Equivariant Hochschild cohomology of group algebras and relative $\operatorname{Ext}$

TL;DR

This paper investigates conditions for non-trivial equivariant Hochschild cohomology of group algebras under finite group actions, linking it to relative Ext groups in homological algebra.

Contribution

It establishes necessary conditions for non-trivial equivariant Hochschild cohomology and relates it to relative Ext groups for $ ext{Ext}$ computations.

Findings

Necessary conditions for non-trivial first $ ext{Hochschild}$ cohomology under group actions.

Isomorphism between $ ext{equivariant Hochschild cohomology}$ and relative $ ext{Ext}$ groups.

Results apply to group algebras over fields with characteristic dividing the group order.

Abstract

For a finite group $\Gamma$, acting on a finite group $G,$ we find necessary conditions for which the first $\Gamma_0$-equivariant Hochschild cohomology of the group algebra $kG$ is non-trivial, where $k$ is a field of characteristic $p$ dividing the order of $G$ and $\Gamma_0$ is the stabilizer subgroup in $\Gamma$ of some element in $G.$ For any field $k$ we show that the $\Gamma$-equivariant Hochschild cohomology of $\Gamma$-algebras with coefficients in a $\Gamma$-equivariant bimodule (Jensen, 1996) is isomorphic with some $k\Gamma$-relative $\operatorname{Ext},$ in the context of relative homological algebra.