### The K-theory of fields in characteristic p, by Thomas Geisser and Marc Levine

The purpose of this paper is to study the p-part of motivic cohomology
and algebraic K-theory in characteristic p (we use higher Chow groups
as our definition of motivic cohomology). The main theorem states
that for a field k of characteristic p, H^i(k,Z(n)) is uniquely
p-divisible for i\not=n. 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 K_n(k) is 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>dim X. Another consequence is Gersten's conjecture with mod p
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 the vanishing conjecture.

Thomas Geisser <geisser@math.uiuc.edu>

Marc Levine <marc@neu.edu>