任意图上的广义样条
Generalized splines on arbitrary graphs
摘要：
设G是一个图，其边由交换环的理想标记.我们引入了一种广义样条，它是G的一个顶点用环上的元素标号，使得任意两个邻顶点的标号之差位于相应的边理想中.广义样条在组合学（Billera等人的em代数样条）和代数拓扑学（某些等变上同调环，由Goresky-Kottwitz-MacPherson等人描述）中自然出现。本文的中心问题是问一个任意边标号图何时具有非平凡的广义样条。答案是“总是”，我们证明了更强的结果，广义样条包含一个自由子模，其秩是G中的顶点数。当G是一棵树时，我们描述了所有的广义样条，并给出了几种将广义样条环描述为广义样条的交的方法。我们还提出了一个新的工具，我们称之为GKM矩阵，一个类似的关联矩阵的图，并结束与开放的问题。
英文摘要：
Let G be a graph whose edges are labeled by ideals of a commutative ring. We introduce a generalized spline, which is a vertex-labeling of G by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the corresponding edge ideal. Generalized splines arise naturally in combinatorics (em algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky-Kottwitz-MacPherson and others). The central question of this manuscript asks when an arbitrary edge-labeled graph has nontrivial generalized splines. The answer is `always', and we prove the stronger result that generalized splines contain a free submodule whose rank is the number of vertices in G. We describe all generalized splines when G is a tree, and give several ways to describe the ring of generalized splines as an intersection of generalized splines for simpler subgraphs of G. We also present a new tool which we call the GKM matrix, an analogue of the incidence matrix of a graph, and end with open questions.
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