Minimal resolutions of some monomial ideals

Eliahou, Shalom; Kervaire, Michel

doi:10.1016/0021-8693(90)90237-i

Cited by 367 publications

(357 citation statements)

References 6 publications

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“…In fact, if I is a Gotzmann ideal generated in degree q, then I lex is also generated in degree q. It follows from the Eliahou-Kervaire resolution [11] (see also [1]) yields that I lex has a linear resolution. By the theorem of Bigatti [5], Hullet [17] as well as Pardue [18], we have β i,i+j (I) ≤ β i,i+j (I lex ).…”

Section: ) §1 Algebraic Properties Of Componentwise Linear Idealsmentioning

confidence: 99%

“…(a) In the theory of monomial ideal, there is the following hierarchy of ideals: lexsegment monomial ideals ⇒ strongly stable monomial ideals ⇒ stable monomial ideals. We refer the reader to [5], [11] and [17] for the definitions and the detailed information about these monomial ideals. See also [2].…”

Section: ) §1 Algebraic Properties Of Componentwise Linear Idealsmentioning

confidence: 99%

“…Some fundamental works have been achieved, from view points of both commutative algebra and combinatorics, for ideals generated by monomials. Eliahou and Kervaire [11] constructed an explicit minimal free resolution and computed Betti numbers of stable monomial ideals (see also [1] and [8]). Their computational result is indispensable for Bigatti [5] and Hullet [17], where upper bounds of Betti numbers of graded ideals with a given Hilbert function are discussed.…”

Section: Introductionmentioning

confidence: 99%

“…Their computational result is indispensable for Bigatti [5] and Hullet [17], where upper bounds of Betti numbers of graded ideals with a given Hilbert function are discussed. Moreover, the squarefree analogues of [11], [5] and [17] are studied in [2] and [3]. On the other hand, in [21] and 142 J. HERZOG AND T. HIBI [22], Betti numbers of squarefree monomial ideals associated with boundary complexes of cyclic polytopes and stacked polytopes are computed.…”

Section: Introductionmentioning

confidence: 99%

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Componentwise linear ideals

Herzog¹,

Hibi²

1999

Nagoya Mathematical Journal

161

187

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Abstract. A componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal I∆ arising from a simplicial complex ∆ is componentwise linear if and only if the Alexander dual of ∆ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the StanleyReisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

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Section: ) §1 Algebraic Properties Of Componentwise Linear Idealsmentioning

confidence: 99%

Section: ) §1 Algebraic Properties Of Componentwise Linear Idealsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Componentwise linear ideals

Herzog¹,

Hibi²

1999

Nagoya Mathematical Journal

161

187

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“…Then, J T is a stable ideal in the sense of [Eliahou and Kervaire 1990] if we reverse the order of variables, i.e., if we list the variables as T d , . .…”

Section: An Application Of Homology Multipliers To Relation Typementioning

confidence: 99%

Homology multipliers and the relation type of parameter ideals

Aberbach¹,

Ghezzi²,

Harbourne³

2006

Pacific J. Math.

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The relation type question, raised by C. Huneke, asks whether for a complete equidimensional local ring R there exists a uniform number N such that the relation type of every ideal I ⊂ R generated by a system of parameters is at most N. Wang gave a positive answer to this question when the non-Cohen-Macaulay locus of R (denoted by NCM(R)) has dimension zero. In this paper, we first present an example, due to the first author, which gives a negative answer to the question when dim NCM(R) ≥ 2. The major part of our work is to investigate the remaining situation, i.e., when dim NCM(R) = 1. We introduce the notion of homology multipliers and show that the question has a positive answer when R/Ꮽ(R) is a domain, where Ꮽ( R) is the ideal generated by all homology multipliers in R. In a more general context, we also discuss many interesting properties of homology multipliers.

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Structure Analysis of Polynomial Modules with Pommaret Bases

Seiler

2003

Proc Appl Math and Mech

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Minimal resolutions of some monomial ideals

Cited by 367 publications

References 6 publications

Componentwise linear ideals

Componentwise linear ideals

Homology multipliers and the relation type of parameter ideals

Structure Analysis of Polynomial Modules with Pommaret Bases

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