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Maximal Cohen–Macaulay Approximations and Serre’s Condition
Hiroki Matsui , Ryo Takahashi
Abstract
This paper studies the relationship between Serre's condition $(\mathsf{R}_n)$ and Auslander--Buchweitz's maximal Cohen--Macaulay approximations. It is proved that a Gorenstein local ring satisfies $(\mathsf{R}_n)$ 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 $n+1$.