Secondary multiplication in Tate cohomology of certain p-groups

TL;DR

This paper explores the properties of a canonical element in the Hochschild cohomology of Tate cohomology for specific p-groups, revealing conditions for module realizability and providing examples of non-realizable modules.

Contribution

It investigates the canonical element in Hochschild cohomology for abelian p-groups and quaternion groups, highlighting its non-triviality and implications for module realization.

Findings

Canonical element's non-triviality in certain p-groups.

Examples of modules not realizable as Tate cohomology modules.

Characterization of modules related to the canonical element.

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 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. We investigate this canonical element in certain special cases, namely that of (finite) abelian p-groups and the quaternion group. In case of non-triviality of the canonical element, we also give examples of non-realizable modules X.