Modules that detect finite homological dimensions

TL;DR

This paper investigates modules that detect finite homological dimensions and generalizes a classical theorem, showing that the existence of a test module of finite Gorenstein dimension implies the ring is Gorenstein.

Contribution

It extends the classical theorem of Auslander and Bridger by linking test modules of finite Gorenstein dimension to the Gorenstein property of the ring.

Findings

If a complete local ring admits a test module of finite Gorenstein dimension, then the ring is Gorenstein.

Generalization of a classical theorem relating test modules and ring properties.

Provides new insights into the homological characterization of Gorenstein rings.

Abstract

We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that, if a commutative Noetherian complete local ring R admits a test module of finite Gorenstein dimension, then R is Gorenstein.