Auslander-type conditions and weakly Gorenstein algebras

TL;DR

This paper explores conditions under which Artin algebras are Gorenstein, providing new characterizations and supporting homological conjectures through properties of Gorenstein projective modules.

Contribution

It offers novel equivalences for (weakly) Gorenstein algebras based on Auslander-type conditions and applies these to support homological conjectures.

Findings

Left quasi Auslander algebras are Gorenstein iff they are weakly Gorenstein.

Algebras satisfying the Auslander condition are Gorenstein iff they are weakly Gorenstein.

Provides reduction of Ausland-Reiten's conjecture regarding Gorenstein algebras.

Abstract

Let $R$ be an Artin algebra. Under certain Auslander-type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander-type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if $R$ is left quasi Auslander, then $R$ is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if $R$ satisfies the Auslander condition, then $R$ is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander--Reiten's conjecture, which states that $R$ is Gorenstein if $R$ satisfies the Auslander condition.