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Stabilité de l’holonomie sur les variétés quasi-projectives

Part of: (Co)homology theory

Published online by Cambridge University Press:  24 August 2011

Daniel Caro
Daniel Caro*
Affiliation:
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Campus 2, 14032 Caen Cedex, France (email: [email protected])
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Abstract

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Let 𝒱 be a mixed characteristic complete discrete valuation ring with perfect residue field k. We solve Berthelot’s conjectures on the stability of the holonomicity over smooth projective formal 𝒱-schemes. Then we build a category of F-complexes of arithmetic 𝒟-modules over quasi-projective k-varieties with bounded and holonomic cohomology. We obtain its stability under Grothendieck’s six operations.

Keywords

p-adic cohomologiesarithmetic 𝒟-modulesholonomicityoverconvergent isocrystalsstability

MSC classification

Secondary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1772 - 1792
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Abe10]Abe, T., Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic 𝒟-modules, Preprint (2010), http://arXiv.org/abs/1105.5796.Google Scholar
[Ber90]Berthelot, P., Cohomologie rigide et théorie des 𝒟-modules, in p-adic analysis (Trento, 1989) (Springer, Berlin, 1990), 80–124.Google Scholar
[Ber96]Berthelot, P., 𝒟-modules arithmétiques. I. Opérateurs différentiels de niveau fini, Ann. Sci. Éc. Norm. Supér. (4) 29 (1996), 185–272.CrossRefGoogle Scholar
[Ber00]Berthelot, P., 𝒟-modules arithmétiques. II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.) 81 (2000).Google Scholar
[Ber02]Berthelot, P., Introduction à la théorie arithmétique des 𝒟-modules, Astérisque 279 (2002), 1–80, Cohomologies p-adiques et applications arithmétiques, II.Google Scholar
[Car04]Caro, D., 𝒟-modules arithmétiques surcohérents. Application aux fonctions L, Ann. Inst. Fourier (Grenoble) 54 (2004), 1943–1996.Google Scholar
[Car05]Caro, D., Comparaison des foncteurs duaux des isocristaux surconvergents, Rend. Sem. Mat. Univ. Padova 114 (2005), 131–211.Google Scholar
[Car06a]Caro, D., Dévissages des F-complexes de 𝒟-modules arithmétiques en F-isocristaux surconvergents, Invent. Math. 166 (2006), 397–456.Google Scholar
[Car06b]Caro, D., Fonctions L associées aux 𝒟-modules arithmétiques. Cas des courbes, Compositio Math. 142 (2006), 169–206.CrossRefGoogle Scholar
[Car07]Caro, D., Overconvergent F-isocrystals and differential overcoherence, Invent. Math. 170 (2007), 507–539.Google Scholar
[Car08]Caro, D., Sur la stabilité par produits tensoriels des F-complexes de 𝒟-modules arithmétiques (2008), http://arXiv.org/abs/math/0605125.Google Scholar
[Car09a]Caro, D., 𝒟-modules arithmétiques associés aux isocristaux surconvergents. Cas lisse, Bull. Soc. Math. France 137 (2009), 453–543.Google Scholar
[Car09b]Caro, D., 𝒟-modules arithmétiques surholonomes, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 141–192.Google Scholar
[Car09c]Caro, D., Une caractérisation de la surcohérence, J. Math. Sci. Univ. Tokyo 16 (2009), 1–21.Google Scholar
[Car11]Caro, D., Pleine fidélité sans structure de Frobenius et isocristaux partiellement surconvergents, Math. Ann. 349 (2011), 747–805.CrossRefGoogle Scholar
[CT08]Caro, D. and Tsuzuki, N., Overholonomicity of overconvergent F-isocrystals over smooth varieties, ArXiv Mathematics e-prints (2008).Google Scholar
[Elk74]Elkik, R., Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. Éc. Norm. Supér. (4) 6 (1974), 553–603.Google Scholar
[Ked02]Kedlaya, K. S., Étale covers of affine spaces in positive characteristic, C. R. Math. Acad. Sci. Paris 335 (2002), 921–926.Google Scholar
[Ked05]Kedlaya, K. S., More étale covers of affine spaces in positive characteristic, J. Algebraic Geom. 14 (2005), 187–192.CrossRefGoogle Scholar
[LeS07]Le Stum, B., Rigid cohomology, Cambridge Tracts in Mathematics, vol. 172 (Cambridge University Press, Cambridge, 2007).CrossRefGoogle Scholar
[MN90]Mebkhout, Z. and Narváez-Macarro, L., Sur les coefficients de de Rham–Grothendieck des variétés algébriques, in p-adic analysis (Trento, 1989), Lecture Notes in Mathematics, vol. 1454 (Springer, Berlin, 1990), 267–308.CrossRefGoogle Scholar
[Noo97]Noot-Huyghe, C., 𝒟†-affinité de l’espace projectif, Compositio Math. 108 (1997), 277–318, With an appendix by P. Berthelot.Google Scholar
[Noo98]Noot-Huyghe, C., 𝒟†(∞)-affinité des schémas projectifs, Ann. Inst. Fourier (Grenoble) 48 (1998), 913–956.Google Scholar
[Noo03]Noot-Huyghe, C., Un théorème de comparaison entre les faisceaux d’opérateurs différentiels de Berthelot et de Mebkhout–Narváez-Macarro, J. Algebraic Geom. 12 (2003), 147–199.CrossRefGoogle Scholar