Effective Redshift

Abstract

The "higher chromatic" Quillen-Lichtenbaum conjecture, as proposed by Ausoni and Rognes, posits that the finite localization map $K(R)\to L_{n+1}^{f}K(R)$ is a $p$-local equivalence in large degrees for suitable ring spectra $R$. We give a simple criterion in terms of syntomic cohomology for an effective version of Quillen-Lichtenbaum, i.e. for identifying the degrees in which the localization map is an isomorphism. Combining our result with recent computations implies that the finite localization map is $(-1)$-truncated in the cases $R=\mathrm{BP}⟨n⟩$, $R=k(n)$, and $R=\mathrm{ko}$.