Finite generation of Tate cohomology

Abstract:

Let $G$ be a finite group and let $k$ be a field of characteristic $p$. Given a finitely generated indecomposable nonprojective $kG$-module $M$, we conjecture that if the Tate cohomology $\hat {H}^*(G, M)$ of $G$ with coefficients in $M$ is finitely generated over the Tate cohomology ring $\hat {H}^*(G, k)$, then the support variety $V_G(M)$ of $M$ is equal to the entire maximal ideal spectrum $V_G(k)$. We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of $k$ in the stable Auslander-Reiten quiver for $kG$, but it is shown to be false in general. It is also shown that all finitely generated $kG$-modules over a group $G$ have finitely generated Tate cohomology if and only if $G$ has periodic cohomology.