AC-Gorenstein rings and their stable module categories

TL;DR

This paper introduces AC-Gorenstein rings, a generalization of Gorenstein rings compatible with specific module classes, and demonstrates their stable module categories are well-behaved and modelled by Quillen equivalent structures.

Contribution

It defines AC-Gorenstein rings, links them to existing classes like Gorenstein and Ding-Chen rings, and constructs their stable module categories with Quillen model structures.

Findings

AC-Gorenstein rings unify Gorenstein and Ding-Chen rings.

Stable module categories for AC-Gorenstein rings are compactly generated.

Existence of two Quillen equivalent abelian model structures for these categories.

Abstract

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring which is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo-Gillespie-Hovey. It is also compatible with the notion of $n$-coherent rings introduced by Bravo-Perez: So a $0$-coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$-coherent AC-Gorenstein ring is precisely a Ding-Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects the Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects the Gorenstein AC-injectives.