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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)

Citations:

Zbl 1060.14028
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References:

[1] N. BOURBAKI, Algèbre [A], chapitres I à VII; Algèbre commutative · Zbl 0498.12001
[2] , chapitres I à X; Topologie Générale [TG], chapitre I.
[3] P. BERTHELOT, Cohomologie rigide et cohomologie rigide à supports propres, 1ère partie, prépublication Rennes 96-03 (1996).
[4] P. BERTHELOT - L. BREEN - W. MESSING, Théorie de Dieudonné cristalline II, LN 930, Springer (1982). Zbl0516.14015 MR667344 · Zbl 0516.14015
[5] P. BERTHELOT - W. MESSING, Théorie de Dieudonné cristalline I, Astéristique 63, Société Math. de France (1979), pp. 17-38. Zbl0414.14014 MR563458 · Zbl 0414.14014
[6] P. BERTHELOT - W. MESSING, Théorie de Dieudonné cristalline III: théorèmes d’équivalence et de pleine fidélité, The Grothendieck Festschrift, Vol. 1, Birkhaüser (1990), pp. 173-247. Zbl0753.14041 MR1086886 · Zbl 0753.14041
[7] R. ELKIK, Solutions d’équations à coefficients dans un anneau hensélien, Annales Scient. Ec. Norm. Sup., 4e série, 6 (1973), pp. 553-604. Zbl0327.14001 MR345966 · Zbl 0327.14001
[8] J.-Y. ETESSE, Descente étale des F-isocristaux surconvergents et rationalité des fonctions L de schémas abéliens, Prépublication Rennes 00-20 (avril 2000). (A paraître aux Annales Scient. Ec. Norm. Sup.) Zbl1060.14028 · Zbl 1060.14028 · doi:10.1016/S0012-9593(02)01099-6
[9] J.-Y. ETESSE, F-isocristaux convergents et fonctions L: la conjecture de Dwork pour la fonction zêta-unité, Preprint Rennes (juin 2000).
[10] A. GROTHENDIECK - J. DIEUDONNÉ, Eléments de Géométrie Algébrique, chap. I (Springer, coll. Grundlehren); chap. II, III, IV (Publ. Math. IHES, n7 8, 11, 17, 20, 24, 28, 32). Zbl0203.23301 · Zbl 0203.23301
[11] S. MATSUDA, Local indices of p-adic differential operators corresponding to Artin-Schreier-Witt coverings, Duke Math. Journal, Vol. 77, n7. 3 (1995). Zbl0849.12013 MR1324636 · Zbl 0849.12013 · doi:10.1215/S0012-7094-95-07719-9
[12] W. MESSING, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, LN 264, Springer 1972. Zbl0243.14013 MR347836 · Zbl 0243.14013 · doi:10.1007/BFb0058301
[13] L. MORET-BAILLY, Un problème de descente, Bull. Soc. Math. France, 124 (1996), pp. 559-585. Zbl0914.13002 MR1432058 · Zbl 0914.13002
[14] P. MONSKY - G. WASHNITZER, Formal Cohomology I, Annals of Math., 88, n7 2 (1968), pp. 181-217. Zbl0162.52504 MR248141 · Zbl 0162.52504 · doi:10.2307/1970571
[15] M. VAN DER PUT, The cohomology of Monsky and Washnitzer, Bulletin de la SMF, mémoire n7 23, tome 114, fasc. 2 (1986), pp. 33-60. Zbl0606.14018 MR865811 · Zbl 0606.14018
[16] M. RAYNAUD, Anneaux locaux henséliens, LN 169, Springer 1970. Zbl0203.05102 MR277519 · Zbl 0203.05102
[17] J.-P. SERRE, Corps locaux, Hermann 1968. MR354618
[18] A. GROTHENDIECK, Revêtements étales et groupe fondamental, LN 224, Springer 1971. MR354651 · Zbl 0234.14002
[19] M. ARTIN - A. GROTHENDIECK - J.-L. VERDIER, Théorie des topos et cohomologie étale des schémas, Tome 2, LN 270, Springer 1972. MR354653
[20] A. GROTHENDIECK, Cohomologie l-adique et fonctions L, LN 589, Springer 1977. MR463174 · Zbl 0356.14004
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