Modules with finitely generated cohomology

Abstract

Let $G$ be a finite group and $\mathsf{k}$ a field of characteristic $p$. It is conjectured in a paper of the first author and John Greenlees that the thick subcategory of the stable module category StMod$(\mathsf{k}G)$ consisting of modules whose cohomology is finitely generated over ${\mathsf{H}}^{\ast }(G,\mathsf{k})$ is generated by finite dimensional modules and modules with no cohomology. If the centraliser of every element of order $p$ in $G$ is $p$-nilpotent, this statement follows from previous work. Our purpose here is to prove this conjecture in two cases with non $p$-nilpotent centralisers. The groups involved are $\mathbb{Z}/3^r\times {\Sigma }_3$ ($r>0$) in characteristic three and $\mathbb{Z}/2\times A_4$ in characteristic two. As a consequence, in these cases the bounded derived category of $C^{\ast }BG$ (cochains on $BG$ with coefficients in $\mathsf{k}$) is generated by $C^{\ast }BS$, where $S$ is a Sylow $p$-subgroup of $G$.

11 pages