On the New Intersection Theorem for totally reflexive modules

Abstract

Let (R,m,k) be a local ring. We establish a totally reflexive analogue of the New Intersection Theorem, provided for every totally reflexive R-module M, there is a big Cohen-Macaulay R-module B_M such that the socle of B_M\otimes_RM is zero. When R is a quasi-specialization of a G-regular local ring or when M has complete intersection dimension zero, we show the existence of such a big Cohen-Macaulay R-module. It is conjectured that if R admits a non-zero Cohen-Macaulay module of finite Gorenstein dimension, then it is Cohen-Macaulay. We prove this conjecture if either R is a quasi-specialization of a G-regular local ring or a quasi-Buchsbaum local ring.