Two-primary algebraic 𝐾-theory of rings of integers in number fields

Abstract

We relate the algebraic K K -theory of the ring of integers in a number field F F to its étale cohomology. We also relate it to the zeta-function of F F when F F is totally real and Abelian. This establishes the 2 2 -primary part of the “Lichtenbaum conjectures.” To do this we compute the 2 2 -primary K K -groups of F F and of its ring of integers, using recent results of Voevodsky and the Bloch–Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory.