Basis Criteria for Extending Generalized Splines

TL;DR

This paper develops a criterion for determining when extending generalized splines over GCD domains are free modules, with implications for equivariant cohomology in geometry.

Contribution

It introduces a basis characterization for extending generalized splines as modules over GCD domains, generalizing previous spline theories.

Findings

Provides a determinant-based criterion for spline module freeness

Characterizes module bases of extending generalized splines

Links spline theory to equivariant cohomology computations

Abstract

Let $R$ be a commutative ring with identity and $G$ a graph. Extending generalized splines are a further extension of generalized splines by allowing vertex labels of $G$ to lie in varying modules rather than in a fixed ring $R$. Geometrically, this corresponds to the construction of equivariant cohomology by Braden and MacPherson (see [5]). Therefore, characterizing such splines has immediate implications in geometry, particularly in the computation of equivariant cohomology. In this paper, we study extending generalized splines as a $R$- module in which each vertex $v$ is labeled by $M_v = m_v R$ and each edge $e$ is labeled by $M_e = R/r_e R$ together with quotient $R$-module homomorphisms $M_v\to M_e$ for each vertex $v$ incident to the edge $e$, where $R$ is a greatest common divisor domain (GCD). We characterize module bases of such splines in terms of determinants so that it provides a criterion for freeness of spline modules.