图上的多元广义样条与合
Multivariate generalized splines and syzygies on graphs
摘要：
给定一个图$G$，其边由一个具有单位元的交换环$R$的理想标记，广义样条是$G$的顶点由$R$的元素标记，使得邻顶点上的标记差是相应边理想的元素.具有基环$R$的图$G$上的所有广义样条的集合具有环和$R$ -模结构。本文研究了一类基环为$R = k[x_1 , \ldots , x_d]$的图上广义样条模的自由性，其中$k$是域.我们首先证明了$k[x,y]$上没有内边的图上的广义样条模的自由性，如圈或具有自由边的圈的不交并。稍后，我们考虑可以分解成不相交的循环，而不改变合合模的同构类的图。然后利用这个分解证明了$k[x , y]$上的广义样条模是自由的，并在一定的限制下将这个结果推广到基环$R = k[x_1 , \ldots , x_d]$上。
英文摘要：
Given a graph $G$ whose edges are labeled by ideals of a commutative ring $R$ with identity, a generalized spline is a vertex labeling of $G$ by the elements of $R$ so that the difference of labels on adjacent vertices is an element of the corresponding edge ideal. The set of all generalized splines on a graph $G$ with base ring $R$ has a ring and an $R$-module structure. In this paper, we focus on the freeness of generalized spline modules over certain graphs with the base ring $R = k[x_1 , \ldots , x_d]$ where $k$ is a field. We first show the freeness of generalized spline modules on graphs with no interior edges over $k[x,y]$ such as cycles or a disjoint union of cycles with free edges. Later, we consider graphs that can be decomposed into disjoint cycles without changing the isomorphism class of the syzygy modules. Then we use this decomposition to show that generalized spline modules are free over $k[x , y]$ and later we extend this result to the base ring $R = k[x_1 , \ldots , x_d]$ under some restrictions.
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