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Cycle classes and Riemann-Roch for crystalline cohomology. (English) Zbl 0651.14014

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.

Reviewer: T.Ekedahl
MSC:
 14F30 $p$-adic cohomology, crystalline cohomology 14C40 Riemann-Roch theorems 14C99 Cycles and subschemes