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