The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in Hermitian K-theory, by A. J. Berrick, M. Karoubi, M. Schlichting, and P. A. Ostvaer

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields.

We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field.

We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its ├ętale version is an isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum conjecture in K-theory.

Applications compute higher Grothendieck-Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.


A. J. Berrick <berrick@math.nus.edu.sg>
M. Karoubi <max.karoubi@gmail.com>
M. Schlichting <m.schlichting@warwick.ac.uk>
P. A. Ostvaer <paularne@math.uio.no>