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The Auslander-Reiten Conjecture, Finite $C$-Injective Dimension of $\operatorname{Hom}$, and vanishing of $\operatorname{Ext}$

Victor D. Mendoza-Rubio, Victor H. Jorge-Pérez

TL;DR

This work extends the Auslander–Reiten conjecture framework by linking Ext vanishing, finite injective dimension of Hom, and finite $C$-injective dimension for semidualizing $C$. Building on Ghosh–Takahashi, it proves that under suitable Ext-vanishing and finiteness hypotheses, modules $M$ must be free and related modules $N$ have finite injective dimensions, with sharpened conclusions when $R$ is Cohen–Macaulay or Gorenstein. It then develops a $C$-version of these results, showing that finite $ ext{I}_C$-id of $ ext{Hom}$ transfers through $C$-tensor products and Ext vanishing, yielding AR conjecture in broader contexts and new canonical-module criteria. The paper further characterizes when a semidualizing module $C$ is canonical via finite $ ext{I}_C$-id conditions on Hom, including implications for CM and Gorenstein structures and extending known results in the literature to the $C$-injective framework.

Abstract

Let $R$ be a Noetherian local ring, and let $C$ be a semidualizing $R$-module. In this paper, we present some results concerning the vanishing of $\operatorname{Ext}$ and finite injective dimension of $\operatorname{Hom}$. Additionally, we extend these results in terms of finite $C$-injective dimension of $\operatorname{Hom}$. We also investigate the consequences of some of these extensions in the case where $R$ is Cohen-Macaulay and $C$ is a canonical module for $R.$ Furthermore, we provide positive answers to the Auslander-Reiten conjecture for finitely generated $R$-modules $M$ such that $\mathcal{I}_C\operatorname{-id}_R(\operatorname{Hom}_R(M,R))<\infty$ or $M \in \mathcal{A}_C(R)$ with $\mathcal{I}_C \operatorname{-id}_R(\operatorname{Hom}_R(M,M))<\infty$. Moreover, we derive a number of criteria for a semidualizing $R$-module $C$ to be a canonical module for $R$ in terms of the vanishing of $\operatorname{Ext}$ and the finite $C$-injective dimension of $\operatorname{Hom}$.

The Auslander-Reiten Conjecture, Finite $C$-Injective Dimension of $\operatorname{Hom}$, and vanishing of $\operatorname{Ext}$

TL;DR

This work extends the Auslander–Reiten conjecture framework by linking Ext vanishing, finite injective dimension of Hom, and finite -injective dimension for semidualizing . Building on Ghosh–Takahashi, it proves that under suitable Ext-vanishing and finiteness hypotheses, modules must be free and related modules have finite injective dimensions, with sharpened conclusions when is Cohen–Macaulay or Gorenstein. It then develops a -version of these results, showing that finite -id of transfers through -tensor products and Ext vanishing, yielding AR conjecture in broader contexts and new canonical-module criteria. The paper further characterizes when a semidualizing module is canonical via finite -id conditions on Hom, including implications for CM and Gorenstein structures and extending known results in the literature to the -injective framework.

Abstract

Let be a Noetherian local ring, and let be a semidualizing -module. In this paper, we present some results concerning the vanishing of and finite injective dimension of . Additionally, we extend these results in terms of finite -injective dimension of . We also investigate the consequences of some of these extensions in the case where is Cohen-Macaulay and is a canonical module for Furthermore, we provide positive answers to the Auslander-Reiten conjecture for finitely generated -modules such that or with . Moreover, we derive a number of criteria for a semidualizing -module to be a canonical module for in terms of the vanishing of and the finite -injective dimension of .
Paper Structure (6 sections, 34 theorems, 23 equations)

This paper contains 6 sections, 34 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Setup and Background
  3. Finite injective dimension of $\operatorname{Hom}$ and vanishing of $\operatorname{Ext}$
  4. Semidualizing modules and $C$-injective dimension
  5. Extension of results to $C$-injective dimension of $\operatorname{Hom}$
  6. Characterizations for a semidualizing module to be canonical module

Key Result

Theorem 1.3

Let $M$ and $N$ be nonzero $R$-modules and let $t=\operatorname{depth}(N)$. Suppose that $\operatorname{Hom}_R(M,N)$ has finite injective dimension, and that $\operatorname{Ext}_R^i(M,N)=\operatorname{Ext}_R^j(M,R)=0$ for all $1\leq i \leq t$ and $1\leq j \leq d$. Then $M$ is free and $N$ has finite

Theorems & Definitions (66)

  • Conjecture 1.1
  • Theorem 1.3: =Theorem \ref{['otrageneralizacionde2.15']}
  • Theorem 1.4: =Theorem \ref{['estasigeneralizacionde215']}
  • Theorem 1.5: =Theorem \ref{['generalizaciondoteo2.5']}
  • Theorem 1.6: =Theorem \ref{['2.15generalizadoIC']}
  • Remark 1.7
  • Theorem 2.1: Bass' Theorem
  • proof
  • Lemma 3.1
  • proof
  • ...and 56 more