Relations between K2 and Galois cohomology
Relations between K2 and Galois cohomology
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DOI：
10.1007/bf01390012
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发表时间：
1976-12
影响因子：
3.1
通讯作者：
J. Tate
中科院分区：
文献类型：
--
作者：
J. Tate
在本文中，我们对于全局域 F，在 K2F 和伽罗瓦上同调群 HI (F,(Q/Z)(2)) 与其最大可分子群的商之间建立了一个自然同构。这种同构最早由利希滕鲍姆猜想，并不新鲜。事实上，它及其一些后果已经被几位作者 I-4-6, 9, 10, 14, 15] 用于研究全局场的 K z 。但到目前为止，仅在[25]和[24]中发布了证明的草图。在这里，我们以稍微简单的方式给出该证明的细节，其中没有使用 Iwasawa 的 Z~-扩展理论。论文的组织结构如下。在 w 2 中，我们回顾了群的连续上链上同调的一些一般事实，特别是 l-adic 系数模。这些事实是众所周知且基本的，但对于续集来说却是基础。因此，为了方便读者，我们对它们进行了总结。在 w 3 中，对于任何域 F 和任何素数 l~: char F，我们构造一个从 KzF 到 H2 (F, Zt (2)) 的同态 h。 h的构造取决于K2F的符号描述，即松本定理。我们证明，如果某个辅助同态h 1 是单射的，则Ker h 是/-整除的并且Coker h 没有/-挠率。由此可见，如果场 F 满足两个条件 (a) K2F 是没有非零可分子群的挠率群，以及 (b) ha 是 F 的内射，则 h 会导致从 KzF 的 /-主部分到 HZ 的挠率子群 (F, Zl (2)) 的同构。已知条件 (a) 对于全局域 F 成立； w167 4 和 5 的目的是表明条件 (b) 也是如此。在 w 4 中，我们给出了 ha 对域 F 是单射的准则。为了证明单射性，可以假设 F 包含 /- 单位根，在这种情况下 h 1: K 2F/IK 2F~ Br~ F 是循环代数理论给出的映射。（Br t F 是 F 的布劳尔群中除 l 阶的元素群。） h 1 是否对每个域都是单射的，这是一个悬而未决的问题。我们证明 hi 的单射性等价于 F'| 的核Br z F 由 a| 形式的元素生成（顺便说一下，h~ 是否满射的问题是经典的
In this paper we establish a natural isomorphism, for a global field F, between K2F and the quotient of the Galois cohomology group HI (F,(Q/Z)(2)) by its maximal divisible subgroup. This isomorphism, first conjectured by Lichtenbaum, is not new; indeed it and some of its consequences have already been used by several authors I-4-6, 9, 10, 14, 15] in studying K z of global fields. But so far only a sketch of a proof has been published, in [25] together with [24]. Here we give the details of that proof, in a slightly simpler arrangement in which no use is made of Iwasawa's theory of Z~-extensions. The organization of the paper is as follows. In w 2, we review some general facts about the continuous cochain cohomology of groups, especially with l-adic coefficient modules. The facts are quite well known and elementary but are basic for the sequel. Therefore we include a summary of them for the convenience of the reader.In w 3 we construct, for any field F, and any prime l~: char F, a homomorphism h from KzF to H2 (F, Zt (2)). The construction of h depends on the description of K2F by symbols, ie, on Matsumoto's theorem. We show that ifa certain auxiliary homomorphism h 1 is injective, then Ker h is/-divisible and Coker h has no/-torsion. It follows that if the field F satisfies the two conditions (a) that K2F is a torsion group with no non-zero divisible subgroup, and (b) that ha is injective for F, then h induces an isomorphism from the/-primary part of KzF to the torsion subgroup of HZ (F, Zl (2)). Condition (a) is known to hold for a global field F; the aim of w167 4 and 5 is to show that condition (b) does also. In w 4 we give a criterion for ha to be injective for a field F. To show injectivity one can assume F contains the/-th roots of unity, in which case h 1: K 2F/IK 2F~ Br~ F is a map given by the theory of cyclic algebras.(Br t F is the group of elements of order dividing l in the Brauer group of F.) It is an open question whether h 1 is injective for every field. We show that the injectivity of hi is equivalent to the kernel of F'| Br z F being generated by the elements of the form a| contained in it.(Incidentally, the question whether h~ is surjective is the classical