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This paper has superceded by a more recent version.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 <>