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A not necesarily commutative connected formal group is called smooth formal k-group? if is a power-series algebra in variables
The coproduct is given by a set af formal power series satisfying the axioms (Ass),(Un) and (Com). Such a set is called a Dieudonné group law.
Let be a (not necessarily commutative) connected formal group of finite type. 1.If then is smooth.
If then the following conditions are equivalent
is an epimorphism.
The previous theorem can be strengthened:
Let , let be a connected (not necessarily commutative) formal -group (realized as the formal spectrum of a k-coring? ).
is the universal enveloping algebra of the Lie algebra of .
The category of connected formal -groups is equivalent to the category of all Lie algebras over .
If is finite dimensional then is smooth.
If is commutative is abelian.
; by duality any unipotent (commutative) -group is a power of the additive group.
(Dieudonné-Cartier-Gabriel) Let , let be a perfect field of characteristic let be a connected (not necessarily commutative) connected formal -group of finite type?, let be a subgroup of , let (this quit ion has not been defined in these lectures).
Then is of the form
This applies for instance to , for an algebraic group .
with smooth and infinitesimal?.
Let be a connected formal group of finite type, let . Then is bounded and
Let be an exact sequence of connected formal groups. Then .
If is a morphism of connected formal groups, with smooth and , then is an epimorphism iff is finite.