Étale fundamental group

Remarks

The group $\pi_1^{\text{ét}}(X, \bar{x})$ is a profinite group, as a closed subgroup of $\prod_Y \operatorname{Sym}(F_{\bar{x}}(Y))$. By Grothendieck's Galois theory, $F_{\bar{x}}$ is an equivalence between $\mathrm{FEt}_X$ and the category of finite sets with a continuous action of $\pi_1^{\text{ét}}(X, \bar{x})$. Example: for $X = \operatorname{Spec} k$ with $k$ a field, the finite étale covers are finite products of finite separable extensions of $k$. Choosing $\bar{x}$ amounts to choosing a separable closure $k^{s}$, and $\pi_1^{\text{ét}}(X, \bar{x}) \cong \Gal(k^{s}/k)$, the absolute Galois group. Non-example: the exponential map $\C \to \C^{*}$ is a covering space with infinitely many sheets, so it is not a finite étale cover and does not enter $\mathrm{FEt}_X$. Therefore $\pi_1^{\text{ét}}$ of the complex punctured line is $\widehat{\Z}$, the profinite completion of the topological fundamental group $\Z$, and not $\Z$ itself. For a variety $X$ over $\C$, the Riemann existence theorem gives $\pi_1^{\text{ét}}(X) \cong \widehat{\pi_1(X(\C))}$. In characteristic $p > 0$ the affine line has nontrivial finite étale covers, such as the Artin–Schreier covers $y^p - y = x$.