Maximal Cohen-Macaulay approximations and Serre's condition

TL;DR

This paper explores the connection between Serre's condition $( ext{R}_n)$ and maximal Cohen-Macaulay approximations, establishing a characterization for Gorenstein local rings based on module decompositions.

Contribution

It provides a new characterization of Gorenstein local rings satisfying $( ext{R}_n)$ through the structure of Cohen-Macaulay modules and their approximations.

Findings

Gorenstein local rings satisfy $( ext{R}_n)$ iff every MCM module is a summand of an approximation of a Cohen--Macaulay module of codimension $n+1$.

Established a precise link between Serre's condition and module decomposition properties.

Enhanced understanding of the structure of Cohen--Macaulay modules in relation to Serre's conditions.

Abstract

This paper studies the relationship between Serre's condition $(\R_n)$ and Auslander--Buchweitz's maximal Cohen--Macaulay approximations. It is proved that a Gorenstein local ring satisfies $(\R_n)$ if and only if every maximal Cohen--Macaulay module is a direct summand of a maximal Cohen--Macaulay approximation of a (Cohen--Macaulay) module of codimension $n+1$.