Finiteness of complete intersection dimensions of RHom complexes and Ext modules

TL;DR

This paper investigates the finiteness properties of complete intersection dimensions in relation to RHom complexes and Ext modules, establishing stability results, introducing CI-perfect modules, and proving the Auslander-Reiten conjecture under new conditions.

Contribution

It introduces the concept of CI-perfect modules, improves existing results on Ext vanishing, and proves the Auslander-Reiten conjecture for modules with finite complete intersection homological dimensions.

Findings

Established stability results for complete intersection dimensions.

Introduced and explored CI-perfect modules.

Proved the Auslander-Reiten conjecture under new conditions.

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.