On Gorenstein homological dimension of groups

Abstract

Let G be a group and R a commutative ring. We define the Gorenstein homological dimension of G over R, denoted by Ghd R G, as the Gorenstein flat dimension of trivial RG-module R. It is proved that for any flat extensionWe show a Gorenstein homological version of Serre's theorem, i.e. Ghd R G = Ghd R H for any subgroup H of G with finite index. As applications, we give a homological characterization for groups by showing that G is a finite group if and only if Ghd R G = 0, and moreover, we use Ghd R G to give an upper bound of Gorenstein flat dimension of modules over the group ring RG.