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Lifting of schemes and Monsky-Washnitzer algebras: equivalence and full faithfulness theorems. (Relèvement de schémas et algèbres de Monsky–Washnitzer: théorèmes d’équivalence et de pleine fidélité.) (French) Zbl 1165.14306
The goal of this paper is to state and prove a number of technical results which are needed for the study of isocrystals and rigid cohomology.
From the introduction (reviewer’s translation): Let \(\mathcal{O}\) be a noetherian ring, \(I \subset \mathcal{O}\) an ideal, \(R\) a noetherian \(\mathcal{O}\)-algebra and \(R_0=R/IR\). If \(R\) is separated and complete for the \(I\)-adic topology or if \(R\) is a Henselian semi-local algebra with radical \(I\), we know that the functor \(B \mapsto B/IB\) is an equivalence from the category of finite étale \(R\)-algebras to the category of finite étale \(R_0\)-algebras.
Here we extend this equivalence to the case of a Monsky-Washnitzer algebra \(R=A^{\dagger}\) (‘weakly complete of finite type’) or to the case of the Henselization in the sense of Raynaud, \(R=\tilde{A}\); the equivalence we obtain answers a question of EGA. This equivalence relies on the fact that \((\text{Spec}\,A^{\dagger}, \text{Spec}\,A_0)\) is a Henselian pair, and one can then apply Elkik’s lifting theorem. We deduce from this the lifting of locally free group schemes of finite type, either étale or of multiplicative type, or of \(p\)-divisible groups, either étale or of multiplicative type.
There were various special cases of these results which were previously known. In addition, the author establishes further results on Monsky-Washnitzer algebras \(A^{\dagger}\) under some restrictive assumptions on \(A\). For example, if \(A\) is either excellent and normal or regular and noetherian or integrally closed or Dedekind, then what can one say about \(A^{\dagger}\) and \(\tilde{A}\)?
This paper is a companion to the author’s works [Ann. Sci. Ec. Norm. Sup., IV. Sér. 35, No. 4, 575–603 (2002; Zbl 1060.14028)] and [“\(F\)-isocristaux convergents et fonctions \(L\): la conjecture de Dwork pour la fonction zêta-unité”, preprint, Rennes (2000); per bibl.] where the paper’s technical results are used in a concrete setting.

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14F35 Homotopy theory and fundamental groups in algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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