Betti numbers of lex ideals over some Macaulay-Lex rings

Mermin, Jeff; Murai, Shinji

doi:10.1007/s10801-009-0192-1

Cited by 9 publications

(6 citation statements)

References 22 publications

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“…Using the notation and terminology introduced in Section 6, we now recover the notion of lex-ideal in quotient rings S := R/M where M is a monomial ideal of R. Definition 7.1. (see [17,20]) A set W of terms of S is called a lex-segment of S if, for all terms u, v ∈ S of the same degree, if u belongs to W and v > lex u then v belongs to W . A monomial ideal U of S is called a lex-ideal if the set of terms in U is a lex-segment of S.…”

Section: Lex-point In Hilbert Schemes Over Macaulay-lex Quotients On ...mentioning

confidence: 99%

Hilbert schemes over quotient rings via relative marked bases

Bertone¹,

Cioffi²,

Orth³

et al. 2022

Preprint

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In this paper, we introduce the notion of relative marked bases over quasi-stable ideals, together with constructive methods and a functorial interpretation. This notion allows to develop computational methods for the study of Hilbert schemes over quotients of polynomial rings for which we obtain the following main results.When the quotient rings are Cohen-Macaulay on quasi-stable ideals (for instance, Clements-Lindström rings), we describe an open cover of the Hilbert schemes by means of suitable changes of variables.When the quotient rings are Macaulay-Lex on quasi-stable ideals (for instance, Clements-Lindström rings), we investigate the lex-point of the Hilbert schemes and find some examples in which this point is singular.

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Section: Lex-point In Hilbert Schemes Over Macaulay-lex Quotients On ...mentioning

confidence: 99%

Hilbert schemes over quotient rings via relative marked bases

Bertone¹,

Cioffi²,

Orth³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In recent years Mermin, Peeva and their collaborators started a systematic investigation of rings R = A/a for which all the Hilbert functions of homogeneous ideals are obtained by Hilbert functions of images in R of lex-segment ideals. They called these rings Macaulay-lex [GaHoPe,Me1,Me2,MeMu1,MeMu2,MePe,MePeSt]. Two typical examples of such rings are the polynomial ring A and the so called Clements-Lindström rings, i.e.…”

Section: Introductionmentioning

confidence: 99%

Distractions of Shakin rings

Caviglia

Sbarra

2014

Journal of Algebra

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We study, by means of embeddings of Hilbert functions, a class of rings\ud which we call Shakin rings, i.e. quotients $K[X_1, . . . , X_n]/a$ of a polynomial ring over a field K by ideals a=$L+P$ which are the sum of a piecewise lex-segment ideal $L$, as defined by Shakin, and a pure powers ideal $P$. Our main results extend Abedelfatah’s recent work on the Eisenbud-Green-Harris Conjecture, Shakin’s generalization of Macaulay and Bigatti-Hulett-Pardue Theorems on Betti numbers and, when $\char(K) = 0$, Mermin-Murai Theorem on the Lex-Plus-Power inequality, from monomial regular sequences to a larger class of ideals. We also prove an extremality property of embeddings induced by distractions in terms of Hilbert functions of local cohomology modules

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“…(See [EGH96, Conjecture CB12] for the original formulation, and [FR07] for a more recent survey.) In a similar vein, V. Gasharov, N. Horwitz, J. Mermin, S. Murai and I. Peeva studied algebras R = A/a (where a is graded A-ideal) for which every possible Hilbert function is attained by the images (in R) of lex-segment A-ideals: quotients by compressedmonomial-plus-powers ideals [MP06], rational normal curves [GHP08], Veronese rings [GMP11] and quotients by coloured square-free monomial ideals [MM10]. In these papers, such rings are called Macaulay-lex, to emphasize the fact that every Hilbert function is attained by the image of a lex-segment ideal, analogous to the theorem of Macaulay.…”

Section: Introductionmentioning

confidence: 99%

Poset embeddings of Hilbert functions

Caviglia

Kummini

2012

Math. Z.

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Abstract. For a standard graded algebra R, we consider embeddings of the poset of Hilbert functions of R-ideals into the poset of R-ideals, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements-Lindström is proved. We exhibit a condition on the embedding that ensures that the classification of Hilbert functions is obtained with images of lexicographic segment ideals.

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Betti numbers of lex ideals over some Macaulay-Lex rings

Cited by 9 publications

References 22 publications

Hilbert schemes over quotient rings via relative marked bases

Hilbert schemes over quotient rings via relative marked bases

Distractions of Shakin rings

Poset embeddings of Hilbert functions

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