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

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.

💡 Research Summary

The paper establishes two parallel “Quillen‑Lichtenbaum‑type” comparison theorems for hermitian K‑theory (also called higher Grothendieck‑Witt theory) on a broad class of schemes. Let X be a noetherian scheme of finite Krull dimension that satisfies three technical hypotheses: (1) 2 is invertible in the structure sheaf, (2) X possesses an ample family of line bundles, and (3) there is a uniform bound on the virtual mod‑2 cohomological dimensions (vcd₂) of all residue fields of X. Under these conditions the authors prove:

1. Homotopy Fixed Point Theorem for Hermitian K‑theory.
   The natural involution on algebraic K‑theory, induced by taking the dual of a vector bundle, yields a Z/2‑action on the K‑theory spectrum K(X). There is a canonical comparison map
   \