### Galois cohomology of certain field extensions and the divisible case of Milnor-Kato conjecture, by Leonid Positselski

[This preprint has been updated by the author.]

We prove the divisible case of the Milnor-Bloch-Kato conjecture (the first
step of Voevodsky's unfinished proof of this conjecture for arbitrary
prime l) in a rather clear and elementary way (without ever using the
motivic cohomology at all). Assuming this conjecture, we prove a number of
exact sequences for Galois cohomology in cyclic, biquadratic, and dihedral
field extensions (including the "relative conjecture" of Bloch-Kato for
cyclic extensions and the conjecture of Merkurjev-Tignol and Kahn in
the biquadratic case). Besides, we introduce a more sophisticated version
of the classical argument known as the "Bass-Tate lemma". Several results
about annihilator ideals in Milnor rings are deduced as corollaries.

Replacement: exposition improved, one corollary and one conjecture added.
Second replacement: more detailed proofs + another dihedral exact sequence.

Leonid Positselski <posic@mccme.ru, posic@mpim-bonn.mpg.de>