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Once again sorry for the formatting, I'm using a phone.

Fix an étale cover of $Y\times S$, where $S$ is connected. We pull-back along inclusions of points into $S$ to get a family of étale covers of $Y$. Are the Galois groups of these covers isomorphic? This seems unlikely to be true but I'd love if it were. A reference would be ideal.

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    $\begingroup$ Unfortunately it's not so simple. Here's one example of what can go wrong: The cover of $\mathbb{P}^1$ defined by $X^4 + X^3 + X^2 + X + t$ has Galois group $S_4$, so by Hilbert irreducibility, we know that most points will have an $S_4$ cover. But the specialization at $t=1$ has Galois group $C_4$, cyclic of degree 4. However, I think what you do get is that all the Galois groups of the points will be subgroups of the Galois group of the cover. $\endgroup$
    – PrimeRibeyeDeal
    Nov 3, 2017 at 13:07
  • $\begingroup$ Thanks, that's a good example if I understand it correctly. If we require everything to be over the complexes is the situation simpler? $\endgroup$
    – EBz
    Nov 3, 2017 at 14:25
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    $\begingroup$ You need to clarify what you mean by "Galois group of cover" for $f:X\to Y\times_{\text{Spec}\ k} S.$ Perhaps you mean, for every geometric point $\text{Spec} \ \kappa \to S$, you compute the Galois group of the finite etale cover of $\kappa$-schemes $X_{\kappa} \to Y\times_{\text{Spec}\ k} \text{Spec}\ \kappa.$ That would eliminate the issue of the previous commenter. If that is what you mean, please confer SGA 1, Exp. X, Cor. 2.4, Thm. 3.8, Cor. 3.9 and Section 2 of Exp. XIII. $\endgroup$
    – Jason Starr
    Nov 3, 2017 at 14:42
  • $\begingroup$ That is indeed what I mean, my apologies to the previous commenter and many thanks for the reference! $\endgroup$
    – EBz
    Nov 3, 2017 at 14:47

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