THE FLAT COHOMOLOGY OF GROUP SCHEMES OF RANK b.
THE FLAT COHOMOLOGY OF GROUP SCHEMES OF RANK b.
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秩群方案的平坦上同调 b．
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发表时间：
1973
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通讯作者：
Lawrence G. Roberts
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作者：
Lawrence G. Roberts
本文的目的是确定定义在局部数域上的整数环谱上的有限平坦群概型的第一平坦上同调群。我们发现第一上同调群同构于模p次幂环中单位群的某些子群。我们的主要结果，定理1，在[M-R，Prop.9.3]中公布。我要感谢巴里马祖尔教授对这项工作的慷慨兴趣和鼓励。在整个过程中，我们将始终使用以下符号：K是具有整数环R的局部数域; U是R中的单位群，ord是R满射到Z的加法赋值; U（M { u C U：ord（1 - u）？i}，R的剩余域k被假定具有特征p，并且我们将把P.= Z/pZ看作是k的子域; kc中的元素的数目是q =- pf; e = e（K/Qp）将表示K在Q上的绝对分歧指数。我们总是假设K包含p次单位根;这意味着-p是R的p -1次幂，m = e/（p-11）是整数。KS将表示K的固定可分闭包。我们所有的组方案在规范（R）上都是平坦的，并且将被视为规范（R）上（fppf）或（fpqf）站点的诱导层[SGA 3，IV 6.3]。
The intent of this paper is to determine the first flat cohomology groups of certain finite fiat group schemes which are defined over the spectrum of the ring of integers in a local number field. We discover that the first cohomology groups are isomorphic to certain subgroups of the group of units in the ring modulo p-th powers. Our main result, Theorem 1, was announced in [M-R, Prop. 9.3]. I would like to express my thanks to Professor Barry Mazur for his generous interest and encouragement in this work. Throughout we will consistently use the following notation: K is a local number field with ring of integers R; U is the group of units in R, ord is the additive valuation which takes R surjectively to Z; U(M { u C U: ord (1 - u) ? i}, the residue field k of R is assumed to have characteristic p, and we shall regard P. = Z/pZ as being a subfield of k; the number of elements in kc is q =- pf; e = e(K/Qp) will denote the absolute ramification index of K over Q,. We will always assume that K contains the p-th roots of unity; among other things this implies that -p is a p - 1-st power in R and that m = e/ (p -11 is an integer. Ks will denote a fixed separable closure of K. All our group schemes will be flat over Spec (R) and will be considered as inducing sheaves for the (fppf)- or (fpqf)-site over Spec(R) [SGA 3, IV 6.3].