Betti numbers of lex-segment ideals
type: Item
instance of: scholarly article
title: Betti numbers of lex-segment ideals
zbMATH Open document ID: 1091.13014
DOI: 10.1016/S0021-8693(03)00383-1
author: Heather Hulett
author: Heath M. Martin
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Consider the polynomial ring \(R = k[x_1,\dots,x_n]\) over a field of arbitrary characteristic. If \(I\) is a lex-segment ideal in \(R\) of degree \(d\), such that \(I_{d'}=0\) for \(d'<d\), then the authors give an explicit basis for the \(k\)-vector space of Koszul homology \(H_i(x_j, \dots, x_n; R/I)_{d+i-1}\). This allows them to write a formula for the lowest degree Betti numbers for lex-segment ideals. In the last section the authors provide a criterion for ideals that are lex-segment in a particular degree to be lex-segment in all degrees. They subsequently use this criterion in order to give another criterion for when a completely lex-segment ideal has a linear resolution, and to compute the Castelnuovo-Mumford regularity.
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 2105389
zbMATH Keywords: Eliahou-Kervaire resolution
zbMATH Keywords: Koszul homology
zbMATH Keywords: Castelnuovo-Mumford regularity
reviewed by: Cornel Baetica
MaRDI profile type: Publication
cites work: Koszul cycles and Eliahou-Kervaire type resolutions
cites work: Squarefree lexsegment ideals
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cites work: Q4274969
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full work available at URL: s0021-8693(03)00383-1
OpenAlex ID: W2021962616
description: scientific article; zbMATH DE number 2105389