On the new intersection theorem for totally reflexive modules

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 ⊗ R M is zero. When R is a quasi-specialization of a G -regular loca...

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Bibliographic Details
Published in:Collectanea mathematica (Barcelona) 2020-09, Vol.71 (3), p.369-381
Main Authors: Divaani-Aazar, Kamran, Mashhad, Fatemeh Mohammadi Aghjeh, Tavanfar, Ehsan, Tousi, Massoud
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Geometry
Mathematics
Mathematics and Statistics
Modules
Theorems
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Summary: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 ⊗ R M 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.
ISSN:0010-0757
2038-4815
DOI:10.1007/s13348-019-00264-3