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

TL;DR

This paper proves the Auslander-Reiten conjecture for modules over commutative Noetherian rings when their duals or self-duals have finite complete intersection dimension, extending previous results in the literature.

Contribution

It establishes the conjecture for modules with duals of finite complete intersection dimension and further generalizes to modules with finite Gorenstein dimension under certain conditions.

Findings

The conjecture holds for modules with duals of finite complete intersection dimension.

The conjecture is validated for modules with self-duals of finite complete intersection dimension and finite Gorenstein dimension.

The results unify and strengthen previous partial results in the literature.

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-P\'{e}rez.