Mathematics Colloquium
Date:Lincoln NE 68508
Affiliation: University of Aberdeen
Local Host: Srikanth Iyengar
Title: Matrix factorisations and modular representations of elementary abelian p-groups
Additional Public Info:
Abstract: When Brauer laid the foundations of modular representation theory, he was interested in characters and their relationship with group structure, and in particular the applications to the classification of finite simple groups. Green and others later gradually moved the emphasis from characters to modules. The work of Quillen, Chouinard, Dade, Carlson and others illustrated the importance of elementary abelian p-groups in understanding modules and cohomology over a general finite group in characteristic p.
Meanwhile, in commutative algebra, the theory of matrix factorisations was developed by Eisenbud and others as a tool for understanding maximal Cohen-Macaulay modules over a hypersurface singularity. More recently, Orlov has shown how to relate the singularity category of any complete intersection to that of a suitable hyper surface. In this lecture I shall describe a modification of Orlov’s construction and
use it to give an equivalence between the derived category D^b(kE) of the group algebra of an elementary abelian p-group E = (Z/p)^r over a field k of characteristic p and the category of reduced graded matrix factorisations of a suitable polynomial in 2r variables, namely
y_1 X^p_1+ … + y_r X^p_r, with deg(y_i) = 1 and deg(X_i) = 0. It is an unnerving exercise even to follow the trivial kE-module through this equivalence. Nonetheless, kE-modules with some desirable properties can be manufactured this way.
Refreshments will be served in 348 Avery 3:30-4pm.
The talk is free and open to the public.
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