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Effective Redshift
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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 $R = \mathrm{ko}$.
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Syntomic cohomology of truncated Brown--Peterson spectra
For every E1 MU-algebra form of BP⟨n⟩, the mod (p,v1,...,v_{n+1}) syntomic cohomology is computed, yielding redshift, telescope, and Lichtenbaum–Quillen for its algebraic K-theory.
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