Multivariate generalized splines and syzygies on graphs

Abstract

Given a graph [Formula: see text] whose edges are labeled by ideals of a commutative ring [Formula: see text] with identity, a generalized spline is a vertex labeling of [Formula: see text] by the elements of [Formula: see text] so that the difference of labels on adjacent vertices is an element of the corresponding edge ideal. The set of all generalized splines on a graph [Formula: see text] with base ring [Formula: see text] has a ring and an [Formula: see text]-module structure. In this paper, we focus on the freeness of generalized spline modules over certain graphs with the base ring [Formula: see text] where [Formula: see text] is a field. We first show the freeness of generalized spline modules on graphs with no interior edges over [Formula: see text] such as cycles or a disjoint union of cycles with free edges. Later, we consider graphs that can be decomposed into disjoint cycles without changing the isomorphism class of the syzygy modules. Then we use this decomposition to show that generalized spline modules are free over [Formula: see text] and later we extend this result to the base ring [Formula: see text] under some restrictions.