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Barsotti-Tate groups and crystals. (English) Zbl 0919.14029
This paper surveys recent results (many still unpublished) on the crystalline Dieudonné module functor, \(\mathbb D\), for \(p\)-divisible groups over schemes of characteristic \(p\), and their application to questions about extensions of homomorphisms of \(p\)-divisible groups. A list of properties of \(\mathbb D\) is given without proofs (but with references) concerning faithfulness, equivalence, and faithfulness up to isogeny.
Using these, a proof is outlined of the fact that the restriction functor \(FC_S\rightarrow FC_\eta\) is fully faithful if \(R\) has a \(p\)-basis. Here \(S=\text{Spec }R\), \(\eta= \text{Spec }K\), \(R\) a discrete valuation ring with quotient field \(K\) of characteristic \(p\), and \(FC_S\) is the category of Dieudonné crystals over \(S\). A consequence of this result is that if \(G\), \(H\) are \(p\)-divisible groups over \(R\) then \(\operatorname{Hom}(G,H)\rightarrow \operatorname{Hom}(G_K,H_K)\) is a bijection, answering a question posed by Grothendieck, Raynaud and Rim in SGA 7 [A. Grothendieck in: Sém. Géom. algébrique, 1967-1969, SGA 7 I, Exposé IX, Lect. Notes Math. 288, 313-523 (1972; Zbl 0248.14006)].

MSC:
14L05 Formal groups, \(p\)-divisible groups
14F30 \(p\)-adic cohomology, crystalline cohomology
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