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Information Journal Paper

Title

Gorenstein Injective Dimension and Cohen-Macaulayness

Pages

  157-164

Abstract

 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 x m\m2 is a non-zerodivisor, then for every finitely generated R/xR-module M, there is id R M= id R/xr M+1. 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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    APA: Copy

    Sazeedeh, Reza, & Savoji, Fatemeh. (2020). Gorenstein Injective Dimension and Cohen-Macaulayness. MATHEMATICAL RESEARCHES, 5(2 ), 157-164. SID. https://sid.ir/paper/407579/en

    Vancouver: Copy

    Sazeedeh Reza, Savoji Fatemeh. Gorenstein Injective Dimension and Cohen-Macaulayness. MATHEMATICAL RESEARCHES[Internet]. 2020;5(2 ):157-164. Available from: https://sid.ir/paper/407579/en

    IEEE: Copy

    Reza Sazeedeh, and Fatemeh Savoji, “Gorenstein Injective Dimension and Cohen-Macaulayness,” MATHEMATICAL RESEARCHES, vol. 5, no. 2 , pp. 157–164, 2020, [Online]. Available: https://sid.ir/paper/407579/en

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