On Gorenstein homological dimension of groups

TL;DR

This paper introduces the Gorenstein homological dimension for groups over rings, explores its properties under ring extensions, and establishes a Gorenstein version of Serre's theorem, linking group finiteness to the dimension.

Contribution

It defines the Gorenstein homological dimension for groups over rings and proves its key properties and relations, including a Gorenstein Serre's theorem and criteria for finiteness.

Findings

Ghd_R G is invariant under flat ring extensions.

Ghd_R G equals Ghd_R H for finite index subgroups.

Finite groups are characterized by Ghd_R G = 0.

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.