On the jacobian module associated to a graph

Abstract

Abstract. We consider the jacobian module of a set f := ff1; : : : ; fmg 2 R:= k[X1; : : : ;Xn] 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[f] to be a complete intersection.