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)
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 \textit{P. Deligne}'s book [''Equations différentielles à points singuliers réguliers'', Lect. Notes Math. 163 (1970; Zbl 0244.14004)] and \textit{P. Robba}'s papers on the index of a p-adic differential operator have had on this paper.
p-adically valued complete extension field
scientific article; zbMATH DE number 3939492
scientific article