Modules with finitely generated cohomology

TL;DR

This paper proves a conjecture about the structure of modules with finitely generated cohomology for certain finite groups, extending previous results to cases with non p-nilpotent centralisers.

Contribution

The authors verify the conjecture for specific groups with non p-nilpotent centralisers, expanding understanding of module categories in modular representation theory.

Findings

The conjecture holds for groups Z/3^r x S_3 in characteristic three.

The conjecture holds for groups Z/2 x A_4 in characteristic two.

The bounded derived category of C^*BG is generated by C^*BS for these groups.

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 $\mathsf{H}^*(G,\mathsf{k})$ 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 $C^*BG$ (cochains on $BG$ with coefficients in $\mathsf{k}$) is generated by $C^*BS$, where $S$ is a Sylow $p$-subgroup of $G$.