On the ideal case of a conjecture of Auslander and Reiten

TL;DR

This paper investigates a special case of the Auslander-Reiten conjecture, proving that certain ideals satisfy the extension condition and providing a new characterization of regularity via injective dimensions.

Contribution

It demonstrates that a broad class of ideals over commutative Noetherian local rings meet the extension condition, and introduces a novel regularity criterion based on injective dimensions.

Findings

A large class of ideals satisfy the extension condition of the conjecture.

Regularity characterized by injective dimensions of specific ideals.

Advances understanding of the conjecture in the context of commutative Noetherian local rings.

Abstract

A celebrated conjecture of Auslander and Reiten claims that a finitely generated module $M$ that has no extensions with $M\oplus \Lambda$ over an Artin algebra $\Lambda$ must be projective. This conjecture is widely open in general, even for modules over commutative Noetherian local rings. Over such rings, we prove that a large class of ideals satisfy the extension condition proposed in the aforementioned conjecture of Auslander and Reiten. Along the way we obtain a new characterization of regularity in terms of the injective dimensions of certain ideals.