Complexity and qoutient categories for group algebras

Abstract

Let k be an algebraically closed field of characteristic p > 0 and let G be a finite group. The complexity of a kG-module is the polynomial rate of growth of the projective resolution of the module. It can also be described as the dimension of the support variety of the cohomology of the module as a module over the cohomology ring of the group. In this paper we consider the thick subcategories M c , of the stable category of all nonprojective kG-modules, consisting of all modules of complexity at most c. Of particular interest are the triangulated quotients Q c = M c M c−1 . It is shown that the set of homomorphisms between two modules M and N in Q c is the localization of Ext ∗ kG(M,N) at an ideal of H ∗(G,k) determined by a general position condition on varieties. One consequence is that if r is the p-rank of G then the endomorphism ring of the trivial module in Q r is the direct sum of t local rings where t is the number of components of the maximal ideal spectrum of H ∗(G,k) of maximum dimension.