Algebraic Geometry – Etale K Theory and Algebraic Theory in High Degrees

Question

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?

Answer 1

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.