The saturation number of \(\mathbf{c}\)-bounded stable monomial ideals and their powers
type: Item
instance of: scholarly article
title: The saturation number of \(\mathbf{c}\)-bounded stable monomial ideals and their powers
zbMATH Open document ID: 1503.13011
DOI: 10.1215/21562261-2022-0013
author: Reza Abdolmaleki
author: Guangjun Zhu
author: Jürgen Herzog
published in: Kyoto Journal of Mathematics
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1909.10663
review text: In this paper, the authors mainly investigate the behavior of the saturation number of \(\mathbf{c}\)-bounded strongly stable ideals and their powers. They obtain relevant results for such a class of monomial ideals. In general, let \(I\) be a monomial ideal of a polynomial ring over a field, \(S=K[x_1,\ldots,x_n]\), it is known that the saturation number of \(I^k\), \(\mathrm{sat}(I^k)\), is a quasilinear function for \(k\gg 0\). The authors prove that \(\mathrm{sat}(I^k)\) is a quasilinear funtion from the beginning for a particular class of monomials ideals. In detail, the authors firstly compute in a simple way the saturation number of strongly stable ideals. Moreover, they compute the socle and the saturation number of equigenerated \(\mathbf{c}\)-bounded strongly stable ideals. We recall that a \(\mathbf{c}\)-bounded ideal is an ideal of \(S\) generated by monomials \(\mathbf{x}^\mathbf{a}\), with \(\mathbf{a}=(a_1,\ldots, a_n)\) bounded by \(\mathbf{c}=(c_1,\ldots, c_n)\). These results lead to an explicit formula for \(\text{sat}(I)\) when \(I\) is an ideal of Veronese type, \textit{i.e.}, a particular type of \(\mathbf{c}\)-bounded strongly stable ideal. Finally, the authors prove that \(\mathrm{sat}(I^k)\) is a quasilinear function from the beginning for a Veronese-type ideal \(I\), and they also provide an explicit formula for such a quasilinear function. In this work interesting examples are given to illustrate the constructions and the necessity of the hypotheses.
reviewed by: Luca Amata
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 7608271
zbMATH Keywords: \(\mathbf{c}\)-bounded strongly stable ideal
zbMATH Keywords: quasilinear function
zbMATH Keywords: saturation number
zbMATH Keywords: Veronese-type ideal
MaRDI profile type: Publication
OpenAlex ID: W4285613170
arXiv ID: 1909.10663
cites work: The asymptotic nature of the analytic spread
cites work: Asymptotic Stability of Ass(M/I n M)
cites work: Q4267400
cites work: Minimal resolutions of some monomial ideals
cites work: Resolutions of \textbf{a}-stable ideals.
cites work: Dirac's theorem on chordal graphs and Alexander duality
cites work: Homological shift ideals
cites work: The stable set of associated prime ideals of a polymatroidal ideal
cites work: Asymptotic behaviour of Castelnuovo-Mumford regularity
description: scientific article