Group schemes of prime order
1970; Société Mathématique de France; Volume: 3; Issue: 1 Linguagem: Inglês
10.24033/asens.1186
ISSN1873-2151
Autores Tópico(s)Homicide, Infanticide, and Child Abuse
ResumoINTRODUCTION.-Our aim in this paper is to study group schemes G of prime order p over a rather general base scheme S. Suppose G===Spec(A), S=:Spec(R),and suppose the augmentation ideal I==Ker(A-^R) is free of rank one over R (so G is of order p = 2), say I = Rrc; then there exist elements a and c in R such that x 2 = ax and such that the group structure on G is defined by sx=x(^)i-{-i(^)x-cx(^)x.One easily checks that ac=i\ conversely any factorization ac = 2 € R defines a group scheme of order 2 over R. In this way all R-group schemes whose augmentation ideal is free of rank one are classified, and an easy sheaf-theoretic globalization yields a classification of group schemes of order 2 over any base S. In case p > 2 the difficulty is to find a good generator for the ideal I. To this end we prove first (theorem 1) that any G of order p is commutative and killed by p, i. e. is a <( module scheme 59 over F^==Z/pZ.In order to exploit the action of F^ on G, we assume in section 2 that the base S lies over Spec (Ap), where ^^'i^-d^'^ being a primitive (p -i)-th root of unity in the ring of p-adic integers Zp.For S over Spec (Ap) we prove (theorem 2) that the S-groups of order p are classified by triples (L, a, &) consisting of an invertible 0s-module L, together with sections a and b of L 0 ^"^ and L 0 ^ such that a(^) b = Wpy where Wp is the product of p and of an invertible element of Ap.Since the p-adic completion of A.p is Zpy this structure theorem
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