Subspace Arrangements over Finite Fields: Cohomological and Enumerative Aspects
Subspace Arrangements over Finite Fields: Cohomological and Enumerative Aspects
复制标题
有限域上的子空间排列：上同调和枚举方面
DOI：
10.1006/aima.1997.1647
复制
发表时间：
1997
影响因子：
1.7
通讯作者：
T. Ekedahl
中科院分区：
文献类型：
--
作者：
A. Björner;T. Ekedahl
在组合学中利用特征多项式处理了有限域上的线性或仿射子空间并上（或离）点的枚举问题，在代数几何中利用ζ函数处理了这一问题。我们讨论了这两种观点之间的基本关系。计数点也与排列的l进上同调有关（作为一种变体）。我们描述了作用在这个上同调上的Frobenius映射的特征值，它对应于zeta函数的更精细的分解。l进上同群及其特征空间的分解是由组合数据完全确定的。最后，在一些重要情况下，证明了zeta函数是由相应复变的拓扑结构决定的。
Abstract The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the l-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The l-adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data. Finally, it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases.