Gorenstein analogues of a projectivity criterion over group algebras

TL;DR

This paper extends classical projectivity criteria to Gorenstein homological algebra over group rings, generalizing prior results from integral to more general commutative base rings.

Contribution

It formulates and answers Gorenstein analogues of a classical projectivity question for group rings over broader base rings, building on recent advances in Gorenstein homological algebra.

Findings

Develops Gorenstein projective, flat, and injective analogues of classical criteria.

Generalizes results from integral group rings to more general commutative rings.

Improves and broadens existing Gorenstein homological algebra results.

Abstract

We formulate and answer Gorenstein projective, flat, and injective analogues of a classical projectivity question for group rings under some mild additional assumptions. Although the original question, that was proposed by Jang-Hyun Jo in 2007, was for integral group rings, in this article, we deal with more general commutative base rings. We make use of the vast developments that have happened in the field of Gorenstein homological algebra over group rings in recent years, and we also improve and generalize several existing results from this area along the way.