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 $gamma_Gin H! H^{3,-1} hat{H}^*(G)$ 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 $ngeq 3$. 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 $n=3$.
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机译：假设$ k $是一个字段，而$ G $是一个有限组。根据D. Benson，H。Krause和S. Schwede的一个定理，泰特（Tate）同调$ gamma_G in H ！中的Hochschild同调存在一个规范元素。 H ^ {3，-1} hat {H} ^ *（G）$具有以下属性：给定任何已分级的$ hat {H} ^ *（G）$-module $ X $，图像$ $ mathrm {Ext} ^ {3，-1} _ { hat {H} ^ *（G）}（X，X）$中的gamma_G $当且仅当$ X $同构为直接求和$ smash { hat {H} ^ *（G，M）} $的价格，用于某些$ kG $模块$ M $。特别是，如果$ gamma_G = 0 $，则每个模块都是可实现的$ hat {H} ^ *（G）$模块的直接加法。通过详细研究广义四元数群的情况，我们证明了最后一个陈述的反面是不正确的。假设$ k $是特征$ 2 $的字段，并且$ G $是阶数为$ 2 ^ n $且具有$ n geq 3 $的广义四元数。我们表明，对于所有$ n $，$ gamma_G $都是不平凡的，但是只有当$ n = 3 $时，有一个$ hat {H} ^ *（G）$模块才能检测到这种不平凡。
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