The Étale Fundamental GroupIntroduction

# 1 Introduction

The fundamental theorem of Galois theory says

Theorem (Galois theory)

Let $k$ be a field and $G = \operatorname{Gal}(\bar{k}/k)$ be the absolute Galois group 1 . Then there is a contravariant equivalence of categories between

• the category of finite separable extensions of $k$; and

• the category of finite sets with a continuous transitive action of $G$.

The second category is also isomorphic to the category of open subgroups of $G$ by sending the subgroup $H \leq G$ to $G/H$, which is how Galois theory is often stated.

There is a very similar theorem in algebraic topology.

Theorem (Covering space theory)

Let $X$ be a path connected, locally path connected and semi-locally simply connected topological space. Then there is an equivalence of categories between

• the category of finite connected covering spaces of $X$; and

• the category of finite sets with a transitive action of $\pi _1(X, x)$, for any base point $x \in X$.

There are some superficial differences between these two isomorphisms. In Galois theory, we have a contravariant equivalence of categories, instead of a usual equivalence, but this is easily explained by the fact that we really should apply $\operatorname{Spec}$ to all our fields, thereby reversing the arrow.

On the other hand, in covering space theory, we have to pick a basepoint of $X$ to talk about the fundamental group, and the correspondence is natural (in the English sense) only after we fix such a basepoint. The analogous situation in Galois theory is that in fact, $G$ is not canonically associated to $k$ either. Instead, we must fix an embedding of $k$ into a separably closed field, which is the same as picking a geometric point of $\operatorname{Spec}k$.

To push the analogy further, there is a sense in which a separable field extension is a “covering space”. By definition, a map $p: Y \to X$ is a covering space iff there is some space $X'$ and a map $f: X' \to X$ such that

1. $f$ is surjective;

2. restricted to each component of $X'$, the map $f$ is the inclusion of an open subspace; and

3. the pullback $Y \times _X X'$ is a disjoint union of copies of $X'$, and the map $f^*p: Y \times _{X} X' \to X'$ is the map sending each copy isomorphically onto $X'$.

This is just the local triviality condition phrased in a fancy way.

The idea is to think of the (surjective!) morphism $\operatorname{Spec}\bar{k} \to \operatorname{Spec}k$ as an inclusion of an “open subset”, and an extension $K/k$ is separable iff the pullback to $\operatorname{Spec}\bar{k}$ is a disjoint union of copies of $\operatorname{Spec}\bar{k}$, i.e. $K \otimes _k \bar{k} = \bar{k} \times \cdots \times \bar{k}$. Of course, we can also replace $\bar{k}$ by some appropriate finite Galois extension if we want to remain in the finite world.

By using an appropriate generalized notion of “open subset”, hence “covering space”, we will generalize the notion of fundamental group to arbitrary schemes, of which Galois theory and covering space theory (of complex algebraic surfaces) are special cases.