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Étale? affine (resp. étale formal) $k$-groups are equivalent to finite (resp. all) Galois modules by
$G$ is etale iff $ker F_G=e$. This implies that $F$ is an isomorphism.
A $k$-formal group $G=Spf A$ is called local (or connected) if the following two equivalent conditions hold:
$A$ is local
$G(k)=\{0\}$ for any field $K$.
A morphism from a connected group to an étale group is zero.
Let $G$ be a $k$-formal group.
where $G^\circ$ is connected and $\pi_\circ(G)$ is étale. If $R\in Mf_k$ 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
If $p$ is not $0$, then
If $k\to k^\prime$ is a field extension then $(G\otimes_k k^\prime)^\circ =G^\circ\otimes_k k^\prime$, and $\pi_0(G\otimes_k k^\prime)=\pi_0(G)\otimes_k k^\prime$.
An affine $k$-group $G$ is called infinitesimal if it one of the followig conditions is satisfied:
$G$ is finite and local.
$G$ is algebraic and $G(cl(k))=e$ (the terminal $k$-group)
A finite group is an extension of an étale group by an infinitesimal group.
This extension splits if $k$ is perfect.
A (not necessarily commutative) connected formal $k$-group $G=Spf A$ is said to be of finite type if $A$ is noetherian. The dimension of $G$ is defined to be the Krull dimension? of $A$.