Auslander-Reiten conjecture and Auslander-Reiten duality

TL;DR

This paper extends Auslander-Reiten duality to Cohen-Macaulay local rings and investigates conditions under which the Auslander-Reiten conjecture holds, linking duality theory with conjectural properties of modules.

Contribution

It generalizes Auslander-Reiten duality to a broader class of rings and identifies conditions that ensure the conjecture's validity in local rings.

Findings

Extended duality to Cohen-Macaulay rings

Identified conditions for the conjecture to hold

Connected duality theory with the conjecture's validity

Abstract

Motivated by a result of Araya, we extend the Auslander-Reiten duality theorem to Cohen-Macaulay local rings. We also study the Auslander-Reiten conjecture, which is rooted in Nakayama's work on finite dimensional algebras. One of our results detects a certain condition that forces the conjecture to hold over local rings of positive depth.