The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Details The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

2010-11-23

arxiv.org/abs/1011.4977v2

Mathematics

K-Theory and Homology

16 pages

Scientific paper

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 \'etale 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.

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