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.
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K. Divaani-Aazar and M. Tousi were supported by Grants from IPM (Nos. 92130212 and 90130211; respectively). The research of E. Tavanfar is also supported by IPM.
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Divaani-Aazar, K., Mashhad, F.M.A., Tavanfar, E. et al. On the new intersection theorem for totally reflexive modules. Collect. Math. 71, 369–381 (2020). https://doi.org/10.1007/s13348-019-00264-3
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DOI: https://doi.org/10.1007/s13348-019-00264-3
Keywords
- Big Cohen–Macaulay module
- Complete intersection dimension
- Deformation
- Gorenstein dimension
- Quasi-deformation
- Totally reflexive module