A MODERN PROOF OF CHEVALLEY’S THEOREM ON ALGEBRAIC GROUPS
A MODERN PROOF OF CHEVALLEY’S THEOREM ON ALGEBRAIC GROUPS
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代数群Chevalley定理的现代证明
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发表时间：
2004
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通讯作者：
B. Conrad
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作者：
B. Conrad
设k是一个域，G是k上的一个代数群，这里我们指的是一个连通光滑k-群概型（不一定是仿射的）。回想一下，这样的G是自动分离的，有限类型的，并且在k上几何积分[4，Exp VIA，0.3，2.1.2，2.4]。最重要的代数群是仿射代数群（也称为线性代数群，因为仿射代数群与矩阵群GL（n）/k [15，Thm 3.4]的闭代数子群重合）和真代数群（也称为交换簇）。其他类型的代数群自然出现。例如，如果X是完美域k上的真（可能是奇异的）方案，则其相对Picard方案的约化连通分量（PicX/k）red是k上的典型非真非仿射代数群;当X是代数曲线时，则这样的代数群是所谓的几何类场论的广义雅可比。在另一个方向上，如果R是一个具有分式域K和剩余域k的离散赋值环，且A是K上的阿贝尔簇且具有R上的Neron模型，则A的闭纤维的连通分支是一个k上的代数群，它通常既不是仿射的，也不是真的（根据EGA IV 3，15.6.7，15.6.8和[1，1.2/8]，闭合纤维的连接分量的适当性等同于良好的还原）。因此，了解一般代数群的一般结构是很重要的。Chevalley定理断言，在一个完美域上的每一个代数群都是由一个线性代数群和一个阿贝尔簇“建立”起来的（在某种程度上，我们很快就会精确化）。这是一个极其重要的结果。例如，证明交换簇的良好约化的Neron-Ogg-Shafarevich准则（参见[14，Thm 1]和[1，7.4/5]）在很大程度上依赖于这个一般结构定理。几何类场论（如[13]），它将有理映射从代数曲线分类到交换代数群，也依赖于Chevalley定理。它来作为一个有点惊讶的作者发现，在出版的文献中没有存在一个证明Chevalley定理在现代语言，即使该定理被广泛使用。本文的目的是基于概型理论而不是Weil的基础[16]给出一个证明。Chevalley定理的已发表的证明，由Chevalley [2]和Rosenlicht [12]，是写非常古老的术语，不再使用。例如，[12]对于那些不熟悉韦尔基础的人来说似乎是不可读的。然而，[2]中的基本方法几乎是可穿透的，所以下面我们给出这个证明的现代翻译。Chevalley的证明有两个技术上的困难。首先，他采用了一种老式的概念“代数家庭的除数”之间的地方坐的概念韦尔除数和相对有效的卡地亚除数。它需要一点照顾翻译成现代术语，而绕过Chevalley的文件对旧理论的除数家庭。关于可逆层的基本结果（如[9]）将提供我们所需的工具。
Let k be a field, and let G be an algebraic group over k, by which we mean a connected smooth k-group scheme (not necessarily affine). Recall that such a G is automatically separated, finite type, and geometrically integral over k [4, Exp VIA, 0.3, 2.1.2, 2.4]. The most important classes of algebraic groups are the affine algebraic groups (also called linear algebraic groups, since affine algebraic groups coincide with the closed algebraic subgroups of the matrix groups GL(n)/k [15, Thm 3.4]) and the proper algebraic groups (also called abelian varieties). Other types of algebraic groups do arise naturally. For example, if X is a proper (possibly singular) scheme over a perfect field k then the reduced connected component (PicX/k)red of its relative Picard scheme is a typically non-proper non-affine algebraic group over k; when X is an algebraic curve then such algebraic groups are the so-called generalized Jacobians of geometric class field theory [13]. In another direction, if R is a discrete valuation ring with fraction field K and residue field k and if A is an abelian variety over K with Neron model A over R, the connected component of the closed fiber of A is an algebraic group over k which usually is neither affine nor proper (properness of the connected component of the closed fiber is equivalent to good reduction, by EGA IV3, 15.6.7,15.6.8 and [1, 1.2/8]). Thus, it is important to understand something about the general structure of general algebraic groups. Chevalley’s Theorem asserts that every algebraic group over a perfect field is ‘built up’ from a linear algebraic group and an abelian variety (in a way we will make precise shortly). This is an extremely important result. For example, the proof of the Neron-Ogg-Shafarevich criterion for good reduction of abelian varieties (see [14, Thm 1] and [1, 7.4/5]) relies heavily on this general structure theorem. Geometric class field theory (as in [13]), which classifies rational maps from an algebraic curve to commutative algebraic groups, also depends on Chevalley’s Theorem. It came as somewhat of a surprise to the author to find that in the published literature there does not exist a proof of Chevalley’s Theorem in modern language, even though the theorem is widely used. The purpose of this note is to present a proof based on scheme theory rather than Weil’s Foundations [16]. The published proofs of Chevalley’s Theorem, by both Chevalley [2] and Rosenlicht [12], are written with very archaic terminology which is no longer used. For example, [12] seems to be unreadable for those not familiar with Weil’s Foundations. However, the underlying method in [2] is almost penetrable, so below we present a modern translation of this proof. There are two sources of technical difficulties with Chevalley’s proof. First, he employs an old-style notion of “algebraic families of divisors” which sits somewhere between the notions of Weil divisor and relative effective Cartier divisor. It requires a little care to translate this into modern terminology while bypassing Chevalley’s papers on the old theory of divisor families. Basic results on invertible sheaves (such as in [9]) will supply the tools we need.