Associated primes and syzygies of linked modules

Celikbas, Olgur; Dibaei, Mohammad T.; Gheibi, Mohsen; Sadeghi, Arash; Takahashi, Ryo

doi:10.1216/jca-2019-11-3-301

Cited by 4 publications

(2 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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].…”

Section: Introductionmentioning

confidence: 99%

Notes on linkage of modules

Sadeghi

2019

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Notes on linkage of modules

Sadeghi

2019

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…The classical linkage theory has been extended to modules by Martin [19], Yoshino and Isogawa [32], Martsinkovsky and Strooker [21], and by Nagel [23], in different ways. Based on these generalizations, several works have been done on studying the linkage theory in the context of modules; see for example [7], [8], [9], [17], [26] and [4]. In this paper, we introduce the notion of linkage with respect to a semidualizing module.…”

Section: Introductionmentioning

confidence: 99%

Linkage of modules with respect to a semidualizing module

Dibaei

Sadeghi

2018

Pacific J. Math.

Self Cite

View full text Add to dashboard Cite

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect to the canonical module. For a linked module M with respect to a semidualizing module, the connection between the Serre condition (S n ) on M with the vanishing of certain local cohomology modules of its linked module is discussed.

show abstract

Notes on linkage of modules with respect to a semidualizing module

Souza

2022

Communications in Algebra

View full text Add to dashboard Cite

Associated primes and syzygies of linked modules

Cited by 4 publications

References 32 publications

Notes on linkage of modules

Notes on linkage of modules

Linkage of modules with respect to a semidualizing module

Notes on linkage of modules with respect to a semidualizing module

Contact Info

Product

Resources

About