Over convergent \(F\)-isocrystals and differential overcoherence. (\(F\)-isocristaux surconvergents et surcohérence différentielle.) (English) Zbl 1203.14025

Caro, D.

Invent. Math. 170, No. 3, 507-539 (2007).

This article is part of author’s successful attempt to build a category of arithmetic coefficients for varieties in car \(p>0\), stable by Grothendieck’s six operations (among which inverse images, direct images, tensor product, duality). The categories contructed by Caro are subcategories of arithmetic \(D\)-modules in the sense of P. Berthelot [\(p\)-adic cohomology and arithmetic applications (II). Paris: Société Mathématique de France. Astérisque 279, 1–80 (2002; Zbl 1098.14010)].
To achieve this goal, D. Caro first contructed a category of overcoherent arithmetic \(D\)-modules with Frobenius structure [J. Math. Sci., Tokyo 16, No. 1, 1–21 (2009; Zbl 1213.14041)], a priori stable by direct, inverse images and local cohomological functors, and then a subcategory of the previous one: the category of overholonomic \(D\)-modules. An essential step in his constructions is to prove that overconvergent \(F\)-isocrystals, which can be seen as elementary pieces in this theory, are overcoherent. This fact is the aim of this article, and is established for a variety \(Y\) smooth over a perfect field \(k\) of car \(p>0\). Caro proves moreover that complexes of arithmetic \(F\)-\(D\)-modules with bounded and overcoherent cohomology can be unscrewed into overconvergent \(F\)-isocrystals.

Reviewer: Christine Noot-Huyghe (Strasbourg)

MSC:

14F30	\(p\)-adic cohomology, crystalline cohomology
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials

Keywords:

overconvergent \(F\)-isocrystals; arithmetic \(D\)-modules; overcoherent modules; Grothendieck’s 6 operations; \(p\)-adic cohomologies

Citations:

Zbl 1098.14010; Zbl 1213.14041

References:

[1]	Berthelot, P.: Cohomologie cristalline des schémas de caractéristique p>0. Lect. Notes Math., vol. 407. Springer, Berlin (1974) · Zbl 0298.14012
[2]	Berthelot, P.: Géométrie rigide et cohomologie des variétés algébriques de caractéristique p. Introductions aux cohomologies p-adiques (Luminy 1984). Mém. Soc. Math. Fr., Nouv. Sér. 23(3), 7–32 (1986)
[3]	Berthelot, P.: Cohomologie rigide et cohomologie rigide à support propre. Première partie. Prépublication IRMAR 96-03. Université de Rennes (1996)
[4]	Berthelot, P.: \(\mathcal{D}\) -modules arithmétiques, I. Opérateurs différentiels de niveau fini. Ann. Sci. Éc. Norm. Supér., IV. Sér. 29(2), 185–272 (1996)
[5]	Berthelot, P.: \(\mathcal{D}\) -modules arithmétiques, II. Descente par Frobenius. Mém. Soc. Math. Fr., Nouv. Sér. 81, vi+136 (2000) · Zbl 0948.14017
[6]	Berthelot, P.: Introduction à la théorie arithmétique des \(\mathcal{D}\) -modules. In: Cohomologies p-adiques et applications arithmétiques, II. Astérisque, vol. 279, pp. 1–80. Soc. Math. Fr., Paris (2002)
[7]	Caro, D.: Dévissages des F-complexes de \(\mathcal{D}\) -modules arithmétiques en F-isocristaux surconvergents. Invent. Math. 166(2), 397–456 (2006) · Zbl 1114.14011 · doi:10.1007/s00222-006-0517-9
[8]	Caro, D.: Fonctions L associées aux \(\mathcal{D}\) -modules arithmétiques. Thèse, Université de Rennes (2002)
[9]	Caro, D.: \(\mathcal{D}\) -modules arithmétiques surcohérents. Application aux fonctions L. Ann. Inst. Fourier 54(6), 1943–1996 (2004) · Zbl 1129.14030
[10]	Caro, D.: \(\mathcal{D}\) -modules arithmétiques associés aux isocristaux surconvergents. Cas lisse. ArXiv Mathematics e-prints (2005)
[11]	Caro, D.: \(\mathcal{D}\) -modules arithmétiques surholonomes. ArXiv Mathematics e-prints (2005)
[12]	Caro, D.: Fonctions L associées aux \(\mathcal{D}\) -modules arithmétiques. Cas des courbes. Compos. Math. 142(01), 169–206 (2006) · Zbl 1167.14012 · doi:10.1112/S0010437X05001880
[13]	de Jong, A.J.: Smoothness, semi-stability and alterations. Publ. Math., Inst. Hautes Étud. Sci. 83, 51–93 (1996) · Zbl 0916.14005 · doi:10.1007/BF02698644
[14]	Elkik, R.: Solutions d’équations à coefficients dans un anneau hensélien. Ann. Sci. Éc. Norm. Supér., IV. Sér. 6, 553–603 (1973) (1974) · Zbl 0327.14001
[15]	Étesse, J.-Y., Le Stum, B.: Fonctions L associées aux F-isocristaux surconvergents. I. Interprétation cohomologique. Math. Ann. 296(3), 557–576 (1993) · Zbl 0789.14015 · doi:10.1007/BF01445120
[16]	Étesse, J.-Y., Le Stum, B.: Fonctions L associées aux F-isocristaux surconvergents. II. Zéros et pôles unités. Invent. Math. 127(1), 1–31 (1997) · Zbl 0911.14011 · doi:10.1007/s002220050112
[17]	Kedlaya, K.S.: Full faithfulness for overconvergent F-isocrystals. In: Geometric Aspects of Dwork Theory, vol. I, II, pp. 819–835. Walter de Gruyter, Berlin (2004) · Zbl 1087.14018
[18]	Mebkhout, Z., Narváez-Macarro, L.: Sur les coefficients de de Rham-Grothendieck des variétés algébriques. In: p-adic Analysis (Trento, 1989). Lect. Notes Math., vol. 1454, pp. 267–308. Springer, Berlin (1990)
[19]	Monsky, P., Washnitzer, G.: Formal cohomology. I. Ann. Math. (2) 88, 181–217 (1968) · Zbl 0162.52504 · doi:10.2307/1970571
[20]	Noot-Huyghe, C.: Un théorème de comparaison entre les faisceaux d’opérateurs différentiels de Berthelot et de Mebkhout-Narváez-Macarro. J. Algebr. Geom. 12(1), 147–199 (2003) · Zbl 1053.14015
[21]	Tsuzuki, N.: Morphisms of F-isocrystals and the finite monodromy theorem for unit-root F-isocrystals. Duke Math. J. 111(3), 385–418 (2002) · Zbl 1055.14022 · doi:10.1215/S0012-7094-02-11131-4
[22]	Virrion, A.: Dualité locale et holonomie pour les \(\mathcal{D}\) -modules arithmétiques. Bull. Soc. Math. Fr. 128(1), 1–68 (2000) · Zbl 0955.14015