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I'm trying to understand unramified Galois cohomology of number fields a bit better.

Set-up: Let $k$ be a number field and $S$ a finite set of places of $k$ which contains all the archimedean places. Fix an algebraic closure $\bar k$ of $k$ and denote by $k_S$ the maximal field extension of $k$ which is unramified outside of $S$. Let $G$ be an algebraic group over $k$, not necessarily finite nor abelian (this is important for the application which I have in mind.)

I shall say that a class $c \in H^1(k,G(\bar k))$ is unramified at a place $v$ of $k$ if it lies in $$\ker\left(H^1(k,G(\bar k)) \to H^1(I_v,G(\bar k)\right),$$ where $I_v$ denotes the inertia group at $v$. I shall denote by $H^1_S(k,G(\bar k))$ the collection of elements of $H^1(k,G(\bar k))$ which are unramified outside of all places of $S$.

This looks like it should be closely related to $H^1(k_S/k, G(k_S))$. My question is as follows

How are $H^1_S(k,G(\bar k))$ and $H^1(k_S/k, G(k_S))$ related? For example, are they equal? Is one naturally contained in the other?

I'm willing to enlarge the set of places $S$ if necessary to include any "bad" primes.

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    $\begingroup$ As for inclusions, I think $H^1(k_S/k, G(k_S)) \subset H^1_S(k, G(\overline{k}))$, no? The pointed set $H^1(k_S/k, G(k_S))$ classifies all $G$-torsors (over $k$) that have a $k_S$-point; the pointed set $H^1_S(k, G(\overline{k}))$ classifies all $G$-torsors that have a point over the maximal unramified extension of $k_v$ for every $v \not\in S$. For every such $v$ a choice of its extension to $k_S$ leads to an unramified extension of $k_v$, which, together with the interpretation in terms of torsors, shows the claimed inclusion. $\endgroup$ Feb 26, 2014 at 2:19
  • $\begingroup$ Hi Kestutis. Yes I think I can see this inclusion. Do you know whether or not this inclusion is strict? $\endgroup$ Feb 26, 2014 at 14:15
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    $\begingroup$ Hi Dan. I think the inclusion is strict in general: take $k = \mathbf{Q}$ and $S = \{ \infty \}$ and let $G$ be an abelian variety with nontrivial Sha. Then $k_S = \mathbf{Q}$ and a nontrivial but everywhere locally trivial $G$-torsor gives an element in the difference of the two sets. $\endgroup$ Feb 26, 2014 at 15:03
  • $\begingroup$ Aha, nice example! If you post this as an answer I will accept it. $\endgroup$ Feb 27, 2014 at 10:05

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