A scheme-theoretic interpretation of generalized splines

TL;DR

This paper develops a scheme-theoretic interpretation of generalized splines on edge-labeled graphs by showing the spline ring $R_G$ is the limit of a spline diagram and can be assembled via finitely many pullbacks or equalizers of surjections, endowing $\text{Spec}(R_G)$ with an $R$-scheme structure. It demonstrates flat base change preserves spline structure and proves local-to-global freeness results, enabling freeness criteria to be checked on standard opens or prime localizations; these results are complemented by computational verifications and concrete examples. The work also describes the geometric counterpart of deletion and contraction through Ferrand pushouts and coequalizers in the category of schemes, linking combinatorial operations to scheme-theoretic gluing. Overall, the paper lays foundational links between generalized splines and scheme theory, providing tools for local-global analysis and a path for extending these ideas to broader categorical contexts and local rings.

Abstract

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.

Figures (31)

• Figure 1: An edge-labeled graph with ideals $I_{uv}, I_{vw}, I_{wu}$.
• Figure 2: Spline diagram of rings associated to Figure \ref{['fig:edge-triangle']}.
• Figure 3: Limit object $P$ with projections to the vertex and edge objects.
• Figure 4: Second-level pullback describing simultaneous GKM-conditions for two edges.
• Figure 5: Case 1: pullback along the common projection to $R_v$, yielding $R_{G} \times_{R_v}\bigl(R_u \times_{R/I_{uv}} R_v\bigr)$.

Theorems & Definitions (53)

• Definition 1
• Theorem 1
• proof
• Theorem 2
• proof
• Definition 2
• Lemma 1
• proof
• Definition 3
• Definition 4