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4:00 pm Tuesday, November 21, 2017 AGNT: Brauer p-dimensions of complete discretely valued fieldsby Nivedita Bhaskhar (UCLA) in HBH 227- The Brauer dimension of a field F is defined to be the least number n such that index(A) divides period(A)^n for every central simple algebra A defined over any finite extension of F. One can analogously define the Brauer-p-dimension of F for p, a prime, by restricting to algebras with period, a power of p. The 'period-index' questions revolve around bounding the Brauer (p) dimensions of arbitrary fields. In this talk, we look at the period-index question over complete discretely valued fields in the so-called 'bad characteristic' case. More specifically, let K be a complete discretely valued field of characteristic 0 with residue field k of characteristic p > 0 and p-rank n (= [k:k^p]). It was shown by Parimala and Suresh that the Brauer p-dimension of K lies between n/2 and 2n. We will investigate the Brauer p-dimension of K when n is small and find better bounds. For a general n, we will also construct a family of examples to show that the optimal upper bound for the Brauer-p-dimension of such fields cannot be less than n+1. These examples embolden us to conjecture that the Brauer p-dimension of K lies between n and n+1. The proof involves working with Kato's filtrations and bounding the symbol length of the second Milnor K group modulo p in a concrete manner, which further relies on the machinery of differentials in characteristic p as developed by Cartier. This is joint work with Bastian Haase.
Submitted by ae22@rice.edu |