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Crystalline conjecture via $$K$$-theory. (English) Zbl 0929.14009
The author gives a new proof of the crystalline conjecture of J.-M. Fontaine comparing étale and crystalline cohomology of smooth projective varieties over $$p$$-adic fields in the good reduction case.
This conjecture has been previously proved by J.-M. Fontaine and W. Messing [in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata 1985, Contemp. Math. 67, 179-207 (1987; Zbl 0632.14016)] and by G. Faltings [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 25-80 (1989; Zbl 0805.14008)].
In the paper under review, it is derived from R. W. Thomason’s comparison theorem between algebraic and étale $$K$$-theory [see “Algebraic K-theory and étale cohomology”, Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)].
The (very) rough idea is as follows: By standard arguments, it suffices to construct a map from étale cohomology to crystalline cohomology which is compatible with Poincaré duality and with several further structures. Modulo powers of the Bott element, étale cohomology may be identified with (a graded piece of) algebraic $$K$$-theory by Thomason’s theorem.
Now, the crystalline Chern class map yields the desired map. This argument proves the crystalline conjecture only for primes which are larger than, roughly speaking, the cube of the dimension, and it proves the rational crystalline conjecture in general.

MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 19E20 Relations of $$K$$-theory with cohomology theories
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