Application of the Generalized Weierstrass Preparation Theorem to the Study of Homogeneous Ideals

Amasaki, Mutsumi

doi:10.2307/2001452

Cited by 4 publications

(6 citation statements)

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“…⊓ ⊔ Lemma 2. 3 With the same notations as in Lemma 2.2, let N = le ≺ H q . If ν ∈ N with cls ν = k, then 2 ν − 1 k + 1 j ∈ N for all k < j ≤ n. Conversely, let N ⊆ (AE n 0 ) q be a set of multi indices of degree q.…”

Section: Lemma 22mentioning

confidence: 99%

“…After Proposition A. 3 Let H define a Rees decomposition of the form (75) for the submodule M ⊆ P r with respect to the basis Y of P 1 and let H and Y be determined with the Sturmfels-White Algorithm 2. With respect to the basis Y we have the inequalities cls h ≥ ccls y h for all h ∈ H.…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Pommaret bases (for ideals in power series rings) implicitly appeared already in the classical work of Hironaka [40] on the resolution of singularities. Later, Amasaki [3,4] followed up this idea and explicitly introduced them under the name Weierstraß bases because of their connection to the Weierstraß Preparation Theorem. In his study of their properties, Amasaki obtained to some extent similar results than we present in this article, however in a different way.…”

Section: Introductionmentioning

confidence: 99%

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A combinatorial approach to involution and δ-regularity II: structure analysis of polynomial modules with pommaret bases

Seiler

2009

AAECC

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Much of the existing literature on involutive bases concentrates on their efficient algorithmic construction. By contrast, we are here more concerned with their structural properties. Pommaret bases are particularly useful in this respect. We show how they may be applied for determining the Krull and the projective dimension, respectively, and the depth of a polynomial module. We use these results for simple proofs of Hironaka's criterion for Cohen-Macaulay modules and of the graded form of the Auslander-Buchsbaum formula, respectively.Special emphasis is put on the syzygy theory of Pommaret bases and its use for the construction of a free resolution. In the monomial case, the arising complex always possesses the structure of a differential algebra and it is possible to derive an explicit formula for the differential. Furthermore, in this case one can give a simple characterisation of those modules for which our resolution is minimal. These observations generalise results by Eliahou and Kervaire.Using our resolution, we show that the degree of the Pommaret basis with respect to the degree reverse lexicographic term order is just the Castelnuovo-Mumford regularity. This approach leads to new proofs for a number of characterisations of this regularity proposed in the literature. This includes in particular the criteria of Bayer/Stillman and Eisenbud/Goto, respectively. We also relate Pommaret bases to the recent work of Bermejo/Gimenez and Trung on computing the Castelnuovo-Mumford regularity via saturations.It is well-known that Pommaret bases do not always exist but only in so-called δ-regular coordinates. Fortunately, generic coordinates are δ-regular. We show that several classical results in commutative algebra, holding only generically, are true for these special coordinates. In particular, they are related to regular sequences, independent sets of variables, saturations and Noether normalisations. Many properties of the generic initial ideal hold also for the leading ideal of the Pommaret basis with respect to the degree reverse lexicographic term order. We further present

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Section: Lemma 22mentioning

confidence: 99%

Section: ⊓ ⊔mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A combinatorial approach to involution and δ-regularity II: structure analysis of polynomial modules with pommaret bases

Seiler

2009

AAECC

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“…There are many examples of homogeneous Buchsbaum integral domains having linear resolutions; see e.g., [1,3,7]. However we have no examples of Buchsbaum homogeneous integral domain over an algebraically closed field k with minimal multiplicity of degree q ≥ 2.…”

Section: The Case Of Integral Domainsmentioning

confidence: 92%

Buchsbaum homogeneous algebras with minimal multiplicity

Gotō

Yoshida

2007

Journal of Pure and Applied Algebra

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In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous k-algebras A in terms of the dimension d, the codimension c, the initial degree q, and the length of the local cohomology modules of A. Next, we introduce the notion of Buchsbaum k-algebras with minimal multiplicity of degree q, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.

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“…From the viewpoint of standard free resolutions (see [2,Section 3]), homogeneous ideals defining curves in P 3 and homogeneous ideals in a polynomial ring k[x 1 , x 2 , x 3 ] defining Artinian graded rings can be treated in the same manner to a certain extent. We have recently found a way to apply the method developed in [5] to the study of the weak Lefschetz property for Artinian Gorenstein graded rings.…”

Section: Introductionmentioning

confidence: 99%

The weak Lefschetz property for Artinian graded rings and basic sequences

Amasaki

2011

Preprint

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The basic sequence of a homogeneous ideal I ⊂ R = k[x 1 , . . . , x r ] defining an Artinian graded ring A = R/I not having the weak Lefschetz property has the property that the first term of the last part is less than the last term of the penultimate part. For a general linear form ℓ in x 1 , . . . , x r , this fact affects in a certain way the behavior of the r − 1 square matrices in k[ℓ] which represent the multiplications of the elements of A by x 1 , . . . , x r−1 through a minimal free presentation of A over k [ℓ]. Taking advantage of it, we consider some modules over an algebra generated over k[ℓ] by the square matrices mentioned above. In this manner, for the case r = 3, we prove that an Artinian Gorenstein graded ring A = k[x 1 , x 2 , x 3 ]/I has the weak Lefschetz property if char(k) = 0 and the number of the minimal generators of 0 : A ℓ over k[x 1 , x 2 , x 3 ] is two.

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Application of the Generalized Weierstrass Preparation Theorem to the Study of Homogeneous Ideals

Cited by 4 publications

References 0 publications

A combinatorial approach to involution and δ-regularity II: structure analysis of polynomial modules with pommaret bases

A combinatorial approach to involution and δ-regularity II: structure analysis of polynomial modules with pommaret bases

Buchsbaum homogeneous algebras with minimal multiplicity

The weak Lefschetz property for Artinian graded rings and basic sequences

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