Let $X$ be an affine variety and $G$ an affine algebraic group (for example $\operatorname{PGL}_n$). How do I compute the Selmer set $$ \operatorname{Sel}_\zeta(\mathbb{Q},G) = \{\tau \in H^1(\mathbb{Q},G) \ | \ \tau_\nu \in \zeta(X(\mathbb{Q}_\nu)) \ \text{for all places} \ \nu\} $$ where $(\zeta \mapsto \zeta(x))$, $H^1(X,G) \to H^1(\mathbb{Q}_\nu,G)$ comes from a point $x: \operatorname{Spec}(\mathbb{Q}_\nu) \to X$ and $(\tau \mapsto \tau_\nu)$ comes from the map $\mathbb{Q} \to \mathbb{Q}_\nu$?
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$\begingroup$ What do you mean by H(X,G)? Is it the etale or Zariski sheaf cohomology? or something else? Do you have any reference where these objects appear? Your Selmer set look looks really similar to Selmer groups which are important groups associated to Galois representations. $\endgroup$– Marsault ChabatCommented Aug 3, 2022 at 16:14
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$\begingroup$ Apparently it is fppf cohomology. These are defined in the Poonen's book Rational Points on Varities. $\endgroup$– Mikko PitkonenCommented Aug 3, 2022 at 16:19
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Assume that $X(\mathbb{Q}_\nu)$ is nonempty for every place $\nu$. Since $X$ is affine, $H^1(X,G)$ is trivial. This means that $\zeta(x) = e$ for every $x \in X(\mathbb{Q}_\nu)$, which means that $e \in \operatorname{Sel}_\zeta(\mathbb{Q},G)$. So the Selmer set is nonempty.