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

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 $\mathrm{Ext},$ in the context of relative homological algebra.