# 4 The Étale Fundamental Group

In Galois theory and covering space theory, a lot of information can be understood by looking at automorphisms of covers. To understand these automorphisms, it is useful to look at what it does to geometric points. Fix a base point $x \in X$.

For any $Y \in \mathscr {FE}t_{X}$, we define $F(Y)$ to be the set of all lifts of $x$ to $Y$.

Let $X$ be a scheme and $x$ a base point. The *étale fundamental group* $\pi _1(X, x)$ is defined to be $\operatorname{Aut}F$, the group of all automorphisms of the functor $F: \mathscr {FE}t_{X} \to \mathrm{Sets}$.

A functor $F: \mathcal{C} \to \mathrm{Sets}$ is *pro-representable* if there is a directed/pro-system $(X_\alpha )_\alpha$ of objects in $\mathcal{C}$ such that

for all $Y \in \mathcal{C}$.

In this case, we have

$\operatorname{Aut}(F) = \lim _\alpha \operatorname{Aut}(X_\alpha )^\mathrm{op},$and in particular, this is a profinite group (since each $\operatorname{Aut}(X_\alpha )$ acts freely on the finite set $F(X_\alpha )$). In the case of Galois theory, the functor $F$ is pro-represented by the pro-system of all Galois extensions. In covering space theory, $F$ is pro-represented by the pro-system of all normal covers. In fact,

$F$ is always pro-representable. Hence, $\pi _1(X, x)$ is profinite, acting continuously on $F$.

$\mathscr {FE}t_{X}$ is equivalent to the category of finite continuous $\pi _1(X, x)$-sets.

In particular, the isomorphism class of $\pi _1(X, x)$ does not depend on the choice of basepoint. Thus, we may write $\pi _1(X)$ instead.

Let $k$ be a field. Then $\pi _1(\operatorname{Spec}k) = \operatorname{Gal}(\bar{k}/k)$.

Let $X$ be a projective variety over $\mathbb {C}$. Then $\pi _1(X)$ is the pro-completion of the topological fundamental group, by the characterization of étale maps and the Riemann existence theorem.

By Riemann–Hurwitz, over any algebraically closed field of characteristic zero, any covering of $\mathbb {P}^1$ is trivial. So $\pi _1(\mathbb {P}^1) = 0$. This agrees with what topology tells us if we work over $\mathbb {C}$.