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

TL;DR

This paper proves a homotopy fixed point theorem in hermitian K-theory for schemes with certain properties, establishing equivalences and isomorphisms that extend the Quillen-Lichtenbaum conjecture to hermitian K-theory, with applications to algebraic varieties and number theory.

Contribution

It extends the Quillen-Lichtenbaum conjecture to hermitian K-theory, proving comparison maps are equivalences under specific conditions and computing higher Grothendieck-Witt groups.

Findings

Comparison map from hermitian K-theory to homotopy fixed points is a 2-adic equivalence.

Comparison map between higher Grothendieck-Witt theory and its étale version is an isomorphism.

Applications include computing higher Grothendieck-Witt groups and values of Dedekind zeta-functions.

Abstract

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.