A Global Duality Theorem for Varieties Over Global Fields
A Global Duality Theorem for Varieties Over Global Fields
复制标题
全局域上簇的全局对偶定理
DOI：
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发表时间：
1989
期刊：
影响因子：
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通讯作者：
S. Saito
中科院分区：
文献类型：
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作者：
S. Saito
关于有限域上的整体域（数域或单变量函数域）的Galois上同调群，Poitou和Tate独立地证明了一个对偶定理，该对偶定理表示为局部紧交换群的某个长正合序列。这个定理被认为是经典算术理论中一些基本结果的改写，如类域理论和整体域的Brauer群理论。本文将该定理推广到高维情形，即给出了整体域上簇的根上同调群的对偶定理。证明主要结果的关键是利用Poincare对偶性将其化为数域上的整数环或有限域上的曲线的余上同调群的对偶定理，这与Poitou和Tate的原定理是等价的.
Concerning Galois cohomology groups of a global field, namely a number field or a function field in one variable over a finite field, Poitou and Tate have proved independently a duality theorem which is expressed in a certain long exact sequence of locally compact abelian groups. The theorem is considered as a rephrasing of some fundamental results in the classical arithmetic theory such as the class field theory and the theory of the Brauer group of a global field. In this paper we generalize the theorem to a higher dimensional case, namely we give a duality theorem for etale cohomology groups of a variety over a global field. The key point in the proof of our main result is that by using Poincare duality for a smooth morphism of schemes we reduce it to a duality theorem for etale cohomology groups of the ring of integers of a number field or a curve over a finite field, which is essentially equivalent to the original theorem of Poitou and Tate.