# Formal gluing along non-linear flags

## Mauro Porta (UPenn)

In this talk I will survey the recent paper arXiv 1607.04503 (joint with joint with B. Hennion and G. Vezzosi). When X = Spec(A) is an affine noetherian scheme and Z = V(I) is a closed subscheme of X defined by the ideal I, a classical theorem of Artin allows to describe coherent sheaves on X as gluing of coherent sheaves on the formal completion XZ∧ and coherent sheaves over X - Z over the category of coherent sheaves over Coh(Spec(A∧I) - V(I)). It is highly non-trivial to generalize this result to a global setting, the difficulty being explained by the appearance of Spec(A∧I) instead of Spf(A∧I). In our joint work we explain how to construct Coh(Z∧ - Z) in general, and along the way we provide an explicit global description of this category.
We apply this result to decompose the category of coherent sheaves on a surface S in terms of a full flag on S. If time will permit, I will survey the possible further developments of this project, which go in the direction of the Geometric Langlands Program for surfaces and of the study of the geometry of the flag Hilbert scheme.

Tuesday, October 18th, at 4:00pm in HBH 227

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