Arithmetical rank of a squarefree monomial ideal whose Alexander dual is of deviation two
type: Item
instance of: scholarly article
title: Arithmetical rank of a squarefree monomial ideal whose Alexander dual is of deviation two
zbMATH Open document ID: 1330.13039
author: Kyouko Kimura
author: Naoki Terai
author: Ken-Ichi Yoshida
published in: Acta Mathematica Vietnamica
publication date: 2015-10-20T00:00:00Z
review text: Let \(S\) be a standard graded polynomial ring over a field. Let \(I\) be a homogeneous ideal of \(S\). The arithmetical rank of \(I\), denoted \(\text{ara} I\), is the minimal number \(r\) such that \(I\) can be generated by \(r\) forms up to radical. \textit{G. Lyubeznik} [Lect. Notes Math. 1092, 214--220 (1984; Zbl 0578.13012)] showed that for a monomial ideal \(I\), the arithmetical rank is bounded below by the projective dimension of \(S/I\). There is no known criteria for when the equality holds. It is expected that the equality holds when the arithmetic degree of \(I\), namely the number of prime components of \(I\), is small. The main result of the paper (Theorem 1.1) is that the equality holds when \(I\) has a linear resolution and the difference between the arithmetic degree and the initial degree is (at most) \(2\). Using Alexander's duality, the main tool is the classification of Cohen-Macaulay squarefree monomial ideals of deviation \(2\) via hypergraphs given by Kimura, Terai, and Yoshida [\textit{K. Kimura} et al., Contemp. Math. 502, 73--112 (2009; Zbl 1194.13025)].
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13D02
zbMATH DE Number: 6496838
zbMATH Keywords: arithmetical rank
zbMATH Keywords: Alexander duality
zbMATH Keywords: arithmetic degree
zbMATH Keywords: initial degree
zbMATH Keywords: linear resolution
zbMATH Keywords: hypergraph
MaRDI profile type: Publication
full work available at URL: s40306-015-0136-x
OpenAlex ID: W626552662
cites work: Arithmetical Ranks of Stanley–Reisner Ideals Via Linear Algebra
cites work: On the Arithmetical Rank of the Edge Ideals of Forests
cites work: Arithmetical Ranks of Stanley–Reisner Ideals of Simplicial Complexes with a Cone
cites work: The Stanley–Reisner Ideals of Polygons as Set-Theoretic Complete Intersections
cites work: Arithmetical rank of the cyclic and bicyclic graphs
cites work: Resolutions of Stanley-Reisner rings and Alexander duality
cites work: Arithmetical rank of lexsegment edge ideals
cites work: Q4939593
cites work: On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals
cites work: Arithmetical rank of Cohen-Macaulay squarefree monomial ideals of height two
cites work: Arithmetical rank of strings and cycles
cites work: Arithmetical Rank of Squarefree Monomial Ideals Generated by Five Elements or with Arithmetic Degree Four
cites work: Binomial arithmetical rank of edge ideals of forests
cites work: Arithmetical rank of Gorenstein squarefree monomial ideals of height three
cites work: Arithmetical rank of squarefree monomial ideals of small arithmetic degree
cites work: Q5852755
cites work: Licci squarefree monomial ideals generated in degree two or with deviation two
cites work: Regularity, depth and arithmetic rank of bipartite edge ideals
cites work: Projective Dimension of String and Cycle Hypergraphs
cites work: A new explicit finite free resolution of ideals generated by monomials in an R-sequence
cites work: The arithmetical rank of the edge ideals of graphs with whiskers
cites work: Upper bounds for the arithmetical ranks of monomial ideals
cites work: Simplicial ideals, 2-linear ideals and arithmetical rank
cites work: On set-theoretic intersections
cites work: Note on set-theoretic intersections of subvarieties of projective space
cites work: Symbolic powers and matroids
cites work: An étale analog of the Goresky-MacPherson formula for subspace arrangements
DOI: 10.1007/S40306-015-0136-X
description: scientific article; zbMATH DE number 6496838