Finite generation of Tate cohomology

TL;DR

This paper investigates the conditions under which Tate cohomology of modules over finite groups is finitely generated, proposing a conjecture relating cohomology finiteness to support varieties and proving related results.

Contribution

It introduces a conjecture linking Tate cohomology finiteness to support varieties and proves partial results, including the equivalence with periodic cohomology.

Findings

Support for the conjecture in certain modules

Counterexamples to the converse in general

Finiteness of Tate cohomology characterizes periodic groups

Abstract

Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M, we conjecture that if the Tate cohomology $\HHHH^*(G, M)$ of G with coefficients in M is finitely generated over the Tate cohomology ring $\HHHH^*(G, k)$, 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.