On the new intersection theorem for totally reflexive modules

Abstract

Let (R, m, k) be a local ring. The celebrated New Intersection Theorem is perceived as a deep result at the interface of homological and local algebra. We establish a complete intersection analogue of this theorem. Also when R is a quasi-specialization of a G-regular local ring, we extend the New Intersection Theorem to totally reflexive R-modules. There are plenty of examples of quasispecializations of G-regular rings which are neither G-regular nor Cohen-Macaulay. It is conjectured that if R admits a nonzero Cohen-Macaulay module of finite Gorenstein dimension, then it is Cohen-Macaulay. We establish this conjecture if either R is a quasi-specialization of G-regular local ring or a quasi-Buchsbaum ring. 2010 Mathematics Subject Classification. 13D05; 13C14; 13D22.