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

2011-05-05T03:31:58Z

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.

Answer by Alex Bartel for Galois descent for K-groups (or for étale cohomology groups)

2011-05-09T11:02:10Z

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.

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

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

Answer by Alex Bartel for Galois descent for K-groups (or for étale cohomology groups)