Characteristic modules and Gorenstein (co-)homological dimension of groups

Abstract

In this paper, we examine the Gorenstein dimension of modules over the group algebra kG of a group G with coefficients in a commutative ring k. As a Gorenstein analogue of the classical case, we bound this dimension in terms of the Gorenstein dimension of the underlying k-module and the Gorenstein dimension of G over k. Our method is based on the notion of a characteristic module for G, introduced by the second author, and uses the stability properties of the Gorenstein categories. We also examine the class of hierarchically decomposable groups defined by Kropholler and use the module of bounded $Z$-valued functions on such a group G to characterize the Gorenstein flat $ZG$-modules, in terms of flat modules, and the Gorenstein injective $ZG$-modules, in terms of injective modules (by complete analogy with the characterization of Gorenstein projective $ZG$-modules, in terms of projective modules, obtained by Dembegioti and the second author). It follows that, for a group G in Kropholler's class, (a) any Gorenstein projective $ZG$-module is Gorenstein flat and (b) a $ZG$-module is Gorenstein flat if its Pontryagin dual module is Gorenstein injective.