The Valabrega–Valla module of monomial ideals

In this paper, we focus on the initial degree and the vanishing of the Valabrega–Valla module of a pair of monomial ideals [Formula: see text] in a polynomial ring over a field [Formula: see text]. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of J. For edge ideal of graphs, a complete characterization of the vanishing of the Valabrega–Valla module is given. For higher degree ideals, we find classes, where the Valabrega–Valla module vanishes. For the case that J is the facet ideal of a clutter [Formula: see text] and I is the defining ideal of singular subscheme of J, the non-vanishing of this module is investigated in terms of the combinatorics of [Formula: see text]. Finally, we describe the defining ideal of the Rees algebra of I/J provided that the Valabrega–Valla module is zero.