Linkage of finite Gorenstein dimension modules

Dibaei, Mohammad T.; Sadeghi, Arash

doi:10.1016/j.jalgebra.2012.11.035

Cited by 14 publications

(22 citation statements)

References 13 publications

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“…The present authors, in [8,Theorem 4.2], extended Schenzel's result for any horizontally linked module of finite G-dimension over a more general ground ring, i.e. over a Cohen-Macaulay local ring.…”

Section: Introductionmentioning

confidence: 53%

“…Theorem 1.6. ( [13,4,8]) For a semidualizing R-module C and an R-module M of finite G Cdimension, the following statements hold true. Throughout, we fix an R-module C and denote (−) ▽ as the dual functor (−) ▽ := Hom R (−, C).…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4, for a semidualizing module C, we study the theory of linkage for modules of finite G C -dimension which is inspired by authors pervious work [8] on the theory of linkage for modules of finite G-dimension. Let R be a Cohen-Macaulay local ring, C a semidualizing R-module and c 1 , c 2 two G C -Gorenstein ideals.…”

Section: Introductionmentioning

confidence: 99%

“…These modules can be viewed as a generalization of the Eilenberg-Maclane module [14]. For a reduced G C -perfect module M of G C -dimension n over a Cohen-Macaulay local ring R, it is shown in Theorem 5.2 that if λM ∈ A C , then depth R (M ) + depth R (λM ) = depth R + depth R (Ext n R (M, C)) which is a generalization of [8,Theorem 3.3]. We end the paper by determining conditions under which an Eilenberg-Maclane horizontally linked R-module is generalized Cohen-Macaulay (see Corollary 5.4).…”

Section: Introductionmentioning

confidence: 99%

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Linkage of modules and the Serre conditions

Dibaei

Sadeghi

2015

Journal of Pure and Applied Algebra

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Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linkage of modules and the Serre conditions

Dibaei

Sadeghi

2015

Journal of Pure and Applied Algebra

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“…Proof. First note that by [8,Theorem 4.2], for a prime ideal p of R we have 13.1). It follows from Corollary 3.12 that…”

Section: Formally Equidimensional Local Ring Of Dimension D Which Is mentioning

confidence: 99%

Notes on linkage of modules

Sadeghi

2019

Proceedings of the Edinburgh Mathematical Society

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Let R be a Cohen-Macaulay local ring. It is shown that under some mild conditions, the Cohen-Macaulayness property is preserved under linkage. We also study the connection of (S n ) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module. INTRODUCTIONThe theory of linkage for subschemes of projective space goes back more than a century in some sense, but the modern study was introduced by Peskine and Szpiro [27] in 1974. Recall that two ideals I and J in a Cohen-Macaulay local ring R are said to be linked if there is a regular sequence α in their intersection such that I = (α : J) and J = (α : I). The first main result in the theory of linkage, due to Peskine and Szpiro, indicates that the Cohen-Macaulayness property is preserved under linkage over Gorenstein rings. They also give a counterexample to show that the above result is no longer true if the base ring is Cohen-Macaulay but not Gorenstein. On the other hand, there are interesting extensions of the Peskine-Szpiro Theorem to the case that R is Cohen-Macaulay and one of the ideals I or J is strongly Cohen-Macaulay (i.e. Koszul homology modules are Cohen-Macaulay) [15] or satisfies the sliding depth condition (on Koszul homology) [14]. In a different direction, Schenzel generalized the Peskine-Szpiro Theorem to the vanishing of certain local cohomology modules. More precisely, for linked ideals I and J over a Gorenstein local ring R with maximal ideal m, the Serre condition (S n ) for R/I is equivalent to the vanishing of the local cohomology modules H i m (R/J) = 0 for all i, dim(R/J) − n < i < dim(R/J) [32, Theorem 4.1]. A few authors advanced the theory to the setting of linkage of modules in different ways, for instance Martin [19], Yoshino and Isogawa [35], Martsinkovsky and Strooker [20], and Nagel [24]. Based on these generalizations, several works have been done on studying the linkage theory in the context of modules; see for example [6]-[10], [16], [26], [28] and [29]. In this paper, we are interested in linkage of modules in the sense of [20]. Martsinkovsky and Strooker generalized the notion of linkage for modules by using the composition of two functors: transpose and syzygy. They showed that ideals a and b are linked by zero ideal if and only if R/a ∼ = λ (R/b) and R/b ∼ = λ (R/a), where λ := ΩTr.Recall that the (S k ) locus of M, denoted by S k (M), is the subset of Spec R consisting of all prime ideals p of R such that M p satisfies the Serre condition (S k ). The Gorenstein locus of R, denoted by Gor(R), is the subset of Spec R consisting of all prime ideals p of R such that R p is a Gorenstein local ring. In the fisrt part of this paper, we study the connection of (S n ) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module. As a consequence, we obtain the following result (see also Theorem 3.3 for a more general case).2010 Mathematics Subject Classification. 13C40, 13D45, 13D05, 13C14.

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Linkage of Ideals in Integral Domains

Kabbaj

Mimouni

2020

Algebr Represent Theor

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Linkage of finite Gorenstein dimension modules

Cited by 14 publications

References 13 publications

Linkage of modules and the Serre conditions

Linkage of modules and the Serre conditions

Notes on linkage of modules

Linkage of Ideals in Integral Domains

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