
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 is a -local equivalence in large degrees for suitable ring spectra . 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 -truncated in the cases , , and .
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