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 $x \in X$ .

Definition

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

By definition,

|F(Y)| = \deg Y

Definition (Étale fundamental group)

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 $F: \mathscr {FE}t_{X} \to \mathrm{Sets}$ .

In the case of covering theory or Galois theory, this functor

F

is “representable”. For Galois groups, this is represented by the separable closure

\bar{k}

, and for topological spaces, this is represented by the universal cover. Hence we can (almost) simply define the fundamental group to be the group of automorphisms of this universal object. However,

\operatorname{Spec}\bar{k} \to \operatorname{Spec}k

is not a finite étale cover, and the universal cover may be an infinite cover. Nevertheless, we can produce arbitrarily good approximations to these universal covers.

Definition (Pro-representable)

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

F(Y) = \operatorname*{colim}_\alpha \operatorname{Hom}_{\mathcal{C}}(X_\alpha , Y)

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,

Theorem (Fundamental Theorem of Galois Theory)

$F$ 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.

The proof of the theorem is the content of the next section. For now, we shall amuse ourselves by computing some fundamental groups.

Example

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

Example

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.

Example

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 $\mathbb {C}$ .