Free arrangements of hyperplanes over an arbitrary field
Free arrangements of hyperplanes over an arbitrary field
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超平面在任意域上的自由排列
DOI：
10.3792/pjaa.59.301
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发表时间：
1983
期刊：
影响因子：
--
通讯作者：
H. Terao
中科院分区：
文献类型：
--
作者：
H. Terao
文[6]证明了复数域C上l维向量空间中超平面的补的Poincar多项式的一个因式分解定理，其中超平面的排列是自由的。这就是Shephard-Todd-Brieskorn定理。本文的主要目的是给出任意域上的广义泛函定理。详细的证明将出现在[3]中。设A是域K上的k维向量空间V中的排列。换句话说，A是V的（1 - 1）维向量子空间的有限族。用V* 表示V的对偶向量空间。设S-S（V*）是V* 的对称代数。修复ba.se {x，...，x}，且S同构于。多项式代数，K[x，...，x]。设QeS是[_JneH]的约化定义方程.则Q是V* 的元素的乘积。f S的导数是K-。线性映射t”S-S对任意f，geS满足]-0和O（fg）-ft（g）+gt（f）.定义1.一个沿A的导子沿着（当我们在复解析范畴中时称为对数向量场[4]）是S的一个导子t满足o（Q）eQs。设D（A）表示沿A的沿着导子的集合。那么D（A）自然是一个S-模。定义2.如果D（A）是一个S-自由模，我们说A是一个自由安排。定义3. f S的一个导数0被称为。是B次齐次的，如果t（x）∈ S（i-1，...，i），其中S是由次数为B的单项式生成的S的向量子空间。我们写为B=degO。我们可以证明D（A）有一个自由基{O，...，O}，如果A是自由排列，则A是齐次导子.整数（deg 0，...，[6] A的广义指数（generalized exponents）。它们只依赖于A。K. Saito [4]当K= C时，对任意K”
In [6], we proved a factorization theorem for the Poincar polynomial of the complement of hyperplnes in an /-dimensional vector space over the complex number field C when the arrangement of the hyperplanes is free. That was called Shephard-Todd-Brieskorn theorem there. Our main aim here is to report generalized fa.ctoriza.tion theorem tor ree rrangement over a.n a.rbitrary field. The detailed proof will appear in [3]. 1o Let A be an arrangement in an/-dimensional vector spce V over a field K. In other words, A is finite fa.mily of (/-1)-dimensional vector subspaces o.f V. Denote the dual vector space of V by V*. Let S-S(V*) be the symmetric algebra of V*. Fix a ba.se {x, ..., x} for V*, and S is isomorphic to. the polynomial algebra, K[x, ..., x]. Let Q e S be a reduced defining equation for [_Jne H. Then Q is a product of elements o.f V*. The derivation o.f S is a K-. linear map t" S-S satisfying ]--0 and O(fg)--ft(g)+gt(f) for any f, geS. Definition 1. A derivation along A (which is called a logarithmic vector field [4] when we are in the complex analytic category) is a derivation t of S satisfying o(Q) e Qs. Let D(A) denote the set o.f derivations along A. Then D(A) is naturally an S-module. Definition 2. If D(A) is an S-free module, we say that A is a free arrangement. Definition 3. A derivation 0 o.f S is said to. be homogeneous of degree b if t(x)e S (i--1, ...,/), where S is the vector subspace of S generated by monomials of degree b. We write b=degO. We can show that D(A) has a free base {O, ..., O} consisting of homogeneous derivations if A is a free arrangement. The integers (degO,..., deg Ot) are called the degree of A (called the generalized exponents of A in [6]). They depend only upon A. The following useful criterion, proved by K. Saito [4] when K= C, remains true for arbitrary K"