On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

Abstract

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and Q a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra RQ, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented. © 2017, Copyright © Taylor & Francis Group, LLC.