On the jacobian module associated to a graph

Abstract:

We consider the jacobian module of a set $\mathbf {f}:=\{f_1,\ldots ,f_m\} \in R:=k[X_1,\ldots ,X_n]$ of squarefree monomials of degree $2$ corresponding to the edges of a connected bipartite graph $G$. We show that for such a graph $G$ the number of its primitive cycles (i.e., cycles whose chords are not edges of $G$) is the second Betti number in a minimal resolution of the corresponding jacobian module. A byproduct is a graph theoretic criterion for the subalgebra $k[G]:=k[\mathbf {f}]$ to be a complete intersection.