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Crystalline Dieudonné module theory via formal and rigid geometry. (English) Zbl 0864.14009
Dieudonné theory associates to a \(p\)-adic divisible group a Dieudonné-crystal, that is (roughly) a module with connection, fibration and Frobenius. The idea is that these are easier to handle than \(p\)-divisible groups. Except for some minor technical restrictions the paper shows that over a field \(R\) of characteristic \(p\) the functor is an equivalence of categories for smooth formal schemes, and still fully faithful up to isogeny for arbitrary reduced schemes. For the first result the author first treats fields, then studies deformations, and finally uses noetherian induction. For this last step one is already forced to work in the additional generality of formal schemes. The proof of the second result is very complicated and uses rigid-analytic methods.
Reviewer: G.Faltings (Bonn)

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14L05 Formal groups, \(p\)-divisible groups
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
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