The K-theory of fields in characteristic p |
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Authors: | Thomas Geisser Marc Levine |
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Institution: | Institute for Experimental Mathematics, Ellernstr. 29, D-45326 Essen, Germany?(e-mail: geisser@exp-math.uni-essen.de), DE Department of Mathematics, Northeastern University, Boston, MA 02115, USA?(e-mail: marc@neu.edu), US
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Abstract: | We show that for a field k of characteristic p, H
i
(k,ℤ(n)) is uniquely p-divisible for i≠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.
Oblatum 21-I-1998 & 26-VII-1999 / Published online: 18 October 1999 |
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Keywords: | Mathematics Subject Classification (1991): 19D50 19D45 19E08 14C25 |
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