Total acyclicity of complexes over group algebras

TL;DR

This paper investigates the properties of modules over group algebras related to Gorenstein homological algebra, establishing a class of groups with stable properties under certain operations and generalizing previous results.

Contribution

It introduces a class of groups satisfying key Gorenstein properties and shows this class is closed under specific algebraic operations, extending known results.

Findings

The class of groups with these properties is closed under the LH operation.

The class is also closed under the { ext{ extbackslash Phi}} operation.

All previously known results are unified within this framework.

Abstract

In this paper, we study group algebras over which modules have a controlled behaviour with respect to the notions of Gorenstein homological algebra, namely: (a) Gorenstein projective modules are Gorenstein flat, (b) any module whose dual is Gorenstein injective is necessarily Gorentein flat, (c) the Gorenstein projective cotorsion pair is complete and (d) any acyclic complex of projective, injective or flat modules is totally acyclic (in the respective sense). We consider a certain class of groups satisfying all of these properties and show that it is closed under the operation LH defined by Kropholler and the operation {\Phi} defined by the second author. We thus generalize all previously known results regarding these properties over group algebras and place these results in an appropriate framework.