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 ( k G ) consisting of modules whose cohomology is finitely generated over H ⁎ ( G , 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 Z / 3 r × Σ 3 ( r ⩾ 1 ) in characteristic three and Z / 2 × A 4 in characteristic two. As a consequence, in these cases the bounded derived category of C ⁎ B G (cochains on BG with coefficients in k ) is generated by C ⁎ B S , where S is a Sylow p -subgroup of G .