The Flat Cohomology of Group Schemes of Rank p

Abstract

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].