Advertisement

On the Reciprocity Sequence in the Higher Class Field Theory of Function Fields

Chapter
Part of the NATO ASI Series book series (ASIC, volume 407)

Abstract

According to a conjecture of Kato (1986), the classical reciprocity sequence for the Brauer group of a function field in one variable over a finite field F should have analogues for higher dimensional function fields. A more precise form of the conjecture is that on smooth projective varieties of dimension d over F, the homology of a certain Bloch-Ogus complex of length d + 1 should be trivial except in the last term, where it should be ℚ/ℤ. For surfaces, the conjecture was established some years ago. In the present paper, I prove that for varieties of arbitrary dimension, the complex has the expected homology in its last four terms, thus settling the case of threefolds (attention is restricted to torsion prime to the characteristic).

Keywords

Exact Sequence Spectral Sequence Finite Field Residue Field Torsion Group 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ab]
    Sh. S. Abhyankar.–Resolutions of Singularities of Arithmetical Surfaces, in Arithmetical Algebraic Geometry, ed. O. F. G. Schilling, Harper’s Series in Modern Mathematics, Harper and Row, New York (1965) 111–152.Google Scholar
  2. [A/K]
    A. B. Altman and S. L. Kleiman.–Bertini theorems for hypersurface sections containing a subscheme, Commun. Algebra 7 (1979), 775–790.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [B/O]
    S. BLOCH and A. Ocus. Gersten’s conjecture and the homology of schemes, Ann. Scient. Ec. Norm. Sup., 4ème série 7 (1974), 181–202.MathSciNetzbMATHGoogle Scholar
  4. [CT]
    J.-L. Colliot-Thélène. Cycles algébriques de torsion et K-théorie algébrique, Notes d’un cours au CIME, juin 1991; prépublication d’Orsay 92–14.Google Scholar
  5. [CT/R1]
    J.-L. Colliot-Thélène and W. Raskind.–K 2 -cohomology and the second Chow group, Math. Ann. 270 (1985), 165–199.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [CT/R2]
    J.-L. Colliot-Thélène and W. Raskind.–On the reciprocity law for surfaces over finite fields, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math. 33 (1986), 283–294.zbMATHGoogle Scholar
  7. [CT/S/S]
    J.-L. Colliot-Thélène, J.-J. SANSUC et C. SOuLÉ. — Torsion dans le groupe de Chow de codimension deux,Duke Math. J. 50 (1983), 763–801.Google Scholar
  8. [G/S]
    M. Gros et N. SuwA. — Application d’Abel-Jacobi p-adique et cycles algébriques, Duke Math. J. 57 (1988), 579–613.MathSciNetzbMATHGoogle Scholar
  9. [Gr]
    A. Grothendieck.–Le groupe de Brauer III: exemples et compléments, in Dix exposés sur la cohomologie des schémas, North-Holland/Masson (1968), 88–188.Google Scholar
  10. [J1]
    U. Jannsen.–Principe de Hasse cohomologique, in Séminaire de théorie des nombres de Paris 1989–90, éd. S. David, Progress in Math. Birkhäuser 102, 121–140.Google Scholar
  11. [J2]
    U. Jannsen. - Geheime Notizen.Google Scholar
  12. [J3]
    U. Jannsen. - Mixed motives and algebraic K-theory, Springer L.N.M. 1400 (1990).Google Scholar
  13. [K]
    K. Kato.–A Hasse principle for two dimensional global fields, J. für die reine und ang. Math. 366 (1986), 142–181.zbMATHGoogle Scholar
  14. [K/S1]
    K. Kato and S. Saito.–Unramified class field theory of arithmetical surfaces, Annals of Math. 118 (1983), 241–275.zbMATHCrossRefGoogle Scholar
  15. [K/S2]
    K. Kato and S. Saito.–Global class field theory of arithmetic schemes, in Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I, Contemp. Math. 55 (1986), 255–331.Google Scholar
  16. [M/S1]
    A. S. Merkur’ev and A. A. Suslin.–K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR, Ser. Mat. 46 (1982), 1011–1046 = Math. USSR Izvestiya 21 (1983), 307–340.zbMATHCrossRefGoogle Scholar
  17. [M/S2]
    A. S. Merkur’ev and A. A. Suslin. On the norm residue homomorphism of degree three, Izv. Akad. Nauk SSSR, Ser. Mat. 54 (1990), 339–356 = Math. USSR Izvestiya 36 (1991), 349–367.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [P]
    I. A. Panin.–Fields whose K2 is zero. Torsion in H 1 (X,K 2 ) and CH 2 (X), Zap. LOMI 116 (1982), 108–118.Google Scholar
  19. [R]
    W. Raskind.–On K 1 of curves over global fields, Math. Ann. 288 (1990), 179–193.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [Sa]
    S. Saito. — Cohomological Hasse principle for a threefold over a finite field,to appear in these Proceedings.Google Scholar
  21. [Sh]
    C. Sherman. — Some theorems on the K-theory of coherent sheaves, Commun. in Algebra 7 (1979), 1489–1508.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [So]
    C. Soulé. — Groupes de Chow et K-théorie de variétés sur un corps fini, Math. Ann. 268 (1984), 317–345.Google Scholar
  23. [Su]
    A. A. Suslin. — Torsion in K2 of fields, Journal of K-theory 1 (1987), 5–29.Google Scholar
  24. [Sw]
    N. Suwa. — A note on Gerten’s conjecture for logarithmic Hodge-Witt sheaves,Prépublication d’Orsay 92–35.Google Scholar
  25. [SGA4 1/2]
    Cohomologie étale,éd. P. Deligne, Springer L.N.M. 569 (1977).Google Scholar
  26. [SGA7 II]
    P. Deligne et N. Katz. — Groupes de rnonodromie en géométrie algébrique, Springer LNM 340 (1973).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  1. 1.C.N.R.S., URA D0752Université de Paris-SudOrsay CédexFrance

Personalised recommendations