Secondy multiplication in Tate cohomology of generalized quaternion groups

Abstract

Let $k$ be a field, and let $G$ be a finite group. By a theorem of D. Benson, H. Krause, and S. Schwede, there is a canonical element in the Hochschild cohomology of the Tate cohomology $\gamma_G\in H\! H^{3,-1} \hat{H}^*(G)$ with the following property: Given any graded $\hat{H}^*(G)$-module $X$, the image of $\gamma_G$ in $\mathrm{Ext}^{3,-1}_{\hat{H}^*(G)} (X,X)$ is zero if and only if $X$ is isomorphic to a direct summand of $\smash{\hat{H}^*(G,M)}$ for some $kG$-module $M$. In particular, if $\gamma_G=0$ then every module is a direct summand of a realizable $\hat{H}^*(G)$-module.

We prove that the converse of that last statement is not true by studying in detail the case of generalized quaternion groups. Suppose that $k$ is a field of characteristic $2$ and $G$ is generalized quaternion of order $2^n$ with $n\geq 3$. We show that $\gamma_G$ is non-trivial for all $n$, but there is an $\hat{H}^*(G)$-module detecting this non-triviality if and only if $n=3$.