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Let $X$ be a connected scheme,$\pi_1(X,\bar{x})$ its étale fundamental group for some geometric point $\bar{x} : Spec(K) \rightarrow X$ and $E = \pi_1(X,\bar{x})/N$ a finite quotient of $\pi_1(X,\bar{x})$

I am looking for book or paper describing the explicit construction of the Galois cover $Y \rightarrow X$ corresponding to $E$ other than Grothendieck's SGA or Tamas Szamuely's book

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    $\begingroup$ What does an explicit construction mean to you? Do you want equations for it? This can be notoriously difficult. If $X$ is $\mathbb{P}^1$ and the cover is only branched above three points then this is the question of "computing belyi maps" (google it!), which is very difficult in general. $\endgroup$ – Will Chen Sep 29 '19 at 23:25
  • $\begingroup$ no just i wonder if the Galois group of this cover arise in some exact sequence; in such way we can compute its etale cohomology of degree 1 or 2 by using this group ; and also im asking what is $\pi_1(Y)$; i know (for the topological case) when $N$ is finite subgroup of $\pi_1$ then the corresponding cover has fundamental group equal to $N$; $\endgroup$ – Moutand Mohammed Sep 30 '19 at 0:09
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    $\begingroup$ If $Y$ corresponds to a quotient of $\pi_1(X)$, then $\pi_1(Y)$ is the kernel of the quotient map (the same as the topological setting). This should be a purely formal consequence of the Galois correspondence. $\endgroup$ – Will Chen Sep 30 '19 at 0:21
  • $\begingroup$ is there a relation between this galois group or $\pi_1(Y)$ and $H^2(Y, f^*\mathcal{F})$ for some given sheaf $\mathcal{F}$ on $X$ ( in the low degrees exact sequence of the Hochschild Serre spectral sequence only the the $H^0(Y)$ appears ) $\endgroup$ – Moutand Mohammed Sep 30 '19 at 0:44

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