j -Multiplicity and depth of associated graded modules

Polini, Claudia; Xie, Yu

doi:10.1016/j.jalgebra.2013.01.001

Cited by 19 publications

(60 citation statements)

References 25 publications

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“…. , x i ), 0 ≤ i ≤ d − 1 [25]. By the weak Artin-Nagata property AN − d−2 , one has that J i : I ∞ = J i : I = J i : x i+1 and (J i : [29].…”

Section: Lemma 21 Let D ⊆ B ⊆ a And D ⊆ C ⊆ A Be Finite Modules Ovementioning

confidence: 97%

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Generalized Hilbert coefficients and Northcott's inequality

Xie

2016

Journal of Algebra

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Let R be a Cohen-Macaulay local ring of dimension d with infinite residue field. Let I be an R-ideal that has analytic spread (I) = d, satisfies the G d condition and the weak Artin-Nagata property AN − d−2 . We provide a formula relating the length λ(I n+1 /JI n ) to the difference P I (n) −H I (n), where J is a general minimal reduction of I, P I (n) and H I (n) are respectively the generalized Hilbert-Samuel polynomial and the generalized Hilbert-Samuel function. We then use it to establish formulas to compute the generalized Hilbert coefficients of I. As an application, we extend Northcott's inequality to non-m-primary ideals. Furthermore, when equality holds, we prove that the ideal I enjoys nice properties. Indeed, if this is the case, then the reduction number of I is at most one and the associated graded ring of I is Cohen-Macaulay. We also recover results of G. Colomé-Nin, C. Polini, B. Ulrich and Y. Xie on the positivity of the generalized first Hilbert coefficient j 1 (I). Our work extends that of S. Huckaba, C. Huneke and A. Ooishi to ideals that are not necessarily m-primary.

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“…. , x i ), 0 ≤ i ≤ d − 1 [25]. By the weak Artin-Nagata property AN − d−2 , one has that J i : I ∞ = J i : I = J i : x i+1 and (J i : [29].…”

Section: Lemma 21 Let D ⊆ B ⊆ a And D ⊆ C ⊆ A Be Finite Modules Ovementioning

confidence: 97%

“…Besides the application in intersection theory and singularity theory, these invariants are also used in the study of the arithmetical properties, like the depth, of the blowup algebras such as the associated graded rings (see for instance, [27,25,21]). …”

Section: Introductionmentioning

confidence: 99%

Generalized Hilbert coefficients and Northcott's inequality

Xie

2016

Journal of Algebra

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“…A similar formula for ideals of analytic spread ℓ(I) = s that satisfy condition G s (see Section 3) is given in the following theorem. In [24], the authors defined an R-ideal I of analytic spread ℓ(I) = d to be of minimal j-multiplicity if the ideal I in Proposition 2.2 is of minimal multiplicity, i.e., I 2 = x d I. They proved that this property is well-defined, i.e., that for general elements x 1 , x 2 , .…”

Section: Preliminariesmentioning

confidence: 99%

“…The j-multiplicity has been a very active research topic in the last few years as several results for m-primary ideals have been shown to hold for arbitrary ideals if the Hilbert-Samuel multiplicity is replaced by the j-multiplicity. For instance, numerical criteria for integral dependence (Rees' criterion, [9]), combinatorial interpretation of the multiplicity of monomial ideals ( [16]), and the relation with the Cohen-Macaulayness of blowup algebras ( [21], [24]) have been generalized this way.…”

Section: Introductionmentioning

confidence: 99%

“…The minimal multiplicity property was successfully extended by C. Polini and Y. Xie in [24] to arbitrary ideals of analytic spread d. They defined the properties of minimal j-multiplicity and almost minimal j-multiplicity, and with these new notions they were able to generalize the classical results on the Cohen-Macaulayness of G(I), under certain residual assumptions for I. In this paper, we introduce Goto-minimal j-multiplicity and almost Goto-minimal j-multiplicity and study the interplay among all these notions and the Cohen-Macaulayness of F (I), G(I), and the Rees algebra R(I) = ∞ n=0 I n , under the same residual assumptions.…”

Section: Introductionmentioning

confidence: 99%

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Artin–Nagata properties, minimal multiplicities, and depth of fiber cones

Montaño

2015

Journal of Algebra

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Minimal values of multiplicities of ideals have a strong relation with the depth of blowup algebras. In this paper, we introduce the notion of Goto-minimal j-multiplicity for ideals of maximal analytic spread. In a Cohen-Macaulay ring, inspired by the work of S. Goto, A. Jayanthan, T. Puthenpurakal, and J. Verma, we study the interplay among this new notion, the notion of minimal j-multiplicity introduced by C. Polini and Y. Xie, and the Cohen-Macaulayness of the fiber cone of ideals satisfying certain residual assumptions. We are also able to provide a bound on the reduction number of ideals of Goto-minimal j-multiplicity having either Cohen-Macaulay associated graded algebra, or linear decay in the depth of their powers.Theorem 4.2. Let R be a Cohen-Macaulay local ring of dimension d, with maximal ideal m, and infinite residue field. Let I be an R-ideal with analytic spread ℓ(I) = s and reduction number r. Assume I satisfies G s and AN − s−2 . Let J be a minimal reduction of I such that r J (I) = r and assume J ∩ I j m = JI j−1 m for every 2 j r, then the following are equivalent:The second result is a generalization of [10, 2.7], here a(F (I)) is the a-invariant of F (I):Theorem 4.8. Let R be a Cohen-Macaulay local ring of dimension d, with maximal ideal m, and infinite residue field. Let I be an R-ideal with analytic spread ℓ(I) = s, grade g, and reduction number r. Assume I satisfies G s , AN − s−2 , and Im = Jm for J a minimal reduction of I. Consider the following statements i) R(I) is Cohen-Macaulay (when g 2), ii) G(I) is Cohen-Macaulay, iii) F (I) is Cohen-Macaulay and a(F (I)) −g + 1, iv) r s − g + 1.

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Generalized multiplicities of edge ideals

2017

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We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show the j -multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j -multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.2010 Mathematics Subject Classification. 13H15, 13D40, 52B20.

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j -Multiplicity and depth of associated graded modules

Cited by 19 publications

References 25 publications

Generalized Hilbert coefficients and Northcott's inequality

Generalized Hilbert coefficients and Northcott's inequality

Artin–Nagata properties, minimal multiplicities, and depth of fiber cones

Generalized multiplicities of edge ideals

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