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Arithmetic 𝒟-modules on the unit disk. With an appendix by Shigeki Matsuda

Part of: (Co)homology theory

Published online by Cambridge University Press:  09 November 2011

Richard Crew
Richard Crew*
Affiliation:
Department of Mathematics, The University of Florida, Gainesville, FL 32601, USA (email: rcrew@ufl.edu)
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Abstract

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Let 𝒱 be a complete discrete valuation ring of mixed characteristic. We classify arithmetic 𝒟-modules on Spf(𝒱[[t]]) up to certain kind of ‘analytic isomorphism’. This result is used to construct canonical extensions (in the sense of Katz and Gabber) for objects of this category.

Keywords

arithmetic 𝒟-modulecanonical extension

MSC classification

Secondary: 14F30: $p$-adic cohomology, crystalline cohomology 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 227 - 268
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Abh64]Abhyankar, S. S., Local analytic geometry (Academic Press, New York, 1964).Google Scholar
[And02]André, Y., Filtrations de type Hasse–Arf et monodromie p-adique, Invent. Math. 148 (2002), 285–317.Google Scholar
[Ber90]Berthelot, P., Cohomologie rigide et théorie des 𝒟-modules, in p-adic analysis, Trento, 29 May–2 June, 1989, Lecture Notes in Mathematics, vol. 1454 eds Baldassari, F., Bosch, S. and Dwork, B. (Springer, New York, 1990).Google Scholar
[Ber96a]Berthelot, P., 𝒟-modules arithmétiques I. Opérateurs différentiels de niveau fini, Ann. Sci. Éc. Norm. Supér. (4) 29 (1996), 185–272.CrossRefGoogle Scholar
[Ber96b]Berthelot, P., Cohomologie rigide et cohomologie rigide à support propre, Preprint 96-03 (1996), IRMAR, Université de Rennes I.Google Scholar
[Ber02a]Berthelot, P., 𝒟-modules arithmétiques II. Descente par Frobenius, Mémoires, vol. 81 (Societé Mathématique de France, Paris, 2002).Google Scholar
[Ber02b]Berthelot, P., Introduction à la théorie arithmétique des 𝒟-modules, in Cohomologies p-adiques et applications arithmétiques (II), Astérisque, vol. 279 eds Berthelot, P.et al. (Societé Mathématique de France, Paris, 2002), 1–80.Google Scholar
[Ber07]Berthelot, P., Letter to Daniel Caro, June 22 (2007).Google Scholar
[Ber10]Berthelot, P., Letter to the author, Sept. 14 (2010).Google Scholar
[Bos84]Bosch, S., Güntzer, U. and Remmert, R., Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, vol. 261 (Springer, Berlin, 1984).Google Scholar
[Car06a]Caro, D., Dévissages des F-complexes de 𝒟-modules arithmetiques en F-isocristaux surconvergents, Invent. Math. 166 (2006), 397–456.CrossRefGoogle Scholar
[Car06b]Caro, D., Fonctions L associées aux 𝒟-modules arithmétiques. Cas des courbes, Compositio Math. 142 (2006), 169–206.CrossRefGoogle Scholar
[CM02]Christol, G. and Mebkhout, Z., Equations différentielles p-adiques et coefficients p-adiques sur les courbes, in Cohomologies p-adiques et applications arithmétiques (II), Astérisque, vol. 279 eds Berthelot, P.et al. (Societé Mathématique de France, Paris, 2002), 125–184.Google Scholar
[Cre87]Crew, R., F-isocrystals and p-adic representations, in Algebraic geometry – Bowdoin 1985, Part 2, Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 1987), 111–138.CrossRefGoogle Scholar
[Cre98]Crew, R., Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 717–763.Google Scholar
[Cre06]Crew, R., Arithmetic 𝒟-modules on a formal curve, Math. Ann. 336 (2006), 439–448.Google Scholar
[Fon94]Fontaine, J.-M., Le corps de périodes p-adiques, in Périodes p-adiques, Astérisque, vol. 223 (Societé Mathématique de France, Paris, 1994).Google Scholar
[Gar95]Garnier, L., Correspondance de Katz et irrégularité des isocristaux surconvergents de rang un, Manuscripta Math. 87 (1995), 327–348.CrossRefGoogle Scholar
[Gro61]Grothendieck, A., Éléments de géometrie algébrique III 0, Publ. Math. Inst. Hautes Études Sci. 11 (1961).Google Scholar
[Ked04]Kedlaya, K., A p-adic local monodromy theorem, Ann. of Math. (2) 160 (2004), 93–184.CrossRefGoogle Scholar
[Lau87]Laumon, G., Transformation de Fourier, constantes locales d’équations fonctionelle et conjecture de Weil, Publ. Math. Inst. Hautes Études Sci. 66 (1987), 131–210.CrossRefGoogle Scholar
[Mal91]Malgrange, B., Equations différentielles à coefficients polynomiaux (Birkhäuser, Basel, 1991).Google Scholar
[Mat95]Matsuda, S., Local indices of p-adic differential operators corresponding to Artin–Schreier–Witt coverings, Duke Math. J. 77 (1995), 607–625.CrossRefGoogle Scholar
[Mat02]Matsuda, S., Katz correspondence for quasi-unipotent overconvergent isocrystals, Compositio Math. 134 (2002), 1–34.Google Scholar
[Meb02]Mebkhout, Z., Analogue p-adique du théorème de Turritin et le théorème de la monodromie p-adique, Invent. Math. 148 (2002), 319–335.Google Scholar
[Noo04]Noot-Hyughe, C., Transformation de fourier des 𝒟-modules arithmétiques, in Geometric aspects of Dwork theory, eds Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N. M. and Loeser, F. (Walter de Gruyter, Berlin, 2004), vol. 2, 857–907.Google Scholar
[Ray70]Raynaud, M., Anneaux locaux henséliens, Lecture Notes in Mathematics, vol. 169 (Springer, New York, 1970).CrossRefGoogle Scholar
[Sch02]Schneider, P., Nonarchimedean functional analysis (Springer, 2002).CrossRefGoogle Scholar
[Vir00]Virrion, A., Dualité locale et holonomomie pour les 𝒟-modules arithmétiques, Bull. Soc. Math. France 128 (2000), 1–68.Google Scholar