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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 smooth
is reduced.
is an epimorphism.
The previous theorem can be strengthened:
(Cartier)
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 .
Let , let be a connected formal group? (=local formal group) of finite type?. Then
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.
Last revised on May 27, 2012 at 13:36:32. See the history of this page for a list of all contributions to it.