The homotopy limit problem for two-primary algebraic \(K\)-theory (Q2568996)

Let \(k\) be a field of characteristic different from \(2,\) \(k^s\) a separable closure of \(k\) and \(G_k\) the absolute Galois group of \(k.\) Let further \(K/2(k)\) and \(K/2(k^s)\) denote the corresponding mod 2 \(K\)-theory spectra. One of the main theorems of the paper is the following: Theorem 1. Let \(k\) be a field of characteristic different from \(2.\) Then \[ K/2(k)\rightarrow K/2(k^s)^{hG_k} \] is a weak equivalence on \((\text{vcd}(k)-2)\)-connected covers. \(K/2(k^s)^{hG_k}\) denotes here homotopy fixed points of \(K/2(k^s)\) under the action of \(G_k\) and \(\text{vcd}(k)\) is the virtual cohomological dimension of \(k.\) Theorem 1 is equivalent to the Quillen-Lichtenbaum conjecture which asserts that the algebraic to étale \(K\)-theory map \(K/2(k)\rightarrow K^{\text{ét}}/2(k)\) is a weak equivalence on some connected cover. The Quillen-Lichtenbaum conjecture at the prime \(2\) for fields of finite cohomological dimension was proven by M. Levine [cf. \textit{M. Levine}, K-theory and motivic cohomology of schemes, UIUC \(K\)-theory preprint server (1999)]. In the current paper the authors prove the conjecture for all fields. They also obtain the corresponding result for schemes: Theorem 2. Let \(X\) be a separated Noetherian regular scheme of finite Krull dimension. Assume all residue fields of \(X\) have characteristic different from \(2.\) Then there is a weak equivalence between Quillen algebraic \(K\)-theory and Jardine étale \(K\)-theory of \(X\) \[ K/2(X)\rightarrow K^{\text{ét}}/2(X) \] on \(\sup\{(\text{vcd}(k(x))-2)\}_{x\in X}\)-connected covers, where \(k(x)\) is the residue field of a point \(x\in X.\)