Multivariate generalized splines and syzygies on graphs

Given a graph $G$ whose edges are labeled by ideals of a commutative ring $R$ with identity, a generalized spline is a vertex labeling of $G$ by the elements of $R$ 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 $G$ with base ring $R$ has a ring and an $R$-module structure. In this paper, we focus on the freeness of generalized spline modules over certain graphs with the base ring $R = k[x_1 , \ldots , x_d]$ where $k$ is a field. We first show the freeness of generalized spline modules on graphs with no interior edges over $k[x,y]$ 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 $k[x , y]$ and later we extend this result to the base ring $R = k[x_1 , \ldots , x_d]$ under some restrictions.