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Let $\mathbf{G}$ be a split connected reductive group scheme over a scheme $X$.

Let $X'\rightarrow X$ an étale Galois cover of group $\Gamma$.

We consider $G$ a quasi-split group scheme over $X$ that splits over $X'$ to $\mathbf{G}$.

How can we describe a $G$-torsor on $X$ in terms of $X'$ and $\mathbf{G}$?

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    $\begingroup$ Is $X$ connected? What do you mean by "quasi-split" in the relative setting? (Just existence of a Borel $X$-subgroup $B$, or also $B$ contains a fiberwise maximal $X$-torus, or...? SGA3, XXIV, 3.9 demands a further condition in terms of the scheme of Dynkin diagrams, automatic for semi-local $X$.) Identifying ${\rm{H}}^1(X',\mathbf{G})$ with ${\rm{H}}^1(X,{\rm{R}}_{X'/X}(\mathbf{G}))$ and taking inspiration from Prop. 36 in section 5.4 of Ch. I of Serre's "Galois cohomology" book, the set of such isom. classes is $\mathbf{G}(X')\backslash ({\rm{R}}_{X'/X}(\mathbf{G})/G)(X)$. $\endgroup$
    – user76758
    Jun 8, 2014 at 2:39
  • $\begingroup$ Sorry, I meant to say at the end above that the set of isom. classes of such torsors on $X$ which also split over $X'$ is given by the indicated "double coset" construction. $\endgroup$
    – user76758
    Jun 8, 2014 at 15:09
  • $\begingroup$ I put it in a general situation but you can suppose that $X$ is a curve over an algebraically cclosed field. quasi-split means that you have a pinning defined over $X$ $\endgroup$
    – prochet
    Jun 8, 2014 at 19:05
  • $\begingroup$ In your comment above I think you meant to write "split" rather than "quasi-split", so it remains unclear what you mean by "quasi-split" (do you follow SGA3 or have another definition in mind)? Maybe some context and motivation would help. $\endgroup$
    – user76758
    Jun 9, 2014 at 1:33
  • $\begingroup$ yes, same as SGA3 $\endgroup$
    – prochet
    Jun 9, 2014 at 9:16

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