Devissage of the \(F\)-complexes of arithmetic \(\mathcal D\)-modules in overconvergent \(F\)-isocrystals. (Dévissages des \(F\)-complexes de \(\mathcal D\)-modules arithmétiques en \(F\)-isocristaux surconvergents.) (French) Zbl 1114.14011

Caro, D.

Invent. Math. 166, No. 2, 397-456 (2006).

Let \(V\) be a discrete valuation ring of characteristics \((0,p)\), \(K\) its fraction field, \(k\) the residue field, \(\mathcal{P}\) a smooth proper formal scheme over \(\text{Spf}(V)\), \(P \) the special fiber of \(\mathcal{P}\), \(U\) the complementary of a divisor, and \(Y\) a smooth closed scheme of \(U\). In the \(\ell\)-adic context, it is known for a long time that constructible \(\mathbb{Q}_\ell\)-adic sheaves over \(Y\) (\(\ell\neq p\)) can be decomposed “by devissage” as direct images along strata of smooth \(\ell\)-adic sheaves.
The aim of this paper is to show that overholonomic arithmetic \(\mathcal{D}\)-modules over \(\mathcal{P}\) with a Frobenius admit a decomposition along strata of \(Y\), as direct images of overconvergent \(F\)-isocrystals. This result is important and has several applications.
One application is to prove that the category of overholonomic \(\mathcal{D}\)-modules, endowed with a Frobenius, is stable by tensor products. This category introduced by the author is stable by other cohomological operations, as proved in [D. Caro, \(\mathcal{D}\)-modules arithmetiques surholonomes, preprint] and unit-root \(F\)-isocrystals are overholonomic \(\mathcal{D}\)-modules.
A second important application of this theorem consists to give a \(p\)-adic Weil II statement based on the analogous statement of Kedlaya for overconvergent \(F\)-isocrystals [K. S. Kedlaya, Compos. Math. 142, No. 6, 1426–1450 (2006; Zbl 1119.14014)].
A third important application is that if \(\mathcal{U}\) is a smooth lifting of \(U\), then the restriction to \(\mathcal{U}\) is fully faithfull for “devissable” \(F\)-complexes of arithmetic \(\mathcal{D}\)-modules. An important part of the paper is devoted to the definition of the direct image over \(\mathcal{P}\) of an overconvergent \(F\)-isocrystal, as an arithmetic \(\mathcal{D}\)-modules, by a functor \(\text{sp}_+\). This was done in the simple case where \(Y\) is the special fiber of the complementary of a divisor in \(\mathcal{P}\) by P. Berthelot [in: \(p\)-adic analysis, Proc. Int. Conf., Trento/Italy 1989, Lect. Notes Math. 1454, 80–124 (1990; Zbl 0722.14008)] and by the author [Bull. Soc. Math. Fr. 137, No. 4, 453–543 (2009; Zbl 1300.14021), preprint arXiv:math/0500422]. The construction here is more general and works in the case of a smooth compactification. It gives the previous construction of the author over an open dense subset of \(Y\) thanks to de Jong’s desingularisation theorem and as exspected, the de Rham cohomology of the direct image by sp\(_+\) of an overconvergent \(F\)-isocrystal is equal to the rigid cohomology of the \(F\)-isocrystal. This construction is an important tool to define “devissable” \(F\)-complexes of arithmetic \(\mathcal{D}\)-modules. This is a conjecture that such complexes are in fact overholonomic, which is true if standard conjectures of Berthelot over holonomy of arithmetic \(\mathcal{D}\)-modules with Frobenius structure, are true.

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
14F40	de Rham cohomology and algebraic geometry

Keywords:

arithmetic \(\mathcal D\)-modules; overconvergent \(F\)-isocrystals; devissage; overholonomic \(\mathcal D\)-modules

Citations:

Zbl 0722.14008; Zbl 1119.14014; Zbl 1300.14021

References:

[1]	Arabia, A.: Relèvements des algèbres lisses et de leurs morphismes. Comment. Math. Helv. 76, 607–639 (2001) · Zbl 1063.13009
[2]	Berthelot, P.: Cohomologie Cristalline des Schémas de Caractéristique p>0, Lecture Notes in Mathematics, vol. 407. Berlin: Springer 1974 · Zbl 0298.14012
[3]	Berthelot, P.: Géométrie rigide et cohomologie des variétés algébriques de caractéristique p. Mém. Soc. Math. France (N.S.) (1986), no. 23, p. 3, 7–32, Introductions aux cohomologies p-adiques (Luminy, 1984)
[4]	Berthelot, P.: Cohomologie rigide et théorie des \(\mathcal{D}\) -modules. p-Adic Analysis, pp. 80–124. (Trento, 1989). Berlin: Springer, 1990
[5]	Berthelot, P.: \(\mathcal{D}\) -modules arithmétiques. I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Super. (4) 29, 185–272 (1996)
[6]	Berthelot, P.: Cohomologie rigide et cohomologie rigide à support propre. Première partie. Prépublication IRMAR 96-03, Université de Rennes, 1996
[7]	Berthelot, P.: \(\mathcal{D}\) -modules arithmétiques. II. Descente par Frobenius. Mém. Soc. Math. Fr. (N.S.), p. vi+136. (2000) · Zbl 0948.14017
[8]	Berthelot, P.: Introduction à la théorie arithmétique des \(\mathcal{D}\) -modules. In: Cohomologies p-adiques et applications arithmétiques, II. Astérisque 279, 1–80 (2002)
[9]	Caro, D.: Fonctions L Associées aux \(\mathcal{D}\) -Modules Arithmétiques. Thèse, Université de Rennes1, 2002
[10]	Caro, D.: Cohérence différentielle des F-isocristaux unités. C. R. Math. Acad. Sci. Paris 338, 145–150 (2004)
[11]	Caro, D.: \(\mathcal{D}\) -modules arithmétiques surcohérents. Application aux fonctions L. Ann. Inst. Fourier, Grenoble 54, 1943–1996 (2004) · Zbl 1129.14030
[12]	Caro, D.: \(\mathcal{D}\) -modules arithmétiques surholonomes. arXiv:math/0502442 (2005)
[13]	Caro, D.: \(\mathcal{D}\) -modules arithmétiques associés aux isocristaux surconvergents. Cas lisse. arXiv:math.AG/0510422 (2005)
[14]	Caro, D.: Fonctions L associées aux \(\mathcal{D}\) -modules arithmétiques. Cas des courbes. Compos. Math. 142, 169–206 (2006) · Zbl 1167.14012
[15]	Deligne, P.: Cohomologie étale. Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 \(\frac{1}{2}\) . Avec la collaboration de J.F. Boutot, A. Grothendieck, L. Illusie et J.L. Verdier, Lecture Notes in Mathematics, vol. 569. Berlin: Springer 1977 · Zbl 0345.00010
[16]	Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980) · Zbl 0456.14014
[17]	de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996) · Zbl 0916.14005
[18]	Elkik, R.: Solutions d’équations à coefficients dans un anneau hensélien. Ann. Sci. École Norm. Sup., IV Ser. 6, 553–603 (1973) (1974) · Zbl 0327.14001
[19]	Étesse, J.-Y., Le Stum, B.: Fonctions L associées aux F-isocristaux surconvergents. I. Interprétation cohomologique. Math. Ann. 296, 557–576 (1993) · Zbl 0789.14015
[20]	Étesse, J.-Y.: Descente étale des F-isocristaux surconvergents et rationalité des fonctions L de schémas abéliens. Ann. Sci. École Norm. Sup., IV. Ser. 35, 575–603 (2002)
[21]	Grothendieck, A.: Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. 4, 228 (1960)
[22]	Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. 24, 231 (1965)
[23]	Grothendieck, A.: Crystals and the de Rham cohomology of schemes. Dix Exposés sur la Cohomologie des Schémas, pp. 306–358. Amsterdam: North-Holland 1968
[24]	Hartshorne, R.: Residues and Duality. Berlin: Springer 1966 · Zbl 0212.26101
[25]	Kedlaya, K.S.: Finiteness of rigid cohomology with coefficients. À paraître dans Duke Math. J.
[26]	Kedlaya, K.S.: Fourier transforms and p-adic ”Weil II”. À paraître dans Compos. Math.
[27]	Kedlaya, K.S.: Semistable reduction for overconvergent F-isocrystals, I: Unipotence and logarithmic extensions. arXiv:math.NT/0405069 (2006)
[28]	Kedlaya, K.S.: Counting points on hyperelliptic curves using Monsky–Washnitzer cohomology. J. Ramanujan Math. Soc. 16, 323–338 (2001) · Zbl 1066.14024
[29]	Kedlaya, K.S.: Computing zeta functions via p-adic cohomology. Algorithmic Number Theory, vol. 3076, p. 1–17. Lecture Notes in Comput. Sci. Berlin: Springer 2004 · Zbl 1125.14300
[30]	Kedlaya, K.S.: Full faithfulness for overconvergent F-isocrystals. Geometric aspects of Dwork theory. Vol. I, II, pp. 819–835. Berlin: Walter de Gruyter GmbH & Co. KG 2004 · Zbl 1087.14018
[31]	Kato, K., Trihan, F.: On the conjectures of Birch and Swinnerton-Dyer in characteristic p>0. Invent. Math. 153, 537–592 (2003) · Zbl 1046.11047
[32]	Meredith, D.: Weak formal schemes. Nagoya Math. J. 45, 1–38 (1972) · Zbl 0207.51502
[33]	Milne, J.S.: Étale Cohomology. Princeton, N.J.: Princeton University Press 1980 · Zbl 0433.14012
[34]	Mebkhout, Z., Narváez-Macarro, L.: Sur les coefficients de de Rham-Grothendieck des variétés algébriques. p-Adic Analysis (Trento, 1989). Lect. Notes Math., vol. 1454, pp. 267–308. Berlin: Springer 1990
[35]	Monsky, P., Washnitzer, G.: Formal cohomology. I. Ann. of Math. 88,181–217 (1968) · Zbl 0162.52504
[36]	Noot-Huyghe, C.: Un théorème de comparaison entre les faisceaux d’opérateurs différentiels de Berthelot et de Mebkhout–Narváez-Macarro. J. Algebr. Geom. 12, 147–199 (2003) · Zbl 1053.14015
[37]	Shiho, A.: Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology. J. Math. Sci. Univ. Tokyo 9, 1–163 (2002) · Zbl 1057.14025
[38]	Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki, vol. 9, pp. 415–440. Paris: Soc. Math. France 1995
[39]	Tsuzuki, N.: Morphisms of F-isocrystals and the finite monodromy theorem for unit-root F-isocrystals. Duke Math. J. 111, 385–418 (2002) · Zbl 1055.14022
[40]	Virrion, A.: Dualité locale et holonomie pour les \(\mathcal{D}\) -modules arithmétiques. Bull. Soc. Math. France 128, 1–68 (2000) · Zbl 0955.14015
[41]	Virrion, A.: Trace et dualité relative pour les \(\mathcal{D}\) -modules arithmétiques. Geometric aspects of Dwork theory. Vol. I, II, pp. 1039–1112. Berlin: Walter de Gruyter GmbH & Co. KG 2004 · Zbl 1083.14017