We prove the divisible case of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary primes 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 biquadratc group Z/2 x Z/2, and the symmetric group Z_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 the "Bass-Tate lemma". Some results about annihilator ideals in Milnor rings are deduced as corollaries.
We prove that the embedding of the derived category of 1-motives up to isogeny into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor LAlb ℚ, commute with the Hodge realization. This result yields a new proof of the rational form of Deligne's conjecture on 1-motives.
We compute the rational K_2 group of an elliptic surface over a finite field in terms of the order of zeros of the associated L-function.