广义四元数群Tate上同调的二次乘法
Secondary multiplication in Tate cohomology of generalized quaternion groups
摘要：
设k是域，G是有限群.根据D.Benson，H.Krause和S.Schwede的一个定理，Tate上同调HH^{3，-1} H*（G）的Hochschild上同调中存在一个标准元，它具有以下性质：给定任意分次H*（G）-模X，Ext^{3，-1}（X，X）中标准元的象为零当且仅当X对某个KG-模M同构于H*（G，M）的直和项。特别地，如果标准元为零，则每个模都是可实现H*（G）-模的直和项。我们证明，最后一句话的逆是不正确的，通过详细研究的情况下，广义四元数群。设k是特征为2的域，G是2^n阶广义四元数，n>2。我们证明了正则元对所有n都是非平凡的，但存在一个H*（G）-模检测这种非平凡性当且仅当n=3。
英文摘要：
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 HH^{3,-1} H*(G) with the following property: Given any graded H*(G)-module X, the image of the canonical element in Ext^{3,-1}(X,X) is zero if and only if X is isomorphic to a direct summand of H*(G,M) for some kG-module M. In particular, if the canonical element vanishes, then every module is a direct summand of a realizable 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>2. We show that the canonical element is non-trivial for all n, but there is an H*(G)-module detecting this non-triviality if and only if n=3.
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