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Displays and formal \(p\)-divisible groups. (English) Zbl 1186.14048

Th. Zink [Astérisque 278, 127–248 (2002; Zbl 1008.14008), see also W. Messing, “Travaux de Zink”, Astérisque 311, 341–364 (2007; Zbl 1115.00012)] has proved, that over \(p\)-adic Nagata rings, formal \(p\)-divisible groups are classified by nilpotent displays. In the paper at hand, this result is extended to arbitrary \(p\)-adic rings.
Let \(p\) be a prime, and let \(R\) be a commutative ring in which \(p\) is nilpotent. A display over \(R\) is a quadruple \((P, Q, F_0, F_1)\), where \(P\) is a finitely generated \(W(R)\)-module, \(Q\) a submodule such that \(P/Q\) is a projective \(R\)-module, \(F_0 \colon P \rightarrow P\), \(F_1\colon Q \rightarrow P\) are Frobenius-linear maps, such that certain conditions are satisfied. Over a perfect field, a display is “the same” as a Dieudonné module \((P, F, V)\), by setting \(Q=V(P)\), \(F_1 = V^{-1}\). In that case, the nilpotence condition on the display corresponds to \(V\) being topologically nilpotent.
Zink has constructed a functor \(BT\) from the category of nilpotent displays over \(R\) to the category of formal \(p\)-divisible groups over \(R\), proved that it is always faithful, fully faithful if the nilradical of \(R\) is nilpotent, and an equivalence of categories, if \(R\) is a Nagata ring, and conjectured that it is an equivalence of categories at least over noetherian rings.
The author proves that \(BT\) is an equivalence of categories whenever \(p\) is nilpotent in \(R\), and hence, by taking projective limits, for every \(p\)-adically complete, separated ring. An important ingredient of the proof is the Grothendieck-Illusie deformation theory of truncated \(p\)-divisible groups.

MSC:

14L05 Formal groups, \(p\)-divisible groups
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