Resolution of unmixed bipartite graphs

摘要

Let $G$ be a graph on the vertex set $V(G)={x_1,ldots, x_n}$ with the edge set $E(G)$, and let $R=K[x_1,ldots,x_n]$ be the polynomial ring over a field $K$. Two monomial ideals are associated to $G$, the extit{edge ideal} $I(G)$ generated by all monomials $x_i x_j$ with ${x_i,x_j}in E(G)$, and the extit{vertex cover ideal} $I_G$ generated by monomials $prod_{x_iin C} x_i$ for all minimal vertex covers $C$ of $G$. A {em minimal vertex cover} of $G$ is a subset $Csubset V(G)$ such that each edge has at least one vertex in $C$ and no proper subset of $C$ has the same property. Indeed, the vertex cover ideal of $G$ is the Alexander dual of the edge ideal of $G$. In this paper, for an unmixed bipartite graph $G$ we consider the lattice of vertex covers $mathcal{L}_G$ and we explicitly describe the minimal free resolution of the ideal associated to $mathcal{L}_G$ which is exactly the vertex cover ideal of $G$. Then we compute depth, projective dimension, regularity and extremal Betti numbers of $R/I(G)$ in terms of the associated lattice.