  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.
