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The Relation between BarsottiTate and Formal
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abelian scheme abelian variety affine scheme algebra assume augmentation ideal B. T. group Barsotti-Tate group base change bi-algebra canonical isomorphism canonical lifting Cartier dual co-algebra commutative diagram compatible completes the proof condition Corollary corresponding Cospec crystals IE defined definition denote direct limit E_(u element epimorphism equivalence of categories etale exact sequence exponential faithfully flat fiber finite and locally finite and locally-free finite presentation finite type formal Lie group formally smooth functor give given group G group scheme hence homomorphism Horn G hypothesis IE G implies ind-etale induces isomorphism kernel Lemma Let G Lie E(G Lie F Lie G Lie H linear locally free locally nilpotent morphism morphism G nilpotent divided powers Noetherian notation obtain obvious open set p-divisible p-torsion phism proposition prove radiciel Remark restriction ring S-group S-scheme satisfies sheaf sheaves Spec surjective theorem topology unique universal extension Zariski topology