The local homological structure of generalized splines

TL;DR

This paper explores the local homological structure of generalized splines, linking algebraic and combinatorial perspectives, and proves conditions under which spline modules are free, especially over polynomial rings.

Contribution

It introduces a homological framework connecting local ring theory with generalized splines and establishes new criteria for spline module freeness.

Findings

Generalized splines are controlled by local homological properties.

Spline modules over certain rings are proven to be free under specific conditions.

The framework applies to polynomial rings with principal edge labels.

Abstract

Generalized splines are a simultaneous generalization of GKM theory -- which studies equivariant cohomology -- and classical splines, which provide piecewise approximations of functions. Generalized splines can also be understood via schemes, with the interpolation constraints -- or so-called GKM-condition -- encoded by gluing along certain closed subschemes. This view provides a local-global principle, with the local pictures retaining the generalized spline structure. Consequently, the behavior of generalized splines over local rings controls certain global phenomena, such as projectivity and often freeness. We introduce an interface between the homological study of local rings and the combinatorial study of generalized splines. We identify precisely how the generalized spline structure coordinates with the existing homological local ring machinery. This is accomplished by two exact sequences that provide a regulatory structure on the local cohomology of a generalized spline module. As an application, we use this to prove that for any edge-labeled graph $G$ with principal ideal labels, and any Cohen-Macaulay ring $R$ of Krull dimension 2 at each maximal ideal, the module of splines $R_G$ is free, provided it has finite projective dimension. As a special case, this implies every generalized spline module over $k[x,y]$ with principal edge labels is free.