Modules with finitely generated cohomology, and singularities of C⁎BG

Abstract

Let G be a finite group and k a field of characteristic p. We conjecture that if M is a kG-module with H⁎(G,M) finitely generated as a module over H⁎(G,k) then as an element of the stable module category StMod(kG), M is contained in the thick subcategory generated by the finitely generated kG-modules and the modules M′ with H⁎(G,M′)=0.We show that this is equivalent to a conjecture of the second author about generation of the bounded derived category of cochains C⁎(BG;k), and we prove the conjecture in the case where the centraliser of every element of G of order p is p-nilpotent. In this case some stronger statements are true, that probably fail for more general finite groups.