THE HOMOTOPY FIXED POINT THEOREM AND THE QUILLEN-LICHTENBAUM CONJECTURE IN HERMITIAN K-THEORY

Abstract. We settle two conjectures for computing higher Grothendieck-Witt groups (also known as Hermitian K-groups) of noetherian schemes X, under some mild conditions. It is shown 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. We also prove that the mod 2ν comparison map between the Hermitian K-theory of X and its étale ver-sion 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. 1