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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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References:

[1] P. BERTHELOT , L. BREEN and W. MESSING , Théorie de Dieudonné cristalline. II , Lect. Notes in Math., vol. 930, Springer-Verlag, Berlin-New York, 1982 . MR 85k:14023 | Zbl 0516.14015 · Zbl 0516.14015
[2] P. BERTHELOT , A. GROTHENDIECK and L. ILLUSIE , Théorie des intersections et théorème de Riemann-Roch , Lect. Notes in Math., vol. 225, Springer-Verlag, Berlin, Heidelberg and New York, 1971 . MR 50 #7133 | Zbl 0218.14001 · Zbl 0218.14001
[3] P. BERTHELOT and A. OGUS , F-isocrystals and de Rham cohomology. I , Inv. Math. 72, 1983 , pp. 159-199. MR 85e:14025 | Zbl 0516.14017 · Zbl 0516.14017 · doi:10.1007/BF01389319
[4] W. DWYER and E. FRIEDLANDER , Algebraic and étale K-theory , Trans. Amer. Math. Soc. 292, 1985 , no. 1, pp. 247-280. MR 87h:18013 | Zbl 0581.14012 · Zbl 0581.14012 · doi:10.2307/2000179
[5] G. FALTINGS , Crystalline cohomology and p-adic Galois representations , Algebraic analysis, geometry and number theory (J. I. Igusa ed.), Johns Hopkins University Press, Baltimore, 1989 , pp. 25-80. MR 98k:14025 | Zbl 0805.14008 · Zbl 0805.14008
[6] G. FALTINGS , p-adic Hodge-theory , J. of the AMS 1, 1988 , pp. 255-299. MR 89g:14008 | Zbl 0764.14012 · Zbl 0764.14012 · doi:10.2307/1990970
[7] J.-M. FONTAINE , Cohomologie de de Rham, cohomologie crystalline et représentations p-adiques , Algebraic Geometry Tokyo-Kyoto, Lect. Notes Math., vol. 1016, Springer-Verlag, Berlin, Heidelberg and New York, 1983 , pp. 86-108. MR 85f:14019 | Zbl 0596.14015 · Zbl 0596.14015
[8] J.-M. FONTAINE , Sur certains types de représentations p-adiques du groupe de Galois d’un corps local, construction d’un anneau de Barsotti-Tate , Ann. of Math. 115, 1982 , pp. 529-577. MR 84d:14010 | Zbl 0544.14016 · Zbl 0544.14016 · doi:10.2307/2007012
[9] J.-M. FONTAINE , Le corps des périodes p-adiques , Astérisque, vol. 223, 1994 , pp. 321-347. Zbl 0873.14020 · Zbl 0873.14020
[10] J.-M. FONTAINE and G. LAFAILLE , Construction de représentations p-adiques , Ann. Sci. Ec. Norm. Sup., IV. Ser., 15, 1982 , pp. 547-608. Numdam | MR 85c:14028 | Zbl 0579.14037 · Zbl 0579.14037
[11] J.-M. FONTAINE and W. MESSING , p-adic periods and p-adic étale cohomology , Current Trends in Arithmetical Algebraic Geometry (K. Ribet, ed.), Contemporary Math., vol. 67, Amer. Math. Soc., Providence, 1987 , pp. 179-207. MR 89g:14009 | Zbl 0632.14016 · Zbl 0632.14016
[12] E. FRIEDLANDER , Étale K-theory. II. Connections with algebraic K-theory , Ann. Sci. École Norm. Sup., IV. Ser., 15, 1982 , pp. 231-256. Numdam | MR 85c:14014 | Zbl 0537.14011 · Zbl 0537.14011
[13] H. GILLET , Riemann-Roch theorems for higher algebraic K-theory , Adv. Math. 40, 1981 , pp. 203-289. MR 83m:14013 | Zbl 0478.14010 · Zbl 0478.14010 · doi:10.1016/S0001-8708(81)80006-0
[14] H. GILLET and W. MESSING , Cycle classes and Riemann-Roch for crystalline cohomology , Duke Math. J., 55, 1987 , pp. 501-538. Article | MR 89c:14025 | Zbl 0651.14014 · Zbl 0651.14014 · doi:10.1215/S0012-7094-87-05527-X
[15] M. GROS , Régulateurs syntomiques et valeurs de fonctions L p-adiques I , Invent. Math., 99, 1990 , pp. 293-320. MR 91e:11070 | Zbl 0667.14006 · Zbl 0667.14006 · doi:10.1007/BF01234421
[16] M. GROS , Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmic , Mém. Soc. Math. France (N.S.) 21, 1985 . Numdam | Zbl 0615.14011 · Zbl 0615.14011
[17] K. KATO , On p-adic vanishing cycles (application of ideas of Fontaine-Messing), Algebraic Geometry, Sendai, 1985 , Adv. Stud. Pure Math. 10, North-Holland, Amsterdam-New York, 1987 , pp. 207-251. Zbl 0645.14009 · Zbl 0645.14009
[18] K. KATO and W. MESSING , Syntomic cohomology and p-adic étale cohomology , Tôhoku Math. J. 44, 1992 , pp. 1-9. Article | MR 93b:14035 | Zbl 0792.14008 · Zbl 0792.14008 · doi:10.2748/tmj/1178227370
[19] D. QUILLEN , Higher algebraic K-theory I , Algebraic K-theory I, Lecture Notes in Math. 341, Springer-Verlag, Berlin-Heidelberg-New-York, 1973 , pp. 85-147. MR 49 #2895 | Zbl 0292.18004 · Zbl 0292.18004
[20] V. V. SHEKHTMAN , Chern classes in algebraic K-theory , Trans. Moscow Math. Soc., Issue I, 1984 , pp. 243-271. Zbl 0541.18008 · Zbl 0541.18008
[21] C. SOULÉ , Operations on étale K-theory . Applications, Algebraic K-theory I, Lect. Notes Math. 966, Springer-Verlag, Berlin, Heidelberg and New York, 1982 , pp. 271-303. MR 85b:18011 | Zbl 0507.14013 · Zbl 0507.14013
[22] C. SOULÉ , K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale , Inv. Math. 55, 1979 , pp. 251-295. MR 81i:12016 | Zbl 0437.12008 · Zbl 0437.12008 · doi:10.1007/BF01406843
[23] C. SOULÉ , Opérations en K-théorie algébrique , Canad. J. Math. 37, no. 3, 1985 , pp. 488-550. MR 87b:18013 | Zbl 0575.14015 · Zbl 0575.14015 · doi:10.4153/CJM-1985-029-x
[24] R. THOMASON , Algebraic K-theory and étale cohomology , Ann. Scient. Ecole Norm. Sup. 18, 1985 , pp. 437-552. Numdam | MR 87k:14016 | Zbl 0596.14012 · Zbl 0596.14012
[25] R. THOMASON , Bott stability in algebraic K-theory , Applications of algebraic K-theory to algebraic geometry and number theory I, II (Boulder, Colo., 1983 , Contemporary Math., vol. 55, Amer. Math. Soc., Providence, 1986 , pp. 389-406. MR 87m:18022 | Zbl 0594.18012 · Zbl 0594.18012
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