Virtual Gorensteinness over group algebras

Abstract

Let $\Gamma$ be a finite group, and let $\Lambda$ be any Artin algebra. It is shown that the group algebra $\Lambda \Gamma$ is virtually Gorenstein if and only if $\Lambda \Gamma \text{'}$ is virtually Gorenstein, for all elementary abelian subgroups $\Gamma \text{'}$ of $\Gamma$. We also extend this result to cover the more general context. Precisely, assume that $\Gamma$ is a group in Kropholler’s hierarchy $\mathbf{H}\mathfrak{F}$, $\Gamma \text{'}$ is a subgroup of $\Gamma$ of finite index, and $R$ is any ring with identity. It is proved that, in certain circumstances, that $R\Gamma$ is virtually Gorenstein if and only if $R\Gamma \text{'}$ is so.