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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.04440 |
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Table of Contents:
- 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.