Sequentially Cohen-Macaulayness of bigraded modules

Rahimi, Ahad

doi:10.1216/rmj-2017-47-2-621

Cited by 8 publications

(12 citation statements)

References 12 publications

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“…As M t is Cohen-Macaulay, it has maximal depth and so depth M t = mdepth M t . Since M is sequentially Cohen-Macaulay, it follows that depth M i = depth M for all i, see for instance [12,Fact 2.3]. Thus depth M = depth M t = mdepth M t = mdepth M, as desired.…”

Section: Preliminariesmentioning

confidence: 88%

Maximal depth property of finitely generated modules

Rahimi

2018

J. Algebra Appl.

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Let (R, m) be a Noetherian local ring and M a finitely generated R-module. We say M has maximal depth if there is an associated prime p of M such that depth M = dim R/p. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen-Macaulay modules with maximal depth are classified. Finally, the attached primes of H i m (M ) are considered for i < dim M .2010 Mathematics Subject Classification. 13C14, 13C15, 13E05, 13D45.

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Section: Preliminariesmentioning

confidence: 88%

Maximal depth property of finitely generated modules

Rahimi

2018

J. Algebra Appl.

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“…Ordinary sequentially Cohen-Macaulay introduced by Stanley results from our definition if we assume P = 0. Note that if M is sequentially Cohen-Macaulay with respect to Q, then the filtration F is uniquely determined and it is just the dimension filtration of M with respect to Q, that is, F = D, see [15].…”

Section: Size Bigsize and Cohen-macaulayness Of Monomial Idealsmentioning

confidence: 99%

“…, y n ). The second author has been studying the algebraic properties of a finitely generated bigraded S-module M and also the local cohomology modules of M with respect to Q, see for instance [12], [13], [14], [15]. In Section 2, we study the sequentially Cohen-Macaulayness of S/I with respect to Q where I ⊂ S is a monomial ideal.…”

Section: Introductionmentioning

confidence: 99%

Cohen–Macaulayness and sequentially Cohen–Macaulayness of monomial ideals

Noormohammadi

Rahimi

2018

Rend. Sem. Mat. Univ. Padova

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In this paper, we give a characterization for Cohen-Macaulay rings R/I where I ⊂ R = K[y1,. .. , yn] is a monomial ideal which satisfies bigsize I = size I. Next, we let S = K[x1,. .. , xm, y1,. .. , yn] be a polynomial ring and I ⊂ S a monomial ideal. We study the sequentially Cohen-Macaulayness of S/I with respect to Q = (y1,. .. , yn). Moreover, if I ⊂ R is a monomial ideal such that the associated prime ideals of I are in pairwise disjoint sets of variables, a classification of R/I to be sequentially Cohen-Macaulay is given. Finally, we compute grade(Q, M) where M is a sequentially Cohen-Macaulay S-module with respect to Q.

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“…Let M be a finitely generated bigraded S-module. The author has been studying the algebraic properties of a finitely generated bigraded S-module M with respect to Q, see for instance [9], [13], [14]. We denote by cd(Q, M) the cohomological dimension of M with respect to Q which is the largest integer i for which H i Q (M) = 0.…”

Section: Introductionmentioning

confidence: 99%

maximal depth property of bigraded modules

Rahimi¹

2020

Preprint

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Let S = K[x 1 , . . . , x m , y 1 , . . . , y n ] be the standard bigraded polynomial ring over a field K. Let M be a finitely generated bigraded S-module and Q = (y 1 , . . . , y n ). We say M has maximal depth with respect to Q if there is an associated prime p of M such that grade(Q, M ) = cd(Q, S/p). In this paper, we study finitely generated bigraded modules with maximal depth with respect to Q. It is shown that sequentially Cohen-Macaulay modules with respect to Q have maximal depth with respect to Q. In fact, maximal depth property generalizes the concept of sequentially Cohen-Macaulayness. Next, we show that if M has maximal depth with respect to Q with grade(Q, M ) > 0, then H grade(Q,M) Q (M ) is not finitely generated. As a consequence, "generalized Cohen-Macaulay modules with respect to Q" having "maximal depth with respect to Q" are Cohen-Macaulay with respect to Q. All hypersurface rings that have maximal depth with respect to Q are classified.

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Sequentially Cohen-Macaulayness of bigraded modules

Cited by 8 publications

References 12 publications

Maximal depth property of finitely generated modules

Maximal depth property of finitely generated modules

Cohen–Macaulayness and sequentially Cohen–Macaulayness of monomial ideals

maximal depth property of bigraded modules

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