# Galois descent for K-groups (or for étale cohomology groups)

Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$.

When is $$K_{2n-1}(\mathcal{O}_F)^G \cong K_{2n-1}(\mathcal{O}_K)?$$

It would be good enough for me to have this on the prime-to-2 parts. So let $p$ be an odd prime. Then $$K_{2n-1}(\mathcal{O}_F)\otimes_{\mathbb{Z}}\mathbb{Z}_p\cong H_{ét}^1(\mathcal{O}_F,\mathbb{Z}_p(n))$$ by the Bloch-Kato conjecture, as proven by Rost, Suslin, Voevodsky and Weibel. So my question becomes:

when is $$H_{ét}^1(\mathcal{O}_F,\mathbb{Z}_p(n))^G \cong H_{ét}^1(\mathcal{O}_K,\mathbb{Z}_p(n))?$$

A concise reference to a place that summarises everything we know about this would be great! Or maybe it is clear to the experts that this is always true? That would be even better.

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Your form of the Bloch-Kato conjecture surprises me.:) As for the basic question, I'm affraid that you just have a spectral sequence that converges to you left hand size, and that you have other non-zero terms in it (besides your right hand side). –  Mikhail Bondarko May 5 '11 at 6:08
@Misha I am not offering an alternative version of the Bloch-Kato conjecture. But it is known that the Bloch-Kato conjecture implies the Quillen-Lichtenbaum conjecture, which is the isomorphism between K-theory and étale cohomlogy that I quoted. I just said "by Bloch-Kato" because that's what Rost,Voevodsky, et aliae have proven. –  Alex B. May 5 '11 at 12:30
The Quillen-Lichtenbaum conjecture: 1. Describes motivic cohomology, that is certainly related with K-theory, but only via certain spectral sequences. 2. Ignores the infinitely divisible part of K-groups. –  Mikhail Bondarko May 5 '11 at 15:32
I don't know what you mean by "ignores the infinitely divisible parts of K-groups". The K-groups in question are finitely generated abelian groups with known ranks and known torsion. There is no infinitely divisible part. –  Alex B. May 5 '11 at 16:03
The following question might be relevant : mathoverflow.net/questions/11209/… (see in particular John Rognes' answer). –  François Brunault May 5 '11 at 22:09

However, you should note that $O_K \to O_F$ is only $G$-Galois if $K \to F$ is unramified. Galois descent for etale cohomology refers to Galois extensions of rings, not of their fraction fields. If you are not willing to invert the ramified primes in $O_K$, you may have to work harder. You can compare the etale cohomology localization sequence for $O_K \to K$ (with third term involving the finite residue fields) with the one for $O_F \to F$. –  John Rognes Jun 23 '11 at 22:01