# Finiteness of de Rham cohomology in rigid analysis

@article{GrosseKlnne2002FinitenessOD,
title={Finiteness of de Rham cohomology in rigid analysis},
author={Elmar Grosse-Kl{\"o}nne},
journal={arXiv: Algebraic Geometry},
year={2002}
}
For a big class of smooth dagger spaces --- dagger spaces are 'rigid spaces with overconvergent structure sheaf' --- we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of Berthelot's rigid cohomology also in the non-smooth case. We need a careful study of de Rham cohomology in situations of semi-stable reduction.
De Rham cohomology of rigid spaces
Abstract.We define de Rham cohomology groups for rigid spaces over non-archimedean fields of characteristic zero, based on the notion of dagger space introduced in [12]. We establish some functorial
Frobenius and monodromy operators in rigid analysis, and Drinfel'd's symmetric space
We define Frobenius and monodromy operators on the de Rham cohomology of K-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction Y , over a complete
Finiteness of rigid cohomology with coefficients
We prove that for any field k of characteristic p>0, any separated scheme X of finite type over k, and any overconvergent F-isocrystal E over X, the rigid cohomology H^i(X, E) and rigid cohomology
Weight decomposition of de Rham cohomology sheaves and tropical cycle classes for non-Archimedean spaces
We construct a functorial decomposition of de Rham cohomology sheaves, called weight decomposition, for smooth analytic spaces over non-Archimedean fields embeddable into $\mathbf{C}_p$, which
The Monsky-Washnitzer and the overconvergent realizations
We construct the dagger realization functor for analytic motives over non-archimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a
Cohomologie analytique des arrangements d'hyperplans
In this article, we study the cohomology of some analytic sheaves on the complementary in the projective space of a suitable inﬁnite collection of hyperplane like the Drinfel’d symetric space. In
The de Rham-Fargues-Fontaine cohomology
• Mathematics
• 2021
We show how to attach to any rigid analytic variety V over a perfectoid space P a rigid analytic motive over the Fargues-Fontaine curve X (P ) functorially in V and P . We combine this construction
Frobenius and monodromy operators in rigid analysis, and Drinfel'd's symmetric space
We define Frobenius and monodromy operators on the de Rham cohomology of $K$-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction $Y$, over a complete
We show that the de Rham cohomology of any separated and smooth rigid variety over a field of Laurent series of characteristic zero carries a natural formal meromorphic connection, which we call the
On base change theorem and coherence in rigid cohomology.
We prove that the base change theorem in rigid coho- mology holds when the rigid cohomology sheaves both for the given morphism and for its base extension morphism are coherent. Apply- ing this

## References

SHOWING 1-10 OF 36 REFERENCES
De Rham cohomology of rigid spaces
Abstract.We define de Rham cohomology groups for rigid spaces over non-archimedean fields of characteristic zero, based on the notion of dagger space introduced in [12]. We establish some functorial
Rigid analytic spaces with overconvergent structure sheaf
We introduce a category of 'rigid spaces with overconvergent structure sheaf' which we call dagger spaces --- this is the correct category in which de Rham cohomology in rigid analysis should be
Étale Cohomology of Rigid Analytic Varieties and Adic Spaces
Summary of the results on the etale cohomology of rigid analytic varieties - Adic spaces - The etale site of a rigid analytic variety and an adic space - Comparison theorems - Base change theorems -
Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant
Summary.This article contains an elementary constructive proof of resolution of singularities in characteristic zero. Our proof applies in particular to schemes of finite type and to analytic spaces
On the de rham cohomology of algebraic varieties
© Publications mathématiques de l’I.H.É.S., 1966, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://
Sur le théorème de finitude de la cohomologie p-adique d'une variété affine non singulière
On démontre dans cet article que le Théorème de l'indice pour une classe d'équations différentielles p-adiques sur la droite projective entraîne le Théorème de finitude de la cohomologie p-adique de
Formal and rigid geometry
• Mathematics
• 1993
Calendario: 10 ore, 26 e 28 aprile 2010, 3, 5, 6 maggio dalle 16:30 alle 18:15. Torre Archimede, Aula 2AB40. Prerequisiti: Familiarità con i primi rudimenti di Geometria Algebrica (ideali primi,
Die de Rham Kohomologie algebraischer Mannigfaltigkeiten über einem bewerteten Körper
• Mathematics
• 1967
© Publications mathématiques de l’I.H.É.S., 1967, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://
Lecture Notes in Math
Une notion très importante pour la géometrie algébrique est celle de fibré projectif. Si f : X → S est un morphisme lisse entre variétés algébriques lisses, dont toute fibre est isomorphe à P, on se