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The local homological structure of generalized splines

Kyle Stoltz

TL;DR

The paper develops a local-homological framework for generalized splines by introducing two exact sequences (potential and copotential) that regulate depth and associated primes of spline modules over local rings. It connects depth, Auslander–Buchsbaum, and local cohomology to derive freeness results, showing that Cohen–Macaulay rings of dimension 2 with principal edge labels yield free spline modules when the projective dimension is finite, with a special case recovering freeness over $k[x,y]$. The approach provides a local-to-global principle: local freeness under flat base change implies global freeness for a broad class of edge-labeled graphs, including 2-variable splines. This work unifies combinatorial spline theory with local ring homology, offering explicit criteria for projectivity and laying groundwork for further graph-depth investigations and cohomological methods.

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.

The local homological structure of generalized splines

TL;DR

The paper develops a local-homological framework for generalized splines by introducing two exact sequences (potential and copotential) that regulate depth and associated primes of spline modules over local rings. It connects depth, Auslander–Buchsbaum, and local cohomology to derive freeness results, showing that Cohen–Macaulay rings of dimension 2 with principal edge labels yield free spline modules when the projective dimension is finite, with a special case recovering freeness over . The approach provides a local-to-global principle: local freeness under flat base change implies global freeness for a broad class of edge-labeled graphs, including 2-variable splines. This work unifies combinatorial spline theory with local ring homology, offering explicit criteria for projectivity and laying groundwork for further graph-depth investigations and cohomological methods.

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 with principal ideal labels, and any Cohen-Macaulay ring of Krull dimension 2 at each maximal ideal, the module of splines is free, provided it has finite projective dimension. As a special case, this implies every generalized spline module over with principal edge labels is free.

Paper Structure

This paper contains 5 sections, 6 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Basic Homology of Local Rings
  3. The Homological Theory of Generalized Splines
  4. Applications
  5. Conclusion

Key Result

Lemma 3.2

Let $R$ be a commutative ring and $G$ a finite edge-labeled graph. There exist canonical exact sequences: \begin{tikzcd} 0 \arrow[r] & R_G \arrow[r] & R^V \arrow[r, "\overline{\rho_G}"] & \overline{R_G} \arrow[r] & 0 \end{tikzcd}and \begin{tikzcd} 0 \arrow[r] & \overline{R_G} \arrow[r] & R^E \arrow[

Figures (1)

  • Figure 1: A diamond graph with principal ideal edge labels over $k[x,y]$. Theorem \ref{['thm:main_theorem']} guarantees the spline module is free.

Theorems & Definitions (13)

  • Definition 3.1
  • Lemma 3.2: The Fundamental Sequences
  • proof
  • Proposition 3.3: Depth Estimates
  • proof
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 4.1
  • ...and 3 more