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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
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The following question might be relevant : mathoverflow.net/questions/11209/… (see in particular John Rognes' answer). –  François Brunault May 5 '11 at 22:09

1 Answer 1

up vote 2 down vote accepted

Sorry to answer my own question, but according to these notes by Manfred Kolster, Proposition 2.9, the étale cohomology groups in fact always satisfy Galois descent. Should have googled harder before asking.

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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
    
@John Thank you for these remarks! –  Alex B. Jun 24 '11 at 2:19

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