Galois descent for K-groups (or for étale cohomology groups) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:28:28Z http://mathoverflow.net/feeds/question/63971 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63971/galois-descent-for-k-groups-or-for-etale-cohomology-groups Galois descent for K-groups (or for étale cohomology groups) Alex Bartel 2011-05-05T03:31:58Z 2011-05-09T11:02:10Z <p>Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let <code>$\mathcal{O}_F$</code> and <code>$\mathcal{O}_K$</code> be the associated rings of integers, and let $n\geq 1$.</p> <blockquote> <p>When is <code>$$K_{2n-1}(\mathcal{O}_F)^G \cong K_{2n-1}(\mathcal{O}_K)?$$</code></p> </blockquote> <p>It would be good enough for me to have this on the prime-to-2 parts. So let $p$ be an odd prime. Then <code>$$K_{2n-1}(\mathcal{O}_F)\otimes_{\mathbb{Z}}\mathbb{Z}_p\cong H_{ét}^1(\mathcal{O}_F,\mathbb{Z}_p(n))$$</code> by the Bloch-Kato conjecture, as proven by Rost, Suslin, Voevodsky and Weibel. So my question becomes:</p> <blockquote> <p>when is <code>$$H_{ét}^1(\mathcal{O}_F,\mathbb{Z}_p(n))^G \cong H_{ét}^1(\mathcal{O}_K,\mathbb{Z}_p(n))?$$</code></p> </blockquote> <p>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.</p> http://mathoverflow.net/questions/63971/galois-descent-for-k-groups-or-for-etale-cohomology-groups/64373#64373 Answer by Alex Bartel for Galois descent for K-groups (or for étale cohomology groups) Alex Bartel 2011-05-09T11:02:10Z 2011-05-09T11:02:10Z <p>Sorry to answer my own question, but according to <a href="http://www.ictp.it/~pub_off/lectures/lns015/Kolster/Kolster_Final.pdf" rel="nofollow">these notes</a> by Manfred Kolster, Proposition 2.9, the étale cohomology groups in fact always satisfy Galois descent. Should have googled harder before asking.</p>