On the new intersection theorem for totally reflexive modules

Abstract

Let $$(R,\mathfrak {m},k)$$ 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 $${\text {G}}$$ -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 $${\text {G}}$$ -regular local ring or a quasi-Buchsbaum local ring.