The overconvergent site
type: Item
instance of: scholarly article
zbMATH Open document ID: 1246.14028
author: Bernard Le Stum
publication date: 2012-05-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/0707.1809
full work available at URL: smf mem-ns 127.php
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 6037362
zbMATH Keywords: rigid cohomology
zbMATH Keywords: overconvergent site
zbMATH Keywords: isocrystal
zbMATH Keywords: fibration theorem
MaRDI profile type: Publication
OpenAlex ID: W2583537366
arXiv ID: 0707.1809
DOI: 10.24033/MSMF.438
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title: The overconvergent site
published in: Mémoires de la Société Mathématique de France. Nouvelle Série
review text: ``Rigid cohomology is a cohomology theory for algebraic varieties over a field \(k\) of positive characteristic \(p\) with values in a vector space over a \(p\)-adic field \(K\) whose residue field is \(k\).''NEWLINENEWLINERigid cohomology is of `de Rham type' in that it is defined in terms of certain de Rham complexes on smooth formal liftings to characteristic \(0\). It is natural to ask if -- like usual de Rham cohomology of algebraic varieties over a field of characteristic \(0\), which is known to be equivalent with the cohomology of the infinitesimal site -- also rigid cohomology has an interpretation as the cohomology of an analogous site. As long as one is interested in smooth proper \(k\)-varieties only, Berthelot's crystalline site is the perfect solution to this problem (in fact this concerns crystalline cohomology rather then rigid cohomology), but for more general varieties this cannot work.NEWLINENEWLINEIn the present book, Bernard Le Stum defines such a site, the `overconvergent site'. By its very definition as given by Pierre Berthelot, rigid cohomology is defined by means of \textit{chosen} embeddings of the \(k\)-varieties in question (or just of sufficiently small open subsets of them) into smooth formal schemes over mixed characteristic valuation rings. It is then a theorem that the definitions do not depend on the choices made, that they glue as desired and make up a completely functorial theory. The point in introducing the overconvergent site is to \textit{intrinsically} build in functoriality into the theory.NEWLINENEWLINE``Of course, this article owes much to Berthelot's previous work on rigid cohomology\dots We only want to rewrite his theory with a slightly different approach.''NEWLINENEWLINEWhile rigid cohomology theory is classically formulated in terms of Tate's theory of rigid spaces, here Berkovich analytic spaces are used instead.NEWLINENEWLINEThe book consists of three chapters. In the first chapter the necessary geometric foundatio…
reviewed by: Elmar Große-Klönne
description: scientific article; zbMATH DE number 6037362