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Jun 23, 2011 at 22:01 comment added John Rognes 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 $O_F \to F$.
May 17, 2011 at 13:12 vote accept Alex B.
May 9, 2011 at 11:02 history answered Alex B. CC BY-SA 3.0