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Motivic cohomology over Dedekind rings

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We study properties of Bloch’s higher Chow groups on smooth varieties over Dedekind rings. We prove the vanishing of for i > n, and the existence of a Gersten resolution for if the residue characteristic is p. We also show that the Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture an identification for m invertible, and a Gersten resolution with (arbitrary) finite coefficients. Over a complete discrete valuation ring of mixed characteristic (0,p), we construct a map from motivic cohomology to syntomic cohomology, which is a quasi-isomorphism provided the Bloch-Kato conjecture holds.

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References

  1. Bloch, S.: Algebraic Cycles and Higher K-theory, Adv. Math. 61, 267–304 (1986)

    MathSciNet  MATH  Google Scholar 

  2. Bloch, S.: A note on Gersten’s conjecture in the mixed characteristic case. Appl. Alg. K-theory Alg. Geometry and Number Theory, Cont. Math. 55 (Part I), 75–78 (1986)

    Google Scholar 

  3. Bloch, S., Kato, K.: p-adic étale cohomology. Publ. Math. IHES 63, 147–164 (1986)

    Google Scholar 

  4. Brown, K., Gersten, S.: Algebraic K-theory as generalized sheaf cohomology. Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 266–292, Lecture Notes in Math., 341, Springer, Berlin, 1973

  5. Colliot-Thélène, J.L., Hoobler, R., Kahn, B.: The Bloch-Ogus-Gabber theorem. Fields Inst. Com. 16, 31–94 (1997)

    Google Scholar 

  6. Fontaine, J.M., Messing, W.: p-adic periods and p-adic étale cohomology. Cont. Math. 67, 179–207 (1987)

    MATH  Google Scholar 

  7. Fujiwara, K.: A proof of the absolute purity conjecture (after Gabber). Algebraic geometry 2000, Azumino (Hotaka), pp. 153–183, Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo, 2002

  8. Geisser, T., Levine, M.: The p-part of K-theory of fields in characteristic p. Inv. Math. 139 , 459–494 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Geisser, T., Levine, M.: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. Reine Angew. Math. 530, 55–103 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Geisser, T.: Motivic cohomology, K-theory and topological cyclic homology. To appear in: K-theory handbook

  11. Gillet, H., Levine, M.: The relative form of Gersten’s conjecture over a discrete valuation ring: The smooth case. J. Pure Appl. Alg. 46, 59–71 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gros, M., Suwa, N.: La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique. Duke Math. J. 57, 615–628 (1988)

    MathSciNet  MATH  Google Scholar 

  13. Illusie, L.: Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. Ecole Norm. Sup. 12 (4), 501–661 (1979)

    MATH  Google Scholar 

  14. Kleiman, S.: The transversality of a general translate. Compositio Math. 28, 287–297 (1974)

    MATH  Google Scholar 

  15. Kurihara, M.: A Note on p-adic Etale Cohomology. Proc. Jap. Acad. 63A, 275–278 (1987)

    MathSciNet  MATH  Google Scholar 

  16. Levine, M.: Techniques of localization in the theory of algebraic cycles. J. Alg. Geom. 10, 299–363 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Levine, M.: K-theory and Motivic Cohomology of Schemes. Preprint 1999

  18. Milne, J.S.: Values of zeta functions of varieties over finite fields. Am. J. Math. 108 (2), 297–360 (1986)

    MATH  Google Scholar 

  19. Milne, J.S.: Motivic cohomology and values of zeta functions. Comp. Math. 68, 59–102 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Milne, J.S.: Etale cohomology. Princeton Math. Series 33

  21. Nesterenko, Y., Suslin, A.: Homology of the general linear group over a local ring, and Milnor’s K-theory. Math. USSR-Izv. 34 (1), 121–145 (1990)

    MATH  Google Scholar 

  22. Nisnevich, Y.A.: The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory. Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279, 241–342 (1989)

    MATH  Google Scholar 

  23. Niziol, W.: Crystalline Conjecture via K-theory. Ann. Sci. Math. ENS 31, 659–682 (1998)

    Article  MATH  Google Scholar 

  24. Quillen, D.: Higher algebraic K-theory. In: Algebraic K-theory, I: Higher K-theories (Proc. Conf. Battelle Memorial Inst. Seattle, Wash. 1972), Lecture Notes in Math. 341, Springer, 1973, pp. 85–147

  25. Roberts, P.: Multiplicities and Chern classes in local algebra. Cambridge Tracts in Mathematics, 133. Cambridge University Press, Cambridge, 1998

  26. Sato, K.: An étale Tate twist with finite coefficients and duality in mixed characteristic. Preprint 2002

  27. Schneider, P.: p-adic points of motives. Proc. Symp. Pure Math. AMS 55 Part 2, 225–249 (1994)

    MATH  Google Scholar 

  28. Spaltenstein, N.: Resolutions of unbounded complexes. Compositio Math. 65, 121–154 (1988)

    MathSciNet  MATH  Google Scholar 

  29. Suslin, A.: Higher Chow group and étale cohomology. Ann. Math. Studies 132, 239–554 (2000)

    Google Scholar 

  30. Voevodsky, V.: Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. Int. Math. Res. Not. 7, 351–355 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Voevodsky, V.: The 2-part of motivic cohomology. To appear in Publ. Math. IHES

  32. Voevodsky, V.: On motivic cohomology with ℤ/l-coefficients. Preprint, 2003

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Authors and Affiliations

  1. Department of Mathematics, KAP108, University of Southern California, 3620 Vermont Av., Los Angeles, CA, 90089, USA

    Thomas Geisser

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  1. Thomas Geisser
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Supported in part by JSPS, NSF Grant. No. 0070850, and the Alfred P.Sloan Foundation

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Geisser, T. Motivic cohomology over Dedekind rings. Math. Z. 248, 773–794 (2004). https://doi.org/10.1007/s00209-004-0680-x

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  • DOI: https://doi.org/10.1007/s00209-004-0680-x

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