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mmr 2019, 5(2): 157-164 |
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Sazeedeh R, Savoji F. Gorenstein Injective Dimension and Cohen-Macaulayness. mmr 2019; 5 (2) :157-164
URL: http://mmr.khu.ac.ir/article-1-2699-en.html
URL: http://mmr.khu.ac.ir/article-1-2699-en.html
1- Urmia University , rsazeedeh@ipm.ir
2- Urmia University
2- Urmia University
Abstract: (2869 Views)
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.
./files/site1/files/52/4.pdf
(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.
./files/site1/files/52/4.pdf
Type of Study: Original Manuscript |
Subject:
alg
Received: 2017/11/5 | Revised: 2019/12/16 | Accepted: 2018/04/23 | Published: 2019/11/18 | ePublished: 2019/11/18
Received: 2017/11/5 | Revised: 2019/12/16 | Accepted: 2018/04/23 | Published: 2019/11/18 | ePublished: 2019/11/18
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