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Concerning http://swc.math.arizona.edu/aws/1999/99RubinES.pdf, especially section I.5:

Can one define the Selmer groups and the unramified cohomology groups as étale cohomology groups of certain schemes, so that one does not have to take a kernel, similar to the Selmer group of Abelian varieties $\mathrm{Sel}^n(A/K) = \ker(H^1(K,A[n]) \to \prod_v H^1(K_v,A))$, which equals (perhaps up to the infinite places) $H^1(\mathrm{Spec} O_K, \mathscr{A}[n])$ with $\mathscr{A}$ the Néron model, and ${\mathrm{III}}(A/K) = H^1(\mathrm{Spec} O_K, \mathscr{A})$?

For example, one has perhaps $H^1_\mathrm{u}(K,A) = H^1(\mathrm{Spec} O_K, A)$ with $O_K$ the valuation ring of a local field $K$ (if $A$ is an unramfied module).

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    $\begingroup$ You can almost do this by treating a Galois module $A$ as an etale sheaf on Spec K, and taking the cohomology of the sheaf $j_* A$ on Spec $O_K$, where $j_*$ is the push-forward of etale sheaves along the inclusion $Spec K \hookrightarrow Spec O_K$. Unfortunately this doesn't work, because you really want to have the primes dividing the order of $A$ being invertible on the base, so if $A$ is a $\mathbf{Z}_p$-module, you need to use $Spec O_K[1/p]$ and then handle the local condition at the prime $p$ separately. $\endgroup$
    – David Loeffler
    Commented Feb 6, 2017 at 13:59
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    $\begingroup$ [cont'd] This is ultimately this is a different language for expressing the same concepts as the Galois-theoretic language that Rubin uses; but the etale-cohomology version is sometimes preferred by some authors. See e.g. Kato's magisterial article in Astersique 295, in particular sections 8.2, 8.5, and 14.9. (I was very confused by this etale-cohomology language at first, as you can see from some of my own questions on MO in the past.) $\endgroup$
    – David Loeffler
    Commented Feb 6, 2017 at 14:02

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