The behavior of depth functions of cover ideals of unimodular hypergraphs
type: Item
instance of: scholarly article
title: The behavior of depth functions of cover ideals of unimodular hypergraphs
zbMATH Open document ID: 1393.13005
author: Nguyen Thu Hang
author: Trân Nam Trung
published in: Arkiv för Matematik
publication date: 2018-01-09T00:00:00Z
review text: We recall some basic definitions from this paper. Let \(\mathcal H=(\mathcal V, \mathcal E)\) be a hypergraph, where \(\mathcal V=\{1,\dots,n\}\) (the set of \textit{vertices} of \(\mathcal H\)), and \(\mathcal E\) is a family of subsets of \(\mathcal V\) (the set of \textit{edges} of \(\mathcal H\)). The hypergraph \(\mathcal H\) is \textit{unimodular} if its incidence matrix is totally unimodular, i.e., every square submatrix of it has determinant equal to \(0,\pm1\). A \textit{vertex cover } of \(\mathcal H\) is a subset of \(\mathcal V\) which meets every edge of \(\mathcal H\). The \textit{cover ideal } \(J(\mathcal H)\) of \(\mathcal H\) is the ideal of the polynomial ring \(k[x_1,\dots,x_n]\) generated by the monomials \textbf{x}\(_{\tau}=x_{i_1}\dots x_{i_n}\), where \(\tau=\{i_1,\dots,i_n\}\) is a minimal vertex cover of \(\mathcal H\). Here are the three main results of this paper: (1) The depth function of the cover ideal of a unimodular hypergraph is non-increasing. (The \textit{depth function} of an ideal \(I\) of the polynomial ring \(R=k[x_1,\dots,x_n]\) is defined as depth \(R/I^s\) for \(s\geq1\).) (2) Let \(\mathcal H=(\mathcal V,\mathcal E)\) be a unimodular hypergraph with vertex set \(\mathcal V=\{1,\dots,n\}\). Then, \[ \text{depth}\, R/J(\mathcal H)^s =n-\ell(\mathcal (J(\mathcal H)) \] for all \(s\geq n\), where \(\ell\) denotes analytic spread. (3) If \(G\) is a bipartite graph with \(n\) vertices, then \[ \text{depth}\, R/J(G)^s =n-\nu_0(G)-1 \] for all \(\geq \nu_0 (G)\), where \(\nu_0(G)\) is the ordered matching number of \(G\). The authors show that this bound is sharp. Here is a quote from the paper: ``Our approach is based on a generalized Hochster's formula for computing local cohomology modules of arbitrary monomial ideals formulated by Takayama. Using this formula we are able to investigate the depth of powers of monomial ideals via the integer solutions of certain systems of linear inequalities''.
reviewed by: Moshe Roitman
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 05E40
zbMATH DE Number: 6823274
zbMATH Keywords: depth function
zbMATH Keywords: cover ideals
zbMATH Keywords: powers of ideals
zbMATH Keywords: analytic spread
zbMATH Keywords: index of depth stability
zbMATH Keywords: monomial ideals
zbMATH Keywords: unimodular hypergraphs
MaRDI profile type: Publication
DOI: 10.4310/ARKIV.2017.V55.N1.A4
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description: scientific article; zbMATH DE number 6823274