The K-theory of fields in characteristic p

We show that for a field k of characteristic p, H-i(k, Z(n)) is uniquely p-divisible for i not equal n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural map K-n(M) (k) --> K-n(k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that K-n(M)(k) and K-n(k) are p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example K-n(X, Z/p(r)) = 0 for n > dimX. Another consequence is Gersten's conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch's cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except possibly the vanishing conjecture.