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 $\mathsf{StMod}(kG)$, $M$ 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.