Auslander-Reiten conjecture and finite injective dimension of Hom

Abstract

For a finitely generated module $M$ over a commutative Noetherian ring $R$, we settle the Auslander-Reiten conjecture when at least one of ${\mathrm{Hom}}_R(M,R)$ and ${\mathrm{Hom}}_R(M,M)$ 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.

arXiv comment11 pages, to appear in Kyoto J. Math

References in corpus (4)

• Complexity and rigidity of Ulrich modules, and some applications
• Characterizations of regular local rings via syzygy modules of the residue field
• Two generalizations of Auslander-Reiten duality and applications
• Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology

Cited by in corpus (4)

• Complexity and rigidity of Ulrich modules, and some applications
• Auslander-Reiten annihilators
• The Auslander-Reiten Conjecture, Finite $C$-Injective Dimension of $\mathrm{Hom}$, and vanishing of $\mathrm{Ext}$
• On modules whose dual is of finite Gorenstein dimension