Algebras that satisfy Auslander's condition on vanishing of cohomology

TL;DR

This paper investigates rings satisfying Auslander's vanishing cohomology condition, proving new properties for AC Artin algebras, including Gorenstein symmetry and the validity of the Auslander-Reiten Conjecture.

Contribution

It introduces the class of AC rings, proves Gorenstein symmetry for AC Artin algebras, and confirms the Auslander-Reiten Conjecture within this class.

Findings

AC Artin algebra that is left-Gorenstein is also right-Gorenstein.

The Auslander-Reiten Conjecture holds for AC rings.

Auslander's G-dimension is functorial for certain AC rings.

Abstract

Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture - by a 2003 counterexample due to Jorgensen and Sega - motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex.