Auslander-Reiten conjecture and finite injective dimension of Hom

TL;DR

This paper proves the Auslander-Reiten conjecture under specific conditions involving finite injective dimensions of certain Hom modules and introduces new characterizations of Gorenstein local rings based on Ext vanishing.

Contribution

It establishes the Auslander-Reiten conjecture for modules where either Hom_R(M,R) or Hom_R(M,M) has finite injective dimension and provides new Gorenstein ring characterizations.

Findings

Proves the Auslander-Reiten conjecture in new cases

Provides new Gorenstein local ring characterizations

Connects finite injective dimension of Hom modules with ring properties

Abstract

For a finitely generated module $ M $ over a commutative Noetherian ring $R$, we settle the Auslander-Reiten conjecture when at least one of ${\rm Hom}_R(M,R)$ and ${\rm Hom}_R(M,M)$ has finite injective dimension. A number of new characterizations of Gorenstein local rings are also obtained in terms of vanishing of certain Ext and finite injective dimension of Hom.