On Two Conjectures in Birational Algebraic Geometry

  • Fedor A. Bogomolov


In this article I want to formulate and prove a synthetic version of two well known conjectures. One of them is the so-called Bloch-Kato conjecture. It provides a description of the torsion cohomology groups for any field in terms of Milnor K-functor. Another one was formulated by A. Grothendieck and concerns only the fields of rational functions on algebraic varieties over number fields. Namely, it claims that the Galois group of the algebraic closure of such field considered as an abstract profinite group defines the field in a functorial way. In fact, the Bloch-Kato conjecture can also be reformulated in terms of some quotient of the Galois group above.


Linear Subspace Galois Group Projective Line Valuation Ring Residue Field 


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  1. 1.
    F.A. Bogomolov “On the structure of abelian subgroups of the Galois groups” Izvestiya of the Academy of Science of the USSR (in russian) Nl, 1991.Google Scholar
  2. 2.
    A.S. Merkuriev, A.A. Suslin “If-cohomology of Severi-Brauer varieties and norm residue homomorphism’Izvestiya of the Academy of Science of USSR 1982 v.46 N.5 p.1011–1061.Google Scholar

Copyright information

© Springer-Verlag Tokyo 1991

Authors and Affiliations

  • Fedor A. Bogomolov
    • 1
  1. 1.Steklov Institute of MathematicsAcademy of Science of USSRMoscowUSSR

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