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?


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.

  • $\begingroup$ 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
  • 2
    $\begingroup$ 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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