推广广义样条的基础准则
Basis Criteria for Extending Generalized Splines
AI总结：
本文提出通过行列式特征化推广广义样条的模基，以判断样条模的自由性，应用于等价同调计算。
摘要：
设R是一个有单位元的交换环，G是一个图。推广广义样条是对广义样条的进一步扩展，允许图G的顶点标签位于变化的模中，而不是固定的环R中。几何上，这对应于通过Braden和MacPherson（见[5]）构造等价同调。因此，特征化此类样条具有立即的几何学意义，特别是等价同调的计算。本文研究将推广广义样条作为R-模，在其中每个顶点v被标记为M_v = m_v R，每个边e被标记为M_e = R/r_e R，并且伴随商R-模同态M_v→M_e，其中每个顶点v与边e相邻。我们通过行列式来特征化此类样条的模基，从而提供了一个判断样条模自由性的准则。
英文摘要：
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.
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