On Gorenstein homological dimension of groups

Abstract

Let $G$ be a group and $R$ be a ring. We define the Gorenstein homological dimension of $G$ over $R$, denoted by ${\mathrm{Ghd}}_{R}G$, as the Gorenstein flat dimension of trivial $RG$-module $R$. It is proved that ${\mathrm{Ghd}}_{S}G\le {\mathrm{Ghd}}_{R}G$ for any flat extension of commutative rings $R\to S$; in particular, ${\mathrm{Ghd}}_{R}G$ is a refinement of ${\mathrm{Ghd}}_{\mathbb{Z}}G$ if $R$ is $\mathbb{Z}$-torsion-free. We show a Gorenstein homological version of Serre's theorem, i.e. ${\mathrm{Ghd}}_{R}G={\mathrm{Ghd}}_{R}H$ for any subgroup $H$ of $G$ with finite index. As an application, $G$ is a finite group if and only if ${\mathrm{Ghd}}_{R}G=0$; this is different from the fact that the homological dimension of any non-trivial finite group is infinity.

Major revision; 10 pages. We appreciate for any comments and suggestions