# Galois cover corresponding to finite quotient of the étale fundamental group

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

• 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. Sep 29, 2019 at 23:25
• 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 $N$; Sep 30, 2019 at 0:09
• 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. Sep 30, 2019 at 0:21
• 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 ) Sep 30, 2019 at 0:44