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Suppose $G$ is a finite linear group, and I have a $G$-torsor $Z \to X$. Suppose also I have a morphism $f : Y \to X$ with some properties $P$. What should these properties $P$ be in order to make the fiber product $Y \times_X Z \to Y $ (with the natural projection) a $G$-torsor over $Y$?

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I think P should be empty. The group $G$ acts on the fibre product by acting on $Z$, pulling back a trivialising étale cover for $Z \to X$ gives a trivial étale cover for $Y \times_X Z \to Y$.

No?

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I think I have a gap in notation - does the superscript tau mean fixed points? What do you mean by $Z^1$? –  user125763 Mar 3 at 12:45
    
@user47386 Wild guess: $Z^1$ means 1-cocycles, and $\tau$ is a twist, but I think it ought to be flat cohomology over $X$ instead of $k$. –  S. Carnahan Mar 3 at 12:47
    
Yes, sorry, $\tau$ is a 1-cocycle, and $f^\tau: Z^\tau \to X$ is the twist of $f : Z \to X$ by $\tau$. –  user47386 Mar 3 at 13:21
    
sorry I'm not familiar with such things. By cocycles I presume you mean group cohomology? I know nothing of this. (originally I thought you meant Weil divisors) –  user125763 Mar 3 at 22:05

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