Direct images. I: Rigid analytic spaces and direct images
type: Item
instance of: scholarly article
title: Direct images. I: Rigid analytic spaces and direct images
zbMATH Open document ID: 1254.14024
author: J.-Y. Etesse
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2012-08-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/0910.4433
reviewed by: Jérôme Poineau
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 6075024
zbMATH Keywords: rigid analytic spaces
zbMATH Keywords: formals schemes
zbMATH Keywords: overconvergent isocrystals
zbMATH Keywords: proper base change
zbMATH Keywords: rigid cohomology
MaRDI profile type: Publication
OpenAlex ID: W1556750849
arXiv ID: 0910.4433
cites work: Cohomologie cristalline des schemas de caractéristique \(p >0\)
cites work: Géométrie rigide et cohomologie des variétés algébriques de caractéristique $p$
cites work: Finiteness and cohomological purity in rigid cohomology (with an appendix by Aise Johan de Jong)
cites work: Q3661717
cites work: Q3698926
cites work: Un théorème de prolongement pour des fonctions analytiques. (A theorem on pralongation of analytic functions)
cites work: Q3325851
cites work: Formal and rigid geometry. I: Rigid spaces
cites work: Formal and rigid geometry. II: Flattening techniques
cites work: Q5344719
cites work: Q4935949
cites work: Cohomological descent of rigid cohomology for étale coverings
cites work: Q5602063
cites work: Lifting of schemes and Monsky-Washnitzer algebras: equivalence and full faithfulness theorems
cites work: Descente étale des F-isocristaux surconvergents et rationalité des fonctions L de schémas abéliens
cites work: Q3154989
cites work: Lifting of abelian schemes, \(F\)-isocrystals and \(L\)-functions
cites work: Rigid analytic spaces with overconvergent structure sheaf
cites work: Q5650791
cites work: Les foncteurs dérivés de lim et leurs applications en théorie des modules
cites work: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin
cites work: Finiteness of rigid cohomology with coefficients
cites work: Q3154994
cites work: Semistable reduction for overconvergent $F$-isocrystals I: Unipotence and logarithmic extensions
cites work: Q5428573
cites work: Formal-algebraic and rigid-analytic geometry
cites work: Image directe supérieure et unipotence
cites work: Relative log convergent cohomology and relative rigid cohomology II
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: On base change theorem and coherence in rigid cohomology
DOI: 10.5802/JTNB.790
review text: Let \(\mathcal{V}\) be a discrete valuation ring with fraction field \(K\) of characteristic 0 and residue field \(k\) of characteristic \(p>0\). Let \(f : X \to S\) be a morphism of \(k\)-varieties. Without going into the details, let us recall that an isocrystal \(E\) over \(X\) is a module with an integrable connection which is defined on some \(K\)-analytic rigid space associated to \(X\) (a strict neighbourhood of its tube in a certain formal scheme). The article under review gives sufficient conditions in order that the direct images \(R^i f_{\mathrm{rig}*}\) of an overconvergent isocrystal on \(X\) remain overconvergent. This will be the case when \(S\) is smooth over \(k\), \(f\) is smooth and projective and either \(X\) lifts to a flat scheme over \(\mathcal{V}\) or \(X\) is a relative complete intersection inside projective spaces over \(S\).NEWLINENEWLINEThe first section of the text is devoted to base change theorems for proper morphisms, which will be a crucial ingredient in the proof. The author first handle the case of formal schemes, then of rigid analytic spaces. The second section gathers a few technical results on strict neighbourhoods, in particular conditions to ensure that a fundamental system of strict neighbourhoods remains so after pull-back. The third section contains a base change theorem for overconvergent modules by a proper morphism and ends with the proofs of the results concerning direct images of overconvergent isocrystals.NEWLINENEWLINETo finish, let us mention that the author announces two articles to follow, in which he will consider isocrystals endowed with a Frobenius structure in the convergent and overconvergent settings.
description: scientific article; zbMATH DE number 6075024