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The Arithmetic and Geometry of Algebraic Cycles pp 117–189Cite as

Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients

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Part of the NATO Science Series book series (ASIC,volume 548)

Abstract

In this paper we show that the Beilinson-Lichtenbaum Conjecture which describes motivic cohomology of (smooth) varieties with finite coefficients is equivalent to the Bloch-Kato Conjecture, relating Milnor K-theory to Galois cohomology. The latter conjecture is known to be true in weight 2 for all primes [M-S] and in all weights for the prime 2 [V 3].

Keywords

  • Spectral Sequence
  • Finite Type
  • Closed Subscheme
  • Distinguished Triangle
  • Open Subscheme

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. H. Bass and J. Tate,Milnor Ring of a Global Field,Lecture Notes in Math. 342 (1973), 349–446.

    MathSciNet  Google Scholar 

  2. S. Bloch,Algebraic Cycles and Higher K-Theory,Adv. in Math. 61 (1986), 267–304.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. S. Bloch, The Moving Lemma for Higher Chow Groups,Journal of Algebraic Geometry 3 (1994), 537–568.

    MathSciNet  MATH  Google Scholar 

  4. S. Bloch and A. Ogus, Gersten’s Conjecture and the Homology of Schemes, Ann. Sci. Ecole Norm. Super. (4) 7 (1975), 181–201.

    MathSciNet  Google Scholar 

  5. A. Dold, Lectures on Algebraic Topology, Springer-Verlag, 1972.

    Google Scholar 

  6. E. Friedlander and V. Voevodsky, Bivariant Cycle Cohomology, Cycles, Transfers and Motivic Homology Theories (V. Voevodsky, A. Suslin and E. Friedlander, eds.), Annals of Math. Studies, 1999.

    Google Scholar 

  7. O. Gabber, Affine Analog of the Proper Base Change Theorem, Israel J. Math. 87 (1994), 325–335.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. A. Grothendieck, Étude Local des Schemas et des Morphisms de Schemas,Publ. Math. IHES 20, 24, 28, 32 (1964–1967.).

    Google Scholar 

  9. A. Grothendieck, M. Artin and J.-L. Verdier, Théorie des Topos et Cohomologie Étale des Schemas,Lecture Notes in Math. 269, 270, 304 (1972–73).

    MathSciNet  Google Scholar 

  10. A.J. de Jong, Smoothness, Semistability and Alterations, Publ. Math. IHES 83 (1996), 51–93.

    CrossRef  MATH  Google Scholar 

  11. A. Merkurjev and A. Suslin, Norm Residue Homomorphism and K-cohomology of Severi-Brauer Varieties., Math USSR Izv. 21 (1983), 307–340.

    CrossRef  Google Scholar 

  12. A. Merkurjev, On the Norm Residue Homomorphism for Fields, Amer. Math. Soc. Transl. 174 (1996).

    Google Scholar 

  13. J. Milne, Étale Cohomology, Princeton University Press, 1980.

    MATH  Google Scholar 

  14. Yu. Nesterenko and A. Suslin, Homology of the General Linear Group over a Local Ring and Milnor K-Theory, Izv. AN SSSR 53 (1989), 121–146.

    MATH  Google Scholar 

  15. Ye. Nisnevich, The Completely Decomposed Topology on Schemes and Associated De-scent Spectral Sequence in Algebraic K-Theory,Algebraic K-Theory: Connections with Geometry and Topology, Kluwer Acad. Publ., 1989.

    Google Scholar 

  16. M. Raynaud and L. Gruson, Critéres de Platitude et de Projectivité,Inv. Math. 13 (1971), 1–89.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. A Suslin Higher Chow Groups and Étale Cohomology, Cycles, Transfers and Motivic Homology Theories (V. Voevodsky, A. Suslin and E. Friedlander, eds.), Annals of Math. Studies, 1999.

    Google Scholar 

  18. A. Suslin and V. Voevodsky, Singular Homology of Abstract Algebraic Varieties, Inv. Math. 123 (1996), 61–94.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. V. Voevodsky, Homology of Schemes, Selecta Math. 2 (1996), 111–153.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. V. VoevodskyCohomological Theory of Presheaves with Transfers, Cycles, Transfers and Motivic Homology Theories (V. Voevodsky, A. Suslin and E. Friedlander, eds.), Annals of Math. Studies, 1999.

    Google Scholar 

  21. V. Voevodsky, Triangulated Category of Motives over a field,Cycles, Transfers and Motivic Homology Theories (V. Voevodsky, A. Suslin and E. Friedlander, eds.), Annals of Math. Studies, 1999.

    Google Scholar 

  22. V. Voevodsky The Milnor Conjecture, Preprint, Max-Planck-Institut fur Math. (1977).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, IL, 60208-2730, USA

    Andrei Suslin

  2. School of Mathematics, The Institute for Advanced Study, Princeton, NJ, 08540, USA

    Vladimir Voevodsky

Authors
  1. Andrei Suslin
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  2. Vladimir Voevodsky
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Editor information

Editors and Affiliations

  1. University of Oklahoma, Norman, Oklahoma, USA

    B. Brent Gordon

  2. University of Alberta, Edmonton, Alberta, Canada

    James D. Lewis

  3. Universität-GH-Essen, Essen, Germany

    Stefan Müller-Stach

  4. Tokyo Institute of Technology, Tokyo, Japan

    Shuji Saito

  5. Queen’s University, Kingston, Ontario, Canada

    Noriko Yui

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Suslin, A., Voevodsky, V. (2000). Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients. In: Gordon, B.B., Lewis, J.D., Müller-Stach, S., Saito, S., Yui, N. (eds) The Arithmetic and Geometry of Algebraic Cycles. NATO Science Series, vol 548. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4098-0_5

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  • DOI: https://doi.org/10.1007/978-94-011-4098-0_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-6194-7

  • Online ISBN: 978-94-011-4098-0

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