Integral closure and other operations on monomial ideals

Quiñonez, Veronica Crispin

doi:10.1216/jca-2010-2-3-359

Cited by 11 publications

(7 citation statements)

References 11 publications

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“…The normality of I can be determined using the program Normaliz [3]. For the normality of monomial ideals of dimension 2 see [6,12] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Gimenez

Martinez-Bernal

Simis

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertexweighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen-Macaulay vertexweighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen-Macaulay and satisfy Alexander duality.

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“…The normality of I can be determined using the program Normaliz [3]. For the normality of monomial ideals of dimension 2 see [6,12] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Gimenez

Martinez-Bernal

Simis

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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“…Let I be an integrally closed monomial ideal of R. It is well known that R is an integrally closed domain (i.e., R is an integral domain that contains every nonzero element of the quotient field of R that is integral over R), and that each principal ideal of R is integrally closed, and the product of an integrally closed ideal of R and a nonzero element of R yields another integrally closed ideal of R. Hence by Lemma 3.1 we may assume that I is (x, y)-primary. Now by [16,Proposition 2.6] there are monomial ideals I 1 = ({x r−i y bi } r i=0 ) and I 2 = ({x…”

Section: ω-Linear Idealsmentioning

confidence: 99%

n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS

Choi¹,

Walker²

2019

International Electronic Journal of Algebra

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“…The first main result of Section 2 is Corollary 2.7, in which we give a complete description of componentwise polymatroidal ideals in K[x, y]. For this we recall the notion of x-tight and y-tight ideals from [9]. Our next main result is Theorem 2.9 which states that the componentwise polymatroidal ideals in K[x, y] have linear quotients.…”

Section: Introductionmentioning

confidence: 99%

Ideals with linear quotients and componentwise polymatroidal ideals

Bandari¹,

Qureshi²

2021

Preprint

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If I is a monomial ideal with linear quotients, then it has componentwise linear quotients. However, the converse of this statement is an open question. In this paper, we provide two classes of ideals for which the converse of this statement holds. First class is the componentwise polymatroidal ideals in K[x, y] and the second one is the componentwise polymatroidal ideals with strong exchange property.

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Integral closure and other operations on monomial ideals

Cited by 11 publications

References 11 publications

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS

Ideals with linear quotients and componentwise polymatroidal ideals

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