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Let a system of -modules? such that for all
is of finite length.
then is a finitely generated -module and the canonical map identifies
(-torsion formal group) A formal group is called -torsion formal group if
There are exact sequences
showing by induction the also is finite for all . Define
is a (contravariant) equivalence between the category of -torsion?formal groups and the category of tuples where M is a finitely generated -module and , to groups of endomorphisms of with
It follows from the lemma that is finitely generated and that
Conversely if is as before we define where
Moreover we have:
is finite iff is finite and in that case is the same as in § 7.
is -divisible iff is torsion-less (= free) and .
For any perfect extension? there is a functorial isomorphism
If is -divisible with Serre dual? then with
(Demazure Theorem p.71-72)
a) The Dieudonné functor
is a contravariant equivalence between the category of -torsion formal groups, and the category of all triples where is a finitely generated -module and , two group endomorphisms of satisfying
It follows from the lemma that is finitely generated and . Conversely if is as before, then we define as where .
Michel Demazure, lectures on p-divisible groups web