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Étale? affine (resp. étale formal) -groups are equivalent to finite (resp. all) Galois modules by
is etale iff . This implies that is an isomorphism.
A -formal group is called local (or connected) if the following two equivalent conditions hold:
for any field .
A morphism from a connected group to an étale group is zero.
Let be a -formal group.
where is connected and is étale. If is a finite dimensional -ring and is the nilradical (i.e. -if is commutative- the set of all nipotent elements) of then
If is not , then
If is a field extension then , and .
An affine -group is called infinitesimal if it one of the followig conditions is satisfied:
is finite and local.
is algebraic and (the terminal -group)
A finite group is an extension of an étale group by an infinitesimal group.
This extension splits if is perfect.
A (not necessarily commutative) connected formal -group is said to be of finite type if is noetherian. The dimension of is defined to be the Krull dimension? of .