Cycle classes and Riemann-Roch for crystalline cohomology
type: Item
instance of: scholarly article
title: Cycle classes and Riemann-Roch for crystalline cohomology
zbMATH Open document ID: 0651.14014
DOI: 10.1215/S0012-7094-87-05527-X
author: William Messing
author: Henri Gillet
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Succinctly put this article proves the existence of a good cycle class map into crystalline cohomology tensored with the rationals, and consequently that the latter is a ``Weil cohomology theory''. (This has also been proved - independently - by Gros.) More precisely the cycle class map is constructed using Chern classes so in order to prove compatability of the cycle class map with direct images the authors need to prove a Riemann-Roch theorem for crystalline cohomology. The proof follows closely the by now standard proof of Baum-Fulton-MacPherson with the difference that as crystalline cohomology only behaves well for smooth and proper varieties some care has to be taken with the ``degeneration to the normal bundle'' argument. During the course of the proof the authors are able to lift the restriction that the scheme possess an ample line bundle in some of the results of SGA 6. Furthermore, in an appendix they give an axiomatic characterization of the product on Chow groups. - In another appendix the second author uses the results of the paper to prove that the crystalline cohomology of the special fibre of a smooth and proper scheme over an absolutely unramified discrete valuation with residue field of positive characteristic only depends on the general fibre.
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 14C99
zbMATH DE Number: 4061388
zbMATH Keywords: Riemann-Roch-Grothendieck
zbMATH Keywords: existence of a good cycle class map into crystalline cohomology
zbMATH Keywords: Weil cohomology theory
reviewed by: Torsten Ekedahl
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4061388