Weakly polymatroidal ideals with applications to vertex cover ideals (Q601281)

The authors extend the notion of weakly polymatroidal ideals [cf. \textit{M. Kokubo} and \textit{T. Hibi}, Algebra Colloq. 13, No. 4, 711--720 (2006; Zbl 1104.13013)]) to arbitrary monomial ideals. Let \(G(I)\) be the minimal set of generators of a monomial ideal \(I\subset K[x_1,\dots,x_n]\). The ideal \(I\) is said to be weakly polymatroidal if for every pair of monomials \(u=x_1^{a_1}\cdots x_n^{a_n}>_{\mathrm{lex}}v=x_1^{b_1}\cdots x_n^{b_n}\) in \(G(I)\) such that \(a_1=b_1,\dots,a_{t-1}=b_{t-1}\) and \(a_t>b_t\), there exists \(j>t\) such that \(x_t(v/x_j)\in I\). \textit{J. Herzog} and \textit{Y. Takayama} [Homology Homotopy Appl. 4, No. 2(2), 277--294, electronic only (2002; Zbl 1028.13008)] introduced ideals with linear quotients. It means that there exists a minimal system of generators \(f_1,\dots,f_m\) of \(I\) such that for all \(2\leq i\leq m\), the colon ideal \((f_1,\dots,f_{i-1}):f_i\) is generated by a subset of \(\{x_1,\dots,x_n\}\). It allows to construct step by step a free resolution of the module \(K[x_1,\dots,x_n]/I\). The authors prove that a weakly polymatroidal ideal \(I\) has linear quotients. Let \(\mathfrak{m}=(x_1,\dots,x_n)\). It is also showmn that \(\mathfrak{m}I\) is again weakly polymatroidal. In the second part of the paper, vertex cover ideals of weighted hypergraphs (introduced in [\textit{J. Herzog, T. Hibi} and \textit{Ngô Viêt Trung}, Adv. Math. 210, No. 1, 304--322 (2007; Zbl 1112.13006)]) are considered. It is proved that the vertex cover ideals of a weighted Cohen-Macaulay bipartite graph are weakly polymatroidal. The authors finally give conditions on the edges of an hypergraph which imply that the vertex cover ideals are (componentwise) weakly polymatroidal.