The Tate-Hochschild cohomology ring of a group algebra

TL;DR

This paper establishes an explicit decomposition of the Tate-Hochschild cohomology ring of a group algebra into a direct sum of Tate cohomology rings of centralizers, providing a practical formula for computations.

Contribution

It introduces a novel isomorphism and explicit cup product formula for the Tate-Hochschild cohomology ring of group algebras, enabling concrete calculations.

Findings

Decomposition of Tate-Hochschild cohomology into centralizer cohomologies

Explicit cup product formula for the cohomology ring

Application to compute cohomology of S_3 in characteristic 3

Abstract

We show that the Tate-Hochschild cohomology ring $HH^*(RG,RG)$ of a finite group algebra $RG$ is isomorphic to a direct sum of the Tate cohomology rings of the centralizers of conjugacy class representatives of $G$. Moreover, our main result provides an explicit formula for the cup product in $HH^*(RG,RG)$ with respect to this decomposition. As an example, this formula helps us to compute the Tate-Hochschild cohomology ring of the symmetric group $S_3$ with coefficients in a field of characteristic 3.