On the Auslander–Reiten conjecture for algebras

Abstract

A recent result of Araya asserts that if the Auslander–Reiten conjecture holds in codimension one for a commutative Gorenstein ring R, then it holds for R. This note extends this result to left Gorenstein R-algebras Λ, whenever R is a commutative Gorenstein ring. This, in particular, implies that any finitely generated self-orthogonal Gorenstein projective Λ-module is projective, provided Λ is an isolated singularity and dim⁡R≥2. Also, some examples of bound quiver algebras satisfying the Auslander–Reiten conjecture are presented.