Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finiteness of complete intersection dimensions of RHom and Ext modules

Paulo Martins, Victor D. Mendoza Rubio, Zachary Nason

TL;DR

The paper advances the understanding of when complete intersection dimensions of $\mathbf{R}\mathrm{Hom}_R(M,N)$ and Ext modules are finite, developing a cohesive framework of stability results, vanishing criteria, and new module-theoretic notions such as CI-perfect modules. It leverages derived-category methods, quasi-deformations, and dualizing complexes to connect CI-dimensions across $M$, $N$, and their Ext/RHom interactions, yielding both Ext-based and Hom-based criteria. Key contributions include stability theorems for CI-dim$(\mathbf{R}\mathrm{Hom}(M,N))$, transfer results from Ext vanishing to finite CI-dimensions, the introduction and basic theory of CI-perfect modules, and new freeness and Gorenstein criteria tied to vanishing Ext and finite CI-injective dimensions of Hom. These results extend prior work on Ext with finite CI-dimension, provide practical freeness and Gorenstein tests, and offer a robust toolset for studying rings and modules via complete intersection homological dimensions.

Abstract

In this paper, we explore the implications of the finiteness of complete intersection dimensions for RHom complexes and Ext modules. We prove various stability results and criteria for detecting finite complete intersection homological dimension of complexes and modules. In addition, we introduce and explore the concept of CI-perfect modules. We also study the vanishing of Ext when certain Hom module have finite complete intersection homological dimension. In this direction, we improve a result by Ghosh and Samanta, prove the Auslander-Reiten conjecture for finitely generated modules $M$ over a Noetherian local ring $R$ such that $\operatorname{Hom}_R(M,R)$ or $\operatorname{Hom}_R(M,M)$ has finite complete intersection injective dimension, and provide Gorenstein criteria.

Finiteness of complete intersection dimensions of RHom and Ext modules

TL;DR

The paper advances the understanding of when complete intersection dimensions of and Ext modules are finite, developing a cohesive framework of stability results, vanishing criteria, and new module-theoretic notions such as CI-perfect modules. It leverages derived-category methods, quasi-deformations, and dualizing complexes to connect CI-dimensions across , , and their Ext/RHom interactions, yielding both Ext-based and Hom-based criteria. Key contributions include stability theorems for CI-dim, transfer results from Ext vanishing to finite CI-dimensions, the introduction and basic theory of CI-perfect modules, and new freeness and Gorenstein criteria tied to vanishing Ext and finite CI-injective dimensions of Hom. These results extend prior work on Ext with finite CI-dimension, provide practical freeness and Gorenstein tests, and offer a robust toolset for studying rings and modules via complete intersection homological dimensions.

Abstract

In this paper, we explore the implications of the finiteness of complete intersection dimensions for RHom complexes and Ext modules. We prove various stability results and criteria for detecting finite complete intersection homological dimension of complexes and modules. In addition, we introduce and explore the concept of CI-perfect modules. We also study the vanishing of Ext when certain Hom module have finite complete intersection homological dimension. In this direction, we improve a result by Ghosh and Samanta, prove the Auslander-Reiten conjecture for finitely generated modules over a Noetherian local ring such that or has finite complete intersection injective dimension, and provide Gorenstein criteria.
Paper Structure (14 sections, 51 theorems, 56 equations)

This paper contains 14 sections, 51 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The Derived Category of a Ring
  4. Dualizing complex and deficiency modules
  5. Restricted flat dimension
  6. Gorenstein Dimensions
  7. Complete Intersection Dimensions
  8. Finiteness of CI-dimensions of RHom
  9. Finiteness of CI-Dimensions of Ext Modules
  10. Finite CI-dimension of Ext modules
  11. Finite CI-injective dimensions of Ext modules
  12. CI-perfect modules
  13. Vanishing of Ext and finite complete intersection homological dimension of Hom
  14. Proof of Lemma \ref{['implemma']}

Key Result

Theorem 1.1

Let $M$ and $N$ be non-acyclic complexes in $\operatorname{\mathsf{D}}_\square^\mathsf{f}(R)$ such that $\operatorname{\mathbf{R}Hom}_R(M,N)$ is in $\operatorname{\mathsf{D}}_\square^\mathsf{f}(R)$. Assume that $\operatorname{H-dim}_R(\operatorname{\mathbf{R}Hom}_R(M,N))<\infty$.

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5: Auslander-Reiten conjecture
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 93 more