Test modules, weakly regular homomorphisms and complete intersection dimension

TL;DR

This paper proves that the existence of a test module with finite complete intersection dimension characterizes complete intersection rings, answering a question in commutative algebra.

Contribution

It establishes a new criterion for identifying complete intersection rings via test modules and explores ascent properties of test complexes under weakly regular homomorphisms.

Findings

A local ring with a (pd-)test module of finite complete intersection dimension is a complete intersection.

Positive resolution of a question by Celikbas, Dao, and Takahashi.

Analysis of ascent properties of (pd-)test complexes under weakly regular homomorphisms.

Abstract

We prove that if a local ring admits a (pd-)test module of finite complete intersection dimension, then it is a complete intersection ring. This answers, positively, a question proposed by Celikbas, Dao and Takahashi. To this aim, we first investigate another question raised by Celikbas and Sather- Wagstaff concerning ascent properties of (pd-)test complexes under weakly regular homomorphisms.