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Auslander-Reiten conjecture and finite injective dimension of Hom

Dipankar Ghosh, Ryo Takahashi

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.

Auslander-Reiten conjecture and finite injective dimension of Hom

Abstract

For a finitely generated module over a commutative Noetherian ring , we settle the Auslander-Reiten conjecture when at least one of and 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.

Paper Structure

This paper contains 3 sections, 18 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Criteria for a module to be free
  3. Characterizations of Gorenstein local rings

Key Result

Theorem \ref{thm:Hom-injdim-finite-consequence}

Let $M$ and $N$ be nonzero $R$-modules such that $\mathop{\mathrm{depth}}\nolimits(N)=d$ and $\mathop{\mathrm{Ext}}\nolimits_R^i(M,N)=0$ for all $1 \leqslant i \leqslant d$. Then $\mathop{\mathrm{Hom}}\nolimits_R(M,N)$ has finite injective dimension if and only if $M$ is free and $N$ has finite inje

Theorems & Definitions (38)

  • Theorem \ref{thm:Hom-injdim-finite-consequence}
  • Theorem \ref{thm:main-2}
  • Conjecture 1.2
  • Corollary 1.3: =\ref{['cor:M-star-ARC']} and \ref{['cor:ARC']}
  • Theorem \ref{thm:characterizations-Gor}
  • Theorem 1.4: Peskine-Szpiro
  • Theorem 1.5: Foxby
  • Theorem 1.6: Bass' Conjecture
  • Remark 2.2
  • Theorem 2.3
  • ...and 28 more