Classes of sequentially Cohen-Macaulay squarefree monomial ideals
type: Item
instance of: scholarly article
zbMATH Open document ID: 1307.13027
DOI: 10.1142/S1005386714000522
author: Oana Olteanu
published in: Algebra Colloquium
publication date: 2014-11-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.1996
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 6366779
zbMATH Keywords: squarefree lexsegment ideals
zbMATH Keywords: primary decomposition
zbMATH Keywords: sequentially Cohen-Macaulay ideals
MaRDI profile type: Publication
arXiv ID: 1103.1996
cites work: Squarefree lexsegment ideals
cites work: Componentwise linear ideals
cites work: Quotient algebras of Stanley-Reisner rings and local cohomology
cites work: Gotzmann ideals of the polynomial ring
OpenAlex ID: W2962942023
title: Classes of sequentially Cohen-Macaulay squarefree monomial ideals
review text: Let \(S=k[x_1,\dots, x_n]\) (\(k\) field) be a polynomial ring where the monomials are ordered lexicographically with \(x_1> \dots > x_n\). If \(u,v \in S\) are squarefree monomials of the same degree \(q\), the set \(L(u,v)=\{w \in S \mid w \text{ squarefree monomial of degree } q \text{ and } u \geq w \geq v \}\) is called a squarefree lexsegment of degree \(q\). For an arbitrary set of monomials \(T\) in \(S\) we denote \(\text{Shad} T = \{ wx_i \mid w \in T, i=1,\dots, n \text{ and } x_i \text{ does not divide } w \}\) and recursively \(\text{Shad}^i T=\text{Shad}^{i-1}(\text{Shad}T)\). A squarefree lexsegment \(L=L(u,v)\) is called completely squarefree lexsegment if \(\text{Shad}^i L\) is a squarefree lexsegment for all \(i \geq 1\).NEWLINENEWLINEIn this paper the author (1) obtains the minimal primary decomposition for a completely squarefree lexsegment \(L\); (2) describes \(\text{dim} S /I\) when \(I\) is a completely squarefree lexsegment ideal; (3) obtains the multiplicity of \(k[\Delta]\) where \(\Delta\) is the simplicial complex with Stanley-Reisner ideal \(I\); (4) gives a description of the completely squarefree lexsegment ideals that are sequentially Cohen-Macaulay; and (5) gives upper and lower bounds for \(\text{depth} S/I\) for an arbitrary squarefree lexsegment ideal \(I\).
reviewed by: Catalin Ciuperca
description: scientific article; zbMATH DE number 6366779