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Auslander-Reiten conjecture for modules whose (self) dual has finite complete intersection dimension

Dipankar Ghosh, Mouma Samanta

Abstract

Over a commutative Noetherian ring, we show that the Auslander-Reiten conjecture holds true for the class of (finitely generated) modules whose dual has finite complete intersection dimension. We provide another result that validates the conjecture for the class of modules whose self dual has finite complete intersection dimension and either the module or its dual has finite Gorenstein dimension. Thus we combine and strengthen a number of results in the literature, due to Auslander-Ding-Solberg, Dey-Ghosh and Rubio-Pérez.

Auslander-Reiten conjecture for modules whose (self) dual has finite complete intersection dimension

Abstract

Over a commutative Noetherian ring, we show that the Auslander-Reiten conjecture holds true for the class of (finitely generated) modules whose dual has finite complete intersection dimension. We provide another result that validates the conjecture for the class of modules whose self dual has finite complete intersection dimension and either the module or its dual has finite Gorenstein dimension. Thus we combine and strengthen a number of results in the literature, due to Auslander-Ding-Solberg, Dey-Ghosh and Rubio-Pérez.
Paper Structure (3 sections, 11 theorems, 15 equations)

This paper contains 3 sections, 11 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Theorem \ref{thm:les-ARC-finite-CI-dimension-M^*}

Let $R$ be a local ring of depth $t$, and $M$ be an $R$-module such that $\mathop{\mathrm{Ext}}\nolimits_R^{1\leqslant i \leqslant t}(M,R) = 0$, and $\mathop{\mathrm{Ext}}\nolimits_R^{2j}(M,M) = 0$ for some $j \geqslant 1$. Suppose $\mathop{\mathrm{CI-dim}}\nolimits_R(M^*)$ is finite. Then $M$ is fr

Theorems & Definitions (26)

  • Conjecture 1.1: Auslander-Reiten
  • Theorem \ref{thm:les-ARC-finite-CI-dimension-M^*}
  • Theorem \ref{thm:les-CI-dim-Hom(M,M)-G-dim-M}
  • Corollary 1.3
  • Lemma 2.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 16 more