Demazure, lectures on p-divisible groups, II.7, étale and connected formal k-groups

This entry is about a section of the text

Étale? affine (resp. étale formal) kk-groups are equivalent to finite (resp. all) Galois modules by

EE k sk = K/kseparableE(k)E\to E\otimes_{k_s}k^\prime=\coprod_{K/k\;\text{separable}}E(k)

GG is etale iff kerF G=eker F_G=e. This implies that FF is an isomorphism.


A kk-formal group G=SpfAG=Spf A is called local (or connected) if the following two equivalent conditions hold:

  1. AA is local

  2. G(k)={0}G(k)=\{0\} for any field KK.

A morphism from a connected group to an étale group is zero.


Let GG be a kk-formal group.

  1. Then there is an exact sequence
0G Gπ (G)00\to G^\circ\to G\to \pi_\circ(G)\to 0

where G G^\circ is connected and π (G)\pi_\circ(G) is étale. If RMf kR\in Mf_k is a finite dimensional kk-ring and n 0n_0 is the nilradical (i.e. -if RR is commutative- the set of all nipotent elements) of RR then

G (R)=ker(G(R)G(R/n 0))G^\circ(R)=ker(G(R)\to G(R/n_0))

If pp is not 00, then

G =lim nker(FG (p n))G^\circ= lim_n \; ker(F\,G^{(p^n)})

If kk k\to k^\prime is a field extension then (G kk ) =G kk (G\otimes_k k^\prime)^\circ =G^\circ\otimes_k k^\prime, and π 0(G kk )=π 0(G) kk \pi_0(G\otimes_k k^\prime)=\pi_0(G)\otimes_k k^\prime.

  1. If kk is perfect, there is a unique isomorphism G=G ×π 0(G)G=G^\circ\times \pi_0(G)

An affine kk-group GG is called infinitesimal if it one of the followig conditions is satisfied:

  1. GG is finite and local.

  2. GG is algebraic and G(cl(k))=eG(cl(k))=e (the terminal kk-group)


A finite group is an extension of an étale group by an infinitesimal group.

This extension splits if kk is perfect.


A (not necessarily commutative) connected formal kk-group G=SpfAG=Spf A is said to be of finite type if AA is noetherian. The dimension of GG is defined to be the Krull dimension? of AA.

Revised on June 6, 2012 12:22:18 by Stephan Alexander Spahn (