Symbolic powers of edge ideals.
type: Item
instance of: scholarly article
title: Symbolic powers of edge ideals.
zbMATH Open document ID: 1084.13002
DOI: 10.1016/S0021-8693(03)00463-0
author: Carlos E. N. Bahiano
published in: Journal of Algebra
publication date: 2004-03-14T00:00:00Z
review text: Given a (finite simple) graph \(G\) with vertex set \(\{X_1,\dots, X_n\}\), a set of indeterminates generating the polynomial ring \(R= k[X_1,\dots, X_r]\), the ideal \(I(G)= (\{X_iX_j|\, X_i\) adjacent to \(X_j\})\) and the ring \(R/I(G)\) becomes of interest in the setting of commutative algebra/algebraic geometry as well as graph theory, since it is one more way to connect an algebraic object in a natural way to a graph. Thus one may relate parameters such as the stability number \(\alpha\) of \(G\) to properties of \(R/I(G)\) for example. A similar idea with \(R= k\langle X_1,\dots, X_n\rangle\) and the variables being non-commutative, including digraphs, has also been considered earlier than items in the reference list. In this case the setting of the paper under review is that of arbitrary ideals \(I\) which are specialized to ideals generated by square-free monomials (polynomials) of degree two (i.e., ideals of type \(I(G)\)). If \(I^{(m)}\) is the intersection of all primary components of \(I^m\) corresponding to minimal primes of \(I\), \(\sum_r(I)=\sum_r \sum^{r-1}_{i=1} I^{(i)} I^{(r- i)},\) and \(F(I)=\bigoplus{\strut}_{r\geq 2}(I^{(r)}/\sum_r (I))\), the essentially symbolic \(R\)-module of \(I\), then an element \(f\in I^{(r)}\setminus\sum_r(I)\), an essential generator of order \(r\) of \(F(I)\), is the prey mostly to be stalked in order to answer questions which in the \(I(G)\) case provide other answers/estimates for suitable types of simple graphs \(G\). A typical result (among several main results) might be that \(n\leq 2h-1\), where \(h\) is the height of \(I(G)\). While recapturing previous results of \textit{A. Simis}, \textit{W. V. Vasconcelos} and \textit{R. H. Villareal} [J. Algebra 167, No. 2, 389--416 (1994; Zbl 0816.13003)] as well, arguments in this well-done paper tend to have the ``free algebra'' flavor with ``side conditions'' leading to specifics as is common to similar contexts including this one.
reviewed by: J. Neggers
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 05C90
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 2055681
zbMATH Keywords: module of edge ideals
zbMATH Keywords: finite simple graph
zbMATH Keywords: essentially symbolic module
zbMATH Keywords: essential generator
zbMATH Keywords: height
describes a project that uses: Macaulay2
MaRDI profile type: Publication
cites work: Q4317713
cites work: Q3831115
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cites work: On the ideal theory of graphs
cites work: Effective computation of symbolic powers by jacobian matrices
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cites work: Cohen-Macaulay graphs
cites work: Q4528984
full work available at URL: s0021-8693(03)00463-0
OpenAlex ID: W2094801277
description: scientific article; zbMATH DE number 2055681