### The Quillen-Lichtenbaum conjecture at the prime 2, by Bruno Kahn

In this paper we prove the Quillen-Lichtenbaum conjecture relating the
algebraic K-theory of a ring of algebraic integers to its etale l-adic
cohomology, when the prime number l is 2. The case F=Q was obtained
earlier by Weibel. We also get as side results the
validity of the Beilinson-Soule vanishing conjecture for all subfields of
\bar Q, after localization at 2, as well as the injectivity of K_3^M(F)\to
K_3(F) and a control of the kernel of K_4^M(F)\to K_4(F) for
any field F of characteristic 0. The main ingredients of the proof are the
Bloch-Lichtenbaum-Friedlander-Suslin-Voevodsky spectral sequence from the
motivic cohomology to the K-theory of a field of characteristic 0, its
degeneration up to torsion due to Soule, the recent results of Suslin and
Voevodsky settling the Beilinson-Lichtenbaum and Kato conjectures at the
prime 2, and Quillen's finiteness theorem for the K-theory of a ring of
algebraic integers.

In addition to providing a proof of the Quillen-Lichtenbaum conjecture,
this paper is intended as preparatory for another one investigating
consequences of the Bass conjecture.

This paper has superceded by a more recent version.

Bruno Kahn <kahn@math.jussieu.fr>