# Universal etale covering, reference request

I want to find some references for the following theorem. However, I do not know the exact statement of the theorem. So if there is anyone who can give me some references for this, it would be very helpful for me. Let me state the theorem to the extent that I know.

Theorem(?). Let $K$ be a field of characteristic $0$ and $\bar{K}$ be an algebraic closure of $K$ (if you want, you may assume that $K$ is a number field). Let $X$ be a geometrically irreducible smooth variety and $\bar{X}:=X\times_K\bar{K}$. Let $\tilde{\bar{X}}$ be the universal (finite) etale cover of $\bar{X}$. Note that $\tilde{\bar{X}}\to \bar{X}\to X$ becomes the universal (finite) etale cover of $X$. Then there exists a (finite) etale cover $Y\to X$ such that $\bar{Y}\to \bar{X}$ is exactly same with $\tilde{\bar{X}}\to \bar{X}$.

First of all, I am not sure whether all the conditions in the above statement are necessary (for example, geometrically irreducibility, smoothness, finiteness of covers). I would like to know the most general form of the theorem.

Second, I don't know the definition of universal (finite) etale covering. In fact, I know one definition of it which describes it by an inverse limit of some good (finite) etale covers. But I am not sure whether it is most common.

If someone can give me an exact statement, I would most appreciate it. However, anything similar would also be very helpful for me.

• Well any definition of a universal etale cover will generally not be finite. In your setting the universal etale cover is probably referring to the inverse limit over all finite etale covers, or perhaps the pro-object described by the inverse system over all finite etale covers, so if your space admits covers of arbitrarily large degree, this "universal cover" won't be finite. This is described in SGA 1, section V (though in french). The universal cover is described as the "pro-objet fondamental". May 9, 2014 at 22:03

It's not clear what you mean by "exactly same with" in your theorem, but even if we generously interpret it to mean that the map $\bar{Y} \to \bar{X}$ factors through $\tilde{\bar{X}}$, the claim does not seem to be true. For example, if $X$ is a positive genus smooth curve over $K$, then the fundamental pro-object (see page 120 of SGA1 - as oxeimon suggests, this may be what you mean by universal finite étale covering) of $\bar{X}$ is a nontrivial torsor under a torsion-free profinite group. Any factorization of a finite étale cover through the fundamental pro-object would imply the existence of a nontrivial homomorphism from a nontrivial finite group to a torsion-free group, so no such factorization exists.