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

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

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

$$E\to E{\otimes }_{k_s}{k}^{\prime }=\coprod _{K/k\text{separable}}E(k)$$E\to E\otimes_{k_s}k^\prime=\coprod_{K/k\;\text{separable}}E(k)

$G$ is etale iff $\ker F_G=e$. 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. $G(k)=\{0\}$ 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

$$0\to G^{\circ }\to G\to {\pi }_{\circ }(G)\to 0$$0\to G^\circ\to G\to \pi_\circ(G)\to 0

where $G^{\circ }$ is connected and ${\pi }_{\circ }(G)$ 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

$$G^{\circ }(R)=\ker (G(R)\to G(R/n_0))$$G^\circ(R)=ker(G(R)\to G(R/n_0))

If $p$ is not $0$, then

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

If $k\to k^{\prime }$ is a field extension then $(G{\otimes }_k{k}^{\prime }{)}^{\circ }=G^{\circ }{\otimes }_k{k}^{\prime }$, and ${\pi }_0(G{\otimes }_k{k}^{\prime })={\pi }_0(G){\otimes }_k{k}^{\prime }$.

1. If $k$ is perfect, there is a unique isomorphism $G=G^{\circ }\times {\pi }_0(G)$

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 $G(\mathrm{cl}(k))=e$ (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 $A$.