The behavior of depth functions of cover ideals of unimodular hypergraphs
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''.
scientific article; zbMATH DE number 6823274
scientific article