The higher K-theory of complex varieties, by Claudio Pedrini and Charles Weibel

Let X be a smooth complex variety, with function field F. We prove that (after localizing at the prime 2) the K-groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-finite above the dimension of X. If X is projective we can improve this range by 3, except of course for K_0 when X is a curve or surface.

We also describe the torsion in the K-groups of F and X. The finite summands in the K-groups of X and the KU-groups of X are the same. The divisible torsion in the K-theory of X and F comes from the free part of their KU-groups, with a degree shift as predicted by the Quillen-Lichtenbaum Conjecture for K-theory with finite coefficients (which is true at the prime 2, as we show).

Much of this paper depends upon the recent developments in motivic cohomology, related to Voevodsky's proof of the norm residue conjecture at the prime 2. With this in mind, we have stated all our results twice: first at the prime 2 and then under the assumption that the norm residue conjecture holds.

These results in this paper were first announced at the Colloque Karoubi in November 1998, before the announcement of the Friedlander-Suslin results (this server, paper number 360). For this reason, we give a short proof of the Quillen-Lichtenbaum conjecture for F that does not invoke any multiplicative properties of the Bloch-Lichtenbaum spectral sequence for F.

This paper has appeared in K-theory 21 (2001), 367-385.


Claudio Pedrini <pedrini@dima.unige.it>
Charles Weibel <weibel@math.rutgers.edu>