Maximal Cohen-Macaulay approximations and Serre's condition

Abstract

This paper studies the relationship between Serre's condition $({\mathbb{R}}_n)$ and Auslander--Buchweitz's maximal Cohen--Macaulay approximations. It is proved that a Gorenstein local ring satisfies $({\mathbb{R}}_n)$ 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 $n+1$.

7 pages. To appear in Acta Math. Vietnam. (Special issue dedicated to Ngo Viet Trung)