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May 14, 2015 at 22:09 vote accept Mikhail Borovoi
May 14, 2015 at 17:56 comment added Daniel Litt @ACL: Just a comment; as remarked in my answer, if $G$ satisfies weak approximation, the image of the map in question is precisely $H^1_{Zar}(\text{Spec}(R), G)$.
May 14, 2015 at 8:33 comment added ACL In my paper with Yuri Tschinkel, Torseurs arithmétiques et espaces fibrés, we have a similar description (Proposition 1.2.6). NB. As remarked by Philippe Gille, there is a slight mistake there, that we consider torsors which are locally trivial for the Zariski topology, without saying so.
May 14, 2015 at 7:28 answer added Daniel Litt timeline score: 7
May 14, 2015 at 6:12 history edited Mikhail Borovoi CC BY-SA 3.0
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May 14, 2015 at 6:04 history asked Mikhail Borovoi CC BY-SA 3.0