{k}Q_{A} (34.11.1)

be a tensor product of C and Q_{A} over k, which may be considered initially as a module over k. Using the action of A on Q_{A}, we get an action of A on C ⨂_{k} Q_{A}, so that C ⨂_{k} Q_{A} may be considered as a module over A on the left or right, as appropriate, as in Section 1.10. Similarly, C ⨂_{k} Q_{A} may be considered as a module over C on the left and right, using the actions of C on itself on the left and right. Note that these actions of C and A on C ⨂_{k} Q_{A} commute with each other.

Using ε_{A} and the identity mapping on C, we get a homomorphism

(34.11.2) ε_{C ⨂_{k}A} from C ⨂_{k}A onto C ⨂_{k}Q_{A},

as modules over k. This may be considered as a homomorphism as in (34.11.2), as left and right modules over C, and left or right modules over A, as appropriate. We may consider (34.11.1) as a left or right module over C ⨂_{k} A, as appropriate, as in Section 4.2. It is easy to see that (34.11.2) is a homomorphism between left or right modules over C ⨂_{k} A, as appropriate, where C ⨂_{k} A is considered as a left and right module over itself in the usual way. Thus

(34.11.1) and (34.11.2) define a left or right

(34.11.3)

augmentation on C ⨂_{k}A, as appropriate,

as on p163 of [50].

Observe that ϕ(a) = e_{C} ⊗ a (34.11.4)

defines a homomorphism ϕ from A into C ⨂_{k} A, as algebras over k. Similarly,

ψ(x) = e_{C} ⊗ x (34.11.5)

defines a homomorphism from Q_{A} into (34.11.1), as modules over k, and left or right modules over A, as appropriate. It is easy to see that

ε_{C ⨂_{k}A} ∘ ϕ = ψ ∘ ε_{A}, (34.11.6)

as on p163f of [50]. In particular, this implies that

ϕ(ker ε_{A}) ⊆ ker ε_{C ⨂_{k}A}. (34.11.7)

This means that ϕ is a homomorphism from A into C ⨂_{k} A as augmented rings, as in Section 33.9.

Remember that C ⨂_{k} A may be considered as a left and right module over A, using ϕ, as in Section 2.9. In this case, the actions of A on C ⨂_{k} A on the left and right using ϕ are the same as the usual actions obtained from the actions of A on itself on the left and right. Similarly, (34.11.1) may be considered as a left or right module over A, as appropriate, using ϕ and the left or right action of C ⨂_{k} A, as appropriate. This action of A is the same as the one defined directly, as before.

Suppose that Q_{A} is a left module over A, and let

34.12. HOMOMORPHISMS INDUCED BY ϕ 769

(ϕ)Q_{A} = (C ⨂_{k}A) ⨂_{A}Q_{A} (34.11.8) be a tensor product of C ⨂_{k} A and Q_{A}, as modules over A. This may be considered as a left module over C ⨂_{k} A, which is the covariant ϕ-extension of Q_{A}, as in Section 2.9. There is a natural isomorphism between (34.11.8) and

C ⨂_{k}(A ⨂_{A}Q_{A}), (34.11.9)

as in Section 1.12. This reduces to (34.11.1), because Q_{A} satisfies the requirements of A ⨂_{A} Q_{A}. This corresponds to the condition (i) on p164 of [50].

Similarly, let X_{A} be any left module over A, and let

(ϕ)X_{A} = (C ⨂_{k}A) ⨂_{A}X_{A} (34.11.10)

be a tensor product of C ⨂_{k} A and X_{A}, as modules over A. This may be considered as a left module over C ⨂_{k} A, which is the covariant ϕ-extension of X_{A}. There is a natural isomorphism between this and

C ⨂_{k}(A ⨂_{A}X_{A}), (34.11.11)

as before. This reduces to C ⨂_{k} X_{A}, because X_{A} satisfies the requirements of A ⨂_{A} X_{A}. Of course, there are analogous statements for right augmentations.

34.12 Homomorphisms induced by ϕ

Let us continue with the same notations and hypotheses as in the previous section, including that Q_{A} be a left module over A. Let V be a module over k that is a right module over C ⨂_{k} A. Equivalently, this means that V is a right module over A and C, where the actions of A and C on V commute with each other. This action of A on V is the same as the one obtained from the action of C ⨂_{k} A on V using ϕ, as in Section 2.9. Using ϕ, we get a homomorphism

(34.12.1) from the homology of A into the homology of C ⨂_{k}A,

as left augmented rings, with coefficients in V,

as in Section 33.9, and on p164 of [50].

Similarly, let Z be a module over k that is a left module over C ⨂_{k} A. This means that Z is a left module over A and C, where the actions of A and C on Z commute with each other, as usual. The action of A on Z is the same as the one obtained from the action of C ⨂_{k} A on Z using ϕ, as before. Using ϕ, we get a homomorphism

(34.12.2) from the cohomology of C ⨂_{k}A into the cohomology of A,

as left augmented rings, with coefficients in Z,

as in Section 33.9 again. This is mentioned on p164 of [50] too.

One may be interested in having the homomorphisms as in (34.12.1) and (34.12.2) be isomorphisms, as in Proposition 1.1 on p164 of [50]. This holds under two additional conditions, as in Section 33.10. These conditions are (33.10.5) and (33.10.8), in the earlier notation, which correspond to (i) and (ii) on p164 of [50]. Note that the augmented ring B in Section 33.10 corresponds to C ⨂_{k} A here. The first condition holds automatically here, as in [50], because of the isomorphism between (34.11.8) and (34.11.1) discussed in the previous section.

If n is a nonnegative integer, then consider

(34.12.3) the nth homology group of A, as a left augmented ring

with coefficients in C ⨂_{k}A, as a right module over A,

as in Section 33.3. The second condition (33.10.8) mentioned in the preceding paragraph is that (34.12.4) (34.12.3) is equal to {0}

when n > 0.

Let X_{A} be a projective resolution of Q_{A}, as a left module over A, as in Section 10.2. A tensor product

(C ⨂_{k}A) ⨂_{A}X_{A} (34.12.5)

of C ⨂_{k} A, as a right module over A, and X_{A} over A, may be considered as a complex over k, as in Section 7.5. Remember that (34.12.3) is given by

H((C ⨂_{k}A) ⨂_{A}X_{A})_{n} (34.12.6)

for each n ≥ 0, as in Section 33.3. Thus (34.12.4) is the same as saying that

(34.12.7) (34.12.6) is equal to {0}

when n > 0.

As in the previous section, (34.12.5) is isomorphic to a tensor product C ⨂_{k} X_{A}, as a module over k. We may consider C ⨂_{k} X_{A} as a complex over k, as before, and the isomorphism just mentioned is in fact an isomorphism between complexes. Thus (34.12.6) is isomorphic to

H(C ⨂_{k}X_{A})_{n} (34.12.8)

for each n ≥ 0. This means that (34.12.4) and (34.12.7) are the same as saying that (34.12.9) (34.12.8) is equal to {0}

when n > 0.

If (34.12.9) holds for every n > 0, then the homomorphisms as in (34.12.1) and (34.12.2) are isomorphisms, as before. In this case, (34.12.5) is a projective resolution of (34.11.1), as a left module over C ⨂_{k} A, as in Section 33.10. This means that

(34.12.10) C ⨂_{k}X_{A} is a projective resolution of (34.11.1),

as a left module over C ⨂_{k} A. More precisely, this uses the fact that (34.12.5) is isomorphic to C ⨂_{k} X_{A} as left modules over A and C, and thus over C ⨂_{k} A. This corresponds to another part of Proposition 1.1 on p164 of [50].

Suppose now that

(34.12.11) A is projective as a module over k.

This implies that projective modules over A are projective as modules over k, as in Section 2.12. Alternatively, a module over A may be identified with its tensor product with A, over A. This permits one to get the projectivity over k from projectivity of tensor products, as in Section 34.1.

If (34.12.11) holds, then it follows that

(34.12.12) X_{A} is a projective resolution of Q_{A}, as a module over k,

as mentioned on p164 of [50]. This implies that (34.12.8) does not depend on the chioce of X_{A}, up to isomorphism, as in Section 10.12. In particular, this means that the condition (34.12.9) does not depend on the choice of X_{A}.

More precisely, let Y_{A} be any projective resolution of Q_{A}, as a module over k. A tensor product C ⨂_{k} Y_{A} of C and Y_{A} over k may be considered as a complex over k, as before, so that

H(C ⨂_{k}Y_{A})_{n} (34.12.13)

is defined for each n. If (34.12.11) holds, and thus (34.12.12) holds, then

(34.12.8) is isomorphic to (34.12.13)

(34.12.14)

for each n, as in Section 10.12. This means that (34.12.9) is the same as saying that (34.12.15) (34.12.13) is equal to {0}

when n > 0.

Suppose now that

(34.12.16) Q_{A} is projective as a module over k.

This permits us to take (Y_{A})_{0} = Q_{A}, (Y_{A})_{j} = {0} when j ≠ 0, as in Section 10.2. This means that

(C ⨂_{k}Y_{A})_{0} = C ⨂_{k}Q_{A} and (C ⨂_{k}Y_{A})_{j} = {0} when j ≠ 0. (34.12.17)

In particular, (34.12.15) holds for every n > 0 under these conditions.

34.13 Semigroups and tensor products

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let Σ be a semigroup, with the semigroup operation expressed multiplicatively, and with an identity element e_{Σ}. Thus we get semigroup algebras k(Σ) and A(Σ) of Σ with coefficients in k and A, respectively, as in Section 4.9.

There is an obvious mapping

(34.13.1) from A × k(Σ) into A(Σ),

defined by multiplying an element of A by an element of k(Σ) to get an element of A(Σ), using scalar multiplication on A. This mapping is bilinear over k, and it is easy to see that A(Σ) satisfies the requirements of a tensor product

A ⨂_{k}k(Σ) (34.13.2)

of A and k(Σ) over k, as a module over k, with respect to this mapping. One can check that multiplication on (34.13.2), as in Section 4.1, corresponds to multiplication in A(Σ). This corresponds to a remark on p188 of [50].

Now let Σ_{1}, Σ_{2} be semigroups, with the semigroup operations expressed multiplicative, and with identity elements e_{1} = e_{Σ_{1}}, e_{2} = e_{Σ_{2}}, respectively. The product Σ_{1} × Σ_{2} (34.13.3)

of Σ_{1} and Σ_{2} is a semigroup as well, where the semigroup operation is defined coordinatewise, and with identity element (e_{1}, e_{2}). Consider the semigroup algebras k(Σ_{1}), k(Σ_{2}), and k(Σ_{1} × Σ_{2}) of these semigroups, with coefficients in k.

There is an obvious mapping

(34.13.4) from k(Σ_{1}) × k(Σ_{2}) into k(Σ_{1} × Σ_{2}),

which is bilinear over k, and is basically the identity mapping on (34.13.3). One can check that k(Σ_{1} × Σ_{2}) satisfies the requirements of a tensor product

k(Σ_{1}) ⨂_{k}k(Σ_{2}) (34.13.5)

of k(Σ_{1}) and k(Σ_{2}) over k, as a module over k, with respect to this mapping. More precisely, one can verify that multiplication on (34.13.5), as in Section 4.1, corresponds to multiplication in k(Σ_{1} × Σ_{2}).

Let V be a module over k. Suppose that V is a left module over Σ_{1} and Σ_{2}, as in Section 4.8, where the actions of Σ_{1} and Σ_{2} commute with each other. Under these conditions, the actions of Σ_{1} and Σ_{2} on V can be combined to get an action of (34.13.3) on V on the left. Conversely, if V is a left module over (34.13.3), then V may be considered as a left module over Σ_{1} and Σ_{2}, where the actions of Σ_{1} and Σ_{2} on V commute with each other. Of course, this uses the obvious embeddings of Σ_{1} and Σ_{2} into (34.13.3). Similarly, if V is a right module over Σ_{1} and Σ_{2}, where the actions of Σ_{1} and Σ_{2} commute with each other, then V may be considered as a right module over (34.13.3). Conversely, if V is a right module over (34.13.3), then V may be considered as a right module over Σ_{1} and Σ_{2}, where the actions of Σ_{1} and Σ_{2} commute with each other.

Part X

Associative and supplemented algebras A family of augmentations

35.1 The enveloping algebra

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Remember that the corresponding opposite algebra A^{op} is the same as A as a module over k, but with multiplication in the opposite order, as in Section 1.11. If x ∈ A, then we may use x^{op} to indicate that x is being considered as an element of A^{op}, as before.

Let A^{en} = A ⨂_{k}A^{op} (35.1.1)

be a tensor product of A and A^{op}, as modules over k. We can define multiplication on A^{en}, in such a way that

(a_{1} ⊗ a_{2}^{op}) (b_{1} ⊗ b_{2}^{op}) = (a_{1} b_{1}) ⊗ (a_{2}^{op} b_{2}^{op}) = (a_{1} b_{1}) ⊗ ((b_{2} a_{2})^{op}) (35.1.2)

for every a_{1}, b_{1} ∈ A and a_{2}^{op}, b_{2}^{op} ∈ A^{op}, as in Section 4.1. This makes A^{en} an associative algebra, with multilicative identity element e_{A} ⊗ e_{A}^{op}, as before. This is called the enveloping algebra of A, as on p167 of [50].

Let us say that a module V over k is a two-sided module over A if V is both a left and right module over A, and the actions of A on V on the left and right commute with each other. This means that

a_{1} · (v · a_{2}) = (a_{1} · v) · a_{2} (35.1.3)

for every a_{1}, a_{2} ∈ A and v ∈ V. Using the action of A on V on the right, we may consider V as a left module over A^{op}, as in Section 1.11. We may also consider V as a left module over A^{en}, with

(a_{1} ⊗ a_{2}^{op}) · v = a_{1} · (v · a_{2}) = (a_{1} · v) · a_{2} (35.1.4)

for every a_{1}, a_{2} ∈ A and v ∈ V, as in Section 4.2. This corresponds to a remark on p167 of [50].

776 CHAPTER 35. A FAMILY OF AUGMENTATIONS

Similarly, we may consider V as a right module over A^{op}, using the action of A on V on the left. We may consider V as a right module over A^{en} too, with v · (a_{1} ⊗ a_{2}^{op}) = a_{2} · (v · a_{1}) = (a_{2} · v) · a_{1} (35.1.5) for every a_{1}, a_{2} ∈ A and v ∈ V, as in Section 4.2 again. This corresponds to another remark on p167f of [50].

In particular, A may be considered as a two-sided module over itself. Thus we may consider A as a left module over A^{en}, with (a_{1} ⊗ a_{2}^{op}) · a = a_{1} a a_{2} (35.1.6) for every a_{1}, a_{2}, a ∈ A, as on p168 of [50].

Note that (a_{1}, a_{2}^{op}) ↦ a_{1} a_{2} is bilinear over k, as a map from A × A^{op} into A. This implies that there is a unique homomorphism (35.1.7) ρ from A^{en} into A, as modules over k, such that ρ(a_{1} ⊗ a_{2}^{op}) = a_{1} a_{2} (35.1.8) for every a_{1}, a_{2} ∈ A. Equivalently, ρ(x) = x · e_{A} (35.1.9) for every x ∈ A^{en}, as on p168 of [50].

More precisely, ρ is a homomorphism from A^{en} into A, as left modules over A^{en}. It is easy to see that ρ(A^{en}) = A. (35.1.10) This means that (35.1.11) A^{en} is a left augmented ring, as in Section 33.1, with augmentation module A, as a left module over A^{en}, and augmentation homomorphism ρ, as on p168 of [50].

35.2 The augmentation ideal J

Let us continue with the same notation and hypotheses as in the previous section, and let J = ker ρ (35.2.1) be the augmentation ideal of A^{en}, as an augmented ring, as on p168 of [50]. This is a left ideal in A^{en}, as an associative algebra over k.

If a ∈ A, then ρ(a ⊗ e_{A}^{op} − e_{A} ⊗ a^{op}) = a e_{A} − e_{A} a = 0, (35.2.2)

35.2. THE AUGMENTATION IDEAL J 777

so that

a ⊗ e_{A}^{op} − e_{A} ⊗ a^{op} ∈ J. (35.2.3)

Because J is a left module over A^{en}, J may be considered as a two-sided module over A. Proposition 3.1 on p168 of [50] states that

(35.2.4) J is generated by elements as in (35.2.3), with a ∈ A, as a left module over A.

To see this, let a_{1}, …, a_{n} and b_{1}, …, b_{n} be elements of A, with

∑_{l=1}^{n} a_{l} ⊗ b_{l}^{op} ∈ J. (35.2.5)

This means that

0 = ρ( ∑_{l=1}^{n} a_{l} ⊗ b_{l}^{op}) = ∑_{l=1}^{n} a_{l} b_{l}. (35.2.6)

Using this, we get that

(35.2.7)  ∑_{l=1}^{n} a_{l} ⊗ b_{l}^{op} = ∑_{l=1}^{n}(a_{l} ⊗ e_{A}^{op}) (e_{A} ⊗ b_{l}^{op})

= ∑_{l=1}^{n}(a_{l} ⊗ e_{A}^{op}) (e_{A} ⊗ b_{l}^{op} − b_{l} ⊗ e_{A}^{op}).

This shows that ∑_{l=1}^{n} a_{l} ⊗ b_{l}^{op} is in the submodule of J, as a left module over A, generated by elements as in (35.2.3), with a ∈ A.

If a ∈ A, then put

j(a) = a ⊗ e_{A}^{op} − e_{A} ⊗ a^{op}. (35.2.8)

This defines j as a homomorphism from A into J, as modules over k. If a, b ∈ A, then

j(a b) = (a b) ⊗ e_{A}^{op} − e_{A} ⊗ (a b)^{op} = (a b) ⊗ e_{A}^{op} − e_{A} ⊗ b^{op} a^{op}
(35.2.9)  = (a ⊗ e_{A}^{op}) (b ⊗ e_{A}^{op}) − (e_{a} ⊗ b^{op}) (e_{A} ⊗ a^{op})
= (a ⊗ e_{A}^{op}) j(b) + (e_{A} ⊗ b^{op}) j(a).

Equivalently, this means that

j(a b) = a · j(b) + j(a) · b, (35.2.10)

when J is considered as a two-sided module over A, as on p168 of [50].

35.3 Crossed homomorphisms

Let us continue with the same notation and hypotheses as in the previous two sections, and let V be a module over k that is a two-sided module over A. A homomorphism f from A into V, as modules over k, is said to be a crossed homomorphism or derivation if

f(a b) = a · f(b) + f(a) · b (35.3.1)

for every a, b ∈ A, as on p168 of [50]. This implies that

f(e_{A}) = 0, (35.3.2)

as in [50]. Note that the set of all crossed homomorphisms from A into V is a submodule of the space Hom_{k}(A, V) of all homomorphisms from A into V, as modules over k.

Let U be another module over k that is a two-sided module over A. A homomorphism h from U into V, as modules over k, is considered to be a homomorphism from U into V as two-sided modules over A, if h is a homomorphism from U into V, as both left and right modules over A. This is the same as saying that h is a homomorphism from U into V, as left modules over A^{en}. The space of these homomorphisms may be denoted Hom_{A^{en}}(U, V), as usual.

If g is a crossed homomorphism from A into U, and h is a homomorphism from U into V, as modules over k and two-sided modules over A, then it is easy to see that (35.3.3) h ∘ g is a crossed homomorphism from A into V.

Remember that J may be considered as a two-sided module over A, as in the previous section. The mapping j from A into J defined in (35.2.8) is a crossed homomorphism, as in (35.2.10). If h is a homomorphism from J into V, as modules over k and two-sided modules over A, then h ∘ j is a crossed homomorphism from A into V. This defines a homomorphism

(35.3.4) from Hom_{A^{en}}(J, V) into the space of crossed homomorphisms from A into V,

as modules over k.

Proposition 3.2 on p168 of [50] states that this homomorphism is an isomorphism. The injectivity of this homomorphism can be obtained from (35.2.4).

Let f be a crossed homomorphism from A into V. We would like to find a homomorphism h from J into V, as modules over k and two-sided modules over A, such that f = h ∘ j. (35.3.5)

Observe that (a, b^{op}) ↦ −a · f(b) (35.3.6) defines a mapping from A × A^{op} into V that is bilinear over k. This leads to a unique homomorphism

(35.3.7) from A^{en} = A ⨂_{k}A^{op} into V,

as modules over k, such that

a ⊗ b^{op} ↦ −a · f(b) (35.3.8)

for every a, b ∈ A. Let h be the restriction of the homomorphism as in (35.3.7) to J, which is a homomorphism from J into V, as modules over k. If a ∈ A, then

h(j(a)) = h(a ⊗ e_{A}^{op} − e_{A} ⊗ a^{op}) = −a · f(e_{A}) + e_{A} · f(a) = f(a), (35.3.9)

because of (35.3.2). This definition of h was motivated by (35.2.7), as mentioned on p168 of [50].

It is easy to see that the homomorphism as in (35.3.7) is a homomorphism

(35.3.10) from A^{en} into V, as left modules over A.

In particular, h is a homomorphism from J into V, as left modules over A.

We would like to show that h is also a homomorphism from J into V, as right modules over A. Let x ∈ A^{en} be given, so that x can be expressed as

x = ∑_{l=1}^{n} a_{l} ⊗ b_{l}^{op}, (35.3.11)

where a_{1}, …, a_{n} and b_{1}, …, b_{n} are elements of A. If c ∈ A, then

x · c = (e_{A} ⊗ c^{op}) x = (e_{A} ⊗ c^{op}) ∑_{l=1}^{n} a_{l} ⊗ b_{l}^{op}

(35.3.12)  = ∑_{l=1}^{n} a_{l} ⊗ (c^{op} b_{l}^{op}) = ∑_{l=1}^{n} a_{l} ⊗ (b_{l} c)^{op}.

Thus

(35.3.13)  h(x · c) = h((e_{A} ⊗ c^{op}) x) = − ∑_{l=1}^{n} a_{l} · f(b_{l} c)

= − ∑_{l=1}^{n} a_{l} · (b_{l} · f(c)) − ∑_{l=1}^{n} a_{l} · (f(b_{l}) · c)

= − ∑_{l=1}^{n}(a_{l} b_{l}) · f(c) − ∑_{l=1}^{n}(a_{l} · f(b_{l})) · c

= −( ∑_{l=1}^{n} a_{l} b_{l}) · f(c) + h(x) · c.

If x ∈ J, then

0 = ρ(x) = ∑_{l=1}^{n} a_{l} b_{l}, (35.3.14)

and we get that h(x · c) = h(x) · c, (35.3.15)

as desired.

35.4 Principal crossed homomorphisms

Let us continue with the same notation and hypotheses as in the previous three sections. If v ∈ V, then x ↦ x · v (35.4.1)

defines a homomorphism from A^{en} into V, as modules over k, and left modules over A^{en}. This is the unique homomorphism from A^{en} into V, as modules over k, such that a ⊗ b^{op} ↦ (a ⊗ b^{op}) · v = a · (v · b) (35.4.2)

for every a, b ∈ A. Let h_{v} be the restriction of this homomorphism to J, which is a homomorphism from J into V, as modules over k, and left modules over A^{en}. Note that v ↦ h_{v} (35.4.3)

is linear over k, as a map from V into the space Hom_{A^{en}}(J, V) of homomorphisms from A^{en} into V, as left modules over A^{en}.

As in the previous section,

f_{v} = h_{v} ∘ j (35.4.4)

is a crossed homomorphism from A into V. If a ∈ A, then

(35.4.5) f_{v}(a) = h_{v}(j(a)) = h_{v}(a ⊗ e_{A}^{op} − e_{A} ⊗ a^{op})
= a · (v · e_{A}) − e_{A} · (v · a) = a · v − v · a.

A crossed homomorphism of this form is said to be principal, or an inner derivation, as on p169 of [50]. Of course,

v ↦ f_{v} (35.4.6)

defines a homomorphism from V into the space of crossed homomorphisms from A into V, as modules over k. This homomorphism maps V onto the set of principal crossed homomorphisms from A into V, which is a submodule of the space of crossed homomorphisms from A into V, as a module over k.

Remember that e_{A} ⊗e_{A}^{op} is the multiplicative identity element of A^{en}. Every homomorphism from A^{en} into V, as left modules over A^{en}, is of the form (35.4.1) for a unique v ∈ V, which is the image of e_{A} ⊗ e_{A}^{op} under the homomorphism. It 35.5. HOMOLOGY AND COHOMOLOGY OF A 781

follows that a crossed homomorphism f from A into V is principal exactly when it can be expressed as h ∘ j, where h is the restriction to J of a homomorphism from A^{en} into V, as left modules over A^{en}.

The space Hom_{A^{en}}(A^{en}, V) of all homomorphisms from A^{en} into V, as left modules over A^{en}, is isomorphic to V, as modules over k, in a natural way, as in the preceding paragraph. There is a natural homomorphism

(35.4.7) from Hom_{A^{en}}(A^{en}, V) into Hom_{A^{en}}(J, V),

as modules over k, which sends a homomorphism from A^{en} into V, as left modules over A^{en}, to its restriction to J. Remember that

h ↦ h ∘ j (35.4.8)

defines an isomorphism from Hom_{A^{en}}(J, V) onto the space of crossed homomorphisms from A into V, as modules over k, as in the previous section. Using this isomorphism, the space of principal crossed homomorphisms from A into V corresponds to the image of Hom_{A^{en}}(A^{en}, V) under the homomorphism as in (35.4.7), as on p169 of [50].

35.5 Homology and cohomology of A

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k, with a multiplicative identity element e_{A}. Using A, we get the enveloping algebra A^{en} = A ⨂_{k} A^{op} of A, as in Section 35.1. Let ρ be the homomorphism from A^{en} onto A, as left modules over A^{en}, defined in Section 35.1. This makes A^{en} a left augmented ring, with augmentation ideal J = ker ρ, as before.

Let V be a two-sided module over A. Remember that we may consider V as a left or right module over A^{en}, as in Section 35.1.

Let us consider V as a right module over A^{en} for the moment. If n is a nonnegative integer, then the nth homology group

H_{n}(A, V) (35.5.1)

of A with coefficients in V may be defined as in Section 33.3, with respect to A^{en} as a left augmented ring. More precisely, this is a module over k for each n.

Let us now consider V as a left module over A^{en}. If n is a nonnegative integer, then the nth cohomology group

H^{n}(A, V) (35.5.2)

of A with coefficients in V may be defined as in Section 33.5, with respect to A^{en} as a left augmented ring. This is also a module over k for each n.

This follows the definitions of homology and cohomology of associative algebras on p169 of [50]. If k is a field, then these cohomology groups agree with those of Hochschild, as in [50].

If we consider V as a right module over A^{en}, then

H_{0}(A, V) = V ⨂_{A^{en}}A, (35.5.3)

where A is considered as a left module over A^{en}, as in Section 33.3. If V is considered as a left module over A^{en}, then

H^{0}(A, V) = Hom_{A^{en}}(A, V), (35.5.4)

where A is considered as a left module over A^{en}, as in Section 33.5. These modules will be discussed further in the next section.

Let X be a projective resolution of A as a left module over A^{en}, as in Section 10.2. If we consider V as a right module over A^{en} again, then

V ⨂_{A^{en}}X (35.5.5)

may be considered as a complex over k, as in Sections 7.5 and 33.3. The homology of this complex can be used to obtain (35.5.1), as in Section 33.3. Namely,

H_{n}(A, V) = H(V ⨂_{A^{en}}X)_{n} = H(V ⨂_{A^{en}}X)^{−n} (35.5.6)

for each n ≥ 0, as on p169 of [50].

Similarly, if we consider V as a left module over A^{en}, then

Hom_{A^{en}}^{gr}(X, V) (35.5.7)

may be defined as a complex over k, as in Sections 8.4 and 33.5. The homology of this complex can be used to obtain (35.5.2), as in Section 33.5. Namely,

H^{n}(A, V) = H(Hom_{A^{en}}^{gr}(X, V))^{n} (35.5.8)

for each n ≥ 0, as on p169 of [50].

35.6 Some basic cases

Let us continue with the same notation and hypotheses as in the previous section. We would like to consider some basic cases of the homology and cohomology groups.

Remember that J is a left ideal in A^{en}, and thus a left module over A^{en} in particular. Let us consider V as a right module over A^{en} again, and let V ⨂_{A^{en}} J be a tensor product of V and J over A^{en}. Using the mapping

(v, x) ↦ v · x (35.6.1)

from V × J into V, we get a unique homomorphism

(35.6.2) from V ⨂_{A^{en}}J into V, 35.6. SOME BASIC CASES 783 as modules over k, with v ⊗ x ↦ v · x (35.6.3) for every v ∈ V and x ∈ J, as usual. Let V ·J be the image of the homomorphism as in (35.6.2), which is a submodule of V, as a module over k. Equivalently, V · J consists of finite sums of elements of the form v · x, with v ∈ V and x ∈ J.

If a ∈ A, then j(a) = a ⊗ e_{A}^{op} − e_{A} ⊗ a^{op} ∈ J, and v · j(a) = v · a − a · v (35.6.4) is an element of V · J for every v ∈ V. This uses the actions of A on V on the left and right on the right side. In fact, (35.6.5) every element of V · J can be expressed as a finite sum

of elements of the form v · a − a · v, with a ∈ A, v ∈ V,

because of (35.2.4). This is mentioned just after (3a) on p170 of [50].

Because V · J is a submodule of V, as a module over k, the quotient V/(V · J) (35.6.6) can be defined as a module over k. We also have that (35.5.3) is isomorphic to (35.6.6), as a module over k, in a natural way, as in Sections 33.1 and 33.2. This corresponds to (2) and the first part of Proposition 4.1 on p170 of [50].

Remember that homomorphisms from A into V, as left modules over A^{en}, correspond to homomorphisms from A^{en} into V, as left modules over A^{en}, whose kernels contain J. If v ∈ V, then x ↦ x·v is a homomorphism from A^{en} into V, as left modules over A^{en}, and every such homomorphism is of this form. Thus (35.5.4) is isomorphic to (35.6.7) {v ∈ V : x · v = 0 for every x ∈ J}, as modules over k, in a natural way. This corresponds to (2a) on p170 of [50].

It is easy to see that (35.6.7) is equal to (35.6.8) {v ∈ V : j(a) · v = 0 for every a ∈ A}, using (35.2.4). If a ∈ A and v ∈ V, then j(a) · v = (a ⊗ e_{A}^{op} − e_{A} ⊗ a^{op}) · v = a · v − v · a, (35.6.9) using the actions of A on V on the left and right on the right side. This means that (35.6.7) is equal to (35.6.10) {v ∈ V : a · v = v · a for every a ∈ A}, as in the second part of Proposition 4.1 on p170 of [50]. The elements of (35.6.10) are said to be invariant elements of V, as in [50].

Consider the natural homomorphism

(35.6.11) from Hom_{A^{en}}(A^{en}, V) into Hom_{A^{en}}(J, V),

as modules over k, that sends a homomorphism from A^{en} into V, as left modules over A^{en}, to its restriction to J. We have that

(35.6.12) H^{1}(A, V) is isomorphic to the cokernel of the homomorphism as in (35.6.11),

as modules over k. This corresponds to (3a) on p170 of [50], and the analogous statement for arbitrary augmented rings was mentioned in Section 33.5.

We have seen that h ↦ h ∘ j defines an isomorphism from Hom_{A^{en}}(J, V) onto the space of crossed homomorphisms from A into V, as modules over k, as in Section 35.3. This isomorphism maps the image of the homomorphism as in (35.6.11) onto the space of principal crossed homomorphisms from A into V, as in Section 35.4. It follows that this isomorphism induces an isomorphism from the cokernel of the homomorphism as in (35.6.11) onto

(35.6.13) the quotient of the space of all crossed homomorphisms from A into V by the submodule consisting of all principal crossed homomorphisms from A into V,

as modules over k. This means that

(35.6.14) H^{1}(A, V) is isomorphic to (35.6.13),

as modules over k, as in Proposition 4.1 on p170 of [50].

35.7 A right augmentation on A^{en}

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. This leads to the enveloping algebra A^{en} of A, as in Section 35.1.

Of course, A may be considered as a two-sided module over itself, and thus a left or right module over A^{en}, as before. If we consider A as a right module over A^{en}, then we get that

a · (a_{1} ⊗ a_{2}^{op}) = a_{2} a a_{1} (35.7.1)

for every a, a_{1}, a_{2} ∈ A, as in Section 35.1.

Observe that (a_{,}a_{2}^{op}) ↦ a_{2} a_{1} is bilinear over k, as a map from A × A^{op} into A. This leads to a unique homomorphism

(35.7.2) ρ′ from A^{en} into A,

as modules over k, such that

ρ′(a_{1} ⊗ a_{2}^{op}) = a_{2} a_{1} (35.7.3)

35.8. SOME MODULES S_{N}(A) 785

for every a_{1}, a_{2} ∈ A. Equivalently,

ρ′(x) = e_{A} · x (35.7.4)

for every x ∈ A^{en}. This defines a homomorphism from A^{en} into A, as right modules over A^{en}. In fact, ρ′(A^{en}) = A, (35.7.5)

so that (35.7.6) A^{en} is a right augmented ring,

as in Section 33.2, with augmentation module A, as a right module over A^{en}, and augmentation homomorphism ρ′.

This augmentation is discussed in Remark 1 on p171 of [50]. It is mentioned there that one gets the same homology and cohomology groups using this right augmentation as for the left augmentation discussed earlier. More precisely, the descriptions of H_{0}(A, V) and H^{0}(A, V) are the same for both augmentations, and the other homology and cohomology groups are satellites of these.

Note that ρ′ = ρ (35.7.7)

when A is commutative, as on p171 of [50].

35.8 Some modules S_{n}(A)

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Thus we get the corresponding enveloping algebra A^{en}, as in Section 35.1. We would like to define a left complex S(A) over A as a left module over A^{en}, as on p174 of [50]. We begin with some modules S_{n}(A) that will be used to get S(A).

If n ≥ −1 is an integer, then we take S_{n}(A) to be a tensor product of n + 2 copies of A over k. This means that

S_{−1}(A) = A, (35.8.1)

and we can take S_{n+1}(A) to be a tensor product A ⨂_{k} S_{n}(A) of A and S_{n}(A), as modules over k, for each n ≥ −1. If n ≥ 1, then the (n+2)-fold tensor products can be arranged into products of pairs in other ways, up to isomorphism, as in Section 1.12. In particular, if a_{0}, a_{1}, …, a_{n}, a_{n+1} ∈ A, then

a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ a_{n+1} (35.8.2)

defines an element of S_{n}(A), which is linear over k in each variable. Note that every element of S_{n}(A) can be expressed as a finite sum of elements of this form.

We may consider S_{n}(A) as a left module over A for each n ≥ −1, using the action of A on itself on the left on the first factor of A in S_{n}(A), as in Section 1.10. If a, a_{0}, a_{1}, …, a_{n}, a_{n+1} ∈ A, then

a · (a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ a_{n+1}) = (a a_{0}) ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ a_{n+1}. (35.8.3)

786 CHAPTER 35. A FAMILY OF AUGMENTATIONS Similarly, we may consider S_{n}(A) as a right module over A, using the action of A on itself on the right on the last factor of A in S_{n}(A). If a_{0}, a_{1}, …, a_{n}, a_{n+1}, b are elements of A, then (a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ a_{n+1}) · b = a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ (a_{n+1} b). (35.8.4)

It is easy to see that the actions of A on S_{n}(A) on the left and on the right commute with each other, so that S_{n}(A) may be considered as a two-sided module over A. Equivalently, S_{n}(A) may be considered as a left module over A^{en}. If a, b, a_{0}, a_{1}, …, a_{n}, a_{n+1} ∈ A, then (a ⊗ b^{op}) · (a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ a_{n+1}) (35.8.5)

= (a a_{0}) ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ (a_{n+1} b),

as on p174 of [50]. Of course, this should be interpreted a bit carefully when n = −1.

Consider the homomorphism (35.8.6) s_{n} from S_{n}(A) into S_{n+1}(A), as modules over k, defined by s_{n}(x) = e_{A} ⊗ x (35.8.7) for each x ∈ S_{n}(A). More precisely, this is a homomorphism as in (35.8.6), as right modules over A, as on p174 of [50].

Of course, (a, x) ↦ a · x defines a mapping from A × S_{n}(A) into S_{n}(A) that is bilinear over k. This leads to a unique homomorphism (35.8.8) t_{n} from S_{n+1}(A) into S_{n}(A), as modules over k, such that t_{n}(a ⊗ x) = a · x (35.8.9) for every a ∈ A and x ∈ S_{n}(A).

Thus t_{n}(s_{n}(x)) = x (35.8.10) for every x ∈ S_{n}(A). In particular, this means that (35.8.11) s_{n} is injective on S_{n}(A), as on p174 of [50]. Note that (35.8.12) S_{n+1}(A) is generated by s_{n}(S_{n}(A)), as a left module over A.

35.9. SOME HOMOMORPHISMS D_{N} 787

35.9 Some homomorphisms d_{n}

We continue with the same notation and hypotheses as in the previous section. If n is a nonnegative integer, then we would like to define a homomorphism

(35.9.1) d_{n} from S_{n}(A) into S_{n−1}(A),

as modules over k, and left modules over A, with the following two properties, as on p174 of [50]. First, d_{0}(a_{0} ⊗ a_{1}) = a_{0} a_{1} (35.9.2)

for every a_{0}, a_{1} ∈ A. Second,

d_{n+1}(s_{n}(x)) + s_{n−1}(d_{n}(x)) = x (35.9.3)

for every x ∈ S_{n}(A) and n ≥ 0.

Of course, (a_{0}, a_{1}) ↦ a_{0} a_{1} is bilinear over k as a mapping from A×A into A, so that there is a unique homomorphism d_{0} from S_{0}(A) into S_{−1}(A), as modules over k, that satisfies (35.9.2). It is easy to see that d_{0} is a homomorphism from S_{0}(A) into S_{−1}(A), as left and right modules over A, as usual.

One can check that such a family of homomorphisms is unique, as on p174 of [50]. More precisely, the restriction of d_{n+1} to s_{n}(S_{n}(A)) is uniquely determined by d_{n}, because of (35.9.3). We also have that d_{n+1} is uniquely determined on S_{n+1}(A) by its restriction to s_{n}(S_{n}(A)), because of (35.8.12), and because d_{n+1} is supposed to be a homomorphism from S_{n+1}(A) into S_{n}(A), as left modules over A.

In fact, we can take

d_{n}(a_{0} ⊗ ⋯ ⊗ a_{n+1}) = ∑_{l=0}^{n}(−1)^{l} a_{0} ⊗ ⋯ ⊗ (a_{l} a_{l+1}) ⊗ ⋯ ⊗ a_{n+1} (35.9.4)

for every n ≥ 0 and a_{0}, …, a_{n+1} ∈ A, as in (3) on p174 of [50]. More precisely, the right side is clearly linear over k in each of the a_{i}’s, i = 0, …, n + 1, and one can use this to get a unique homomorphism d_{n} from S_{n}(A) into S_{n−1}(A), as modules over k, that satisfies (35.9.4). Note that (35.9.4) reduces to (35.9.2) when n = 0. One can verify directly that (35.9.4) satisfies (35.9.3) for every n ≥ 0.

It is easy to see from (35.9.4) that d_{n} is a homomorphism from S_{n}(A) into S_{n−1}(A), as left modules over A. Similarly, d_{n} is a homomorphism from S_{n}(A) into S_{n−1}(A), as right modules over A, as on p174 of [50]. Thus d_{n} may be considered as a homomorphism from S_{n}(A) into S_{n−1}(A), as left modules over A^{en}.

We would like to check that

d_{n−1} ∘ d_{n} = 0 (35.9.5)

for every n ≥ 1, as on p174 of [50]. If n = 1, then this can be verified using associativity of multiplication on A. Suppose that (35.9.5) holds for some n ≥ 1, and let us show that the analogous statement holds for n + 1. Using (35.9.3) and its analogue for n − 1, we get that

d_{n} ∘ d_{n+1} ∘ s_{n} = d_{n} − d_{n} ∘ s_{n−1} ∘ d_{n} = s_{n−2} ∘ d_{n−1} ∘ d_{n} = 0. (35.9.6)

This implies that d_{n} ∘ d_{n+1} = 0, because of (35.8.12), as desired.

Of course, A^{en} = A ⨂_{k} A^{op} is a left module over itself. Let us identify

S_{0}(A) = A ⨂_{k}A (35.9.7)

with A^{en}, as a module over k, in the obvious way. More precisely, we can identify S_{0}(A) with A^{en} as a left module over A^{en}, as on p175 of [50].

Using this identification,

(35.9.8) d_{0} corresponds to the augmentation homomorphism ρ,

as a homomorphism from A^{en} into A, as left modules over A^{en}, as in Section 35.1. This is also mentioned on p175 of [50].

35.10 The standard complex S(A)

We continue with the same notation and hypotheses as in the previous two sections. We would like to define a left complex S(A) over A, as a left module over A^{en}, as mentioned at the beginning of Section 35.8. We take S(A) to be the direct sum of S_{n}(A), n ≥ 0, as a module over k and a left module over A^{en}, with the grading defined by

(35.10.1)  S(A)_{n} = S(A)^{−n} = S_{n}(A) when n ≥ 0
= {0}  when n < 0.

Note that S(A) is negative as a graded module, as in Section 5.9.

Put

(35.10.2)  d_{S(A),n} = d_{n} when n ≥ 1
= 0 when n ≤ 0,

which is a homomorphism from S(A)_{n} into S(A)_{n−1}, as left modules over A^{en}. Observe that d_{S(A),n−1} ∘ d_{S(A),n} = 0 (35.10.3)

for every n, using (35.9.5) when n ≥ 2. This defines a differentiation operator d_{S(A)} on S(A), which makes S(A) into a complex, as in Section 5.10.

In order for S(A) to be a left complex over A, as a left module over A^{en}, we need to choose an augmentation ε = ε_{S(A)}, as in Section 10.1. This is determined by ε_{0} = ε_{S(A),0}, which is a homomorphism from S(A)_{0} into A, as left modules over A^{en}, as before. Here we take

ε_{0} = d_{0}, (35.10.4)

35.10. THE STANDARD COMPLEX S(A) 789

which is a homomorphism from S(A)_{0} = S_{0}(A) into A = S_{−1}(A), as left modules over A^{en}, as on p175 of [50]. Thus

ε_{0} ∘ d_{S(A),1} = d_{0} ∘ d_{1} = 0, (35.10.5)

by (35.9.5). This shows that S(A) is a left complex over A, with respect to the augmentation ε.

In fact, (35.10.6) S(A) is acyclic as a left complex over A,

as on p175 of [50]. This is equivalent to the exactness of the sequence

⋯ ⟶ S_{n}(A) ⟶^{d_{n}} S_{n−1}(A) ⟶ ⋯ (35.10.7)

⟶ S_{1}(A) ⟶^{d_{1}} S_{0}(A) ⟶^{d_{0}} S_{−1}(A) ⟶ 0,

as in Section 10.1. Of course, this also uses the way that S(A) and ε are defined here. Note that d_{0} maps S_{0}(A) onto S_{−1}(A). The exactness of the rest of this sequence can be obtained from (35.9.3).

Let S̃_{n}(A) be a tensor product of n copies of A over k when n ≥ 1, and put S̃_{0}(A) = k. We may consider S_{n}(A) as a tensor product of the form

S_{n}(A) = A ⨂_{k}S̃_{n}(A) ⨂_{k}A (35.10.8)

for each n ≥ 0, as on p175 of [50]. The actions of A on the left and right on S_{n}(A) are obtained from the actions of A on the left and right on the first and last factors of A on the right side of (35.10.8), respectively, as in Section 35.8.

We can identify (35.10.8) with a tensor product of the form

A ⨂_{k}A ⨂_{k}S̃_{n}(A), (35.10.9)

as a module over k, by interchanging the second and third factors on the right side of (35.10.8). Using this identification, the actions of A on the left and on the right on S_{n}(A) are obtained from the actions of A on the left and on the right on the first and second factors of A in (35.10.9), respectively.

We can identify (35.10.9), and thus S_{n}(A), with

A^{en} ⨂_{k}S̃_{n}(A), (35.10.10)

as a module over k, as on p175 of [50]. We may consider (35.10.10) as a left module over A^{en}, where the action of A^{en} on the left on (35.10.10) is obtained from the action of A^{en} on the left on itself, as the first factor in (35.10.10). This corresponds exactly to the actions of A^{en} on the left on S_{n}(A) or (35.10.9) obtained from the actions of A on the left and right mentioned before.

If (35.10.11) A is projective as a module over k, 790 CHAPTER 35. A FAMILY OF AUGMENTATIONS then it follows that (35.10.12) S̃_{n}(A) is projective as a module over k for every n ≥ 0, as in Section 34.1. In this case, we get that (35.10.13) (35.10.10) is projective as a left module over A^{en}, as in Section 34.1 again. Using the identifications mentioned in the previous paragraphs, we obtain that (35.10.14) S_{n}(A) is projective as a left module over A^{en} for every n ≥ 0, as on p175 of [50].

Combining this with (35.10.6), we get that (35.10.15) S(A) is a projective resolution of A, as a left module over A^{en}, as in Section 10.2. This is called the standard complex of A, as on p175 of [50].

35.11 Using the standard complex

We continue with the same notation and hypotheses as in the previous three sections. Let V be a two-sided module over A. If we consider V as a right module over A^{en}, then a tensor product V ⨂_{A^{en}}S(A) (35.11.1) of V and S(A) over A^{en} may be considered as a complex over k, as in Section 7.5. In particular, this is a graded module over k, with

(V ⨂_{A^{en}}S(A))_{n} = V ⨂_{A^{en}}S(A)_{n} = V ⨂_{A^{en}}S_{n}(A) when n ≥ 0
(35.11.2) = {0} when n < 0.

If A is projective as a module over k, then H_{n}(A, V) = H(V ⨂_{A^{en}}S(A))_{n} (35.11.3) for every n ≥ 0, because of (35.10.15), as in Section 35.5.

Using the identification of S_{n}(A) with (35.10.10), we get that V ⨂_{A^{en}}S_{n}(A) (35.11.4) may be identified with V ⨂_{A^{en}}(A^{en} ⨂_{k}S̃_{n}(A)), (35.11.5) as a module over k, for each n ≥ 0. This is isomorphic to

(V ⨂_{A^{en}}A^{en}) ⨂_{k}S̃_{n}(A) (35.11.6)

in a natural way, as modules over k, as in Section 1.12. Of course, V satisfies the requirements of a tensor product of V with A^{en} over A^{en}, so that (35.11.4) may be identified with V ⨂_{k}S̃_{n}(A), (35.11.7)

as a module over k, for each n ≥ 0. This corresponds to a remark on p175 of [50].

Let v ∈ V and a_{1}, …, a_{n} ∈ A be given, for some n ≥ 1. If we consider (35.11.7) as a tensor product of V and n copies of A over k, then

v ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} (35.11.8)

may be considered as an element of (35.11.7). Using the identification of (35.11.4) and (35.11.7), the differentiation operator on (35.11.1) sends (35.11.8) to

(v · a_{1}) ⊗ a_{2} ⊗ ⋯ ⊗ a_{n} (35.11.9)

+ ∑_{j=1}^{n−1}(−1)^{j} v ⊗ a_{1} ⊗ ⋯ ⊗ (a_{j} a_{j+1}) ⊗ ⋯ ⊗ a_{n}

+(−1)^{n} (a_{n} · v) ⊗ a_{1} ⊗ ⋯ ⊗ a_{n−1},

as on 175 of [50].

If we consider V as a left module over A^{en}, then

Hom_{A^{en}}^{gr}(S(A), V) (35.11.10)

may be defined as a complex over k, as in Section 8.4. Note that

(Hom_{A^{en}}^{gr}(S(A), V))^{n} = Hom_{A^{en}}(S(A)_{n}, V)
(35.11.11)  = Hom_{A^{en}}(S_{n}(A), V) when n ≥ 0
= {0}  when n < 0.

If A is projective as a module over k, then

H^{n}(A, V) = H(Hom_{A^{en}}^{gr}(S(A), V))^{n} (35.11.12)

for every n ≥ 0, because of (35.10.15), as in Section 35.5.

Using the identification of S_{n}(A) with (35.10.10) again, we get that

Hom_{A^{en}}(S_{n}(A), V) (35.11.13)

may be identified with

Hom_{A^{en}}(A^{en} ⨂_{k}S̃_{n}(A), V ) (35.11.14) as a module over k, for each n ≥ 0. This may be identified with

Hom_{k}(S̃_{n}(A), Hom_{A^{en}}(A^{en}, V)), (35.11.15)

as in Section 1.13. Of course, Hom_{A^{en}}(A^{en}, V) may be identified with V, as usual. This means that (35.11.15) may be identified with

Hom_{k}(S̃_{n}(A), V). (35.11.16)

Thus (35.11.13) may be identified with (35.11.16), as on p175 of [50].

The elements of (35.11.16) are called n-dimensional cochains, as on p175 of [50]. They can be identified with functions on the Cartesian product A^{n} of n copies of A with values in V that are linear over k in each variable when n ≥ 1, as in [50]. If n = 0, then (35.11.16) can be identified with V.

Remember that (35.11.10) has a differentiation operator, obtained from the one on S(A). Let f be a function on A^{n} with values in V that is linear over k in each variable for some n ≥ 1, or an element of V when n = 0. Thus f corresponds to an element of (35.11.13), and we let δ(f) be the “coboundary” of f, which is the function on A^{n+1} with values in V that is linear over k in each variable, and which corresponds to the differentiation operator on (35.11.10) acting on f. If a_{1}, …, a_{n+1} ∈ A, then

(δ(f))(a_{1}, …, a_{n+1}) = a_{1} · f(a_{2}, …, a_{n+1})

+ ∑_{j=1}^{n}(−1)^{j} f(a_{1}, …, a_{j} a_{j+1}, …, a_{n+1}) (35.11.17)

+(−1)^{n+1} f(a_{1}, …, a_{n}) · a_{n+1},

as on p175 of [50]. This corresponds to the classical definition of Hochschild, as in [50].

35.12 Some modules N_{n}(A)

Let us continue with the same notation and hypotheses as in the previous four sections. Of course, t ↦ t e_{A} defines a natural homomorphism from k into A. Let A′ be the cokernel of this homomorphism, as a module over k. Thus A′ is given by the quotient A′ = A/{t e_{A} : t ∈ k}, (35.12.1)

as a module over k.

Let Ñ_{n}(A) be a tensor product of n copies of A′ over k when n ≥ 1, and put Ñ_{0}(A) = k. If n is a nonnegative integer, then we take

N_{n}(A) = A^{en} ⨂_{k}Ñ_{n}(A), (35.12.2)

as a module over k, as on p176 of [50]. This may be considered as a left module over A^{en}, where the action of A^{en} on the left is obtained from the action of A^{en} on the left on itself, in the first factor on the right side of (35.12.2).

35.12. SOME MODULES N_{N}(A) 793

Let q be the natural quotient mapping from A onto A′. This leads to a natural homomorphism

(35.12.3) q_{n} from S̃_{n}(A) onto Ñ_{n}(A),

as modules over k, for each n ≥ 0, which is the identity mapping on k when n = 0.

Remember that S_{n}(A) can be identified with (35.10.10) for each n ≥ 0, as a module over k, and as a left module over A^{en}. Using the identity mapping on A^{en} and q_{n}, we get a homomorphism

(35.12.4) from S_{n}(A) onto N_{n}(A),

as modules over k, and as left modules over A^{en}, for every n ≥ 0, as on p176 of [50]. Note that N_{0}(A) is the same as S_{0}(A) with this identification, and that the homomorphism as in (35.12.4) is the identity mapping when n = 0.

As in Section 35.10, we can identify N_{n}(A) with a tensor product of the form

A ⨂_{k}A ⨂_{k}Ñ_{n}(A), (35.12.5)

as a module over k, for each n ≥ 0. This may be considered as a two-sided module over A, where the actions of A on the left and right are obtained from the actions of A on the left and right on the first and second factors of A in (35.12.5), respectively. Using this, (35.12.5) may be considered as a left module over A^{en}, which corresponds exactly to N_{n}(A) as a left module over A^{en}.

Similarly, we can identify (35.12.5) with a tensor product of the form

A ⨂_{k}Ñ_{n}(A) ⨂_{k}A, (35.12.6)

as a module over k, by interchanging the second and third factors in (35.12.5). Of course, the actions of A on the left and right on (35.12.5) mentioned in the preceding paragraph correspond to those obtained from the actions of A on the left and right on the first and last factors of A in (35.12.6).

We may consider a tensor product

A′ ⨂_{k}Ñ_{n}(A) ⨂_{k}A (35.12.7)

as a right module over A, using the action of A obtained from the action of A on the right on the last factor of A. There is a natural homomorphism

(35.12.8) from (35.12.6) onto (35.12.7),

as modules over k, corresponding to q in the first factor, and the identity mappings on Ñ_{n}(A) and A in the other two factors. More precisely, this is a homomorphism between right modules over A.

We may identify (35.12.7) with a tensor product

Ñ_{n+1}(A) ⨂_{k}A, (35.12.9) as modules over k, in an obvious way. As usual, this tensor product may be considered as a right module over A, using the action of A obtained from the action of A on the right on the second factor of A. This corresponds to the action of A on the right on (35.12.6) mentioned before.

We may identify N_{n+1}(A) with a tensor product

A ⨂_{k}Ñ_{n+1}(A) ⨂_{k}A, (35.12.10)

as modules over k, in the same way that N_{n}(A) is identified with (35.12.6). As usual, this may be considered as a two-sided module over A, where the actions of A on the left and right are obtained from the actions of A on the left and right on the first and last factors of A, respectively. Thus (35.12.10) may be considered as a left module over A^{en}, which corresponds exactly to N_{n+1}(A) as a left module over A^{en}.

35.13 Analogues of s_{n}, d_{n}

We continue with the same notation and hypotheses as in the previous sections. Let us identify (35.12.10) with a tensor product of A and (35.12.9) over k. Using this identification, we get a homomorphism

(35.13.1) from (35.12.9) into (35.12.10),

which sends an element x of (35.12.9) to e_{A} ⊗ x, as an element of (35.12.10). This is a homomorphism as in (35.13.1), as modules over k, and right modules over A.

We can compose the homomorphisms as in (35.12.8) and (35.13.1) to get a homomorphism from (35.12.6) into (35.12.10), as modules over k, and right modules over A. This uses the identification of (35.12.7) with (35.12.9) mentioned earlier. This homomorphism may be considered as a homomorphism

(35.13.2) from N_{n}(A) into N_{n+1}(A),

as modules over k, and right modules over A, using identifications mentioned before.

The homomorphism as in (35.13.2) is analogous to the homomorphism s_{n} from S_{n}(A) into S_{n+1}(A) defined in Section 35.8, as on p176 of [50]. More precisely, this homomorphism corresponds exactly to s_{n} with respect to the homomorphism as in (35.12.4), and the analogous homomorphism from S_{n+1}(A) onto N_{n+1}(A).

Suppose that a_{0}, a_{1}, …, a_{n+1} ∈ A for some n ≥ 1, so that

d_{n}(a_{0} ⊗ ⋯ ⊗ a_{n+1}) ∈ S_{n−1}(A) (35.13.3)

can be expressed as in (35.9.4). Consider

(35.13.4) the image of d_{n}(a_{0} ⊗ ⋯ ⊗ a_{n+1}) in N_{n−1}(A), 35.14. THE NORMALIZED STANDARD COMPLEX N(A) 795

under the homomorphism from S_{n−1}(A) onto N_{n−1}(A) as in (35.12.4). If

(35.13.5) a_{r} = t_{r} e_{A} for some r = 1, …, n and t_{r} ∈ k,

then one can check that

(35.13.6) (35.13.4) is equal to 0.

More precisely, the terms on the right side of (35.9.4) corresponding to l ≠ r−1, r are mapped to 0 in N_{n−1}(A), because q(a_{r}) = 0, by hypothesis. One can verify that the terms corresponding to l = r − 1, r on the right side of (35.9.4) cancel in this case.

This shows that (35.13.4) depends only on a_{0}, q(a_{1}), …, q(a_{n}), a_{n+1}. Thus we get a mapping

(35.13.7) from A × A′ × ⋯ × A′ × A into N_{n−1}(A),

with n factors of A′ in the domain, that is linear over k in each variable. This leads to a homomorphism

(35.13.8) from (35.12.6) into N_{n−1}(A),

as modules over k, in the usual way. This may be considered as a homomorphism

(35.13.9) from N_{n}(A) into N_{n−1}(A),

using the identification of N_{n}(A) with (35.12.6) mentioned earlier. It is easy to see that this is a homomorphism between left and right modules over A, or equivalently left modules over A^{en}.

This homomorphism is the analogue of d_{n} in Section 35.9 for N_{n}(A), as on p176 of [50]. One can check that this homomorphism corresponds exactly to d_{n} with respect to the homomorphism as in (35.12.4), and the analgous homomorphism from S_{n−1}(A) onto N_{n−1}(A). We can include n = 0 here by taking N_{−1}(A) = S_{−1}(A) = A, (35.13.10)

and using d_{0} for the homomorphism as in (35.13.9).

Observe that the analogue of (35.9.5) holds on N_{n}(A) for every n ≥ 1, for the homomorphisms as in (35.13.9) in place of the d_{n}’s. Similarly, the analogue of (35.9.3) holds on N_{n}(A) for every n ≥ 0, for the homomorphisms as in (35.13.2) and (35.13.9) in place of the s_{n}’s and d_{n}’s, as on p176 of [50]. More precisely, this uses s_{−1} from Section 35.8, considered as a mapping from (35.13.10) into N_{0}(A).

35.14 The normalized standard complex N(A)

Let us continue with the same notation and hypotheses as in the last few sections. Let N(A) be the direct sum of N_{n}(A) over n ≥ 0, as a module over k and a left module over A^{en}, as on p176 of [50]. This is a graded module over A^{en}, with grading defined by

(35.14.1)  N(A)_{n} = N(A)^{−n} = N_{n}(A) when n ≥ 0
= {0}  when n < 0.

Put

d_{N(A),n} = the homomorphism as in (35.13.9) when n ≥ 1
(35.14.2)  = 0  when n ≤ 0.

This is a homomorphism from N(A)_{n} into N(A)_{n−1}, as left modules over A^{en}. Note that d_{N(A),n−1} ∘ d_{N(A),n} = 0 (35.14.3) for every n, as mentioned near the end of the previous section. This defines a differentiation operator d_{N(A)} on N(A), which makes N(A) into a negative complex, as in Sections 5.9 and 5.10.

We can make N(A) a left complex over A, as a left module over A^{en}, by choosing a suitable augmentation ε_{N(A)}, as in Section 10.1. This is determined by ε_{N(A),0}, which is a homomorphism

(35.14.4) from N(A)_{0} = N_{0}(A) = S_{0}(A) into A,

as left modules over A^{en}. As in Section 35.10, we take

ε_{N(A),0} = d_{0}. (35.14.5)

One can check that ε_{N(A),0} ∘ d_{N(A),1} = 0, (35.14.6) as in (35.10.5). This implies that N(A) is a left complex over A, with respect to the augmentation ε_{N(A)}, as on p176 of [50].

More precisely,

(35.14.7) N(A) is acyclic as a left complex over A,

as on p176 of [50]. This is analogous to the corresponding statement (35.10.6) for S(A).

Suppose now that

(35.14.8) A′ is projective as a module over k.

This implies that

(35.14.9)  Ñ(A) is projective as a module over k

for every n ≥ 0, as in Section 34.1. Using this, we get that

(35.14.10) N_{n}(A) is projective as a left module over A^{en}

for every n ≥ 0, as in Section 34.1 again. It follows that

(35.14.11) N(A) is a projective resolution of A, as a left module over A^{en},

because of (35.14.7). This is called the normalized standard complex of A, as on p176 of [50].

35.15 Some notation

We continue with the same notation and hypotheses as in the previous sections. Let n be a nonnegative integer. In this section, we shall use the identification of N_{n}(A) with a tensor product of the form (35.12.6), which is more compatible with the initial definition of S_{n}(A) in Section 35.8. More precisely, this is analogous to the identification of S_{n}(A) with a tensor product of the form (35.10.8). Using these identifications, the homomorphism from S_{n}(A) onto N_{n}(A), as modules over k, as in (35.12.4), is obtained using the identity mapping on A in the first and last factors in the tensor products, and the homomorphism q_{n} from S̃_{n}(A) onto Ñ_{n}(A) as in (35.12.3) in the middle factors.

It is a bit simpler to consider S_{n}(A) as a tensor product of n + 2 copies of A over k, as in Section 35.8. Similarly, N_{n}(A) may be considered as a tensor product of n+2 modules over k, where the first module is A, the next n modules are copies of A′ as in (35.12.1), and the last module is another copy of A. Using these identifications, the homomorphism from S_{n}(A) onto N_{n}(A), as modules over k, as in (35.12.4), is obtained using the identity mapping on the first and last copies of A, and the natural quotient mapping q′ from A onto A′ on the n copies of A in the middle.

If a_{0}, a_{1}, …, a_{n}, a_{n+1} ∈ A, then

a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ a_{n+1} (35.15.1)

defines an element of S_{n}(A), as in Section 35.8. Similarly,

a_{0} ⊗ q(a_{1}) ⊗ ⋯ ⊗ q(a_{n}) ⊗ a_{n+1} (35.15.2)

defines an element of N_{n}(A), using the identification mentioned in the preceding paragraph. Of course, the homomorphism from S_{n}(A) onto N_{n}(A) as in (35.12.4) sends (35.15.1) to (35.15.2), with these identifications.

As on p176 of [50], one may use the notation

a_{0} [a_{1}, …, a_{n}] a_{n+1} (35.15.3)

for (35.15.2). If a_{0} = e_{A}, then this may be expressed as

[a_{1}, …, a_{n}] a_{n+1}. (35.15.4)

Similarly, if a_{n+1} = e_{A}, then (35.15.3) may be expressed as

a_{0} [a_{1}, …, a_{n}]. (35.15.5)

If a_{0} = a_{n+1} = e_{A}, then (35.15.3) may be expressed as

[a_{1}, …, a_{n}]. (35.15.6)

If n = 0, then (35.15.3) becomes

a_{0} [ ] a_{1}, (35.15.7) which is the same as a_{0} ⊗ a_{1} as an element of S_{0}(A) = N_{0}(A).

Suppose now that n ≥ 1, and let d_{N(A),n} be as in the previous section. Using the notation in the preceding paragraph, we have that

d_{N(A),n}(a_{0} [a_{1}, …, a_{n}] a_{n+1}) = (a_{0} a_{1}) [a_{2}, …, a_{n}] a_{n+1}

+ ∑_{l=1}^{n−1}(−1)^{l} a_{0} [a_{1}, …, a_{l} a_{l+1}, …, a_{n}] a_{n+1} (35.15.8)

+(−1)^{n} a_{0} [a_{1}, …, a_{n−1}] (a_{n} a_{n+1}).

This is mentioned on p176 of [50], with a_{0} = a_{n+1} = e_{A}. Note that d_{N(A),n} is determined by its values in that case, because d_{N(A),n} is a homomorphism as in (35.13.9), as left and right modules over A.

Of course, if n = 1, then the sum in the middle of the right side of (35.15.8) should be interpreted as being equal to 0. In this case, (35.15.8) reduces to

d_{N(A),1}(a_{0} [a_{1}] a_{2}) = (a_{0} a_{1}) [ ] a_{2} − a_{0} [ ] (a_{1} a_{2}). (35.15.9)

This is mentioned on p176 of [50] too, with a_{0} = a_{2} = e_{A}, as before.

It may be convenient to sometimes use the notation (35.15.3) for (35.15.1), as an element of S_{n}(A), as mentioned on p176 of [50]. It should always be clear from the particular discussion which interpretation is intended.

A family of augmentations, 2

36.1 Homomorphisms between algebras

Let k, k_{1} be commutative rings with multiplicative identity elements 1_{k}, 1_{k_{1}}, respectively. Also let A, B be associative algebras over k, k_{1}, with multiplicative identity elements e_{A}, e_{B}, respectively. Suppose that ψ is a ring homomorphism from k into k_{1}, with ψ(1_{k}) = 1_{k_{1}}.

Let ϕ be a ring homomorphism from A into B, with ϕ(e_{A}) = e_{B}. Suppose that ϕ(t a) = ψ(t) ϕ(a) (36.1.1)

for every a ∈ A and t ∈ k. If k = k_{1} and ψ is the identity mapping on k, then this means that ϕ is a homomorphism from A into B, as algebras over k. Alternatively, one may consider B as an algebra over k, using ψ.

Remember that A^{op}, B^{op} are the opposite algebras corresponding to A, B, respectively, and let

A^{en} = A ⨂_{k}A^{op}, B^{en} = B ⨂_{k_{1}}B^{op} (36.1.2)

be enveloping algebras of A, B, as algebras over k, k_{1}, respectively, as in Section 35.1. Note that B^{en} may be considered as an algebra over k, using ψ.

Of course, ϕ may be considered as a ring homomorphism from A^{op} into B^{op}. Consider the mapping

(a_{1}, a_{2}^{op}) ↦ ϕ(a_{1}) ⊗_{k_{1}} ϕ(a_{2})^{op} (36.1.3)

from A × A^{op} into B^{en}. It is easy to see that this mapping is bilinear over k, when B^{en} is considered as a module over k using ψ. Thus we get a unique homomorphism (36.1.4) ϕ^{en} from A^{en} into B^{en}, as modules over k, such that

ϕ^{en}(a_{1} ⊗_{k} a_{2}^{op}) = ϕ(a_{1}) ⊗_{k_{1}} ϕ(a_{2})^{op} (36.1.5)

for every a_{1}, a_{2} ∈ A. One can check that ϕ^{en} is a ring homomorphism.

Let ρ_{A}, ρ_{B} be the homomorphisms from A^{en}, B^{en} onto A, B, as left modules over A^{en}, B^{en}, respectively, as in Section 35.1. One can verify that

ρ_{B} ∘ ϕ^{en} = ϕ ∘ ρ_{A}, (36.1.6)

as on p171 of [50]. In particular,

ϕ^{en}(ker ρ_{A}) ⊆ ρ_{B}. (36.1.7)

This shows that ϕ^{en} is a homomorphism from A^{en} into B^{en}, as left augmented rings with respect to the augmentations defined by ρ_{A} and ρ_{B}, respectively, as in Section 33.9.

Let V be a module over k_{1} that is a two-sided module over B, as an algebra over k_{1}. Thus V may be considered as a module over k using ψ, and as a two-sided module over A using ϕ, as in Section 2.9. Alternatively, V may be considered as a left or right module over B^{en}, as an algebra over k_{1}, and thus as a left or right module over A^{en}, as an algebra over k, using ϕ^{en}.

Using ϕ^{en}, we get induced homomorphisms

(36.1.8) from H_{n}(A, V) into H_{n}(B, V)

and (36.1.9) from H^{n}(B, V) into H^{n}(A, V)

for each n ≥ 0, as in Section 33.9. These homomorphisms correspond to those in (1), (2) on p171 of [50].

36.2 The case where B = k_{1} ⨂_{k} A

Let us continue with the same notation and hypotheses as in the previous section, with the following additions. In this section, we consider k_{1} to be an algebra over k, with ψ(t) = t · 1_{k_{1}} for every t ∈ k. We also take B to be a tensor product B = k_{1} ⨂_{k}A (36.2.1)

of k_{1} and A over k, which may be considered as an algebra over k and k_{1}. Thus

ϕ(a) = 1_{k_{1}} ⊗ a (36.2.2)

defines a homomorphism from A into B, as algebras over k, where ϕ(e_{A}) = 1_{k_{1}} ⊗ e_{A} is the multiplicative identity element in B.

Consider B^{en} = (k_{1} ⨂_{k}A) ⨂_{k_{1}}(k_{1} ⨂_{k}A)^{op}. (36.2.3)

36.2. THE CASE WHERE B = K_{1} ⨂_{K} A 801

As a module over k, (k_{1} ⨂_{k}A)^{op} is the same as k_{1} ⨂_{k}A^{op}, and in fact they are the same as algebras over k and k_{1}. Thus B^{en} can be expressed as

B^{en} = (k_{1} ⨂_{k}A) ⨂_{k_{1}}(k_{1} ⨂_{k}A^{op}). (36.2.4)

This is isomorphic in a natural way to a tensor product of the form

((k_{1} ⨂_{k}A) ⨂_{k_{1}}k_{1}) ⨂_{k}A^{op}, (36.2.5)

as in Section 1.12. This reduces to

(k_{1} ⨂_{k}A) ⨂_{k}A^{op}, (36.2.6)

because any module over k_{1} satisfies the requirements of its tensor product with k_{1}, over k_{1}. It follows that B^{en} can be identified with a tensor product of the form k_{1} ⨂_{k}A^{en}, (36.2.7)

up to a natural isomorphism, as in Section 1.12 again. This corresponds to some remarks near the top of p172 of [50].

Of course, k_{1} ⨂_{k} A may be considered as a two-sided module over k_{1} and A, because k_{1} and A are two-sided modules over themselves. This corresponds to considering B as a two-sided module over itself, and thus as a left or right module over B^{en}. Alternatively, we may consider k_{!} ⨂_{k} A as a left or right module over k_{1} and A^{en}, where the actions of k_{1} and A^{en} commute with each other. This means that k_{1} ⨂_{k} A may be considered as a left or right module over (36.2.7). This is equivalent to considering k_{1} ⨂_{k} A as a left or right module over B^{en}, respectively, with respect to the identification between B^{en} and (36.2.7) mentioned in the preceding paragraph.

Remember that A^{en}, B^{en} may be considered as left augmented rings, with augmentation modules A, B and augmentation homomorphisms ρ_{A}, ρ_{B}, as in Section 35.1. Alternatively, using the identification of B^{en} with (36.2.7), we can get a left augmentation with augmentation module k_{1} ⨂_{k} A, and augmentation homomorphism obtained from the identity mapping on k_{1} and ρ_{A}, as in Section 34.11. These are equivalent descriptions of the same augmentation, because of the remarks in the preceding paragraph.

Similarly, x ↦ 1_{k_{1}} ⊗ x (36.2.8)

defines a homomorphism from A^{en} into (36.2.7), as algebras over k. This corresponds to the homomorphism ϕ^{en} from A^{en} into B^{en} mentioned in the previous section, with respect to the identification between B^{en} and (36.2.7), because ϕ is defined here as in (36.2.2).

This means that we are in the same type of situation as in Section 34.11, with A, Q_{A}, and ε_{A} taken to be A^{en}, A, and ρ_{A}, respectively, and with C = k_{1}, as mentioned on p172 of [50]. The homomorphism considered in Section 34.11 is given by (36.2.8), which corresponds to ϕ^{en}, as in the preceding paragraph.

Let V be a module over k_{1} that is a two-sided module over B, as an algebra over k_{1}, as in the previous section. Thus V may be considered as a module over k, because k_{1} is an algebra over k, and as a two-sided module over A, as an algebra over k. Let us emphasize that the actions of k_{1} on B on the right and on the left, coming from the actions of B on V on the right and on the left, are supposed to be the same, and the same as the action of k_{1} on B as a module over k_{1}, as on p172 of [50].

We may consider V as a left or right module over B^{en}, as an algebra over k_{1}, and as a left or right module over A^{en}, as an algebra over k, as in the previous section. We may also consider V as a left or right module over (36.2.7), as an algebra over k_{1}. This is equivalent to considering V as a left or right module over B^{en}, with respect to the identification of B^{en} with (36.2.7) mentioned earlier.

Under these conditions, we get induced homomorphisms as in (36.1.8) and (36.1.9) for each n ≥ 0, using ϕ^{en}. These are the same as the homomorphisms from the homology of A^{en} into the homology of B^{en}, as left augmented rings, with coefficients in V, and from the cohomology of B^{en} into the cohomology of A^{en} with coefficients in V, as in Section 33.9.

Using the identification of B^{en} with (36.2.7), we may identify these homomorphisms with the homomorphisms from the homology of A^{en} into the homology of (36.2.7), as left augmented rings, with coefficients in V, and from the cohomology of (36.2.7) into the cohomology of A^{en} with coefficients in V, corresponding to the homomorphism from A^{en} into (36.2.7) defined by (36.2.8). These homomorphisms were discussed in Section 34.12.

Suppose now that

(36.2.9) A is projective as a module over k.

This corresponds to the condition that Q_{A} be projective as a module over k near the end of Section 34.12. This also implies that A^{en} is projective as a module over k, as in Section 34.1. In this case, it follows that the homomorphisms mentioned in the preceding paragraph are isomorphisms, as in Section 34.12.

This means that

the homomorphisms as in (36.1.8) and (36.1.9) are isomorphisms

(36.2.10)

when (36.2.9) holds. If X is a projective resolution of A, as a left module over A^{en}, then we get that

(36.2.11) k_{1} ⨂_{k}X is a projective resolution of B,

as a left module over B^{en},

when (36.2.9) holds, as in Section 34.12 again. This corresponds to Proposition 5.1 on p172 of [50].

36.3 Projectivity of A over A^{en}

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k, with a multiplicative identity element e_{A}.

36.3. PROJECTIVITY OF A OVER A^{EN} 803

Remember that A^{op} is the opposite algebra of A, and let A^{en} = A ⨂_{k} A^{op} be a tensor product of A and A^{op}, as in Section 35.1.

Of course, A may be considered as a two-sided module over itself, and thus as a left module over A^{en}, as before. One may wish to know more about conditions under which (36.3.1) A is projective as a left module over A^{en},

as on p179 of [50].

Let ρ be the homomorphism from A^{en} onto A, as modules over k, and left modules over A^{en}, defined in Section 35.1. If (36.3.1) holds, then

(36.3.2) there is a homomorphism f from A into A^{en}, as left modules over A^{en}, such that ρ ∘ f = I_{A},

where I_{A} is the identity mapping on A.

Conversely, suppose that (36.3.2) holds, so that

(36.3.3) A^{en} corresponds to the direct sum of f(A) and ker ρ, as a left module over A^{en}.

This implies that (36.3.1) holds, because A^{en} is projective as a module over itself, and f is injective, as in Section 2.7.

Let A ⨂_{k} A be a tensor product of A with itself, as a module over k. Of course, this is the same as A^{en}, as a module over k, but it is convenient to refer to it a bit differently. We may consider A ⨂_{k} A as a two-sided module over A, where the action of A on the left is obtained from the action of A on the left on the first factor of A, and the action of A on the right is obtained from the action of A on the right on the second factor of A. This corresponds to A^{en} as a left module over itself.

Using multiplication on A, we get a unique homomorphism ρ̃ from A ⨂_{k} A into A, as modules over k, such that

ρ̃(a_{1} ⊗ a_{2}) = a_{1} a_{2} (36.3.4)

for every a_{1}, a_{2} ∈ A. More precisely, ρ̃ is a homomorphism from A ⨂_{k} A onto A, as left and right modules over A. This corresponds to the homomorphism ρ from A^{en} onto A, as left modules over A^{en}, from Section 35.1.

Thus (36.3.2) is the same as saying that

(36.3.5) there is a homomorphism f̃ from A into A ⨂_{k}A,

as left and right modules over A, such that ρ̃∘ f̃ = I_{A}.

This is equivalent to (36.3.1), as before.

Remember that an element x of A ⨂_{K} A, as a two-sided module over A, is said to be invariant if a · x = x · a (36.3.6) for every a ∈ A, as in Section 35.6. Proposition 7.7 on p179 of [50] states that (36.3.1) holds if and only if

(36.3.7) there is an invariant element x of A ⨂_{k}A such that ρ̃(x) = e_{A}.

If (36.3.1) holds, then (36.3.5) holds, and one can take

x = f̃(e_{A}). (36.3.8)

Conversely, suppose that (36.3.7) holds, and put

f̃(a) = a · x = x · a (36.3.9)

for every a ∈ A. This defines a homomorphism from A into A ⨂_{k} A, as modules over k, and in fact as left and right modules over A. If a ∈ A, then

ρ̃(f̃(a)) = ρ̃(a · x) = a ρ̃(x) = a e_{A} = a. (36.3.10)

This means that (36.3.5) holds, so that (36.3.1) holds.

36.4 Some examples with matrices

Let k be a commutative ring with a multiplicative identity element 1_{k}, and let n be a positive integer. The space M_{n}(k) of n × n matrices with entries in k is an associative algebra over k with respect to matrix multiplication, and with the usual identity matrix I_{n} as the multiplicative identity element.

If 1 ≤ j, l ≤ n, then let a_{j,l} be the element of M_{n}(k) whose (j, l) entry is equal to 1_{k}, and whose other entries are equal to 0. Thus

∑_{j=1}^{n} a_{j,j} = I_{n}. (36.4.1)

Of course, M_{n}(k) is freely generated by a_{j,l}, 1 ≤ j, l ≤ n, as a module over k. If 1 ≤ j, l, m, r ≤ n, then

(36.4.2)  a_{j,l} a_{m,r} = a_{j,r} when l = m
= 0  when l ≠ m.

Let us take A = M_{n}(k) (36.4.3)

in the previous section. We would like to verify that (36.3.7) holds, as in Proposition 7.8 on p179 of [50]. To do this, we take

x = ∑_{j=1}^{n} a_{j,1} ⊗ a_{1,j}. (36.4.4)

Let us first check that x is invariant as an element of A ⨂_{k} A.

If 1 ≤ m, r ≤ n, then

a_{m,r} · x = ∑_{j=1}^{n}(a_{m,r} a_{j,1}) ⊗ a_{1,j} = a_{m,1} ⊗ a_{1,r}. (36.4.5)

Similarly,

x · a_{m,r} = ∑_{j=1}^{n} a_{j,1} ⊗ (a_{1,j} a_{m,r}) = a_{m,1} ⊗ a_{1,r}. (36.4.6)

It follows that x is invariant in A ⨂_{k} A, because M_{n}(k) is generated by the a_{m,r}’s, 1 ≤ m, r ≤ n, as a module over k.

Observe that

ρ̃(x) = ∑_{j=1}^{n} a_{j,1} a_{1,j} = ∑_{j=1}^{n} a_{j,j} = I_{n}. (36.4.7)

Thus (36.3.7) holds, as desired.

36.5 Some remarks about polynomial algebras

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let T_{1}, …, T_{n} be n distinct commuting indeterminates. Thus the space k[T_{1}, …, T_{n}] of formal polynomials in T_{1}, …, T_{n} with coefficients in k may be defined as in Section 4.3, and is a commutative associative algebra over k. Remember that k may be identified with the subalgebra of k[T_{1}, …, T_{n}] consisting of formal polynomials for which the coefficient of T^{α} is 0 when α ≠ 0. The multiplicative identity element 1_{k} in k corresponds to the multiplicative identity element of k[T_{1}, …, T_{n}] in this way.

Let ε_{0} be the mapping from k[T_{1}, …, T_{n}] that sends a formal polynomial f(T) to its constant term f_{0}. This is a homomorphism from k[T_{1}, …, T_{n}] onto k, as algebras over k, with ε_{0}(1_{k}) = 1_{k}.

Let ε be any homomorphism from k[T_{1}, …, T_{n}] into k, as algebras over k, such that ε(1_{k}) = 1_{k}. (36.5.1)

Note that ε is uniquely determined by its vales at T_{1}, …, T_{n}, which may be arbitrary elements of k.

It is easy to see that there is a unique homomorphism ϕ from k[T_{1}, …, T_{n}] into itself, as an algebra over k, such that ϕ(1_{k}) = 1_{k} and

ϕ(T_{j}) = T_{j} − ε(T_{j}) (36.5.2)

for each j = 1, …, n. More precisely, ϕ is an automorphism of k[T_{1}, …, T_{n}], whose inverse is the analogous homomorphism that sends T_{j} to T_{j} + ε(T_{j}) for each j = 1, …, n.

By construction, ε(ϕ(T_{j})) = ε(T_{j}) − ε(T_{j}) = 0 (36.5.3) for j = 1, …, n. This implies that

ε ∘ ϕ = ε_{0} (36.5.4)

on k[T_{1}, …, T_{n}]. This corresponds to some remarks on p180 of [50].

36.6 More on polynomial algebras

Let k be a commutative ring with a multiplicative identity element, let n be a positive integer, and let X_{1}, …, X_{n} and Y_{1}, …, Y_{n} be distinct commuting indeterminates. Consider A = k[X_{1}, …, X_{n}], (36.6.1)

which is a commutative associative algebra over k, as before. Of course, A is isomorphic to k[Y_{1}, …, Y_{n}] in an obvious way.

Because A is commutative, A^{op} = A, and we may take

A^{en} = A ⨂_{k}A^{op} = A ⨂_{k}A (36.6.2)

to be a tensor product of A with itself over k, as in Section 35.1. This is isomorphic to a tensor product

(k[X_{1}, …, X_{n}]) ⨂_{k}(k[Y_{1}, …, Y_{n}]) (36.6.3)

of k[X_{1}, …, X_{n}] and k[Y_{1}, …, Y_{n}] over k, using the isomorphism mentioned in the preceding paragraph.

Consider the algebra

k[X_{1}, …, X_{n}, Y_{1}, …, Y_{n}] (36.6.4)

of formal polynomials in X_{1}, …, X_{n}, Y_{1}, …, Y_{n} with coefficients in k. There is an obvious mapping

(36.6.5) from (k[X_{1}, …, X_{n}]) × (k[Y_{1}, …, Y_{n}]) into (36.6.4)

defined by multiplication, and which is bilinear over k. Using this mapping, (36.6.4) satisfies the requirements of a tensor product as in (36.6.3). More precisely, multiplication in (36.6.4) corresponds to multiplication in (36.6.3), as a tensor product of associative algebras, as in Section 4.1.

Let ρ be the mapping from A^{en} onto A defined in Section 35.1. This is an algebra homomorphism in this case, because A is commutative. If A^{en} is identified with (36.6.4) as before, then ρ corresponds to the mapping

(36.6.6) η from (36.6.4) into k[X_{1}, …, X_{n}]

in which Y_{j} is replaced with X_{j} for each j = 1, …, n, as on p180 of [50].

The space A[Y_{1}, …, Y_{n}] = (k[X_{1}, …, X_{n}])[Y_{1}, …, Y_{n}] (36.6.7) of formal polynomials in Y_{1}, …, Y_{n} with coefficients in (36.6.1) may be defined as in Section 4.3. This is a commutative associative algebra over A, as well as over k. This can be identified with (36.6.4), as an algebra over k, in a reasonable way. With this identification, η corresponds to the homomorphism

from (36.6.7) into (36.6.1)

(36.6.8)

in which an element of (36.6.7) is evaluated at (X_{1}, …, X_{n}), as an element of A^{n}.

We also have a homomorphism

(36.6.9) η_{0} from (36.6.7) into (36.6.1)

which sends an element of (36.6.7) to its constant term. Observe that there is an automorphism Φ on (36.6.7), (36.6.10)

as an algebra over A, such that

η ∘ Φ = η_{0}, (36.6.11)

as in the previous section. Here η is considered as a homomorphism as in (36.6.8).

This tells us a lot about A^{en} as a left augmented ring with respect to ρ, as on p180 of [50].

36.7 Some remarks about free algebras

Let k be a commutative ring with a multiplicative identity element, and let

E = {x_{1}, …, x_{m}} (36.7.1)

be a finite set with m distinct elements x_{1}, …, x_{m} for some positive integer m. Using E, we can get the free semigroup Σ(E) generated by E, as in Section 4.10. This leads to the corresponding semigroup algebra k(Σ(E)), as in Sections 4.9 and 4.10, which is an associative algebra over k. This is the same as the free algebra F_{k}(x_{1}, …, x_{m}) (36.7.2)

over k generated by x_{1}, …, x_{m}, as on p146, 148 of [50].

Let V be a module over k that is a two-sided module over (36.7.2). Suppose that ϕ is a crossed homomorphism from (36.7.2) into V, as in Section 35.3. It is easy to see that

(36.7.3) ϕ is uniquely determined by its values at x_{1}, …, x_{m}.

One can check that

(36.7.4) the values of ϕ at x_{1}, …, x_{m} may be arbitrary elements of V.

This corresponds to the first part of Exercise 2 on p181 of [50].

Let A be (36.7.2), and let A^{en} be a corresponding enveloping algebra, as in Section 35.1. Also let ρ be the homomorphism from A^{en} onto A, as left modules over A^{en}, defined previously, and put J = ker ρ, as in Section 35.2. We may consider J as a two-sided module over A, because it is a left ideal in A^{en}, and thus a left module over A^{en}. Remember that j(a) = a ⊗ e_{A}^{op} − e_{A} ⊗ a^{op} defines a crossed homomorphism from A into J, as a two-sided module over A. Here e_{A} is the multiplicative identity element in A, which corresponds to the identity element in Σ(E).

If h is a homomorphism from J into V, as two-sided modules over A, then h∘j is a crossed homomorphism from A into V, as in Section 35.3. We have also seen that h is uniquely determined by h ∘ j, because J is generated by j(A) as a left module over A, as in Section 35.2. It follows that h is uniquely determined by its values at the elements

j(x_{l}) = x_{l} ⊗ e_{A}^{op} − e_{A} ⊗ x_{l} (36.7.5)

of J, with l = 1, …, m, because of (36.7.3).

Remember that every crossed homomorphism from A into V can be expressed as h ∘ j for some homomorphism h from J into V, as two-sided modules over A, as in Section 35.3. This and (36.7.4) imply that the values of such a homomorphism h at the elements (36.7.5) of J may be arbitrary elements of V.

The second part of Exercise 2 on p181 of [50] states that J is freely generated, as a left module over A^{en}, by the elements of the form (36.7.5). To see that J is generated by these elements, as a left module over A^{en}, one can consider the quotient of J by the submodule, as a left module over A^{en}, generated by these elements. The corresponding quotient mapping is equal to 0 at these elements, by construction, and is thus equal to 0, because it is uniquely determined by its values at these elements, as before. Alternatively, one can check that j(A) is contained in this submodule, because j is a crossed homomorphism from A into J.

Of course, A^{en} is a left module over itself, and one can use this to consider A^{en} as a two-sided module over A. The values of a homomorphism h from J into A^{en}, as two-sided modules over A, or equivalently as left modules over A^{en}, at the elements (36.7.5) of J may be arbitrary elements of A^{en}, as before. In particular, for each r = 1, …, m, there is such a homomorphism h_{r} that is equal to e_{A} ⊗ e_{A}^{op} at (36.7.5) when l = r, and equal to 0 otherwise. This implies that J is freely generated by these elements, as a left module over A^{en}.

36.8 More on semigroup algebras

Let k be a ring with a multiplicative identity element, and let Σ be a semigroup, with the semigroup operation expressed multiplicatively, and with an identity element e_{Σ}. Also let k(Σ) be the corresponding semigroup algebra of Σ with coefficients in k, as in Section 4.9, and let V be a module over k.

Remember that an action of Σ on V on the left or right with suitable properties makes V into a left or right module over Σ, as in Section 4.8. This corresponds to V being a left or right module over k(Σ), as appropriate, as in Section 4.9.

Let us say that V is a two-sided module over Σ if V is both a left and right module over Σ, and that the actions of Σ on V on the left and on the right commute with each other. This corresponds exactly to V being a two-sided module over k(Σ).

Let Σ^{op} be the opposite semigroup of Σ, as in Section 4.8. Also let A be an asociative algebra over k with a multiplicative identity element e_{A}, and let A^{op} be the corresponding opposite algebra. It is easy to seee that the opposite algebra (A(Σ))^{op} (36.8.1)

of A(Σ) can be identified with the semigroup algebra

A^{op}(Σ^{op}) (36.8.2)

of Σ^{op} with coefficients in A^{op}. In particular, the opposite algebra (k(Σ))^{op} of k(Σ) can be identified with the semigroup algebra

k(Σ^{op}) (36.8.3)

of Σ^{op} with coefficients in k, because k is commutative.

If V is a two-sided module over Σ, then V may be considered as a left module over Σ and Σ^{op}, where the actions of Σ and Σ^{op} on V commute with each other. These actions can be combined to get an action of the product semigroup

Σ × Σ^{op} (36.8.4)

on V on the left, as in Section 34.13. Similarly, V may be considered as a right module over Σ and Σ^{op}, where these actions on V commute with each other. These actions can be combined, to get an action of (36.8.4) on V on the right. Conversely, if V is a left or right module over (36.8.4), then we get commuting actions of Σ and Σ^{op} on the left or right, as appropriate, and V may be considered as a two-sided module over Σ.

Remember that an enveloping algebra k(Σ)^{en} of k(Σ) is obtained from a tensor product of k(Σ) and k(Σ)^{op} over k, as in Section 35.1. This may be identified with a tensor product of k(Σ) and k(Σ^{op}), as before. The semigroup algebra k(Σ × Σ^{op}) (36.8.5)

of (36.8.4) with coefficients in k satisfies the requirements of such a tensor product, as in Section 34.13.

Let V be a two-sided module over Σ, and thus over k(Σ). Let us say that v ∈ V is invariant as an element of V, as a two-sided module over Σ, if

x · v = v · x (36.8.6) for every x ∈ Σ. This is equivalent to the condition that v be invariant as an element of V, as a two-sided module over k(Σ), as in Section 35.6.

Note that a homomorphism f from k(Σ) into any module V over k, as modules over k, is uniquely determined by its restriction to Σ, and that any V -valued function on Σ occurs in this way. If V is a two-sided module over Σ, and thus over k(Σ), then f is a crossed homomorphism, as in Section 35.3, if and only if f(x y) = x · f(y) + f(x) · y (36.8.7)

for every x, y ∈ Σ.

36.9 Some remarks about tensor products

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra with a multiplicative identity element e_{A}, and let V be a left or right module over A. Also let A ⨂_{k} V be a tensor product of A and V over k, and consider the mapping σ from V into A ⨂_{k} V defined by

σ(v) = e_{A} ⊗ v. (36.9.1)

Of course, σ is a homomorphism from V into A ⨂_{k} V, as modules over k.

Using the action of A on V on the left or right, as appropriate, we get a homomorphism τ from A ⨂_{k} V into V, as modules over k, such that

τ(a ⊗ v) = a · v (36.9.2)

for all a ∈ A and v ∈ V when V is a left module over A, and

τ(a ⊗ v) = v · a (36.9.3)

when V is a right module over A. In both cases, we get that

τ(σ(v)) = v (36.9.4)

for every v ∈ V. In particular, this implies that

(36.9.5) σ is injective on V.

This is analogous to an argument in Section 35.8.

Consider the mapping ρ from k into A defined by

ρ(t) = t e_{A} (36.9.6)

for every t ∈ k. This is a homomorphism from k into A, as modules over k, and in fact as associative algebras over k. One can use ρ and the identity mapping on V to get a homomorphism from k ⨂_{k} V into A ⨂_{k} V, as modules over k, as in Section 1.3. Remember that we can take k ⨂_{k} V to be V, as in Section 1.4. Using this, the mapping from k ⨂_{k} V into A ⨂_{k} V just mentioned corresponds to σ.

36.10. MORE ON THE ANALOGUE OF S_{N} 811

Let A′ = A/ρ(k) (36.9.7)

be the cokernel of ρ, as a module over k, and let q be the natural quotient mapping from A onto A′. Also let A′ ⨂_{k} V be a tensor product of A′ and V over k. Using q and the identity mapping on V, we get a homomorphism

(36.9.8) from A ⨂_{k}V onto A′ ⨂_{k}V,

as modules over k. Note that the composition of σ with this homomorphism is equal to 0.

This leads to an exact sequence

V ⟶^{σ} A ⨂_{k}V ⟶ A′ ⨂_{k}V ⟶ 0, (36.9.9)

as in Section 2.5. This means that the kernel of the homomorphism as in (36.9.8) is equal to σ(V). (36.9.10)

We may consider A′ ⨂_{k} V as a left or right module over A, as appropriate, using the action of A on V, as in Section 1.10. Let A ⨂_{k} A′ ⨂_{k} V be a tensor product of A and A′ ⨂_{k} V over k. This leads to a homomorphism σ′

(36.9.11) from A′ ⨂_{k}V into A ⨂_{k}A′ ⨂_{k}V,

as modules over k, as before. Note that

(36.9.12) σ′ is injective on A′ ⨂_{k}V,

as before.

We can compose the mapping as in (36.9.8) with σ′ to get a homomorphism

(36.9.13) from A ⨂_{k}V into A ⨂_{k}A′ ⨂_{k}V,

as modules over k. The kernel of this homomorphism is equal to (36.9.10), because of (36.9.12), and the fact that the kernel of the homomorphism as in (36.9.8) is equal to (36.9.10).

36.10 More on the analogue of s_{n}

Let k be a commutative ring with a multiplicative identity element, let A be an associative algebra over k with a multiplicative identity element e_{A}, and let A^{en} be the corresponding enveloping algebra, as in Section 35.1. We would like to consider some additional properties of the analogue of the mapping s_{n} for the normalized standard complex N(A) associated to A, as in Section 35.13. This corresponds to the first part of Exercise 3 on p181 of [50].

As in Section 35.12, we take A′ = A/{t e_{A} : t ∈ k} to be the cokernel of the natural homomorphism t ↦ t e_{A} from k into A, as a module over k. We also take Ñ_{n}(A) to be a tensor product of n copies of A′ over k for each positive integer n, and Ñ_{0}(A) = k. Instead of the initial definition of N_{n}(A) in Section 35.12, we use an identification of it with a tensor product of the form

A ⨂_{k}Ñ_{n}(A) ⨂_{k}A, (36.10.1)

as a module over k, as before. We may identify this with a tensor product of the form A ⨂_{k} A, as a module over k, when n = 0. We take N_{−1}(A) = S_{−1}(A) = A, as in Section 35.13.

The analogue of s_{n} for N(A) is a homomorphism from N_{n}(A) into N_{n+1}(A), as modules over k, and in fact as right modules over A, as in Section 35.13, and will be denoted s′_{n}. Using the notation as in Section 35.15, we have that

(36.10.2)  s′_{n}(a_{0}[a_{1}, …, a_{n}] a_{n+1}) = e_{A} [a_{0}, a_{1}, …, a_{n}] a_{n+1}
= [a_{0}, a_{1}, …, a_{n}] a_{n+1}

for n ≥ 0 and a_{0}, a_{1}, …, a_{n}, a_{n+1} ∈ A, as mentioned in Exercise 3 on p181 of [50]. If n = 0, then this means that

s′_{0}(a_{0} ⊗ a_{1}) = e_{A} [a_{0}] a_{1} = [a_{0}] a_{1} (36.10.3)

for a_{0}, a_{1} ∈ A. We take s′_{−1} to be the same as s_{−1}, as in Section 35.8, and as mentioned at the end of Section 35.13, so that

s′_{−1}(x) = s_{−1}(x) = e_{A} ⊗ x = e_{A} [ ] x (36.10.4)

for each x ∈ A.

It is easy to see that s′_{n} ∘ s′_{n−1} = 0 (36.10.5)

for each n ≥ 0. This uses the fact that a_{0} [a_{1}, …, a_{r}] a_{r+1} = 0 when a_{l} = e_{A} for some l = 1, …, r, by the definition of N_{r}(A), and in particular when l = 1. Equivalently, this means that

s′_{n−1}(N_{n−1}(A)) ⊆ ker s′_{n} (36.10.6)

for each n ≥ 0.

In fact,

0 ⟶ N_{−1}(A) ⟶^{s′_{−1}} N_{0}(A) ⟶^{s′_{0}} ⋯ (36.10.7)

⟶ N_{n}(A) ⟶^{s′_{n}} N_{n+1}(A) ⟶ ⋯

is exact, as in Exercise 3 on p181 of [50]. The injectivity of s′_{−1} = s_{−1} was mentioned in Section 35.8, and also follows from (36.9.5). Thus we would like to check that s′_{n−1}(N_{n−1}(A)) = ker s′_{n} (36.10.8) for each n ≥ 0.

Let q be the natural quotient mapping from A onto A′, as in Section 35.12. If n ≥ 0, then s′_{n} was obtained by composing two mappings, as in Section 35.13. The first mapping corresponds to the homomorphism from (36.10.1) onto

A′ ⨂_{k}Ñ_{n}(A) ⨂_{k}A, (36.10.9)

as modules over A, obtained using q in the first factor, and the identity mappings on Ñ_{n}(A) and A in the other two factors. We may identify (35.12.7) with a tensor product Ñ_{n+1}(A) ⨂_{k}A, (36.10.10)

as before. The second mapping corresponds to the homomorphism from this tensor product into A ⨂_{k}Ñ_{n+1}(A) ⨂_{k}A, (36.10.11)

as modules over k, that sends an element x of (36.10.10) to e_{A} ⊗x, as an element of (36.10.11).

The homomorphism from (36.10.10) into (36.10.11) mentioned in the preceding paragraph is injective, as in (36.9.5). This uses the fact that (36.10.10) may be considered as a module over A, as in Section 1.10. This implies that

ker s_{N_{n}(A),n} (36.10.12)

is the same as the kernel of the first homomorphism

(36.10.13) from (36.10.1) onto (36.10.9).

The surjectivity of this homomorphism implies that

s′_{n}(N_{n}(A)) (36.10.14)

is the same as the image of (36.10.10) under the second homomorphism

(36.10.15) from (36.10.10) into (36.10.11).

If n ≥ 1, then we get that

s′_{n−1}(N_{n−1}(A)) (36.10.16)

is the same as the image of Ñ_{n}(A) ⨂_{k}A (36.10.17)

into (36.10.1) under the analogous homomorphism x ↦ e_{A} ⊗ x. Using these descriptions of (36.10.12) and (36.10.16), (36.10.8) follows from the fact that the kernel of the homomorphism as in (36.9.8) is equal to (36.9.10). Essentially the same argument works when n = 0 too.

36.11 More on the analogue of d_{n}

Let us continue with the same notation and hypotheses as in the previous section. The analogue of d_{n} for the normalized standard complex N(A) associated to A is a homomorphism from N_{n}(A) into N_{n−1}(A), as left and right modules over A, for each n ≥ 0, as in Section 35.13, and will be denoted d′_{n}. If n ≥ 1, then d′_{n−1} ∘ d′_{n} = 0 (36.11.1)

on N_{n}(A), as mentioned in Section 35.13. We also have that

d′_{n+1}(s′_{n}(x)) + s′_{n−1}(d′_{n}(x)) = x (36.11.2)

for every x ∈ N_{n}(A) and n ≥ 0, as before.

If x ∈ ker s′_{n}, then (36.11.2) implies that

s′_{n−1}(d′_{n}(x)) = x. (36.11.3)

In particular, (36.11.4) d′_{n} is injective on ker s′_{n}.

Using (36.11.1) and (36.11.2), we get that

d′_{n}(s′_{n−1}(d′_{n}(x))) = d′_{n}(x) (36.11.5)

for every x ∈ N_{n}(A) and n ≥ 0. This implies that

d′_{n}(s′_{n−1}(N_{n−1}(A))) = d′_{n}(N_{n}(A)). (36.11.6)

It follows that d′_{n}(ker s′_{n}) = d′_{n}(N_{n}(A)), (36.11.7)

because of (36.10.6). Thus the restriction of d′_{n} to ker s′_{n} is a one-to-one mapping onto d′_{n}(N_{n}(A)), as in Exercise 3 on p181 of [50].

Note that one can use (36.10.6), (36.11.4), and (36.11.6) to obtain (36.10.8). Remember that N_{0}(A) is the same as S_{0}(A), with a suitable identification,

as in Section 35.12. Similarly, d′_{0} is defined to be the analogous mapping d_{0} from N_{0}(A) = S_{0}(A) into N_{−1}(A) = S_{−1}(A) = A defined in Section 35.9, as in Section 35.13. Some of the earlier properties of d′_{n} are more direct when n = 0, which is presumably why one takes n > 0 in Exercise 3 on p181 of [50].

36.12 Adjoining an identity element

Let k be a commutative ring with a multiplicative identity element, and let A be an algebra over k in the strict sense, as in Section 20.13. Put A^{+} = k × A, which is a module over k with respect to coordinatewise addition and scalar multiplication. If t_{1}, t_{2} ∈ k and a_{1}, a_{2} ∈ A, then put

(t_{1}, a_{1}) (t_{2}, a_{2}) = (t_{1} t_{2}, t_{1} a_{2} + t_{2} a_{1} + a_{1} a_{2}), (36.12.1) as in Exercise 4 on p181 of [50]. This makes A^{+} an algebra over k in the strict sense too. Of course, a ↦ (0, a) is an injective algebra homomorphism from A into A^{+}, by construction.

It is easy to see that e_{A^{+}} = (1, 0) (36.12.2)

is the multiplicative identity element of A^{+}. If A is commutative or associative, then A^{+} has the same property. If A is projective as a module over k, then

(36.12.3) A^{+} is projective as a module over k.

Note that the opposite algebra A^{op} associated to A may be defined as an algebra in the strict sense over k as in Section 1.11. The opposite algebra (A^{+})^{op} associated to A^{+} corresponds to (A^{op})^{+} in a natural way.

Suppose from now on in this section that A is an associative algebra over k with a multiplicative identity element e_{A}. Let V be a module over k. If V is a left module over A, then V may be considered as a left module over A^{+}, with

(t, a) · v = t v + a · v (36.12.4)

for every t ∈ k, a ∈ A, and v ∈ V. Similarly, if V is a right module over A, then V may be considered as a right module over A^{+}, with

v · (t, a) = t v + v · a. (36.12.5)

If V is a two-sided module over A, as in Section 35.1, then V may be considered as a two-sided module over A^{+}, as mentioned in [50].

As usual, t ↦ t e_{A^{+}} defines a natural homomorphism from k into A^{+}. Let (A^{+})′ be the cokernel of this homomorphism, as a module over k, as in Section 35.12. This means that (A^{+})′ is given by the quotient

(A^{+})′ = A^{+}/{t e_{A^{+}} : t ∈ k}, (36.12.6)

as a module over k, as before. In this case, (A^{+})′ may be identified with A, as a module over k, in an obvious way, using the mapping

(t, a) ↦ a (36.12.7)

from A^{+} onto A.

Let S̃_{n}(A) be a tensor product of n copies of A over k when n ≥ 1, and put S̃_{0}(A) = k, as in Section 35.10. Similarly, we would like to take Ñ_{n}(A) to be a tensor product of n copies of (A^{+})′ over k when n ≥ 1, and Ñ_{0}(A^{+}) = k, as in Section 35.12. More precisely, we may as well take

Ñ_{n}(A^{+}) = S̃_{n}(A), (36.12.8)

using the identification mentioned in the preceding paragraph.

Let A^{en}, (A^{+})^{en} be as in Section 35.1. We can take

S_{n}(A) = A^{en} ⨂_{k}S̃_{n}(A) (36.12.9) for each n ≥ 0, using the identifications mentioned in Section 35.10. Similarly, we take N_{n}(A^{+}) = (A^{+})^{en} ⨂_{k}Ñ_{n}(A^{+}) = (A^{+})^{en} ⨂_{k}S̃_{n}(A) (36.12.10)

for each n ≥ 0, as in Section 35.12, using (36.12.8) in the second step.

Let V be a two-sided module over A, so that V may be considered as a right module over A^{en}, as in Section 35.1. Remember that

V ⨂_{A^{en}}S_{n}(A) = V ⨂_{k}S̃_{n}(A), (36.12.11)

as modules over k, with suitable identifications, as in Section 35.11. Similarly, V may be considered as a two-sided module over A^{+}, and thus a right module over (A^{+})^{en}. As before,

V ⨂_{(A^{+})^{en}}N_{n}(A^{+}) = V ⨂_{k}Ñ_{n}(A^{+}) = V ⨂_{k}S̃_{n}(A), (36.12.12)

as modules over k, with suitable identifications. This is related to Exercise 4 on p181 of [50].

We may also consider V as a left module over A^{en}, as in Section 35.1. Using this, we have that

Hom_{A^{en}}(S_{n}(A), V) = Hom_{k}(S̃_{n}(A), V), (36.12.13)

as modules over k, with suitable identifications, as in Section 35.11. Similarly, V may be considered as a left module over (A^{+})^{en}, and

(36.12.14) Hom_{(A^{+})^{en}}(N_{n}(A^{+}), V) = Hom_{k}(Ñ_{n}(A^{+}), V) = Hom_{k}(S̃_{n}(A), V),

as modules over k, with suitable identifications. This is related to Exercise 4 on p181 of [50] as well.

Supplemented algebras

37.1 Basic notions

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Of course, k may be considered as an algebra over itself.

Suppose that ε is a homomorphism from A into k, as algebras over k, with ε(e_{A}) = 1. Under these conditions, A together with ε is called a supplemented algebra, as on p182 of [50].

Note that ε(t e_{A}) = t ε(e_{A}) = t (37.1.1)

for every t ∈ k. In particular, ε(A) = k, so that A may be considered as a left and right augmented ring, with augmentation homomorphism ε, as in Sections 33.1 and 33.2.

Using (37.1.1), we get that t ↦ t e_{A} is a one-to-one mapping from k into A. Let I be the kernel of ε, which is a two-sided ideal in A. Observe that A corresponds to the direct sum of

{t e_{A} : t ∈ k} (37.1.2)

and I, as a module over k, as on p182 of [50].

The augmentation module of A as a left or right augmented ring is k, considered as a left or right module over A, as appropriate, using ε. If V is any module over k, then V may be considered as a left module _{ε}V over A, with

a · v = ε(a) v (37.1.3)

for every a ∈ A and v ∈ V, as on p183 of [50]. Similarly, V may be considered as a right module V_{ε} over A, with

v · a = ε(a) v (37.1.4) for every a ∈ A and v ∈ V. In both cases, we say that the corresponding actions or operators of A on V are trivial, as on p183 of [50]. Thus the actions of A on k mentioned earlier are trivial in this sense.

If V is already a right module over A, then _{ε}V is a two-sided module over A, as in Section 35.1. Similarly, if V is already a left module over A, then V_{ε} is a two-sided module over A. More precisely, the actions of A on V should be linear over k in both cases, and thus commute with the trivial actions of A on V. It follows that _{ε}V or V_{ε} may be considered as a module over an enveloping algebra A^{en} of A, as appropriate, as in Section 35.1. This corresponds to some remarks on p183 of [50].

One can use this to reinterpret previous definitions for two-sided modules in the case of one-sided modules, as on p183 of [50]. Some instances of this will be discussed in the next sections.

37.2 Some related modules

Let us continue with the same notation and hypotheses as in the previous section. Let W be a module over k that is a left module over A, so that W_{ε} is a two-sided module over A. Let us say that w ∈ W is invariant if w is invariant as an element of W_{ε}, as a two-sided module over A, as in Section 35.6. This means that for every a ∈ A,

a · w = w · a = ε(a) w, (37.2.1)

as before.

Let W^{A} be the collection of invariant elements of W. It is easy to see that this is a submodule of W, as a module over k, and in fact as a module over A, as on p183 of [50]. More precisely, A acts trivially on W^{A}, in the sense described in the previous section, by construction. This is the largest submodule of W on which A acts trivially.

Observe that I = {a − ε(a) e_{A} : a ∈ A}. (37.2.2)

One can use this to check that

(37.2.3) W^{A} = {w ∈ W : a · w = 0 for every a ∈ I},

as on p183 of [50].

Let Hom_{A}(k, W) (37.2.4)

be the space of homomorphisms from k into W, as modules over k, and left modules over A. These homomorphisms correspond to homomorphisms from A into W, as modules over k, and left modules over A, that are equal to 0 on I. Of course, homomorphisms from A into W, as left modules over A, correspond to elements of W in a natural way. It follows that (37.2.4) corresponds to W^{A} in a natural way, because of (37.2.3), as on p183 of [50]. This could also be considered as a version of a remark in Section 33.1, as in [50].

37.3. CROSSED HOMOMORPHISMS INTO W_{ε} 819

Let H^{0}(A, W_{ε}) be as in Section 35.5. This is isomorphic to W^{A}, as a module over k, as in Section 35.6. This corresponds to (1) on p183 of [50].

Now let V be a module over k that is a right module over A. Consider the subset V · I of V consisting of finite sums of elements of the form v · a, where v ∈ V and a ∈ I. This is a submodule of V, as a module over k, and in fact as a module over A. The quotient

V_{A} = V/(V · I) (37.2.5)

is a module over k, and a right module over A.

More precisely, A acts trivially on V_{A}, in the sense described in the previous section, by construction. In fact, V · I is the smallest submodule of V, as a module over A, such that A acts trivially on the quotient, as on p183 of [50]. Let V ⨂_{A}k (37.2.6)

be a tensor product of V and k over A, where k is considered as a left module over A, as in the previous section. Note that (37.2.6) is isomorphic to V_{A} as a module over k, as in Sections 33.1 and 33.2.

Remember that _{ε}V is a two-sided module over A, as in the previous section. Thus V may be considered as a right module over an enveloping algebra A^{en} of A, as in Section 35.1. Let J be the augmentation ideal of A^{en}, as an augmented ring, as in Section 35.2. One can check that

V · I = (_{ε}V) · J, (37.2.7)

as on p183 of [50]. This uses (37.2.2), and the description of J in Section 35.2. Let H_{0}(A, _{ε}V) be as in Section 35.5. This is isomorphic to

(_{ε}V)/((_{ε}V) · J), (37.2.8)

as a module over k, as in Section 35.6. Note that (37.2.8) is isomorphic to (37.2.5), as a module over k, because of (37.2.7). This corresponds to (2) on p183 of [50].

37.3 Crossed homomorphisms into W_{ε}

We continue with the same notation and hypotheses as in the previous two sections. Let W be a module over k that is a left module over A again, so that W_{ε} is a two-sided module over A. A homomorphism f from A into W_{ε}, as modules over k, is a crossed homomorphism if

f(a b) = a · f(b) + f(a) · b = a · f(b) + ε(b) f(a) (37.3.1)

for every a, b ∈ A, as in Section 35.3. We may now consider this as a crossed homomorphism from A into W, as on p183 of [50].

Let U be another module over k that is a left module over U, so that U_{ε} is a two-sided module over A. If g is a homomorphism from U into W, as modules over k and left modules over A, then g is a homomorphism from U_{ε} into W_{ε}, as two-sided modules over A. If f_{0} is a crossed homomorphism from A into U, then (37.3.2) g ∘ f_{0} is a crossed homomorphism from A into W,

as in Section 35.3.

If a ∈ A, then put p(a) = a − ε(a) e_{A}, (37.3.3)

which is an element of I. Note that p(a) = a when a ∈ I. If a, b ∈ I, then

p(a b) = a b − ε(a b) e_{A} = a b − ε(a) ε(b) e_{A} = a p(b) + ε(b) p(a). (37.3.4)

Of course, I may be considered as a two-sided module over A, because I is a two-sided ideal in A. Let I^{L} be I, considered as a module over k, and a left module over A. Thus (I^{L})_{ε} is a two-sided module over A, although the action of A on the right is not normally the usual one.

It follows from (37.3.4) that

(37.3.5) p is a crossed homomorphism from A into I^{L}.

Let g be a homomorphism from I^{L} into W, as left modules over A. Observe that (37.3.6) g ∘ p is a crossed homomorphism from A into W,

by (37.3.2).

If f is a crossed homomorphism from A into W, then f(e_{A}) = 0, as in Section 35.3. This implies that

f(p(a)) = f(a) − ε(a) f(e_{A}) = f(a) (37.3.7)

for every a ∈ A. In particular,

(37.3.8) f is uniquely determined by its restriction to I.

If f is a crossed homomorphism from A into W, then

(37.3.9) the restriction of f to I defines a homomorphism from I^{L} into W,

as modules over k, and left modules over A. Combining this with (37.3.7), we get that f can be expressed as

f = g ∘ p, (37.3.10)

where g is a homomorphism from I^{L} into W, as modules over k, and left modules over A. This representation is unique, because it implies that

g(a) = g(p(a)) = f(a) (37.3.11)

for every a ∈ I. This corresponds to a remark on p183 of [50].

If w ∈ W, then a ↦ a · w defines a homomorphism from A into W, as modules over k, and left modules over A. This implies that

a ↦ p(a) · w = a · w − ε(a) w (37.3.12)

defines a crossed homomorphism from A into W. A crossed homomorphism of this form is said to be principal, as in Section 35.4. This is the same as a principal crossed homomorphism from A into W_{ε}, as a two-sided module over A.

Remember that a homomorphism from A into W, as left modules over A, is of the form a ↦ a · w for a unique w ∈ W. Thus a crossed homomorphism from A into W is principal exactly when it can be expressed as in (37.3.10), where g is a homomorphism from A into W, as left modules over A. This corresponds to another remark on p183 of [50].

The set of all crossed homomorphisms from A into W is a submodule of the space Hom_{k}(A, W) of all homomorphisms from A into W, as modules over k, as in Section 35.3. We have an isomorphism from

Hom_{A}(I^{L}, W) (37.3.13)

onto the space of crossed homomorphisms from A into W, as modules over k, defined by composition with p, as before. Similarly, there is an isomorphism from Hom_{A^{en}}(J, W_{ε}) (37.3.14)

onto the space of crossed homomorphisms from A into W_{ε}, as modules over k, as in Section 35.3. It follows that (37.3.13) and (37.3.14) are isomorphic as modules over k, as on p184 of [50].

The map from w ∈ W to (37.3.12) is clearly linear over k. In particular, the set of all principal crossed homomorphisms from A into W is a submodule of the space of all crossed homomorphisms from A into W, as a module over k.

Consider the obvious homomorphism

(37.3.15) from Hom_{A}(A, W) into Hom_{A}(I^{L}, W),

which sends a homomorphism from A into W, as left modules over A, to its restriction to I. The image of this homomorphism corresponds to the space of all principal crossed homomorphisms from A into W, under the isomorphism from (37.3.13) onto the space of all crossed homomorphisms from A into W mentioned earlier.

This leads to an isomorphism from the cokernel of the homomorphism as in (37.3.15) onto

(37.3.16) the quotient of the space of all crossed homomorphisms from A into W by the submodule consisting of all principal crossed homomorphisms from A into W, as a module over k. Note that the cokernel of the homomorphism as in (37.3.15) is also isomorphic to

(37.3.17) the first cohomology group of A as a left augmented ring with coefficients in W,

as in Section 33.5. Thus (37.3.16) is isomorphic to (37.3.17), as modules over k, as mentioned on p184 of [50].

Let H^{1}(A, W_{ε}) be as in Section 35.5. This is isomorphic to (37.3.16), as modules over k, as in Section 35.6. It follows that (37.3.17) is isomorphic to H^{1}(A, W_{ϵ}), as modules over k, as in (3) on p184 of [50].

37.4 Modules with trivial operators

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k, with multiplicative identity element e_{A}. Suppose that A is a supplemented algebra over k, with corresponding homomorphism ε from A onto k, as in Section 37.1. Let I be the kernel of ε, as before.

Let V be a module over k that is a right module over A with trivial operators, so that v · a = ε(a) v for every a ∈ A and v ∈ V, as in Section 37.1. Let V ⨂_{A} k be a tensor product of V and k over A, where k is considered as a left module over A using ε, as before. This is isomorphic to V_{A} = V/(V · I), as a module over k, as in Section 37.2. In this case,

V · I = {0}, (37.4.1)

by hypothesis. This means that V ⨂_{A} k is isomorphic to V, as a module over k. This is the same as saying that the 0th homology group of A as a left augmented ring with coefficients in V is isomorphic to V. This corresponds to a remark on p184 of [50].

Similarly, let W be a module over k that is a left module over A with trivial operators, so that the submodule W^{A} of invariant elements of W is equal to W. This implies that the space Hom_{A}(k, W) of homomorphisms from k into W, as modules over k, and left modules over A, is isomorphic to W in a natural way, as a module over k, as in Section 37.2. This means that the 0th cohomology group of A as a left augmented ring with coefficients in W is isomorphic to W, as in Section 33.5. This corresponds to another remark on p184 of [50].

In this case, a homomorphism f from A into W, as modules over k, is a crossed homomorphism if

f(a b) = ε(a) f(b) + ε(b) f(a) (37.4.2)

for every a, b ∈ A. The only principal crossed homomorphism is equal to 0 on A. Thus the quotient (37.3.16) is the same as the space of all crossed homomorphisms from A into W, as on p184 of [50].

Let V be a module over k that is a right module over A with trivial operators again, and let V ⨂_{A}I (37.4.3)

be a tensor product of V and I over A, where I is considered as a left module over A. There is a natural homomorphism from (37.4.3) into V, as modules over k, that comes from the mapping (v, b) ↦ v · b from V × I into V, as in Section 33.3. This homomorphism is equal to 0 here, because V is a right module over A with trivial operators. This means that the one-dimensional homology of A as a left augmented ring with coefficients in V is isomorphic to (37.4.3), as in Section 33.3.

Remember that V satisfies the requirements of a tensor product V ⨂_{k} k of V and k over k, as a module over k. One can check that this is compatible with the action of A on the right on V ⨂_{k} k that is obtained from the action of A on k using ε. Thus (37.4.3) may be identified with a tensor product of the form

(V ⨂_{k}k) ⨂_{A}I. (37.4.4)

There is a natural isomorphism between (37.4.4) and a tensor product of the form V ⨂_{k}(k ⨂_{A}I), (37.4.5)

where k is considered as a right module over A, as in Section 1.12. Remember that I^{2} = I I consists of finite sums of products of the form a b, with a, b ∈ I. This is a submodule of I, as a left and right module over A, and we have seen that there is an isomorphism

(37.4.6) from I/I^{2} onto k ⨂_{A}I,

as left and right modules over A, as in Section 33.4. Thus we get that (37.4.3) is isomorphic to V ⨂_{k}(I/I^{2}), (37.4.7)

as on p184 of [50].

37.5 Homomorphisms and supplemented algebras

Let k, k_{1} be commutative rings with multiplicative identity elements 1_{k}, 1_{k_{1}}, respectively, and let A, B be associative algebras over k, k_{1}, with multiplicative identity elements e_{A}, e_{B}, respectively. Suppose that ψ is a ring homomorphism from k into k_{1}, with ψ(1_{k}) = 1_{k_{1}}, and that ϕ is a ring homomorphism from A into B, with ϕ(e_{A}) = e_{B}, and

ϕ(t a) = ψ(t) ϕ(a) (37.5.1) for every a ∈ A and t ∈ k.

Suppose that A, B are in fact supplemented algebras, with corresponding homomorphisms ε_{A}, ε_{B} from A, B onto k, k_{1}, respectively, as in Section 37.1. If ε_{B} ∘ ϕ = ψ ∘ ε_{A}, (37.5.2)

then the pair ϕ, ψ is considered as a map or homomorphism between supplemented algebras, as on p184 of [50]. Of course, this implies that

ϕ(ker ε_{A}) ⊆ ker ε_{B}. (37.5.3)

This means that we get a homomorphism from A into B as left or right augmented rings, with respect to the augmentations defined by ε_{A}, ε_{B}, as in Section 33.9. This corresponds to a remark on p184 of [50].

If V is a right module over B, then V may be considered as a right module over A, using ϕ. In this case, we can use ϕ to get a homomorphism from the homology of A into the homology of B, as left augmented rings, with coefficients in V, as in Section 33.3. Similarly, if Z is a left module over B, then Z may be considered as a left module over A, using ϕ. This leads to a homomorphism from the cohomology of B into the cohomology of A, as left augmented rings, with coefficients in Z, as before. This corresponds to another remark on p184 of [50].

In particular, we may be interested in the case where k = k_{1} and ψ is the identity mapping on k, as on p184 of [50].

37.6 Homomorphisms from tensor products

Let k be a commutative ring with a multiplicative identity element, and let A, C be associative algebras over k with multiplicative identity elements e_{A}, e_{C}, respectively. Suppose that A is a supplemented algebra, with corresponding homomorphism ε_{A} from A onto k, as in Section 37.1.

Let C ⨂_{k} A be a tensor product of C and A over k, which may be considered as an associative algebra over k, as in Section 4.1. Remember that C satisifes the requirements of a tensor product of C and k over k, as a module over k. One can check that this is also compatible with multiplication on C and on C ⨂_{k} k, as a tensor product of associative algebras over k. Using this, the identity mapping on C, and ε_{A}, we get a homomorphism

(37.6.1) ε_{C ⨂_{k}A} from C ⨂_{k}A onto C,

as algebras over k. Alternatively,

ε_{A} e_{C} (37.6.2)

may be considered as a homomorphism from A into the center of C, as algebras over k. One can use the identity mapping on C and (37.6.2) to get a unique homomorphism as in (37.6.1), as algebras over k, such that

ε_{C ⨂_{k}A}(c ⊗ a) = ε_{A}(a) c (37.6.3) for every a ∈ A and c ∈ C, as in Section 4.1. Remember that e_{C} ⊗ e_{A} is the multiplicative identity element in C ⨂_{k} A, and observe that

ε_{C ⨂_{k}A}(e_{C} ⊗ e_{A}) = e_{C}, (37.6.4)

because ε_{A}(e_{A}) = 1.

We may consider C ⨂_{k} A as a left or right augmented ring, with augmentation homomorphism ε_{C ⨂_{k}A} and augmentation module C, as in Section 34.11. This is mentioned at the bottom of p184 of [50].

As before, ϕ(a) = e_{C} ⊗ a (37.6.5)

defines a homomorphism from A into C ⨂_{k} A, as algebras over k. Of course,

ψ(t) = t e_{C} (37.6.6)

defines a homomorphism from k into C, as algebras over k. If a ∈ A, then

ε_{C ⨂_{k}A}(ϕ(a)) = ε_{C ⨂_{k}A}(e_{C} ⊗ a) = ε_{A}(a) e_{C} = ψ(ε_{A}(a)). (37.6.7)

In particular, ϕ maps the kernel of ε_{A} into the kernel of ε_{C ⨂_{k} A}. This implies that ϕ is a homomorphism from A into C ⨂_{k} A as augmented rings, as in Section 33.9.

Let V be a module over k that is a right module over C ⨂_{k} A. This means that V is a right module over A and C, where the actions of A and C commute with each other. We can use ϕ to get a homomorphism

(37.6.8) from the homology of A into the homology of C ⨂_{k}A,

as left augmented rings, with coefficients in V,

as in Section 34.12. This corresponds to (5) on p185 of [50].

Similarly, let Z be a module over k that is a left module over C ⨂_{k} A, which means that Z is a left module over A and C, where the actions of A and C on Z commute with each other. We can use ϕ to get a homomorphism

(37.6.9) from the cohomology of C ⨂_{k}A into the cohomology of A,

as left augmented rings, with coefficients in Z,

as in Section 34.12 again. This corresponds to (5a) on p185 of [50]. Suppose now that

(37.6.10) A is projective as a module over k.

Note that the augmentation module ε_{A}(A) = k of A is automatically projective as a module over k in this case. This implies that

(37.6.11) the homomorphisms as in (37.6.8) and (37.6.9) are isomorphisms, as in Section 34.12. If X is a projective resolution of k, as a left module over A using ε_{A}, then we also get that

(37.6.12) C ⨂_{k}X is a projective resolution of C,

as a left module over C ⨂_{k}A,

as before. This corresponds to Proposition 1.1 on p185 of [50].

If C is a commutative algebra over k, then C ⨂_{k} A may be considered as a commutative algebra over C. In this case,

(37.6.13) C ⨂_{k}A may be considered as a supplemented algebra,

as an algebra over C, using the homomorphism ε_{C ⨂_{k} A}, as on p185 of [50].

37.7 Connection with A^{en}

Let k be a commutative ring with a multilpicative identity element, and let A be an associative algebra over k, with a multiplicative identity element e_{A}. Suppose that A is a supplemented algebra over k, with corresponding homomorphism ε from A onto k, as in Section 37.1.

Remember that A^{op} is the opposite algebra of A, and let A^{en} = A ⨂_{k} A^{op} be a tensor product of A and A^{op} over k, considered as an algebra over k, as in Section 35.1. Also let ρ be the homomorphism from A^{en} onto A, as left modules over A^{en}, defined in Section 35.1.

Observe that (a_{1}, a_{2}^{op}) ↦ ε(a_{2}) a_{1} (37.7.1)

defines a mapping from A × A^{op} into A that is bilinear over k. This leads to a unique homomorphism (37.7.2) ϕ from A^{en} into A,

as modules over k, such that

ϕ(a_{1} ⊗ a_{2}^{op}) = ε(a_{2}) a_{1} (37.7.3)

for every a_{1}, a_{2} ∈ A. It is easy to see that this defines a homomorphism as in (37.7.2), as algebras over k.

If a_{1}, a_{2} ∈ A, then

(37.7.4)  ε(ϕ(a_{1} ⊗ a_{2}^{op})) = ε(ε(a_{2}) a_{1}) = ε(a_{1}) ε(a_{2})
= ε(a_{1} a_{2}) = ε(ρ(a_{1} ⊗ a_{2}^{op})).

This implies that ε ∘ ϕ = ε ∘ ρ, (37.7.5)

as mappings from A^{en} into k. It follows that

ϕ(ker ρ) ⊆ ker ε. (37.7.6)

37.7. CONNECTION WITH A^{EN} 827

This means that ϕ is a homomorphism as in (37.7.2), as augmented rings with respect to the augmentations defined by ρ and ε, as in Section 33.9. This corresponds to some remarks on p185 of [50].

Let V be a module over k that is a right module over A. We may consider V as a right module over A^{en}, using ϕ, as in Section 2.9. The action of A^{en} on V on the right corresponds to commuting actions of A and A^{op} on V on the right, as in Section 4.2. It is easy to see that the action of A on V on the right obtained from this action of A^{en} on V on the right is the same as the action of A on V as a right module over A. The action of A^{op} on V on the right obtained from this action of A^{en} on V on the right is the same as the trivial action on V defined using ε, as in Section 37.1.

This action of A^{op} on V on the right corresponds exactly to an action of A on V on the left, as usual. This action of A on V on the left is the same as the trivial action defined using ε, as before. This makes V into a two-sided module over A, corresponding to V as a right module over A^{en}, as in Section 35.1. This is the same as _{ε}V as a two-sided module over A, as in Section 37.1, with the trivial action of A on V on the left, and the action of A on V as a right module over A. Thus we may use _{ε}V to refer to V as a right module over A^{en} in this way.

If n is a nonnegative integer, then we can use ϕ to get a homomorphism

(37.7.7) from H_{n}(A,_{ε} V) into the nth homology group of A, as a left augmented ring, with coefficients in V,

where H_{n}(A,_{ε} V) is as in Section 35.5. This follows from the discussion in Section 33.9, because H_{n}(A,_{ε} V) is the same as the nth homology group of A^{en}, as a left augmented ring, with coefficients in _{ε}V, as a right module over A^{en}. This corresponds to some more remarks on p185 of [50].

Similarly, let Z be a module over k that is a left module over A. We may consider Z as a left module over A^{en}, using ϕ, as in Section 2.9 again. The action of A^{en} on Z on the left corresponds to commuting actions of A and A^{op} on Z on the left, as before. The action of A on Z on the left obtained from the action of A^{en} on Z on the left is the same as the action of A on Z as a left module over A. The action of A^{op} on Z on the left obtained from the action of A^{en} on Z on the left is the same as the trivial action on Z defined using ε, because of the way that ϕ is defined.

This action of A^{op} on Z on the left corresponds to an action of A on Z on the right. This action of A on Z on the right is the same as the trivial action of A on V defined using ε. This makes Z into a two-sided module over A, which corresponds to Z as a left module over A^{en}. This is the same as Z_{ε} as a two-sided module over A, as in Section 37.1. We may use Z_{ε} to refer to Z as a left module over A^{en} in this way, as before.

We can use ϕ to get a homomorphism

(37.7.8) from the nth cohomology group of A, as a left augmented ring, with coefficients in Z, into H^{n}(A, Z_{ε}), for each n ≥ 0, where H^{n}(A, Z_{ε}) is as in Section 35.5. This follows from the discussion in Section 33.9 again, as on p185 of [50].

If (37.7.9) A is projective as a module over k,

then Theorem 2.1 on p185 of [50] states that

(37.7.10) the homomorphisms as in (37.7.7) and (37.7.8) are isomorphisms.

We shall discuss this further in the next two sections.

37.8 A helpful isomorphism

Let us continue with the same notation and hypotheses as in the previous section. Also let _{ε}A be as in Section 37.1, where A is considered as a module over k, and a right module over itself. Thus _{ε}A is a two-sided module over A, where the left action of A on _{ε}A is the trivial action. Of course, we may consider _{ε}A as a left or right module over A^{en}, as in Section 35.1.

Suppose that W is another module over k that is a two-sided module over A, so that W may be considered as a left or right module over A^{en} as well. Let

εA ⨂_{A^{en}}W (37.8.1)

be a tensor product of _{ε}A, as a right module over A^{en}, and W, as a left module over A^{en}, over A^{en}. Remember that k may be considered as a left or right module over A, using ε, as in Section 37.1. Let

W ⨂_{A}k (37.8.2)

be a tensor product of W, as a right module over A, and k, as a left module over A, over A.

Consider the mapping from _{ε}A × W into W ⨂_{A} k defined by

(a, w) ↦ (a ·_{A} w) ⊗_{A} 1_{k}, (37.8.3)

where 1_{k} is the multiplicative identity element in k. More precisely, if a ∈ _{ε}A and w ∈ W, then a ·_{A} w ∈ W is defined using the action of a as an element of A on W on the left. The mapping defined by (37.8.3) is clearly bilinear over k. If x ∈ A^{en}, then we would like to check that

((a ·_{A^{en}} x) ·_{A} w) ⊗_{A} 1_{k} = (a ·_{A} (x ·_{A^{en}} w)) ⊗_{A} 1_{k}, (37.8.4)

where a ·_{A^{en}} x is defined using the action of A^{en} on A on the right, and x ·_{A^{en}} w is defined using the action of A^{en} on W on the left.

It suffices to verify (37.8.4) when x ∈ A^{en} is of the form x_{1} ⊗_{k} x_{2}^{op}, where x_{1}, x_{2} ∈ A. In this case, (37.8.4) is the same as saying that

((x_{2} ·_{A} (a ·_{A} x_{1})) ·_{A} w) ⊗_{A} 1_{k} = (a ·_{A} ((x_{1} ·_{A} w) · x_{2})) ⊗_{A} 1_{k}, (37.8.5)

37.8. A HELPFUL ISOMORPHISM 829 using the various actions of A on _{ε}A and W on the left and right. The left side is equal to

(37.8.6) (ε(x_{2}) (a x_{1}) ·_{A} w) ⊗_{A} 1_{k} = ((a x_{1}) ·_{A} w) ⊗_{A} ε(x_{1})
= ((a x_{1}) ·_{A} w) ⊗_{A} (x_{1} ·_{A} 1_{k}),

because the actions of A on _{ε}A and k on the left are trivial. This is equal to (((a x_{1}) ·_{A} w) ·_{A} x_{1}) ⊗_{A} 1_{k}, (37.8.7) because of the way that W ⨂_{A} k is defined. It is easy to see that this is equal to the right side of (37.8.5), as desired.

It follows that there is a unique homomorphism (37.8.8) τ from _{ε}A ⨂_{A^{en}}W into W ⨂_{A}k, as modules over k, such that τ(a ⊗_{A^{en}} w) = (a ·_{A} w) ⊗_{A} 1_{k} (37.8.9) for every a ∈ _{ε}A and w ∈ W. Lemma 2.2 on p186 of [50] states that (37.8.10) τ is an isomorphism. To show this, we shall find an inverse mapping σ.

Consider the mapping from W × k into _{ε}A ⨂_{A^{en}} W defined by (w, t) ↦ (t e_{A}) ⊗_{A^{en}} w. (37.8.11) Note that this mapping is bilinear over k. We would like to check that (t e_{A}) ⊗_{A^{en}} (w ·_{A} a) = ((a ·_{A} t) e_{A}) ⊗_{A^{en}} w (37.8.12) for every a ∈ A, w ∈ W, and t ∈ k.

Of course, e_{A} ⊗_{k} a^{op} ∈ A^{en}, and w ·_{A} a = (e_{A} ⊗_{k} a^{op}) ·_{A^{en}} w. (37.8.13) Using this, we get that the left side of (37.8.12) is equal to (t e_{A}) ⊗_{A^{en}} ((e_{A} ⊗_{k} a^{op}) · w) = ((t e_{A}) · (e_{A} ⊗_{k} a^{op})) ⊗_{A^{en}} w, (37.8.14) because of the way that _{ε}A ⨂_{A^{en}} W is defined. This is equal to (t ε(a) e_{A}) ⊗_{A^{en}} w, (37.8.15) because the action of A on _{ε}A on the left is trivial. This is the same as the right side of (37.8.12), as desired.

Thus there is a unique homomorphism

(37.8.16) σ from W ⨂_{A}k into _{ε}A ⨂_{A^{en}}W,

as modules over k, such that

σ(w ⊗_{A} t) = (t e_{A}) ⊗_{A^{en}} w (37.8.17)

for every w ∈ W and t ∈ k. If a ∈ _{ε}A and w ∈ W, then

σ(τ(a ⊗_{A^{en}} w)) = σ((a ·_{A} w) ⊗_{A} 1_{k}) = e_{A} ⊗_{A^{en}} (a ·_{A} w). (37.8.18)

Observe that a ⊗ e_{A}^{op} ∈ A^{en}, and that

a ·_{A} w = (a ⊗_{k} e_{A}^{op}) ·_{A^{en}} w. (37.8.19)

It follows that (37.8.18) is equal to

(37.8.20) e_{A} ⊗_{A^{en}} ((a ⊗_{k} e_{A}^{op}) ·_{A^{en}} w) = (e_{A} ·_{A^{en}} (a ⊗ e_{A}^{op})) ⊗_{A^{en}} w
= a ⊗_{A^{en}} w.

This implies that σ ∘ τ is the identity mapping on _{ε}A ⨂_{A^{en}} W.

Similarly, if w ∈ W and t ∈ k, then

(37.8.21)  τ(σ(w ⊗_{A} t)) = τ((t e_{A}) ⊗_{A^{en}} w)
= ((t e_{A}) ·_{A} w) ⊗_{A} 1_{k} = w ⊗_{A} t.

This means that τ ∘ σ is the identity mapping on W ⨂_{A} k. Thus σ and τ are inverses of each other, as on p186 of [50].

37.9 Some verifications

We continue with the same notation and hypotheses as in the previous two sections. In other to show (37.7.10), we want to use the Mapping Theorem, as in Section 33.10. In this discussion, A^{en} plays the role of A in Sections 33.9 and 33.10, and A plays the role of B before. Similarly, A plays the role of Q_{A} before, as a left module over A^{en}, and k plays the role of Q_{B}, as a left module over A.

The homomorphism ϕ from A^{en} into A in Section 37.7 plays the same role as the homomorphism ϕ from A into B in Section 33.3. The mapping ψ from Q_{A} into Q_{B} before corresponds to ε here.

In Sections 33.9 and 33.10, B was sometimes considered as a right module over A, using the homomorphism ϕ, as in Section 2.9. This corresponds to considering A as a right module over A^{en} using ϕ here. It is easy to see that this is the same as _{ε}A, considered as a right module over A^{en}, as in the previous section. Of course, this is not normally the same as considering A as a right module over A^{en} in the usual way.

Thus the analogue of B ⨂_{A} Q_{A} in Section 33.9 is a tensor product

εA ⨂_{A^{en}}A, (37.9.1)

where A is considered as a left module over A^{en} in the usual way. If we take W = A in the previous section, then we get an isomorphism τ from (37.9.1) onto A ⨂_{A} k, which can be identified with k as a left module over A in the usual way. This corresponds to the first condition that we need from Section 33.10, as on p186 of [50].

To deal with the second condition that we need from Section 33.10, we let X be a projective resolution of A, as a left module over A^{en}. This corresponds to X_{A} in Section 33.9. Also let

εA ⨂_{A^{en}}X (37.9.2)

be a tensor product of _{ε}A and X over A^{en}, which corresponds to B ⨂_{A} X_{A} in Section 33.9. This may be considered as a complex over k, as in Section 7.5. The second condition that we need from Section 33.10 is that

H(_{ε}A ⨂_{A^{en}}X)_{n} = {0} (37.9.3)

for every n > 0.

Of course, we may consider X as a two-sided module over A, because it is a left module over A^{en}. Let X ⨂_{A}k (37.9.4)

be a tensor product of X, as a right module over A, and k, as a left module over A, over A. This may be considered as a complex over k too, as in Section 7.5.

If we take W = X in the previous section, then we get an isomorphism τ from (37.9.2) onto (37.9.4), as modules over k. It is easy to see that τ is of degree 0, and in fact an isomorphism from (37.9.2) onto (37.9.4), as complexes over k. This induces an isomorphism between the homology groups of these complexes, so that (37.9.3) is equivalent to

H(X ⨂_{A}k)_{n} = {0} (37.9.5)

for every n > 0, as on p186 of [50].

Note that X^{j} is projective as a left module over A^{en} for each j, by hypothesis. If (37.9.6) A is projective as a module over k,

then it follows that X^{j} is projective as a left module over A^{op} for each j, as in Section 34.7, because A^{en} = A ⨂_{k} A^{op}. Equivalently, this means that

(37.9.7) X^{j} is projective as a right module over A

for each j, as on p186 of [50].

By hypothesis, X is an acyclic left complex over A, as a left module over A^{en}. It is easy to see that X may be considered as an acyclic left complex over A, as a left module over A^{op}, and thus as a right module over A. It follows from this and (37.9.7) that X may be considered as a projective resolution of A, as a right module over itself.

Of course, A is projective as a right module over itself, and so we can get another projective resolution Y of A, as a right module over itself, by taking Y^{0} = A and Y^{j} = {0} when j ≠ 0. Under these conditions, the homology groups of X ⨂_{A} k are isomorphic to those of Y ⨂_{A} k, as in Section 10.12. Clearly

H(Y ⨂_{A}k)_{n} = {0} (37.9.8)

for every n > 0. This implies that the second condition that we need from Section 33.10 holds when (37.9.6) holds, as on p186 of [50].

Thus (37.7.10) holds when (37.9.6) holds, as in Section 33.10.

37.10 Some related projective resolutions

Let us continue with the same notation and hypotheses as in the previous three sections. Let Ã be A, considered as a module over k in the usual way, as well as a left module over A and a right module over A^{en}, as follows. The action of A on itself on the left is taken to be the usual one, using multiplication in the algebra. The action of A^{en} on the right is obtained using the homomorphism ϕ from A^{en} onto A defined in Section 37.7, as in Section 2.9. This corresponds to εA as a right module over A^{en}, as in the previous section.

One can check that these actions of A on Ã on the left, and of A^{en} on Ã on the right, commute with each other. Of course, the usual actions of A on itself on the left and right commute with each other, by associativity. The commutativity of the two actions on Ã just mentioned basically involves the commutativity of the usual left action of A on itself with the trivial action of A on itself defined using ε as well. The latter can be verified using the linearity over k of multiplication on A.

Let X be a projective resolution of A, as a left module over A^{en} in the usual way, as in the previous section. A tensor product

Ã⨂_{A^{en}}X (37.10.1)

of Ã, as a right module over A^{en}, and X, as a left module over A^{en}, over A^{en}, is the same as (37.9.2), as a module over k. We may consider (37.10.1) as a left module over A, using the action of A on Ã on the left, because that commutes with the action of A^{en} on Ã on the right, as in the preceding paragraph. This is not normally the same as the analogous action of A on (37.9.2) on the left.

Remember that X may be considered as a two-sided module over A, and that X ⨂_{A} k is a tensor product of X, as a right module over A, and k, as a left module over A, over A, as in (37.9.4). We may consider X ⨂_{A} k as a left module over A, using the left action of A on X.

As in the previous section, we can take W = X in Section 37.8, to get an isomorphism τ from (37.10.1) onto X ⨂_{A} k, as modules over k. One can check that τ is an isomorphism from (37.10.1) onto X ⨂_{A} k, as left modules over A.

More precisely, (37.10.1) and X ⨂_{A} k may be considered as complexes over k, as in the previous section, and in fact as complexes over A. We also have that τ is an isomorphism from (37.10.1) onto X ⨂_{A} k as complexes over k, as before, and thus as complexes over A.

Let us suppose from now on in this section that A is projective as a module over k, as in (37.9.6). This means that the second as well as the first of the usual conditions from Section 33.10 are satisfied, as in the previous section. It follows that

(37.10.2) (37.10.1) is a projective resolution of k, as a left module over A,

as in Section 33.10. One can use this and the fact that τ is an isomorphism from (37.10.1) onto X ⨂_{A} k, as complexes over A, to get that

(37.10.3) X ⨂_{A}k is a projective resolution of k, as a left module over A.

This corresponds to the second part of Theorem 2.1 on p185 of [50].

One can also get (37.10.3) a bit more directly, using the same type of arguments. Remember that k satisfies the requirements of A ⨂_{A} k, as a module over k, and as a left module over A, as in Section 1.10. This implies that X ⨂_{A} k is a left complex over k, as a left module over A, because X is a left complex over A, as a left module over A^{en}, and thus as a left module over A.

If (37.9.6) holds, then one can check that X ⨂_{A} k is acyclic as a left complex over k, as a left module over A. This uses (37.9.5) and the acyclicity of X, as a left complex over A, as a left module over A^{en}, and thus over A.

Remember that X^{j} is projective as a left module over A^{en} for each j, by hypothesis. We may consider X^{j} as a two-sided module over A for each j, or equivalently a right module over A^{en}. One can check that X^{j} is projective as a right module over A^{en} for each j as well.

It follows that (X ⨂_{A}k)^{j} = X^{j} ⨂_{A}k (37.10.4)

is projective as a right module over A^{op} ⨂_{k} k = A^{op} for each j, as in Sections 34.6 and 34.8, because k is projective as a module over itself. Equivalently, this means that (37.10.4) is projective as a left module over A.

37.11 Standard complexes and resolutions

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that A is a supplemented algebra over k, with corresponding homomorphism ε from A onto k, as in Sections 37.1 and 37.7. Let A^{en} = A ⨂_{k} A^{op} be a tensor product of A and the opposite algebra A^{op} over k, considered as an algebra over k, as in Section 35.1.

Remember that A corresponds to the direct sum of

{t e_{A} : t ∈ k} (37.11.1)

and the kernel I of ε, as a module over k, as in Section 37.1. Put

A′ = A/{t e_{A} : t ∈ k}, (37.11.2)

considered as a module over k, as in Section 35.12. It follows that

(37.11.3) A′ is isomorphic to I,

as a module over k.

Suppose for the rest of the section that

(37.11.4) A is projective as a module over k.

This implies that I is projective as a module over k, as in Section 2.7. Equivalently, this means that

(37.11.5) A′ is projective as a module over k,

by (37.11.3). This corresponds to a remark on p186 of [50].

If X is a projective resolution of A, as a left module over A^{en}, then a tensor product X ⨂_{A} k of X, as a right module over A, and k, as a left module over A, over A, is a projective resolution of k, as a left module over A, as in the previous section. Remember that the standard complex S(A) of A is a projective resolution of A, as a left module over A^{en}, as in Section 35.10, because of (37.11.4). Similarly, the normalized standard complex N(A) of A is a projective resolution of A, as a left module over A^{en}, as in Section 35.14, because of (37.11.5). Let

S(A, ε) = S(A) ⨂_{A}k (37.11.6)

and N(A, ε) = N(A) ⨂_{A}k (37.11.7)

be tensor products of S(A), N(A), as right modules over A, and k, as a left module over A, over A, respectively, as on p186 of [50]. Of course, we would like to say more about how these tensor products behave, as in [50].

In particular, S(A) is a graded left module over A^{en}, with S(A)_{n} = S_{n}(A) when n ≥ 0, and S(A)_{n} = {0} when n ≤ 0, as in Section 35.10, where S_{n}(A) is as in Section 35.8. Let S̃_{n}(A) be a tensor product of n copies of A over k when n ≥ 1, and put S̃_{0}(A) = k, as in Section 35.10. We may consider S_{n}(A) as a tensor product of the form

S_{n}(A) = A ⨂_{k}S̃_{n}(A) ⨂_{k}A (37.11.8) for each n ≥ 0, where the actions of A on the left and right on S_{n}(A) are obtained from the actions of A on the left and on the right on the first and last factors of A on the right side of (37.11.8), as before.

More precisely, S(A, ε) is a graded left module over A, with

(37.11.9) S(A, ε)_{n} = S(A)_{n} ⨂_{A}k = S_{n}(A) ⨂_{A}k when n ≥ 0
= {0}  when n < 0,

as in Section 7.5. If n ≥ 0, then S(A, ε)_{n} is isomorphic to

A ⨂_{k}S̃_{n}(A) ⨂_{k}(A ⨂_{A}k) (37.11.10)

in a natural way, as in Section 1.12. Thus we may take S(A, ε)_{n} to be a tensor product of the form

S(A, ε)_{n} = A ⨂_{k}S̃_{n}(A) (37.11.11)

when n ≥ 0, because k satisfies the requirements of A ⨂_{A} k, and similarly for tensor products with k, over k. Note that the action of A on the left on S(A, ε)_{n} corresponds to the action of A on the left on the first factor of A on the right side of (37.11.11).

Similarly, N(A) is a graded left module over A^{en}, with N(A)_{n} = N_{n}(A) when n ≥ 0, and N(A)_{n} = {0} when n ≤ 0, as in Section 35.14. As before, we let Ñ_{n}(A) be a tensor product of n copies of A′ over k when n ≥ 1, and we take Ñ_{0}(A) = k. We can identify N_{n}(A) with a tensor product of the form

N_{n}(A) = A ⨂_{k}Ñ_{n}(A) ⨂_{k}A, (37.11.12)

where the actions of A on the left and on the right on N_{n}(A) are obtained from the actions of A on the left and on the right on the first and last factors of A on the right side of (37.11.12), as in Section 35.12.

We also have that N(A, ε) is a graded left module over A, with

N(A, ε)_{n} = N(A)_{n} ⨂_{A}k = N_{n}(A) ⨂_{A}k when n ≥ 0
(37.11.13)  = {0}  when n < 0,

as in Section 7.5. If n ≥ 0, then we can take N(A, ε) to be a tensor product of the form N(A, ε)_{n} = A ⨂_{k}Ñ_{n}(A), (37.11.14)

as before. With this identification, the action of A on the left on N(A, ε)_{n} corresponds to the action of A on the left on the first factor of A on the right side of (37.11.14). This corresponds to the description of N(A, ε) on p186 of

37.12 Differentiation on these resolutions

Let us continue with the same notation and hypotheses as in the previous section. Remember that the differentiation operators d_{S(A)} and d_{N(A)} on S(A) and N(A) were defined in Sections 35.10 and 35.14, respectively. This leads to differentiation operators d_{S(A,ε)} and d_{N(A,ε)} on S(A, ε) and N(A, ε), respectively, as in Section 7.5.

More precisely, for each n ∈ Z, d_{S(A,ε),n} is the homomorphism

(37.12.1) from S(A, ε)_{n} into S(A, ε)_{n−1},

as left modules over A, that corresponds to d_{S(A),n} and the identity mapping on k. Similarly, d_{N(A,ε),n} is the homomorphism

(37.12.2) from N(A, ε)_{n} into N(A, ε)_{n−1},

as left modules over A, that corresponds to d_{N(A),n} and the identity mapping on k. Note that these homomorphisms are equal to 0 when n ≤ 0. To describe these differentiation operators more precisely when n ≥ 1, we shall use some notation analogous to that of Section 35.15.

If n ≥ 0, then it will be helpful to consider S(A, ε)_{n} as a tensor product of n + 1 copies of A over k. Similarly, it will be helpful to consider N(A, ε)_{n} as a tensor product of n + 1 modules over k, where the first module is A, and the next n modules are copies of A′. Thus, if a_{0}, a_{1}, …, a_{n} ∈ A, then

a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} (37.12.3)

defines an element of S(A, ε)_{n}. We also get that

a_{0} ⊗ q(a_{1}) ⊗ ⋯ ⊗ q(a_{n}) (37.12.4)

defines an element of N(A, ε)_{n}, where q is the natural quotient mapping from A onto A′, as in Section 35.12.

We may use the notation

a_{0} [a_{1}, …, a_{n}] (37.12.5)

for (37.12.4) as on p186f of [50]. If a_{0} = e_{A}, then this may be expressed as

[a_{1}, …, a_{n}]. (37.12.6)

If n = 0, then (37.12.5) becomes a_{0} [ ], (37.12.7)

which is the same as a_{0} as an element of S(A, ε)_{0} = N(A, ε)_{0} = A. We may sometimes multiply expressions like these on the right by elements of k as well.

If n ≥ 1, then

d_{N(A,ε),n}(a_{0} [a_{1}, …, a_{n}]) = (a_{0} a_{1}) [a_{2}, …, a_{n}]

+ ∑_{l=1}^{n−1}(−1)^{l} a_{0} [a_{1}, …, a_{l} a_{l+1}, …, a_{n}] (37.12.8)

+(−1)^{n} a_{0} [a_{1}, …, a_{n−1}] ε(a_{n}).

This is mentioned on p186 of [50], with a_{0} = e_{A}. Of course, one can reduce to that case, because d_{N(A,ε),n} is a homomorphism as in (37.12.2), as left module over A. If n = 1, then the sum in the middle of the right side of (37.12.8) should be interpreted as being 0, and we get that

d_{N(A,ε),1}(a_{0} [a_{1}]) = (a_{0} a_{1}) [ ] − a_{0} [ ]ε(a_{1}) = a_{0} a_{1} − a_{0} ε(a_{1}). (37.12.9)

This is also mentioned on p186 of [50], with a_{0} = e_{A}.

If we are working with S(A, ϵ)_{n}, then we may use the notation (37.12.5) for (37.12.3), as in Section 35.15. In this case, we get the same expressions for d_{S(A,ε),n} as in the preceding paragraph, as mentioned on p187 of [50].

One could also consider k as a right module over A, and A as a right augmented ring, as mentioned on p187 of [50]. If A is projective as a module over k, then tensor products S(ε, A) = k ⨂_{A}S(A) (37.12.10)

and N(ε, A) = k ⨂_{A}N(A) (37.12.11)

are projective resolutions of k as a right module over A, as in [50].

37.13 Some related complexes

Let k be a commutative ring with a multiplicative identity element, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Suppose that A is a supplemented algebra over k, with corresponding homomorphism ε from A onto k, as in Section 37.1.

Let X be a projective resolution of k, as a left module over A. Consider a tensor product \overline{X} = k ⨂_{A}X (37.13.1)

of k, as a right module over A, and X, over A, as in the statement of Proposition 2.3 on p187 of [50]. Note that \overline{X} may be considered as a complex over k, as in Section 7.5.

Let V be a module over k, which leads to a right module V_{ε} over A with trivial action of A on the right, as in Section 37.1. Alternatively, a tensor product V ⨂_{k}k (37.13.2)

of V and k over k may be considered as a right module over A, using the action of A on k on the right defined by ε. Of course, V satisifes the requirements of a tensor product as in (37.13.2), as a module over k, in the usual way. If V is considered as a tensor product (37.13.2), as a module over k, in this way, then it is easy to see that V_{ε} is the same as (37.13.2), considered as a right module over A, using the action of A on k on the right.

Consider a tensor product V_{ε} ⨂_{A}X (37.13.3) of V_{ε} and X over A. This corresponds to a tensor product of the form

(V ⨂_{k}k) ⨂_{A}X, (37.13.4)

where k is considered as a right module over A, as in the preceding paragraph. This is isomorphic to a tensor product of the form

V ⨂_{k}(k ⨂_{A}X) = V ⨂_{k}\overline{X}, (37.13.5)

in a natural way, as in Section 1.12. More precisely, these tensor products may be considered as complexes over k, as in Section 7.5, and they are isomorphic as complexes.

It follows that H(V_{ε} ⨂_{A}X)_{n} (37.13.6)

is isomorphic to H(V ⨂_{k}\overline{X})_{n} (37.13.7)

for each n. Note that (37.13.6) corresponds to the nth homology of A, as a left augmented ring, with coefficients in V_{ε} for every n ≥ 0, as in Section 33.3. This corresponds to the first part of Proposition 2.3 on p187 of [50].

Similarly, let W be a module over k, so that _{ε}W is a left module over A with trivial action of A on the left, as in Section 37.1. Alternatively, the space

Hom_{k}(k, W) (37.13.8)

of homomorphisms from k into W, as modules over k, may be considered as a left module over A, using the action of A on k on the right, as in Section 1.8. If we identify (37.13.8) with W, as modules over k, in the usual way, then one can check that _{ε}W corresponds to (37.13.8), considered as a left module over A in this way.

Remember that Hom_{A}^{gr}(X,_{ε}W) (37.13.9)

may be defined as a complex over k as in Section 8.4. We may consider

Hom_{k}^{gr}(\overline{X}, W) = Hom_{k}^{gr}(k ⨂_{A}X, W ) (37.13.10)

as a complex over k as well. This is isomorphic to

Hom_{A}^{gr}(X, Hom_{k}(k, W)) (37.13.11)

in a natural way, where (37.13.8) is considered as a left module over A, as in the preceding paragraph. This uses isomorphisms as in Section 1.13 for modules without gradings. Of course, we can identify (37.13.11) with (37.13.9), by identifying (37.13.8) with _{ε}W, as left modules over A, as in the preceding paragraph.

37.14. STANDARD COMPLEXES AND \overline{X} 839

More precisely, (37.13.11) may be considered as a complex over k, which can be identified with (37.13.9), as a complex over k. We also have that (37.13.10) and (37.13.11) are isomorphic as complexes, so that (37.13.9) and (37.13.10) are isomorphic as complexes. This means that

H(Hom_{A}^{gr}(X,_{ε}W))^{n} (37.13.12)

is isomorphic to H(Hom_{k}^{gr}(\overline{X}, W))^{n} (37.13.13)

for each n. Remember that (37.13.12) corresponds to the nth cohomology of A, as a left augmented ring, with coefficients in _{ε}W for every n ≥ 0, as in Section 33.5. This corresponds to the second part of Proposition 2.3 on p187 of [50].

37.14 Standard complexes and \overline{X}

Let us continue with the same notation and hypotheses as in the previous section, and suppose now that

(37.14.1) A is projective as a module over k.

Remember that the complexes S(A, ε) and N(A, ε) defined in Section 37.11 are proejective resolutions of k, as a left module over A, under these conditions. Thus we can take X to be either of these two complexes in the previous section. The corresponding complexes \overline{X} as in (37.13.1) are

S(ε, A, ε) = k ⨂_{A}S(A, ε) (37.14.2)

and N(ε, A, ε) = k ⨂_{A}N(A, ε), (37.14.3)

as on p187 of [50].

The grading on S(ε, A, ε) is given by

S(ε, A, ε)_{n} = k ⨂_{A}S(A, ε)_{n} (37.14.4)

for each n, as in Section 7.5. This is {0} when n < 0, and it can be identified with a tensor product of the form

k ⨂_{A}(A ⨂_{k}S̃_{n}(A)) (37.14.5)

when n ≥ 0, using the description of S(A, ε)_{n} in Section 37.11. It follows that we can take S(ε, A, ε)_{n} = S̃_{n}(A) (37.14.6)

when n ≥ 0, with suitable identifications. This uses an isomorphism as in Section 1.12, and the fact that k satisfies the requirements of k ⨂_{A} A, as a module over k.

Similarly, N(ε, A, ε)_{n} = k ⨂_{A}N(A, ε)_{n} (37.14.7) for each n, which is equal to {0} when n < 0. If n ≥ 0, then we can take

N(ε, A, ε)_{n} = Ñ_{n}(A), (37.14.8)

with suitable identifications, as on p187 of [50]. This uses the description of N(A, ε)_{n} in Section 37.11, and the same type of isomorphisms as in the preceding paragraph.

Using the differentiation operators

d_{S(A,ε)} and d_{N(A,ε)} (37.14.9)

on S(A, ε) and N(A, ε), respectively, as in Section 37.12, we get differentiation operators d_{S(ε,A,ε)} and d_{N(ε,A,ε)} (37.14.10)

on S(ε, A, ε) and N(ε, A, ε), respectively, as in Section 7.5. More precisely, for each n ∈ Z, we get homomorphisms

(37.14.11) d_{S(ε,A,ε),n} from S(ε, A, ε)_{n} into S(ε, A, ε)_{n−1}

and (37.14.12) d_{N(ε,A,ε),n} from N(ε, A, ε)_{n} into N(ε, A, ε)_{n−1},

as modules over k. These homomorphisms are obtained from the identity mapping on k and the corresponding homomorphisms

d_{S(A,ε),n} and d_{N(A,ε),n}, (37.14.13)

respectively, in the usual way. Of course, (37.14.11) and (37.14.12) are equal to 0 when n ≤ 0. We can describe these homomorphisms more precisely when n ≥ 1 using the same type of notation as in Sections 35.15 and 37.12.

Remember that S̃_{n}(A), Ñ_{n}(A) are tensor products of n copies of A, A′, respectively, when n ≥ 1, and that they are equal to k when n = 0, as in Section 37.11. If a_{1}, …, a_{n} ∈ A for some n ≥ 1, then

a_{1} ⊗ ⋯ ⊗ a_{n} (37.14.14)

defines an element of S̃_{n}(A), and

q(a_{1}) ⊗ ⋯ ⊗ q(a_{n}) (37.14.15)

defines an element of Ñ_{n}(A), where q is the natural quotient mapping from A onto A′, as in Section 35.12. We may use the notation

[a_{1}, …, a_{n}] (37.14.16)

for (37.14.15), as in Sections 35.15 and 37.12. If we are working with S̃_{n}(A), then we may use (37.14.16) for (37.14.14), as before. In both cases, [ ] may be interpreted as being the multiplicative identity element in k, as on p176 of [50].

If n ≥ 1, then

d_{N(ε,A,ε),n}([a_{1}, …, a_{n}]) = ε(a_{1}) [a_{2}, …, a_{n}]

+ ∑_{l=1}^{n−1}(−1)^{l} [a_{1}, …, a_{l} a_{l+1}, …, a_{n}] (37.14.17)

+(−1)^{n}[a_{1}, …, a_{n−1}] ε(a_{n}),

as on p187 of [50]. If n = 1, then the sum in the middle of the right side should be interpreted as being 0, and we get that

d_{N(ε,A,ε),1}([a_{1}]) = ε(a_{1}) [ ] − [ ] ε(a_{1}) = 0, (37.14.18)

as in [50].

If we are working with S(ε, A, ε)_{n}, then we may use the notation (37.14.16) for (37.14.14), and we get the same expressions for d_{S(ε,A,ε),n} as in the preceding paragraph.

Semigroups and augmentations

38.1 Semigroup rings and augmentations

Let A be a ring with a multiplicative identity element e_{A}, and let Σ be a semigroup, with the semigroup operation expressed multiplicatively, and with an identity element e_{Σ}. Remember that the corresponding semigroup ring A(Σ) may be defined as in Section 4.9. The element of A(Σ) corresponding to e_{Σ} is the multiplicative identity element of A(Σ), which may be expressed as e_{A} e_{Σ}, to be precise.

Let ε be a ring homomorphism from A(Σ) onto A, so that

ε(e_{Σ}) = e_{A}. (38.1.1)

Using ε, A may be considered as a left and right module over A(Σ), as in Section 2.9. Thus A(Σ) may be considered as a left and right augmented ring with respect to ε, as in Sections 33.1 and 33.2.

Put μ(x) = ε(x) (38.1.2)

for every x ∈ Σ, where x is considered as an element of A(Σ) on the right side, which could be expressed as e_{A} x, to be more precise. Thus μ defines a mapping from Σ into A, with μ(x y) = μ(x) μ(y) (38.1.3)

for every x, y ∈ Σ. Of course,

μ(e_{Σ}) = e_{A}, (38.1.4)

by (38.1.1).

Let us say that ε is a multiplicative augmentation of A(Σ) if

ε(a e_{Σ}) = a (38.1.5) for every a ∈ A, as on p148 of [50]. We shall normally be concerned only with multiplicative augmentations of A(Σ), as in [50]. In this case,

ε(a x) = ε(a e_{Σ} x) = ε(a e_{Σ}) ε(x) = a ε(x) (38.1.6)

for every a ∈ A and x ∈ Σ. This implies that

(38.1.7) ε is uniquely determined on A(Σ) by μ,

as on p148 of [50].

Let (38.1.8) Z(A) = {a ∈ A : a b = b a for every b ∈ A}

be the center of A as a ring, which is a subring of A that contains e_{A}. If ε is a multiplicative augmentation of A(Σ), then

(38.1.9) μ(x) ∈ Z(A) for every x ∈ Σ,

as on p148 of [50]. This follows from the fact that elements of A and Σ commute with each other in A(Σ), by construction.

Conversely, suppose that μ is a mapping from Σ into A that satisfies (38.1.3), (38.1.4), and (38.1.9). Consider the mapping ε from A(Σ) into A defined by

ε( ∑_{j=1}^{n} a_{j} x_{j}) = ∑_{j=1}^{n} a_{j} μ(x_{j}) (38.1.10)

for every a_{1}, …, a_{n} ∈ A and x_{1}, …, x_{n} ∈ Σ. It is easy to see that ε defines a ring homomorphism from A(Σ) onto A that is a multiplicative augmentation of A(Σ) under these conditions, as on p148 of [50].

The semigroup Σ together with a mapping μ from Σ into A that satisfies (38.1.3), (38.1.4), and (38.1.9) may be called an augmented semigroup (with multiplicative identity element), as on p188 of [50].

38.2 Semigroup algebras and augmentations

Let k be a commutative ring with a multiplicative identity element 1_{k}, and let Σ be a semigroup, with semigroup operation expressed multiplicatively, and with an identity element e_{Σ}. The corresponding semigroup ring k(Σ), as in Section 4.9, may be considered as an algebra over k.

Suppose that ε_{k} is a ring homomorphism from k(Σ) onto k, so that ε_{k}(e_{Σ}) = 1_{k}. We can define a mapping μ_{k} from Σ into k by (38.1.2), which satisfies (38.1.3) and μ_{k}(e_{Σ}) = 1_{k}, (38.2.1)

as before.

The condition that ε_{k} be a multiplicative augmentation of k(Σ) is equivalent to asking that ε_{k} be a homomorphism from k(Σ) onto k, as algebras over k. If μ_{k} is any mapping from Σ into k that satisfies (38.1.3) and (38.2.1), then (38.1.10) defines a homomorphism from k(Σ) onto k, as algebras over k, as before.

Let A be an associative algebra over k, with multiplicative identity element e_{A}, and let A(Σ) be the corresponding semigroup algebra of Σ over A, as in Section 4.9. Suppose that μ_{k} is a mapping from Σ into k that satisfies (38.1.3) and (38.2.1), and put μ_{A}(x) = μ_{k}(x) e_{A} (38.2.2)

for every x ∈ Σ. This defines a mapping from Σ into A that satisfies (38.1.3), (38.1.4), and (38.1.9). It follows that (38.1.10) defines a ring homomorphism ε_{A} from A(Σ) onto A that is a multiplicative augmentation of A(Σ), as before. Note that ε_{A} is also a homomorphism from A(Σ) into A, as algebras over k.

Any ring with a multiplicative identity element may be considered as an associative algebra over its center. Using this, a multiplicative augmentation of the corresponding group ring may be considered to be as in the preceding paragraph, as on p188 of [50].

Of course, k(Σ) is freely generated by elements of Σ, as a module over k. It is easy to see that A(Σ) corresponds to a tensor product

A ⨂_{k}k(Σ) (38.2.3)

of A and k(Σ) over k, as an algebra over k, as in Section 4.1. This was mentioned in Section 34.13, and corresponds to a remark on p188 of [50].

Let ε_{k} be a homomorphism from k(Σ) onto k, as algebras over k, and let μ_{k} be the corresponding mapping from Σ into k, as in (38.1.2). This leads to a mapping μ_{A} from Σ into A as in (38.2.2), and a ring homomorphism ε_{A} from A(Σ) onto A that is a multiplicative augmentation of A(Σ), as in (38.1.10). Remember that A satisfies the requirements of A ⨂_{k} k, as a module over k. One can check that multiplication on A is compatible with multiplication on A ⨂_{k} k, as a tesor product of algebras over k, as in Section 4.1. Using this identification, one can verify that ε_{A} corresponds to the homomorphism from (38.2.3) into A ⨂_{k} k obtained from the identity mapping on A and ε_{k}, as on p188 of [50].

Alternatively, ε_{k} e_{A} (38.2.4)

defines a homomorphism from k(Σ) into the center of A, as algebras over k. It is easy to see that ε_{A} corresponds to the homomorphism from (38.2.3) into A obtained from the identity mapping on A and (38.2.4) as in Section 4.1. Of course, this is basically the same as the description in the preceding paragraph.

This means that we are in the same type of situation as in Section 37.6, with k(Σ), A in the roles of A, C before. Similarly, C ⨂_{k} A in the earlier discussion corresponds to (38.2.3) here, which we identify with A(Σ). The homomorphism ϕ from A into C ⨂_{k} A before now corresponds to the homomorphism

(38.2.5) ϕ from k(Σ) into A(Σ),

as algebras over k, that sends an element of k(Σ) to its product with e_{A}.

Of course, ψ(t) = t e_{A} (38.2.6)

defines a homomorphism from k into A, as algebras over k, which corresponds to the analogous homomorphism from k into C in Section 37.6. Note that the homomorphisms ε_{A}, ε_{C ⨂_{k} A} before correspond to ε_{k}, ε_{A}, respectively, here. It is easy to see that ε_{A} ∘ ϕ = ψ ∘ ε_{k}, (38.2.7)

as before. In particular, ϕ(ker ε_{k}) ⊆ ker ε_{A}, (38.2.8)

so that ϕ is a homomorphism from k(Σ) into A(Σ), as augmented rings, as in Section 33.9.

Let V be a module over k that is a right module over A(Σ). We may consider V as a right module over k(Σ), using the identification of A(Σ) with (38.2.3), or equivalently using ϕ. We can use ϕ to get a homomorphism

(38.2.9) from the homology of k(Σ) into the homology of A(Σ), as left augmented rings, with coefficients in V,

as in Section 37.6.

Similarly, let Z be a module over k that is a left module over A(Σ), and which may be considered as a left module over k(Σ), as before. We can use ϕ to get a homomorphism

(38.2.10) from the cohomology of A(Σ) into the cohomology of k(Σ), as left augmented rings, with coefficients in Z,

as in Section 37.6 again.

Note that k(Σ) is projective as a module over k, because it is free as a module over k. It follows that

(38.2.11) the homomorphisms as in (38.2.9) and (38.2.10) are isomorphisms,

as in Section 37.6. This corresponds to (1) and (1a) on p188 of [50].

Observe that k(Σ) is a supplemented algebra over k with respect to ε_{k}, as on p188 of [50].

38.3 Semigroups and homomorphisms

Let Σ, Σ′ be semigroups, with the semigroup operations expressed multiplicatively, and with identity elements e_{Σ}, e_{Σ′}, respectively. Also let ϕ be a homomorphism from Σ′ into Σ, as semigroups, with

ϕ(e_{Σ′}) = e_{Σ}. (38.3.1)

Let k be a commutative ring with a multiplicative identity element, and let k(Σ), k(Σ′) be the semigroup algebras of Σ, Σ′ with coefficients in k, as in Section 4.9. If we identify Σ, Σ′ with subsets of k(Σ), k(Σ′) in the obvious way, then ϕ can be extended to a homomorphism

(38.3.2) from k(Σ′) into k(Σ),

as modules over k, and in fact as algebras over k.

Suppose that μ = μ_{Σ}, μ′ = μ_{Σ′} are augmentations of Σ, Σ′, respectively, as semigroups, with values in k, as in Section 38.1. If

μ′ = μ ∘ ϕ (38.3.3)

on Σ′, then ϕ is said to be a map or homomorphism of augmented semigroups, as on p189 of [50].

Let ε = ε_{Σ}, ε′ = ε_{Σ′} be the homomorphisms from k(Σ), k(Σ′) onto k, as algebras over k, corresponding to μ, μ′, respectively, as in (38.1.10). Suppose that (38.3.3) holds, which implies that

ε′ = ε ∘ ϕ (38.3.4)

on k(Σ′). This means that ϕ defines a homomorphism from k(Σ′) into k(Σ), as supplemented algebras with respect to ε′, ε, respectively, as in Section 37.5. This corresponds to a remark on p189 of [50].

Let V be a module over k that is a right module over k(Σ). This means that V may be considered as a right module over k(Σ′), using ϕ, as in Section 2.9. We can use ϕ to get a homomorphism

(38.3.5) from the homology of k(Σ′) into the homology of k(Σ), as left augmented rings, with coefficients in V,

as in Section 37.5.

Similarly, let Z be a module over k that is a left module over k(Σ), so that Z may be considered as a left module over k(Σ′), using ϕ. We can use ϕ to get a homomorphism

(38.3.6) from the cohomology of k(Σ), into the cohomology of k(Σ′), as left augmented rings, with coefficients in Z,

as in Section 37.5 again. These homomorphisms are mentioned on p189 of [50].

One may be interested in conditions under which these homomorphisms are isomorphisms, as in Section 33.10. We have seen that the combination of two conditions is sufficient for this, and we would like to restate these conditions for this case. These conditions are necessary to get an isomorphism as in (38.3.5) when V = k(Σ), considered as a right module over itself, as before.

Note that the augmentation modules of k(Σ), k(Σ′) are equal to k, considered as left modules over k(Σ), k(Σ′) using ε, ε′, respectively. The augmentation module of k(Σ) may be considered as a left module over k(Σ′), using ϕ, as in Section 33.9. This is the same as considering k as a left module over k(Σ′) using ε′, because of (38.3.4).

Similarly, k(Σ) may be considered as a right module over itself, and thus a right module over k(Σ′), using ϕ, as in Section 33.9. Let

k(Σ) ⨂_{k(Σ′)}k (38.3.7)

be a tensor product of k(Σ), as a right module over k(Σ′), and k, as a left module over k(Σ′), over k(Σ′).

The mapping ψ from the augmentation module of k(Σ′) into the augmentation module of k(Σ), as in Section 33.9, corresponds to the identity mapping on k here, because of (38.3.4). There is an obvious mapping

(38.3.8) from k(Σ) × k into k,

corresponding to the action of k(Σ) on k on the left using ε. This mapping is bilinear over k, and satisfies the appropriate compatibility condition with the actions of k(Σ′) on k(Σ) on the right, and on k on the left, to get a homomorphism (38.3.9) from (38.3.7) into k,

as modules over k. More precisely, this compatibility condition uses (38.3.4). This homomorphism corresponds to the one called g in Section 33.9.

The first of the two conditions that we need from Section 33.10 is that

(38.3.10) the homomorphism as in (38.3.9) is an isomorphism.

This corresponds to condition (i) in Proposition 3.1 on p189 of [50]. Remember that

(38.3.11) the nth homology group of k(Σ′), as a left augmented ring with coefficients in k(Σ), as a right module over k(Σ′) using ϕ

is as in Section 33.3 for each n ≥ 0. The second condition that we need from Section 33.10 is that (38.3.12) (38.3.11) is equal to {0}

when n > 0. This corresponds to condition (ii) in Proposition 3.1 on p189 of [50].

Suppose that (38.3.10) holds, and that (38.3.12) holds for every n > 0. This implies that the homomorphisms as in (38.3.5) and (38.3.6) are isomorphisms, as in Section 33.10. This is part of Proposition 3.1 on p189 of [50].

Let X be a projective resolution of k, as a left module over k(Σ′). Also let

k(Σ) ⨂_{k(Σ′)}X (38.3.13)

be a tensor product of k(Σ), as a right module over k(Σ′), and X, as a left module over k(Σ′), over k(Σ′). This may be considered as a complex over k, as in Section 7.5. Remember that (38.3.11) is isomorphic to the nth homology group of (38.3.13) for each n ≥ 0, as in Section 33.3.

More precisely, (38.3.13) may be considered as a left module over k(Σ), and a complex over k(Σ). In fact,

(38.3.14) (38.3.13) is a projective resolution of k, as a left module over k(Σ)

under these conditions, as in Section 33.10. This is another part of Proposition 3.1 on p189 of [50].

38.4 Some related surjectivity conditions

Let us continue with the same notation and hypotheses as in the previous section. Observe that

(38.4.1) the homomorphism as in (38.3.9) is a surjection.

This can be seen by considering e_{Σ} as an element of k(Σ), and using the fact that ε(e_{Σ}) = μ(e_{Σ}) is the multiplicative identity element in k.

Consider a tensor product

k(Σ′) ⨂_{k(Σ′)}k (38.4.2)

of k(Σ′), as a right module over itself, and k, as a left module over k(Σ′) using ε′, as in the previous section, over k(Σ′). Of course, this is isomorphic to k, as a modul over itself, in the usual way, and we shall use this in a moment.

Remember that k(Σ) may be considered as a right module over k(Σ′), as in the previous section. Using this, it is easy to see that ϕ is a homomorphism from k(Σ′) into k(Σ), as right modules over k(Σ′). This was also mentioned in Section 2.9, for homomorphisms between arbitrary associative algebras.

Using ϕ and the identity mapping on k, we get a homomorphism

(38.4.3) from (38.4.2) into (38.3.7),

as modules over k. The composition of this homomorphism with the one as in (38.3.9) is a homomorphism

(38.4.4) from (38.4.2) into k,

as modules over k. This homomorphism is the same as the one that can be obtained from the mapping from k(Σ′) × k into k corresponding to the action of k(Σ′) on k on the left using ε′. More precisely, this uses the analogous description of the homomorphism as in (38.3.9), and (38.3.4).

It follows that

(38.4.5) the homomorphism as in (38.4.4) is an isomorphism,

because k satisfies the requirements of a tensor product as in (38.4.2). In particular, this implies (38.4.1).

Suppose now that ϕ(Σ′) = Σ. (38.4.6)

This implies that ϕ(k(Σ′)) = k(Σ). (38.4.7)

This means that

the homomorphism as in (38.4.3) is a surjection.

(38.4.8)

One can check that (38.3.10) holds in this case, because of (38.4.5). This corresponds to a remark after the statement of Proposition 3.1 on p189 of [50].

38.5 Some augmentations and homomorphisms

Let k be a commutative ring with a multiplicative identity element 1_{k}, and let Σ be a semigroup, with semigroup operation expressed multiplicatively, and with an identity element e_{Σ}. Also let k(Σ) be the corresponding semigroup algebra, with coefficients in k, as in Section 4.9.

If we take (38.5.1) μ_{k}(x) = 1_{k} for every x ∈ Σ,

then μ_{k} satisfies (38.1.3) and (38.2.1), and thus defines a k-valued augmentation of Σ. This is called the unit augmentation of Σ, as on p188 of [50].

Suppose for the moment that if x, y ∈ Σ and

x y = e_{Σ}, (38.5.2)

then x = y = e_{Σ}. (38.5.3)

Let μ_{k} be the k-valued function on Σ defined by

(38.5.4)  μ_{k}(x) = 1_{k} when x = e_{Σ}
= 0 when x ≠ e_{Σ}.

It is easy to see that this satisfies (38.1.3) and (38.2.1), and thus defines an augmentation of Σ. This is called the zero augmentation of Σ, as on p188 of [50].

Let Σ be any semigroup with an identity element again, and let θ be any k-valued augmentation of Σ. There is a unique homomorphism Θ from k(Σ) into itself, as a module over k, such that

Θ(x) = θ(x) x (38.5.5)

for every x ∈ Σ. It is easy to see that Θ is a homomorphism from k(Σ) into itself, as an algebra over k, under these conditions.

Let ε be the augmentation of k(Σ) that corresponds to the unit augmentation of Σ, and let ε_{θ} be the augmentation of k(Σ) that corresponds to θ as in Section 38.1. Observe that ε ∘ Θ = ε_{θ} (38.5.6)

on k(Σ). This corresponds to a remark on p189 of [50].

If θ(x) has a multiplicative inverse in k for each x ∈ Σ, then 1/θ defines an augmentation on Σ too. In this case, Θ is an automorphism of k(Σ).

Note that this condition holds when Σ is a group. Because of this, one often restricts one’s attention to the unit augmentation on Σ when Σ is a group, as in [50].

Let Σ be any semigroup with identity element again. It is easy to see that the kernel of ε is freely generated by elements of the form

x − e_{Σ}, (38.5.7)

with x ∈ Σ and x ≠ e_{Σ}, as a module over k, as on p189 of [50].

38.6 More on the unit augmentation

Let k be a commutative ring with a multiplicative identity element, and let Σ be a semigroup, with the semigroup operation expressed multiplicatively, and with an identity element e_{Σ}. Also let k(Σ) be the corresponding semigroup algebra again, with coefficients in k, as in Section 4.9. In this section, we take k(Σ) to be equipped with the augmentation ε that corresponds to the unit augmentation of Σ.

Let V be a module over k. Remember that an action of Σ on V on the left or right makes V into a left or right module over Σ, as appropriate, and under suitable conditions, as in Section 4.8. This corresponds exactly to V being a left or right module over k(Σ), as appropriate, as in Section 4.9.

We may always consider V as a left or right module over Σ, where the action of each element of Σ is the identity mapping. In this case, we may say that V is trivial as a left or right module over Σ, as appropriate, or that the corresponding action of Σ on V is trivial. This corresponds exactly to trivial actions of k(Σ) on V with respect to ε, as in Section 37.1.

Let W be a module over k that is a left module over Σ, and thus a left module over k(Σ). We may consider W as a right module over Σ, using the trivial action of Σ on W. This makes W into a two-sided module over Σ, because the trivial action of Σ on W commutes with any other action on W. This corresponds to W_{ε} as a two-sided module over k(Σ), as in Section 37.1.

Let us say that w ∈ W is invariant if w is invariant as an element of W, as a two-sided module over Σ, as in Section 36.8. This means that

x · w = w (38.6.1)

for every x ∈ Σ, as before. This is the same as saying that w is invariant in W in the sense of Section 37.2.

A homomorphism f from k(Σ) into W, as modules over k, is uniquely determined by its restriction to Σ, and any W -valued function on Σ can occur in this way. It is easy to see that f is a crossed homomorphism, as in Section 37.3, if and only if f(x y) = x · f(y) + f(x) (38.6.2)

for every x, y ∈ Σ.

If W is trivial as both a left and right module over Σ, then (38.6.2) reduces to f(x y) = f(x) + f(y) (38.6.3)

for every x, y ∈ Σ.

38.7 Free semigroups and augmentations

Let E be a nonempty set, and let Σ(E) be the free semigroup generated by E, as in Section 4.10. Also let k be a commutative ring with a multiplicative identity element, and let k(Σ(E)) be the corresponding semigroup algebra with coefficients in k, as in Section 4.9.

Let μ be a k-valued augmentation of Σ(E), and let ε be the corresponding augmentation of k(Σ(E)), as in Section 38.1. Note that μ is uniquely determined by its values on E, which may be arbitrary elements of k.

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Index

A(Σ), 103 A[[T_{1}, …, T_{n}]], 93 A[d], 121 A[T_{1}, …, T_{n}], 94 A[u], 629 A[u_{1}, …, u_{n}], 630 A[X^{E}], 634 A[{u_{j}}_{j∈I}], 632 A^{op}, 20 A^{en}, 775 acyclic left complexes, 273 acyclic right complexes, 280 affine domains, 500 affine rings, 500 algebra representations, 100 algebraic closures, 477 algebraic dependence on sets, 580 algebraic elements, 472, 629 algebraic field extensions, 472 algebraic independence, 578, 630, 632 algebraically closed fields, 477 algebraically complete fields, 477 algebraically dependent sets, 580 algebras

affine, 500 associative, 498 centers, 499 commutative, 498 homomorphisms, 499 identity elements, 498 in the strict sense, 498 quotient algebras, 526 two-sided ideals, 526

isomorphisms, 499 subalgebras, 499

annihilator ideal, 326 Artin ring, 448 Artinian modules, 446 Artinian rings, 446 associated gradings, 137 associated prime ideals, 426 associates

in commutative rings, 381

in commutative semigroups, 529 augmentation ideals, 726, 728 augmentation modules, 726, 728 augmentations of left complexes, 272 augmentations of right complexes, 279 augmented rings, 726, 728 homomorphisms, 737 augmented semigroups, 843 homomorphisms, 846

automorphisms of fields over a subfield, 480

bases for free modules, 4 belonging to an ideal, 426

embedded prime ideals, 426 isolated prime ideals, 426 minimal prime ideals, 426

bi-gradings, 137

bihomogeneous elements, 137 bihomogeneous submodules, 138 bimodules, 15

c(X, Z), 4, 684 c_{00}(X, Z), 4 C_{A}(E), 567, 568 C_{A}^{B}(E), 567 C_{A}^{V} (E), 568 cancellation laws left, 528 right, 528

center of a ring, 843 of an algebra, 499

centralizers, 567, 568 chains of prime ideals, 368 chains of submodules, 595 characters, 471 cochains, 792 cohomology of algebras, 781 cohomology of augmented rings, 732 coimage, 114 cokernel, 83, 114 comaximal ideals, 418 compatible gradings, 269 complementary ideals, 349 complexes, iii, 132

maps between them, 133, 152 composition series of modules, 595 compositum of subfields, 473 content of a polynomial, 541 content of a polynomial

p-content, 541 content of a polynomial, 535 contractions of ideals, 325 contravariant ϕ-extensions, 47 coprime ideals, 418 covariant ϕ-extensions, 44 crossed homomorphisms, 554, 778, 819

principal, 555, 780, 821

decomposable ideals, 424 Dedekind domains, iii, 313, 391 Dedekind rings, iii, 313 degree of inseparability, 511 degrees of field extensions, 472 degrees of polynomials, 472, 485, 625 degrees of rational functions, 654 δ(V), 297 Der(A), 523 Der(A, B), 552 Der(A, V), 554 derivations

module-valued, 554, 778 inner, 555, 780 on algebras, 523, 552

quotient algebras, 526 on associative algebras

inner, 523, 553 on rings, 525 determinants, 647 dimension of a commutative ring, 368 direct products, 2 direct families of homomorphisms, 27 direct limits, 60 of sets, 700

direct sums, 2

direct systems, 60, 238 homomorphisms between, 62 of sets, 699 discrete valuations, 377 discrete valuation rings, 379 distinguished extensions of fields, 474 divisible elements, 297 divisible modules, 297 division algebras, 263 division rings, 263 domain (integral), 296 double complexes, iii, 140

maps between them, 141, 155 double gradings, iii, 137 negative, 138 positive, 138 dual numbers, iii, 121, 553

E_{A}(x_{1}, …, x_{n}), 110 E_{V} (x_{1}, …, x_{n}), 111 embeddings of fields, 476 over a subfield, 476 over other embeddings, 476

entire rings, 296, 495 enveloping algebra, 775 equivalent valuations, 375 extended formal power series, 659 extension of scalars, 44 extensions of ideals, 325 exterior rings, 110

F_{k}(x_{1}, …, x_{m}), 807 factorial rings, 530 faithful modules, 400 finite field extensions, 472 finitely generated fields, 474 first uniqueness theorem, 426 formal derivatives, 661

partial derivatives, 98, 524 formal polynomials, 93, 471, 500, 574

compositions, 651 formal power series, 92, 500 formal rational functions, 521, 579 fractional ideals, 307, 386 free algebras, 807 free modules, 4, 12 free rings, 105, 268 free semigroups, 105 functors, ii

derived, ii

G(A), 647 G(Σ), 529 Gal(K/k), 480 Galois extensions, 491 Galois groups, 480 Gauss’ lemma, 535, 543, 545 Gaussian integral domains, 530 Gaussian semigroups, 530 general linear groups, 647, 648 GL(n, A), 647 GL(V), 647 going-up theorem, 415 graded associative algebras, 267 graded modules, iii, 130

negative, 130

over graded rings, 269

positive, 130 graded rings, 267 Grassmann rings, 110 greatest common divisors, 383, 531

hereditary rings, iii, 252, 312 Hilbert’s basis theorem, 667 Hom(ϕ, ψ), 147 Hom(V, W) = Hom_{A}(V, W), 13 Hom(V, Z) = Hom_{k}(V, Z), 3 Hom^{gr}(ϕ, ψ), 149 Hom_{A}^{gr}(V, W), 147 homogeneous components, 130 homogeneous elements, 130 homogeneous formal polynomials, 268, homogeneous submodules, 130 homology module, iii, 115 homology of associative algebras, 781 homology of augmented rings, 729 homomorphisms, 116

augmentation, 726, 728

bidegree, 138

components, 131, 138

degree, 131

of augmented rings, 737

of augmented semigroups, 846 of supplemented algebras, 824 homotopies, 117, 134, 141, 153, 155

ideal quotients, 327

idempotent elements, 350 identity matrices, 101, 674 inductive limits, 60 of sets, 700

inductive systems, 60, 238

of sets, 699 injective modules, ii, 41 injective resolutions, ii, 280 injective right complexes, 279 inseparable degree, 511 inseparable polynomials, 485 integral closures of subrings, 402 integral domains, 296

in the strict sense, 528

normal, 497 integral elements of larger rings, 399 integral ideals, 307 integrality over a subring, 402 integrally closed integral domains, 399 integrally closed subrings, 399 invariant elements, 568, 783, 809, 818,

850 inversible ideals, 308 invertible ideals, 308, 387 involutions, 21 irreducible elements

of a commutative semigroup, 529 of an integral domain, 382

irreducible ideals, 421

isolated primary components, 439 isolated sets of ideals, 438 Jacobson radical, 338

(k : M), 307 k(X^{E}), 639 k(X_{1}, …, X_{n}), 521 k({X_{j}}_{j∈I}), 579 k_{I}, 322 Krull–Akizuki theorem, 605

left augmentations, 726 left augmented rings, 726 left complexes, iii, 272 left exactness, 30, 33 left modules, 10 left vector spaces, 263 Leibniz formula, 551 lengths

of chains of prime ideals, 368 of chains of submodules, 595 of modules, 595

liftings of field extensions, 473 linear dependence, 263 linear independence, 263, 578 linear mappings, 11 local rings, 108

M(V, W), 148 M^{r}(V, W), 148 M_{m,n}(A), 673 M_{n}(A), 21, 101, 647, 673 maps between complexes, 133, 152

double complexes, 141, 155 maps of augmented rings, 737 maps of augmented semigroups, 846 maps of supplemented algebras, 824 maps over homomorphisms, 275, 282 minimal polynomials, 473 modular law, 417 module homomorphisms, 11 module isomorphisms, 12 modules

over associative algebras, 10, 11 over semigroups, 102 with differentiation, iii, 114

monic polynomials, 472 Mult(I), 573 multi-indices, 92, 573 multiple roots, 501 multiplication operators, 551 multiplicative augmentations, 842 multiplicatively closed sets, 316

saturated, 433 multiplicities of roots, 501

Nakayama’s lemma, 341 nilpotent elements, 323 nilradical, 323 Noetherian modules, 257 Noetherian rings, 258, 313 normal field extensions, 480 normal field extensions, 480 normalized standard complex, 796

opposite algebra automorphisms, 20 homomorphisms, 20 isomorphisms, 20 opposite algebras, 20 opposite semigroup

homomorphisms, 102 opposite semigroups, 102

P (V), 648

partial derivatives, 98, 524

partial orderings, 59 perfect fields, 501 P GL(n, k), 650 P GL(V), 649 ϕ-injective modules, 51 ϕ-projective modules, 49 pre-directed sets, 59 pre-directed system, 59 pre-orderings, 59 primary decompositions, 424

minimal, 425 primary ideals, 363 prime elements

of commutative rings, 382

of commutative semigroups, 529 primitive polynomials, 543 primitive elements, 516 primitive polynomials, 536 principal ideal domains, 383 principal rings, 547 products of ideals, 326 products of ideals and modules, 327 projective general linear groups, 649,

650 projective isomorphisms, 649 projective left complexes, 273 projective modules, ii, 38 projective resolutions, ii, 274 projective spaces, 648 proper modules, 351 Prüfer rings, 311 purely inseparable elements, 506 purely inseparable extensions, 508 purely transcendental extensions, 581

Q, 399 Q_{k}, 301 quotient modules, 11

with differentiation, 118

R, 632 r(E), 430 r(I), 328 radicals of ideals, 328 radicals of sets, 430 ranks of modules, 598 reduced commutative rings, 500 reducible elements of integral domains,

382 restriction of scalars, 44 right augmentations, 728 right augmented rings, 728 right complexes, iii, 279 right exactness, 36 right modules, 11 right vector spaces, 263 rings of fractions, 317 root fields, 479 roots of polynomials, 472

S^{−1} f, 318 S^{−1} k, 317 S^{−1} V, 318 satellites, 710, 714, 720 saturations of ideals, 435 second uniqueness theorem, 438 semi-hereditary rings, 255, 311 semigroup algebras, 103 semigroup representations, 102 semisimple modules, 57 separable degree, 488 separable elements, 486 separable field extensions, 487, 489 separable polynomials, 485 simple finite filtrations of modules, 595 simple modules, 57, 595 simple roots, 501 SL(n, A), 647 SL(V), 648 special linear groups, 647, 648 splitting fields, 479 standard complexes, 790 submodules, 10

with differentiation, 118 supplemented algebras, 817 homomorphisms, 824

supports of functions, 4

τ(V), 296

tensor products, ii, 6, 16 torsion elements, 296 torsion modules, 296 torsion-free modules, 296 total differentiation operators, 140 total ring of fractions, 434 traces on field extensions, 493 transcendence bases, 580 transcendence degree, 581 transcendental elements, 629 translations of field extensions, 473 trivial modules, 850 trivial operators, 818 trivial valuation, 375 two-sided modules, 554, 775, 809 homomorphisms, 555, 778

unique factorization domains, 530 unit augmentation, 849

V ((T)), 659 V [[T_{1}, …, T_{n}]], 92 V [T_{1}, …, T_{n}], 93 V [X^{E}], 632 V [{T_{j}}_{j∈I}], 574 V_{I}, 322 valuation ring of a valuation, 376 valuation rings, 397 valuations, 374

on integral domains, 377 value group of a valuation, 375 vanishing to some order, 96

Z, 130 Z(A), 499, 843 Z_{+}, 92 zero augmentation, 849