A scheme-theoretic interpretation of generalized splines

Classical splines feature prominently in approximation theory and numerical analysis, while GKM theory arises in the study of equivariant cohomology. More recently, generalized splines have been studied which simultaneously generalize both classical splines and GKM theory. We show that generalized splines can be understood as a scheme-theoretic phenomenon, and establish the foundations for working with generalized splines via scheme theory. To do this, we first prove that the ring of generalized splines over an edge-labeled graph is isomorphic to the limit of a diagram associated to the edge-labeled graph. This is used to establish the connection to scheme theory, and in particular to secure a local--global principle for generalized splines. The secondary aim of the paper is to catalog a variety of initial local--global results. We prove a module of multivariate generalized splines over $k[x,y]$ is free when an edge-labeled graph is locally trivial or determined by a cycle on a standard cover, provide criteria for generalized splines over UFDs to be finite projective, for multivariate generalized splines to be free, along with other results. The appendix includes a variety of examples which demonstrate precise experimental control over when a module of splines is free, with computer verification by Macaulay2. We conclude with an account of how deletion and contraction affect the spectrum of a ring of splines.