Cohomology of p-adic Analytic Groups
Cohomology of p-adic Analytic Groups
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p-adic 解析群的上同调
DOI：
10.1007/978-1-4612-1380-2_12
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
T. Weigel
中科院分区：
文献类型：
--
作者：
P. Symonds;T. Weigel
本文的目的是给出紧p-进解析群的上同调的一个证明。有限群的上同调理论是由J.Tate首创，J-P发展起来的。Serre[23]，及其在数论中的应用。M.Lazard在他关于p-进解析群[17]的非凡工作中也考虑了它们的上同调，并证明了两个显著的定理：Lazard第一定理证明了紧p-进解析群G是一个虚拟Poincare对偶群；他的第二个定理证明了G的有理上同调与它的伴随的{{mathbb{q}_P}-李代数L(G)的G-稳定上同调重合。我们的主要目的是本着文献[10]中处理p-进解析群结构的精神来讨论Lazard的这些结果。我们还希望强调与离散对偶群理论的密切相似之处。为了实现这一目标，我们需要建立适当的同调代数。
The purpose of this article is to give an exposition on the cohomology of compact p-adic analytic groups. The cohomology theory of profinite groups was initiated by J. Tate and developed by J-P. Serre [23] in the sixties, with applications to number theory. In his extraordinary work on p-adic analytic groups [17], M. Lazard also considered their cohomology and proved two striking theorems: Lazard’s first theorem states that a compact p-adic analytic group G is a virtual Poincare duality group; his second theorem states that the rational cohomology of G coincides with the G-stable cohomology of its associated \( {\mathbb{Q}_P}\) -Lie algebra L (G). Our main goal is to discuss these results of Lazard in the spirit of the treatment of the structure of p-adic analytic groups in [10]. We also wish to emphasize the close parallels with the theory of discrete duality groups. In order to achieve this goal we need to set up the appropriate homological algebra.