Gorenstein Injective Dimension and Cohen-Macaulayness
Throughout this paper, (R, m) is a commutative Noetherian local ring with the maximal ideal m. The following conjecture proposed by Bass [1], has been proved by Peskin and Szpiro [2] for almost all rings: (B) If R admits a finitely generated R-module of finite injective dimension, then R...
| Main Authors: | , |
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| Format: | Article |
| Language: | fas |
| Published: |
Kharazmi University
2019-12-01
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| Series: | پژوهشهای ریاضی |
| Subjects: | |
| Online Access: | http://mmr.khu.ac.ir/article-1-2699-en.html |
| Summary: | Throughout this paper, (R, m) is a commutative Noetherian local ring with the maximal ideal m. The following conjecture proposed by Bass [1], has been proved by Peskin and Szpiro [2] for almost all rings:
(B) If R admits a finitely generated R-module of finite injective dimension, then R is Cohen-Macaulay.
The problems treated in this paper are closely related to the following generalization of Bass conjecture which is still wide open:
(GB) If R admits a finitely generated R-module of finite Gorenstein-injective dimension, then R is Cohen-Macaulay.
Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a finitely generated R-module of injective dimension 1.
Levin and Vasconcelos indicate that if is a non-zerodivisor, then for every finitely generated R/xR-module M, there is . Using this fact, they construct a finitely generated R-module of finite injective dimension in the case where R is Cohen-Macaulay (the converse of Conjecture B).
In this paper we study the Gorenstein injective dimension of local cohomology. We also show that if R is Cohen-Macaulay with minimal multiplicity, then every finitely generated module of finite Gorenstein injective dimension has finite injective dimension.
We prove that a Cohen-Macaulay local ring has a finitely generated module of finite Gorenstein injective dimension.
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| ISSN: | 2588-2546 2588-2554 |