Improving the bounds of the multiplicity conjecture: The codimension 3 level case
type: Item
instance of: scholarly article
title: Improving the bounds of the multiplicity conjecture: The codimension 3 level case
zbMATH Open document ID: 1116.13023
author: Fabrizio Zanello
published in: Journal of Pure and Applied Algebra
publication date: 2007-02-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0511308
review text: The Multiplicity Conjecture (MC) of Herzog, Huneke and Srinivasan gives a lower bound (resp.\ an upper bound) for the multiplicity of a Cohen-Macaulay algebra \(A\) in terms of the smallest (resp.\ largest) shift appearing in each free module in the minimal free resolution of \(A\). Many papers have appeared in recent years either proving interesting special cases of this conjecture, providing stronger bounds, or extending to the non-Cohen-Macaulay situation. This paper falls into the first two categories, exclusively in the case of codimension 3. First, the author proves the MC for compressed algebras, i.e.\ algebras which have the biggest possible Hilbert function for a given socle type. The second part of the paper is based on the observation that the Hilbert function (and hence the multiplicity) can be read from the graded Betti numbers, and that redundant terms can be cancelled (numerically) and hence do not contribute to the Hilbert function. (By redundant, we mean summands \(R(-t)\) that are repeated in consecutive free modules.) The idea, then, is to formulate a stronger variant of the MC by first formally removing all redundant summands and then looking for the smallest (resp.\ largest) shifts that remain, and bounding the multiplicity in the analogous way with these numbers. A similar idea was put forward by \textit{C. A. Francisco} [J.\ Algebra 299, No. 1, 309--328 (2006; Zbl 1105.13003)]. Unfortunately, in general there are choices for this ``numerical splitting'', and it is not uniquely determined. However, in the current paper the author had the clever idea of restricting to the case of level algebras in codimension 3. In this case, the numerical splitting is uniquely determined, and the author conjectures that the improved MC holds. Furthermore, this stronger bound can be read immediately from the Hilbert function. He proves a few initial cases of his conjecture. In a subsequent paper jointly written by the author with \textit{U. Nagel} and the review…
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 5124459
zbMATH Keywords: multiplicity conjecture
zbMATH Keywords: level algebra
zbMATH Keywords: \(h\)-vector
zbMATH Keywords: Hilbert function
Wikidata QID: Q123004377
reviewed by: Juan C. Migliore
describes a project that uses: CoCoA
MaRDI profile type: Publication
OpenAlex ID: W2056188633
arXiv ID: math/0511308
cites work: Some features of \(CoCoA 3\)
cites work: Q4175681
cites work: New approaches to bounding the multiplicity of an ideal
cites work: Q3592296
cites work: Q3222956
cites work: Q4882175
cites work: The Hilbert function of a level algebra
cites work: Powers of complete intersections: Graded Betti numbers and applications
cites work: Bounds for multiplicities
cites work: Notes on the multiplicity conjecture
cites work: A Note on the Multiplicity of Cohen-Macaulay Algebras with Pure Resolutions
cites work: Compressed Algebras: Artin Algebras Having Given Socle Degrees and Maximal Length
cites work: Power sums, Gorenstein algebras, and determinantal loci. With an appendix `The Gotzmann theorems and the Hilbert scheme' by Anthony Iarrobino and Steven L. Kleiman
cites work: The multiplicity conjecture in low codimensions
cites work: Extensions of the Multiplicity conjecture
cites work: Note on bounds for multiplicities
cites work: Extending the idea of compressed algebra to arbitrary socle-vectors.
cites work: Extending the idea of compressed algebra to arbitrary socle-vectors. II: Cases of non-existence
cites work: Level Algebras of Type 2
cites work: A non-unimodal codimension 3 level \(h\)-vector
cites work: When Is There a Unique Socle-Vector Associated to a Givenh-Vector?
DOI: 10.1016/J.JPAA.2006.05.017
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description: scientific article