Abstract
Let \({\mathcal{V}}\) be a complete discrete valued ring of mixed characteristic (0, p), K its field of fractions, k its residue field which is supposed to be perfect. Let X be a separated k-scheme of finite type and Y be a smooth open of X. We check that the equivalence of categories sp (Y, X),+ (from the category of overconvergent isocrystals on (Y, X)/K to that of overcoherent isocrystals on (Y, X)/K) commutes with tensor products. Next, in Berthelot’s theory of arithmetic \({\mathcal{D}}\) -modules, we prove the stability under tensor products of the devissability in overconvergent isocrystals. With Frobenius structures, we get the stability under tensor products of the overholonomicity.
References
Abe T.: Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic \({\mathcal{D}}\) -modules. Rend. Semin. Mat. Univ. Padova 131, 89–149 (2014)
Berthelot, P.: Cohomologie rigide et cohomologie rigide à support propre. Première partie. Prépublication IRMAR 96-03, Université de Rennes (1996)
Berthelot P.: \({{\mathcal{D}}}\) -modules arithmétiques. I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Sup. (4) 29(2), 185–272 (1996)
Berthelot, P.: Introduction à la théorie arithmétique des \({\mathcal{D}}\) -modules. Astérisque (2002), no. 279, pp. 1–80, Cohomologies p-adiques et applications arithmétiques II
Caro D.: \({\mathcal{D}}\) -modules arithmétiques surcohérents. Application aux fonctions L. Ann. Inst. Fourier, Grenoble 54(6), 1943–1996 (2004)
Caro D.: Dévissages des F-complexes de \({\mathcal{D}}\) -modules arithmétiques en F-isocristaux surconvergents. Invent. Math. 166(2), 397–456 (2006)
Caro D.: Fonctions L associées aux \({\mathcal{D}}\) -modules arithmétiques. Cas des courbes. Compositio Mathematica 142(01), 169–206 (2006)
Caro D.: Overconvergent F-isocrystals and differential overcoherence. Invent. Math. 170(3), 507–539 (2007)
Caro D.: Arithmetic \({\mathcal{D}}\) -modules associated with overconvergent isocrystals. Smooth case. (\({\mathcal{D}}\) -modules arithmétiques associés aux isocristaux surconvergents. Cas lisse.). Bull. Soc. Math. Fr. 137(4), 453–543 (2009)
Caro D.: \({\mathcal{D}}\) -modules arithmétiques surholonomes. Ann. Sci. École Norm. Sup. (4) 42(1), 141–192 (2009)
Caro D.: Pleine fidélité sans structure de Frobenius et isocristaux partiellement surconvergents. Math. Ann. 349, 747–805 (2011)
Caro D.: Stability of holonomicity over quasi-projective varieties. Compos. Math. 147(6), 1772–1792 (2011)
Caro, D.: Systèmes inductifs surcohérents de \({\mathcal{D}}\) -modules arithmétiques. ArXiv (2012)
Caro, D.: Sur la préservation de la surconvergence par l’image directe d’un morphisme propre et lisse. A paraître aux Ann. Sci. École Norm. Sup. (2015)
Caro D., Tsuzuki N.: Overholonomicity of overconvergent F-isocrystals over smooth varieties. Ann. Math. (2) 176(2), 747–813 (2012)
Elkik, R.: Solutions d’équations à coefficients dans un anneau hensélien. Ann. Sci. École Norm. Sup. (4) 6, (1973) 553–603 (1974)
Kedlaya K.S.: Semistable reduction for overconvergent F-isocrystals. I. Unipotence and logarithmic extensions. Compos. Math. 143(5), 1164–1212 (2007)
Kedlaya K.S.: Semistable reduction for overconvergent F-isocrystals. II. A valuation-theoretic approach. Compos. Math. 144(3), 657–672 (2008)
Kedlaya K.S.: Semistable reduction for overconvergent F-isocrystals. III: local semistable reduction at monomial valuations. Compos. Math. 145(1), 143–172 (2009)
Kedlaya K.S.: Semistable reduction for overconvergent F-isocrystals, IV: local semistable reduction at nonmonomial valuations. Compos. Math. 147(2), 467–523 (2011)
Meredith D.: Weak formal schemes. Nagoya Math. J. 45, 1–38 (1972)
Shiho A.: Crystalline fundamental groups. I. Isocrystals on log crystalline site and log convergent site. J. Math. Sci. Univ. Tokyo 7(4), 509–656 (2000)
Shiho A.: Crystalline fundamental groups II. Log convergent cohomology and rigid cohomology. J. Math. Sci. Univ. Tokyo 9(1), 1–163 (2002)
Tsuzuki N.: Morphisms of F-isocrystals and the finite monodromy theorem for unit-root F-isocrystals. Duke Math. J. 111(3), 385–418 (2002)
Virrion A.: Dualité locale et holonomie pour les \({\mathcal{D}}\) -modules arithmétiques. Bull. Soc. Math. Fr. 128(1), 1–68 (2000)
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L’auteur a bénéficié du soutien du réseau européen TMR Arithmetic Algebraic Geometry (contrat numéro UE MRTN-CT-2003-504917).
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Caro, D. Sur la stabilité par produit tensoriel de complexes de \({\mathcal{D}}\)-modules arithmétiques. manuscripta math. 147, 1–41 (2015). https://doi.org/10.1007/s00229-014-0716-4
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DOI: https://doi.org/10.1007/s00229-014-0716-4