Slopes in rigid cohomology and unipotent F-isocrystals
type: Item
instance of: scholarly article
title: Slopes in rigid cohomology and unipotent F-isocrystals
zbMATH Open document ID: 0980.14016
author: Bruno Chiarellotto
author: Bernard Le Stum
published in: Manuscripta Mathematica
publication date: 2002-02-24T00:00:00Z
review text: In this article, the authors give some general results concerning the slopes of the Frobenius acting on rigid cohomology. They also establish the existence of a slope filtration for unipotent \(F\)-isocrystals on a smooth variety over an algebraically closed field. Let us recall that a unipotent \(F\)-isocrystal is a successive extension of the trivial isocrystal that is endowed with a Frobenius structure. The authors already published an article concerning these questions [\textit{B.~Chiarellotto} and \textit{B. Le Stum}, Compos. Math. 116, No 1, 81-110 (1999; Zbl 0936.14017)], where they established the existence of the slope filtration for unipotent \(F\)-isocrystals in the case of an open subset of the affine line. In the first two parts of the article, the authors give some technical stuff concerning rigid cohomology. In particular, they prove that the Poincaré duality morphism, the Gysin morphism, and the trace morphism are compatible with Frobenius. Denote by \(K=W(k)[p^{1/e}]\), where \(e\) is a nonnegative integer. The statement concerning the slopes of rigid cohomology is the following. Let \(k\) be an algebraically closed field of characteristic \(p\), and \(X\) an algebraic variety of dimension \(d\) over \(k\). Then the slopes of the rigid cohomology with compact support \(H^i_{\text{rig},c}(X)\) are \(\geq 0\) and \(\leq d\). Moreover these numbers are \(\geq i-d\) and \(\leq i\). The results are the same for the groups \(H^i_{\text{rig}}(X)\), if \(X\) is smooth. The key point to define the existence of the slope filtration of unipotent F-isocrystals is to use a tannakian interpretation of unipotent isocrystals. Suppose that \(X\) is connected and has a rational point \(x\). Then the category of overconvergent unipotent isocrystals over \(X\) is a tannakian category that is neutral (the fiber functor is given by \(E\mapsto E_x\)). Let \(G\) be its fundamental group, \(\mathfrak{g}\) its Lie algebra, \({\mathbb{U}}\) the enveloping algebra of \(\mathfr…
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 19E20
zbMATH DE Number: 1399923
zbMATH Keywords: slopes in rigid cohomology
zbMATH Keywords: action of Frobenius in rigid cohomology
zbMATH Keywords: slope filtration
zbMATH Keywords: unipotent F-isocrystals
zbMATH Keywords: tannakian category
reviewed by: Christine Noot-Huyghe
MaRDI profile type: Publication
full work available at URL: s002290050212
OpenAlex ID: W2004843962
DOI: 10.1007/S002290050212
description: scientific article; zbMATH DE number 1399923