# nLab 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) $k$-groups are equivalent to finite (resp. all) Galois modules by

$E\to E{\otimes }_{{k}_{s}}{k}^{\prime }=\coprod _{K/k\phantom{\rule{thickmathspace}{0ex}}\text{separable}}E\left(k\right)$E\to E\otimes_{k_s}k^\prime=\coprod_{K/k\;\text{separable}}E(k)

$G$ is etale iff $\mathrm{ker}{F}_{G}=e$. This implies that $F$ is an isomorphism.

###### Definition

A $k$-formal group $G=\mathrm{Spf}A$ is called local (or connected) if the following two equivalent conditions hold:

1. $A$ is local

2. $G\left(k\right)=\left\{0\right\}$ for any field $K$.

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

###### Proposition

Let $G$ be a $k$-formal group.

1. Then there is an exact sequence
$0\to {G}^{\circ }\to G\to {\pi }_{\circ }\left(G\right)\to 0$0\to G^\circ\to G\to \pi_\circ(G)\to 0

where ${G}^{\circ }$ is connected and ${\pi }_{\circ }\left(G\right)$ is étale. If $R\in {\mathrm{Mf}}_{k}$ is a finite dimensional $k$-ring and ${n}_{0}$ is the nilradical (i.e. -if $R$ is commutative- the set of all nipotent elements) of $R$ then

${G}^{\circ }\left(R\right)=\mathrm{ker}\left(G\left(R\right)\to G\left(R/{n}_{0}\right)\right)$G^\circ(R)=ker(G(R)\to G(R/n_0))

If $p$ is not $0$, then

${G}^{\circ }={\mathrm{lim}}_{n}\phantom{\rule{thickmathspace}{0ex}}\mathrm{ker}\left(F\phantom{\rule{thinmathspace}{0ex}}{G}^{\left({p}^{n}\right)}\right)$G^\circ= lim_n \; ker(F\,G^{(p^n)})

If $k\to {k}^{\prime }$ is a field extension then $\left(G{\otimes }_{k}{k}^{\prime }{\right)}^{\circ }={G}^{\circ }{\otimes }_{k}{k}^{\prime }$, and ${\pi }_{0}\left(G{\otimes }_{k}{k}^{\prime }\right)={\pi }_{0}\left(G\right){\otimes }_{k}{k}^{\prime }$.

1. If $k$ is perfect, there is a unique isomorphism $G={G}^{\circ }×{\pi }_{0}\left(G\right)$
###### Definition

An affine $k$-group $G$ is called infinitesimal if it one of the followig conditions is satisfied:

1. $G$ is finite and local.

2. $G$ is algebraic and $G\left(\mathrm{cl}\left(k\right)\right)=e$ (the terminal $k$-group)

###### Corollary

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

This extension splits if $k$ is perfect.

###### Definition

A (not necessarily commutative) connected formal $k$-group $G=\mathrm{Spf}A$ is said to be of finite type if $A$ is noetherian. The dimension of $G$ is defined to be the Krull dimension? of $A$.

Revised on June 6, 2012 12:22:18 by Stephan Alexander Spahn (178.195.231.138)