Some homological properties of ideals with cohomological dimension one

Pirmohammadi, Gholamreza; Amoli, Khadijeh Ahmadi; Bahmanpour, Kamal

doi:10.4064/cm6939-11-2016

Cited by 16 publications

(8 citation statements)

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In this paper we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the results given in [3,5,9,10,11,13,17,19,20,21,22,26,28].

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confidence: 89%

“…Moreover, with respect to the question (ii), Kawasaki in [21] proved that if an ideal I of a Noetherian ring R is principal, up to radical, then the category C (R, I) cof is Abelian. Pirmohammadi et al in [28] as a generalization of Kawasaki's result proved that if I is an ideal of a Noetherian local ring R with cd(I, R) ≤ 1, then C (R, I) cof is Abelian. Also, more recently, Divaani-Aazar et al in [13] have removed the local condition on the ring.…”

Section: Introductionmentioning

confidence: 98%

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A study of cofiniteness through minimal associated primes

Bahmanpour

2019

Communications in Algebra

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confidence: 89%

Section: Introductionmentioning

confidence: 98%

A study of cofiniteness through minimal associated primes

Bahmanpour

2019

Communications in Algebra

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“…Our final goal in this section is to show that given an ideal a of R with cd(a, R) ≤ 1, the subcategory M(R, a) cof of M(R) is abelian. As mentioned earlier, this fact is proved in [PAB,Theorem 2.4], under the extra assumption that R is local. The tool deployed here is the local homology functor.…”

Section: Cofiniteness Of Modulesmentioning

confidence: 64%

“…a) cof is abelian.Proof. (i): See[Me2, Corollary 3.14] and[PAB, Theorem 2.4]. (ii): See [Me2, Theorem 7.10] and [Me2, Theorem 7.4].…”

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confidence: 99%

Gorenstein Homology and Finiteness Properties of Local (Co)homology

Faridian

2020

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This thesis is comprised of three chapters. The first chapter deals with bounded complexes of Gorenstein projective and Gorenstein injective modules. Deploying methods of relative homological algebra, we approximate such complexes with bounded complexes of projective and injective modules, respectively. As an application, we investigate the Gorenstein version of the New Intersection Theorem.The second chapter studies the notion of cofiniteness for modules and complexes set forth by Hartshorne. Recruiting techniques of derived category, we study this notion thoroughly, obtain novel results, and extend some of the results to stable under specialization sets.The third chapter delves into the Greenlees-May Duality Theorem which is widely thought of as a far-reaching generalization of the Grothendieck's Local Duality Theorem. This theorem is not addressed in the literature as it merits and its proof is indeed a tangled web in a series of scattered papers. By carefully scrutinizing the requisite tools, we present a clear-cut well-documented proof of this theorem.Mathematics, rightly viewed, possesses not only truth, but supreme beauty; a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

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“…Kawasaki proved that if ara(I) = 1 then the category C (R, I) cof is Abelian (see [17]). Pirmohammadi et al in [22] as a generalization of Kawasaki's result proved that if I is an ideal of a Noetherian local ring with cd(I, R) ≤ 1, then C (R, I) cof is Abelian. Recently, Divaani-Aazar et al in [9] have removed the local condition on the ring.…”

Section: Introductionmentioning

confidence: 98%

A Note on Abelian Categories of Cofinite Modules

Bahmanpour

2019

Communications in Algebra

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Let R be a commutative Noetherian ring and I be an ideal of R. In this article we answer affirmatively a question raised by the present author in [1]. Also, as an immediate consequence of this result it is shown that the category of all I-cofinite Rmodules C (R, I) cof is an Abelian subcategory of the category of all R-modules, whenever q(I, R) ≤ 1. These assertions answer affirmatively a question raised by R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9(1970), 145-164], in some special cases. Hartshorne in [14] defined an R-module X to be I-cofinite, if Supp X ⊆ V (I) and Ext i R (R/I, X) is a finitely generated R-module for each integer i ≥ 0. Then he posed the following question: Question 1: Whether the category C (R, I) cof of I-cofinite modules is an Abelian subcategory of the category of all R-modules? That is, if f : M −→ N is an R-homomorphism

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Some homological properties of ideals with cohomological dimension one

Abstract: Abstract. Let R denote a commutative Noetherian ring and let I be an ideal of R such that H i I (R) = 0, for all integers i ≥ 2. In this paper we shall prove some results concerning the homological properties of I.

Cited by 16 publications

References 15 publications

A study of cofiniteness through minimal associated primes

A study of cofiniteness through minimal associated primes

Gorenstein Homology and Finiteness Properties of Local (Co)homology

A Note on Abelian Categories of Cofinite Modules

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