Mathematics – K-Theory and Homology
Scientific paper
2002-09-04
K-Theory 36 (2005), no.1-2, p.33-50
Mathematics
K-Theory and Homology
V2-V4: Small changes, some details added. LaTeX 2e, 14 pages. Submitted also to K-theory electronic preprint archives at htt
Scientific paper
10.1007/s10977-005-3119-1
We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's unfinished proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term exact sequence of Galois cohomology with cyclotomic coefficients for any finite extension of fields whose Galois group has an exact quadruple of permutational representations over it. Examples include cyclic groups, dihedral groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4. Several exact sequences conjectured by Bloch-Kato, Merkurjev-Tignol, and Kahn are proven in this way. In addition, we introduce a more sophisticated version of the classical argument known as "Bass-Tate lemma". Some results about annihilator ideals in Milnor rings are deduced as corollaries.
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