## Comparaison entre la cohomologie algébrique et la cohomologie p-adique rigide à coefficients dans un module différentiel. I: Cas des courbes. (Comparison between the algebraic cohomology and the p-adic rigid cohomology with coefficients in a differential module. I: Case of curves).(French)Zbl 0586.14009

We consider a nonsingular algebraic variety $$X_ 0$$ defined over an algebraic number field $$K_ 0$$ and a locally free $${\mathcal O}_{X_ 0}$$- module of finite type $${\mathcal V}_ 0$$ equipped with an integrable connection $(1)\quad \nabla_ 0:\quad {\mathcal V}_ 0\to \Omega^ 1_{X_ 0/K_ 0}\otimes_{{\mathcal O}_{X_ 0}}{\mathcal V}_ 0.$ From (1) we can construct the de Rham complex of ($${\mathcal V}_ 0,\nabla_ 0):$$ $(2)\quad {\mathcal D}{\mathcal R}({\mathcal V}_ 0,\nabla_ 0):\quad 0\quad \to \quad {\mathcal V}_ 0\quad \to^{\nabla_ 0}\quad \Omega^ 1\otimes {\mathcal V}_ 0\quad \to \quad \Omega^ 2\otimes {\mathcal V}_ 0\quad \to \quad...$ and define the (algebraic de Rham cohomology groups of ($${\mathcal V}_ 0,\nabla_ 0)$$ as: $(3)\quad H^ q_{DR}({\mathcal V}_ 0,\nabla_ 0)={\mathbb{H}}^ q(X_ 0,{\mathcal D}{\mathcal R}({\mathcal V}_ 0,\nabla_ 0)).$ In (3) one must compute the hypercohomology of a complex of abelian sheaves on $$X_ 0$$ equipped with its Zariski topology. Suppose K is a p-adically valued complete extension field of $$K_ 0$$ and denote by X, $${\mathcal V}, \nabla$$ the objects obtained from $$X_ 0, {\mathcal V}_ 0, \nabla_ 0$$ by the scalar extension $$K_ 0\to K$$. The set of closed points of X carries a natural structure of a rigid analytic K-space $$(X_{rig},{\mathcal O}_{X_{rig}})$$ and $${\mathcal V}_{rig}={\mathcal V}\otimes_{{\mathcal O}_ X}{\mathcal O}_{X_{rig}}$$ carries a natural extension $$\nabla_{rig}$$ of the connection $$\nabla$$. As in (2), (3) we obtain the (rigid analytic) de Rham cohomology groups of ($${\mathcal V}_{rig},\nabla_{rig})$$ as: $(4)\quad H^ q_{DR}({\mathcal V}_{rig},\nabla_{rig})={\mathbb{H}}^ q(X_{rig},{\mathcal D}{\mathcal R}({\mathcal V}_{rig},\nabla_{rig})).$ In this paper we conjecture that, in full generality: $(5)\quad H^ q_{DR}({\mathcal V}_{rig},\nabla_{rig})=K\otimes_{K_ 0}H^ q_{DR}({\mathcal V}_ 0,\nabla_ 0).$ This is not the case in the classical situation, where $$K={\mathbb{C}}$$ and one compares $$H^ q_{DR}({\mathcal V},\nabla)$$ with $$H^ q_{DR}({\mathcal V}_{an},\nabla_{an})$$, $$X_{an}, {\mathcal V}_{an}, \nabla_{an}$$ denoting the complex-analytic objects associated to X, $${\mathcal V}, \nabla$$, respectively. There, the difference in dimension between the algebraic and complex-analytic cohomology groups measures the irregularity of $$\nabla$$ at the divisor at infinity in any projective completion $$\bar X$$ of X.
In this paper we reduce our conjecture to a local problem of comparison of cohomology with meromorphic coefficients versus cohomology with essentially singular coefficients. We then prove our conjecture when X is a curve. The reader will recognize the strong influence that P. Deligne’s book [”Equations différentielles à points singuliers réguliers”, Lect. Notes Math. 163 (1970; Zbl 0244.14004)] and P. Robba’s papers on the index of a p-adic differential operator have had on this paper.

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology 14H99 Curves in algebraic geometry 14G20 Local ground fields in algebraic geometry

Zbl 0244.14004
Full Text:

### References:

 [1] Adolphson, A.: An index theorem forp-adic differential operators. Trans. Am. Math. Soc.216, 279-293 (1976) · Zbl 0297.47039 [2] Baldassarri, F.: Differential modules and singular points ofp-adic differential equations. Adv. Math.44, 155-179 (1982) · Zbl 0493.12030 [3] Berger, R., Kiehl, R., Kunz, E., Nastold, H.J.: Differentialrechnung in der analytischen Geometrie. Lecture Notes in Math., vol. 38. Berlin-Heidelberg-New York: Springer 1967 · Zbl 0163.03202 [4] Berthelot, P.: Article en préparation sur la cohomologie rigide. [5] Bosch, S., Güntzer, U., Remmert, R.: Non-archimedean analysis. Grundlehren der math. Wissenschaften, vol. 261. Berlin-Heidelberg-New York-Tokyo: Springer 1984 · Zbl 0539.14017 [6] Clark, D.: A note on thep-adic convergence of solutions of linear differential equations. Proc. Am. Math. Soc.17, 262-269 (1966) · Zbl 0147.31101 [7] Deligne, P.: Equations différentielles à points singuliers réguliers. Lecture Notes in Math., vol. 163. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0244.14004 [8] Grothendieck, A.: On the De Rham cohomology of algebraic varieties. Publ. Math. IHES29, 95-103 (1966) · Zbl 0145.17602 [9] Katz, N.: Nilpotent connections and the monodromy theorem: applications of a result of Turrittin. Publ. Math. IHES39, 355-412 (1970) · Zbl 0221.14007 [10] Kiehl, R.: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. math.2, 191-214 (1967) · Zbl 0202.20101 [11] Kiehl, R.: Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie. Invent. math.2, 256-273 (1967) · Zbl 0202.20201 [12] Kiehl, R.: Die De Rham-Kohomologie algebraischer Mannigfaltigkeiten über einem bewerteten Körper. Publ. Math. IHES33, 5-20 (1967) · Zbl 0159.22404 [13] Köpf, U.: Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen. Schriftenreihe. Math. Inst. Univ. Münster, 2. Serie, Heft 7 (1974) · Zbl 0275.14006 [14] Malgrange, B.: Sur les points singuliers des équations différentielles. L’enseignement mathématique XX, 147-176 (1974) · Zbl 0299.34011 [15] Robba, Ph.: Index ofp-adic differential operators III. Applications to twisted exponential sums. Soc. Math. de France, Astérisque119-120, 191-266 (1984) [16] Robba, Ph.: Indice d’un opérateur différentielp-adique IV. Cas des systémes. Mesure de l’irrégularité dans un disque. Ann. Inst. Fourier, Grenoble35, (2) 13-55 (1985) [17] Serre, J-P.: Géométrie algébrique et géométrie analityque. Ann. Inst. Fourier, Grenoble6, 1-42 (1956)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.