Modules with finitely generated cohomology

Let G be a finite group and 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(kG) consisting of modules whose cohomology is finitely generated over H & lowast;(G, & lowast; (G, k) is generated by finite dimensional modules and modules with no cohomology. If the centraliser of every element of order pin 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 Z/3 r x Sigma 3 3 (r 1) in characteristic three and Z/2 x A4 4 in characteristic two. As a consequence, in these cases the bounded derived category of C & lowast;BG & lowast; BG (cochains on BG with coefficients in k) is generated by C & lowast;BS, & lowast; BS, where S is a Sylow p-subgroup of G. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).