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By a similar discussion (replacing by ) as in § 8, we have:
If is is a connected finite type formal group, define
This is a module over the -completion of .
The Dieudonné-functor is an equivalence
between the category of connected formal groups of finite type and the category of -modules such that has finite length. Moreover we have:
is finite iff has finite length iff for large .
is smooth iff is injective. In that case .