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Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique. (Chern classes and cycle classes in logarithmic Hodge-Witt cohomology). (French) Zbl 0615.14011

The author constructs cycle classes in p-adic cohomology theories; crystalline cohomology and Hodge-Witt cohomology. The technical difficulties usually encountered when trying to do so are resolved by introducing the so called logarithmic cohomology, the fixed points under the Frobenius map of the Hodge-Witt cohomology. It turns out that for the latter theory one can prove purity results strong enough to reduce, in the usual way, the construction of cycle classes to the case of smooth cycles, a case which follows from duality. It should be noted that the needed purity results are not true for crystalline nor for Hodge-Witt cohomology. The article further computes the Hodge-Witt cohomology for a projective bundle and a smooth blowing up in terms of the cohomology of the base variety thereby obtaining Chern classes and proves the usual formulas for the cycle classes up to and including the ”projection formula” and Riemann-Roch ”without denominators” for a closed embedding.
Note that in particular the article provides the missing results enabling us to declare crystalline cohomology (tensored with \({\mathbb{Q}})\) a ”Weil cohomology theory” [cf. S. L. Kleiman in: Dix Exposés Cohomologies Schémas, Adv. Stud. Math. 3, 359-386 (1986; Zbl 0198.259)].
Reviewer: T.Ekedahl

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
57R20 Characteristic classes and numbers in differential topology

Citations:

Zbl 0198.259
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References:

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