The Hilbert Coefficients of the Fiber Cone and the a-Invariant of the Associated Graded Ring

Abstract.Let (A,m) be a Noetherian local ring with infinite residue field and let I be an ideal in A and let be the fiber cone of I. We prove certain relations among the Hilbert coefficients f0(I), f1(I), f2(I) of F(I) when the a-invariant of the associated graded ring G(I) is negative.

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The structure of sally modules and Buchsbaumness of associated graded rings

Nagoya Mathematical Journal ◽

10.1215/00277630-2351002 ◽

2013 ◽

Vol 212 ◽

pp. 97-138 ◽

Cited By ~ 2

Author(s):

Kazuho Ozeki

Keyword(s):

Local Ring ◽

Maximal Ideal ◽

Primary Ideal ◽

Graded Rings ◽

Associated Graded Ring ◽

Graded Ring ◽

Noetherian Local Ring ◽

Hilbert Coefficients ◽

Associated Graded Rings

AbstractLet A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

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The structure of sally modules and Buchsbaumness of associated graded rings

Nagoya Mathematical Journal ◽

10.1017/s002776300002239x ◽

2013 ◽

Vol 212 ◽

pp. 97-138 ◽

Cited By ~ 1

Author(s):

Kazuho Ozeki

Keyword(s):

Local Ring ◽

Maximal Ideal ◽

Primary Ideal ◽

Graded Rings ◽

Associated Graded Ring ◽

Graded Ring ◽

Noetherian Local Ring ◽

Hilbert Coefficients ◽

Associated Graded Rings

AbstractLetAbe a Noetherian local ring with the maximal ideal m, and letIbe an m-primary ideal inA. This paper examines the equality on Hilbert coefficients ofIfirst presented by Elias and Valla, but without assuming thatAis a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring ofI.

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Hilbert-Samuel function and Grothendieck group

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500020708 ◽

2000 ◽

Vol 43 (1) ◽

pp. 73-94

Author(s):

Koji Nishida

Keyword(s):

Local Ring ◽

Grothendieck Group ◽

Residue Field ◽

Primary Ideal ◽

Hilbert Polynomial ◽

Associated Graded Ring ◽

Graded Ring ◽

Noetherian Local Ring ◽

Graded Module ◽

Primary Ideals

AbstractLet (A, m) be a Noetherian local ring such that the residue field A/m is infinite. Let I be arbitrary ideal in A, and M a finitely generated A-module. We denote by ℓ(I, M) the Krull dimension of the graded module ⊕n≥0InM/mInM over the associated graded ring of I. Notice that ℓ(I, A) is just the analytic spread of I. In this paper, we define, for 0 ≤ i ≤ ℓ = ℓ(I, M), certain elements ei(I, M) in the Grothendieck group K0(A/I) that suitably generalize the notion of the coefficients of Hilbert polynomial for m-primary ideals. In particular, we show that the top term eℓ (I, M), which is denoted by eI(M), enjoys the same properties as the ordinary multiplicity of M with respect to an m-primary ideal.

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Homology, mixed multiplies and Hilbert coefficients of the fiber cone

Journal of Algebra and Its Applications ◽

10.1142/s0219498819500610 ◽

2019 ◽

Vol 18 (04) ◽

pp. 1950061

Author(s):

Clare D’Cruz ◽

Anna Guerrieri

Keyword(s):

Upper Bound ◽

Associated Graded Ring ◽

Graded Ring ◽

Hilbert Coefficients ◽

Fiber Cone

In this paper, we compare the depth of the fiber cone and the associated graded ring. To achieve this, we construct a bi-graded complex corresponding to a bi-graded, Noetherian, Hilbert filtration. The vanishing of the homology modules of this complex helps us to compare the depth of the fiber cone of the filtration and the depth of the corresponding associated graded ring. We also give a formula for the fiber coefficients in terms of the lengths of certain homology modules. We give an upper bound for the first fiber coefficient and show that when this bound is attained, the fiber cone has good depth.

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Buchsbaumness of the associated graded rings of filtration

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500396 ◽

2020 ◽

pp. 2250039

Author(s):

Kumari Saloni

Keyword(s):

Local Ring ◽

Alternative Proof ◽

Sufficient Condition ◽

Graded Rings ◽

Associated Graded Ring ◽

Graded Ring ◽

Dimension Formula ◽

Hilbert Coefficients ◽

Associated Graded Rings ◽

Good Filtration

Let [Formula: see text] be a Noetherian local ring of dimension [Formula: see text] and [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text]. In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring [Formula: see text] to be passed onto the associated graded ring of filtration. Let [Formula: see text] denote an [Formula: see text]-good filtration. We prove that if [Formula: see text] is Buchsbaum and the [Formula: see text] -invariant, [Formula: see text] and [Formula: see text], coincide then the associated graded ring [Formula: see text] is Buchsbaum. As an application of our result, we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions for Hilbert coefficients.

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On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

Algebra Colloquium ◽

10.1142/s1005386721000031 ◽

2021 ◽

Vol 28 (01) ◽

pp. 13-32

Author(s):

Nguyen Tien Manh

Keyword(s):

Local Ring ◽

Maximal Ideal ◽

Primary Ideal ◽

Graded Algebra ◽

Finitely Generated ◽

Ideal Formula ◽

Noetherian Local Ring ◽

Graded Modules ◽

Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

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Linearity defect of the residue field of short local rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501638 ◽

2016 ◽

Vol 16 (09) ◽

pp. 1750163

Author(s):

Rasoul Ahangari Maleki

Keyword(s):

Local Ring ◽

Maximal Ideal ◽

Residue Field ◽

Positive Answer ◽

Local Rings ◽

Finitely Generated ◽

Ideal Formula ◽

Noetherian Local Ring ◽

Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

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Annihilator of local cohomology of homogeneous parts of a graded module

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500924 ◽

2020 ◽

pp. 2150092

Author(s):

Tran Do Minh Chau ◽

Nguyen Thi Kieu Nga ◽

Le Thanh Nhan

Keyword(s):

Asymptotic Stability ◽

Local Ring ◽

Local Cohomology ◽

Prime Ideals ◽

Finitely Generated ◽

Graded Ring ◽

Noetherian Local Ring ◽

Graded Module

Let [Formula: see text] be a homogeneous graded ring, where [Formula: see text] is a Noetherian local ring. Let [Formula: see text] be a finitely generated graded [Formula: see text]-module. For [Formula: see text] set [Formula: see text]. Denote by [Formula: see text] the set of all prime ideals of [Formula: see text] containing [Formula: see text]. For [Formula: see text], let [Formula: see text] be the set of all [Formula: see text] such that [Formula: see text] In this paper, we prove that the sets [Formula: see text] and [Formula: see text] do not depend on [Formula: see text] for [Formula: see text]. We show that the annihilators [Formula: see text], [Formula: see text] are eventually stable, where [Formula: see text] for [Formula: see text]. As an application, we prove the asymptotic stability of some loci contained in the non-Cohen–Macaulay locus of [Formula: see text].

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