Q1071829
type: Item
instance of: scholarly article
title: 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)
zbMATH Open document ID: 0586.14009
DOI: 10.1007/BF01389154
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143414
review text: 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 o…
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 3939492
zbMATH Keywords: de Rham complex
zbMATH Keywords: de Rham cohomology groups
zbMATH Keywords: p-adically valued complete extension field
author: Francesco Baldassarri
MaRDI profile type: Publication
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OpenAlex ID: W2314402911
label: 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)
description: scientific article; zbMATH DE number 3939492