Hilbert function of local Artinian level rings in codimension two
type: Item
instance of: scholarly article
title: Hilbert function of local Artinian level rings in codimension two
zbMATH Open document ID: 1168.13013
author: Valentina Bertella
published in: Journal of Algebra
publication date: 2009-04-29T00:00:00Z
review text: The present paper deals with the problem of characterizing the possible Hilbert functions of local Artinian level rings of codimension two. The central theorem in the paper provides, given an ``admissible'' numerical function \(h\), an effective method to produce a zero-dimensional ideal of \(kx,y\) such that \(R/I\) is a local Artinian level ring having \(h\) as a Hilbert function. As a consequence, the author obtains the main result of the paper, which gives a characterization of the possible Hilbert functions of codimension two local Artinian level rings of given type and socle degree. The techniques used in the paper include the use of a similar notion to that of Gröbner bases in the formal power series ring, namely enhanced standard bases, for which the author gives a characterization in the case of ideals of height two. The author uses also the good properties of the generic initial ideal (Gin) of ideals in \(kx_1,\dots, x_n\). The results in the paper extend previous work by \textit{J. Briançon} [Invent. Math. 41, 45--89 (1977; Zbl 0353.14004)], \textit{A. Iarrobino} [Proc. Symp. Pure Math. 40, Part 1, 593--608 (1983; Zbl 0563.13010), Lect. Notes Math. 1124, 146--165 (1985; Zbl 0567.14001), J. Algebra 272, No. 2, 530--580 (2004; Zbl 1119.13015)] and \textit{J. V. Chipalkatti} and \textit{A. V. Geramita} [Mich. Math. J. 51, No. 1, 187--207 (2003; Zbl 1097.13514)].
reviewed by: Eduardo Saenz-de-Cabezon
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 13E10
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 5549367
zbMATH Keywords: Hilbert functions
zbMATH Keywords: level local artinian rings
zbMATH Keywords: monomial standard bases
zbMATH Keywords: minimal number of generators
zbMATH Keywords: generic initial ideals
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.11.023
OpenAlex ID: W1983326248
cites work: Ideals with stable Betti numbers
cites work: Standard bases and some computations in rings of power series
cites work: Description de Hilb sup(n) C{X,Y}
cites work: Q4274969
cites work: Koszul homology and extremal properties of Gin and Lex
cites work: Rigid resolutions and big Betti numbers
cites work: On parameter spaces for Artin level algebras.
cites work: Q4317713
cites work: Q4050885
cites work: Hilbert schemes of zero-dimensional subschemes of smooth varieties
cites work: The leading ideal of a complete intersection of height two
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Ancestor ideals of vector spaces of forms, and level algebras.
cites work: Compressed Algebras: Artin Algebras Having Given Socle Degrees and Maximal Length
cites work: Q3676251
cites work: Punctual Hilbert schemes
cites work: Q3762428
cites work: The Family GTof Graded Artinian Quotients ofk[x,y] of Given Hilbert Function
cites work: The Local Hilbert Function of a Pair of Plane Curves
cites work: Q4507801
cites work: Q3685189
cites work: Q3755558
cites work: Q3222965
DOI: 10.1016/J.JALGEBRA.2008.11.023
Recommended article: Q3105506
Recommended article: Q4364032
Recommended article: Hilbert-Kunz multiplicity of two-dimensional local rings
Recommended article: A note on the Hilbert-Samuel function in a two-dimensional local ring
Recommended article: Hilbert functions of graded algebras over Artinian rings
Recommended article: NORMAL HILBERT FUNCTIONS OF ONE-DIMENSIONAL LOCAL RINGS
Recommended article: Hilbert–Kunz Functions over Rings Regular in Codimension One
Recommended article: UPPER BOUNDS FOR LOCAL COHOMOLOGY FOR RINGS WITH GIVEN HILBERT FUNCTION
Recommended article: A finiteness theorem for the Hilbert functions of complete intersection local rings
Recommended article: Hilbert-Kunz functions of \(2\times 2\) determinantal rings
description: scientific article; zbMATH DE number 5549367