Maximal Cohen–Macaulay Approximations and Serre’s Condition
Hiroki Matsui, Ryo Takahashi



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.