Effective Redshift

TL;DR

This paper provides a criterion based on syntomic cohomology to determine when the finite localization map in algebraic K-theory is an isomorphism in certain degrees, advancing understanding of the Quillen-Lichtenbaum conjecture.

Contribution

It introduces a simple syntomic cohomology criterion for an effective version of the Quillen-Lichtenbaum conjecture, identifying degrees where the localization map is an isomorphism.

Findings

Finite localization map is $(-1)$-truncated for BP⟨n⟩, k(n), and ko.

The criterion simplifies identifying isomorphism degrees in algebraic K-theory.

Results connect syntomic cohomology with localization properties in algebraic K-theory.

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 $R = \mathrm{BP} \langle n \rangle$, $R = k(n)$, and $R = \mathrm{ko}$.