The Arithmetic and Geometry of Algebraic Cycles pp 117-189

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

Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients

  • Andrei Suslin
  • Vladimir Voevodsky
Chapter

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].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-T]
    H. Bass and J. Tate,Milnor Ring of a Global Field,Lecture Notes in Math. 342 (1973), 349–446.MathSciNetGoogle Scholar
  2. [B]
    S. Bloch,Algebraic Cycles and Higher K-Theory,Adv. in Math. 61 (1986), 267–304.MathSciNetMATHCrossRefGoogle Scholar
  3. [B 1]
    S. Bloch, The Moving Lemma for Higher Chow Groups,Journal of Algebraic Geometry 3 (1994), 537–568.MathSciNetMATHGoogle Scholar
  4. [B-O]
    S. Bloch and A. Ogus, Gersten’s Conjecture and the Homology of Schemes, Ann. Sci. Ecole Norm. Super. (4) 7 (1975), 181–201.MathSciNetGoogle Scholar
  5. [D]
    A. Dold, Lectures on Algebraic Topology, Springer-Verlag, 1972.Google Scholar
  6. [F-V]
    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. [G]
    O. Gabber, Affine Analog of the Proper Base Change Theorem, Israel J. Math. 87 (1994), 325–335.MathSciNetMATHCrossRefGoogle Scholar
  8. [EGA-4]
    A. Grothendieck, Étude Local des Schemas et des Morphisms de Schemas,Publ. Math. IHES 20, 24, 28, 32 (1964–1967.).Google Scholar
  9. [SGA-4]
    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).MathSciNetGoogle Scholar
  10. [J]
    A.J. de Jong, Smoothness, Semistability and Alterations, Publ. Math. IHES 83 (1996), 51–93.MATHCrossRefGoogle Scholar
  11. [M-S]
    A. Merkurjev and A. Suslin, Norm Residue Homomorphism and K-cohomology of Severi-Brauer Varieties., Math USSR Izv. 21 (1983), 307–340.CrossRefGoogle Scholar
  12. [Me]
    A. Merkurjev, On the Norm Residue Homomorphism for Fields, Amer. Math. Soc. Transl. 174 (1996).Google Scholar
  13. [M]
    J. Milne, Étale Cohomology, Princeton University Press, 1980.MATHGoogle Scholar
  14. [N-S]
    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.MATHGoogle Scholar
  15. [Nis]
    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. [R-G]
    M. Raynaud and L. Gruson, Critéres de Platitude et de Projectivité,Inv. Math. 13 (1971), 1–89.MathSciNetMATHCrossRefGoogle Scholar
  17. [Su]
    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. [S-V]
    A. Suslin and V. Voevodsky, Singular Homology of Abstract Algebraic Varieties, Inv. Math. 123 (1996), 61–94.MathSciNetMATHCrossRefGoogle Scholar
  19. [V 0]
    V. Voevodsky, Homology of Schemes, Selecta Math. 2 (1996), 111–153.MathSciNetMATHCrossRefGoogle Scholar
  20. [V 1]
    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 2]
    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 3]
    V. Voevodsky The Milnor Conjecture, Preprint, Max-Planck-Institut fur Math. (1977).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Andrei Suslin
    • 1
  • Vladimir Voevodsky
    • 2
  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.School of MathematicsThe Institute for Advanced StudyPrincetonUSA

Personalised recommendations