Characteristic modules over a local ring

TL;DR

This paper introduces characteristic and cocharacteristic modules over a local ring to characterize Cohen–Macaulay and Gorenstein rings, extending homological methods inspired by Vasconcelos and Briggs.

Contribution

It defines new modules, $ ext{T}_M$ and $ ext{E}_M$, and uses them to provide novel characterizations of important classes of local rings.

Findings

Characterization of Cohen–Macaulay rings via $ ext{T}_M$ and $ ext{E}_M$

Characterization of Gorenstein rings using the new modules

Extension of homological characterization methods

Abstract

Let $R$ be a commutative noetherian local ring, and let $M$ be a finitely generated $R$-module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the conormal module, in this paper, we define the characteristic module $\mathrm{T}_M$ and the cocharacteristic module $\mathrm{E}_M$ of $M$, and investigate their properties. Our main results include characterizations of Cohen--Macaulay and Gorenstein local rings.