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

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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.

Last revised on June 6, 2012 at 12:22:18. See the history of this page for a list of all contributions to it.