Ideals with stable Betti numbers
type: Item
instance of: scholarly article
title: Ideals with stable Betti numbers
zbMATH Open document ID: 0968.13008
author: Jürgen Herzog
author: Takayuki Hibi
author: Annetta G. Aramova
published in: Advances in Mathematics
publication date: 2001-09-19T00:00:00Z
review text: Let \(\kappa\) be a field of characteristic \(0\) and \(S=\kappa [x_1, \dots,x_n]\) the polynomial ring over \(\kappa\) with each \(\deg x_i=1\) and with the reverse lexicographic term order induced by \(x_1> \cdots >x_n\). For a graded ideal \(I\) of \(S\), let \(\text{Gin} (I)\) denote the corresponding generic initial ideal and \(I_{\langle j\rangle}\) the ideal generated by all homogeneous polynomials of degree \(j\) belonging to \(I\). In this paper the authors prove that the graded Betti numbers of \(I\) and \(\text{Gin}(I)\) are equal if and only if \(I_{\langle j\rangle}\) has a linear resolution for all \(j\).
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F55
zbMATH DE Number: 1496431
zbMATH Keywords: Stanley-Reisner ideal
zbMATH Keywords: reverse lexicographic term order
zbMATH Keywords: generic initial ideal
zbMATH Keywords: graded Betti numbers
reviewed by: Yves Félix
MaRDI profile type: Publication
full work available at URL: aima.1998.1888
OpenAlex ID: W2027364674
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DOI: 10.1006/AIMA.1998.1888
description: scientific article; zbMATH DE number 1496431