Abstract. We discuss the relation between crystalline Dieudonné theory and Dieudonné displays of p-divisible groups. The theory of Dieudonné displays is extended to the prime 2 without restriction, which implies that the classification of finite locally free group schemes by Breuil-Kisin modules holds for the prime 2 as well.
We show that to every
p
p
-divisible group over a
p
p
-adic ring one can associate a display by crystalline Dieudonné theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated displays, which is a smooth morphism of smooth algebraic stacks. As an application we obtain a new proof of the equivalence between infinitesimal
p
p
-divisible groups and nilpotent displays over
p
p
-adic rings, and a new proof of the equivalence due to Berthelot and Gabber between commutative finite flat group schemes of
p
p
-power order and Dieudonné modules over perfect rings.
Abstract. Let k be a perfect field of characteristic p > 2 and K an extension of F = Frac W (k) contained in some F (µpr ). Using crystalline Dieudonné theory, we provide a classification of p-divisible groups overclassification is a consequence of (a special case of) the theory of Kisin-Ren; in this setting, our construction gives an independent proof of this result, and moreover allows us to recover the Dieudonné crystal of a p-divisible group from the Wach module associated to its Tate module by Berger-Breuil or by Kisin-Ren.
The Dieudonné crystal of a p-divisible group over a semiperfect ring R can be endowed with a window structure. If R satisfies a boundedness condition, this construction gives an equivalence of categories. As an application we obtain a classification of p-divisible groups and commutative finite locally free p-group schemes over perfectoid rings by Breuil-Kisin-Fargues modules if p ≥ 3.such that ϕ • ψ and ψ • (1 ⊗ ϕ) are the multiplication maps. If R is torsion free then ξ is M-regular and ψ is dermined by ϕ.Corollary 1.6. (Theorem 10.12) If p ≥ 3, for each perfectoid ring R the category of commutative finite locally free p-group schemes over R is equivalent to the category of torsion Breuil-Kisin-Fargues modules for R.Acknowledgements. The author thanks Peter Scholze and Thomas Zink for helpful discussions.
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