Generalized stretched ideals and Sally's conjecture

Mantero, Paolo; Xie, Yu

doi:10.1016/j.jpaa.2015.08.013

Cited by 10 publications

(13 citation statements)

References 28 publications

(85 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We recall the following lemma from [MX16] which guarantees that the socle degree and the Hilbert function of an Artinian reduction of A are the same for every general minimal reduction of m: In [EI87, §1] authors defined general minimal reductions in a different way. It is not clear whether their definition and Definition 2.6 are the same.…”

Section: Hence It Follows That a Is Level If And Only If A/(a) Is Levelmentioning

confidence: 99%

“…Following [MX16] we define the index of nilpotency of A with respect to a reduction J of m as In [EI87, Page 346] authors obtained an analogous result which does not depend upon the characteristic of R. Since it is not clear whether the definition of general elements given in [EI87] and Definition 2.4 are the same, it is not known whether Proposition 2.11 is true if R is not equicharacteristic.…”

Section: Hence It Follows That a Is Level If And Only If A/(a) Is Levelmentioning

confidence: 99%

See 1 more Smart Citation

The structure of the inverse system of level K-algebras

Masuti

Tozzo

2018

Collect. Math.

View full text Add to dashboard Cite

Macaulay's inverse system is an effective method to construct Artinian K-algebras with additional properties like, Gorenstein, level, more generally with any socle type. Recently, Elias and Rossi [ER17] gave the structure of the inverse system of d-dimensional Gorenstein K-algebras for any d > 0. In this paper we extend their result by establishing a one-to-one correspondence between d-dimensional level K-algebras and certain submodules of the divided power ring. We give several examples to illustrate our result.

show abstract

Section: Hence It Follows That a Is Level If And Only If A/(a) Is Levelmentioning

confidence: 99%

Section: Hence It Follows That a Is Level If And Only If A/(a) Is Levelmentioning

confidence: 99%

The structure of the inverse system of level K-algebras

Masuti

Tozzo

2018

Collect. Math.

View full text Add to dashboard Cite

show abstract

“…Therefore by Theorem 4.2, the reduction number of the ideal is 1 and the associated graded ring is Cohen-Macaulay. This example is taken from [21]. Example 4.3.…”

Section: Generalized Northcott's Inequalitymentioning

confidence: 99%

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…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%

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

Montaño

2015

Journal of Algebra

View full text Add to dashboard Cite

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.

show abstract

Generalized stretched ideals and Sally's conjecture

Cited by 10 publications

References 28 publications

The structure of the inverse system of level K-algebras

The structure of the inverse system of level K-algebras

Generalized Hilbert coefficients and Northcott's inequality

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

Contact Info

Product

Resources

About