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I have seen the claim that Beilinson Lichtenbaum implies that higher algebraic $K$ groups coincides with etale ones integrally in high enough degrees. Is this statement accurate? What conditions are required and how to derive it?

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To my knowledge the most general known statement has been proven by Clausen and Mathew in their paper Hyperdescent and étale K-theory as Theorem 1.2. The precise conditions on your commutative ring (or more generally algebraic space) are a bit technical to summarize, but are very general and give explicit bounds. Away from the residue chacteristics one can apply the Voevodsky-Rost norm residue theorem and at the residue characteristics they manage to apply a reduction to topological cyclic homology ($TC$). I recommend looking at their paper for a more detailed explanation.

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  • $\begingroup$ I see, one need to prove it for different primes and then rationally. This might be a silly question does this imply that the groups are integrally isomorphic? I am a little bit fuzzy on group theory but I can see how this implies integral isomorphism with the finite generation assumption on the groups but that might not be necessary. $\endgroup$
    – user127776
    Aug 24 at 12:46
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    $\begingroup$ This is in a sense actually easier on the derived level (in this case in the setting of spectra). There, $p$-completion is localization with respect to $\mathbb{S}/p$ and rationalization with respect to $\mathbb{S} \otimes \mathbb{Q}$. If a map is an equivalence after $p$-completion, its cofiber is $\mathbb{S}/p$-acyclic and hence $p$ acts invertibly. If this is true for all $p$ and the cofiber vanishes after tensoring with $\mathbb{Q}$, it must be zero itself. $\endgroup$ Aug 24 at 15:54
  • $\begingroup$ Sorry another question. The conditions seem to be some sort of bound on the Galois cohomological dimension of the residue fields. Do you know if one has such a bound only in mod $l$ Galois cohomology (base field is char 0) does that imply that such an isomorphism holds in high degrees for mod-$l$, $K$-theories? $\endgroup$
    – user127776
    Aug 24 at 18:13

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