Arithmetical rank of the cyclic and bicyclic graphs
type: Item
instance of: scholarly article
zbMATH Open document ID: 1242.13004
DOI: 10.1142/S0219498811005634
author: Margherita Barile
author: Dariush Kiani
author: Fatemeh Mohammadi
author: Siamak Yassemi
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/0805.1657
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 6037686
zbMATH Keywords: arithmetical rank
zbMATH Keywords: projective dimension
zbMATH Keywords: edge ideals
zbMATH Keywords: set-theoretic complete intersection ideals
MaRDI profile type: Publication
arXiv ID: 0805.1657
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cites work: On the arithmetical rank of monomial ideals
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cites work: On the ideal theory of graphs
cites work: Q4528984
published in: Journal of Algebra and its Applications
title: Arithmetical rank of the cyclic and bicyclic graphs
review text: The aim of this article is to study the arithmetical rank of a monomial ideal which is the edge ideal of a cycle or a bicycle graph, a bycicle graph is the union of two cycles with only one common vertex. More precisely, the edge ideal \(I\) of a cycle is generated by the monomials \(x_1x_2, x_2x_3,\dots,x_nx_1\) on the polynomial ring in \(n\) variables \(S\). Recall that the arithmetical rank of \(I\) (ara\((I)\)) is the minimal number of generators of \(I\) up to radical, and there is not a systematic way to compute this invariant. The main result of the paper under review is that ara\((I)\) coincides with the projective dimension of the ring \(S/I\), which was previously computed by Jacques in his thesis work.
reviewed by: Marcel Morales
description: scientific article; zbMATH DE number 6037686