One of the early achievements of Grothendieck’s theory of schemes was the (partial) computation of the étale fundamental group of a smooth projective curve in characteristic ${p}$. The result is that if ${X_0}$ is a curve of genus ${g}$ over an algebraically closed field of characteristic ${p}$, then its fundamental group is topologically generated by ${2g}$ generators. This is analogous to the characteristic zero case, where the topological fundamental group generated by ${2g}$ generators (subject to a single relation) by the theory of surfaces. Since this result uses many interesting techniques, I thought I would devote a few posts to discussing it.

To motivate étale ${\pi_1}$, let’s recall the following statement:

Theorem 1 Let ${X}$ be a nice topological space, and ${x_0 \in X}$. Then the functor ${p^{-1}(x_0)}$ establishes an equivalence of categories between covering spaces ${p: \overline{X} \rightarrow X}$ and ${\pi_1(X, x_0)}$-sets.

This is one way of phrasing the Galois correspondence between subgroups of ${\pi_1(X, x_0)}$ and connected covering spaces of ${X}$, but which happens to be more categorical and generalizable. The interpretation of ${\pi_1(X, x_0)}$ as classifying covering spaces is ultimately the one that will work in an algebraic context. One can’t talk about homotopy classes of loops in an algebraic variety. However, Grothendieck showed:

Theorem 2 Let ${X}$ be a connected scheme, and ${\overline{x_0} \rightarrow X}$ a geometric point. Then there is a unique profinite group ${\pi_1(X, \overline{x_0})}$ such that the fiber functor of liftings ${\overline{x_0} \rightarrow \overline{X}}$ establishes an equivalence of categories between (finite) étale covers ${p: \overline{X} \rightarrow X}$ and finite continuous ${\pi_1(X, x_0)}$-sets.

The intuition is that étale covers are supposed to be analogous to covering spaces (because, for complex varieties, étale morphisms are local isomorphisms on the analytifications). However, one wishes to retain finite type hypotheses in algebraic geometry, so it makes sense to classify finite étale covers. (Hereafter, the word étale cover will implicitly mean finite.)

I don’t actually want to get into the proof of this theorem! Grothendieck did it by inventing his own formalism for Galois theory, which encompasses both classical Galois theory (of either field extensions or covering spaces) and the étale ${\pi_1}$. The idea is that any suitably nice (by which one means artinian, and satisfying some other conditions) category with an appropriate “fiber functor” ${F}$ to the category of finite sets (here, the pre-images of one point) becomes equivalent (via ${F}$) to the category of finite continuous ${G}$-sets for ${G}$ a uniquely determined profinite group. In fact, ${G}$ is the group of automorphisms of the fiber functor. It is also the projective limit of the automorphism groups of Galois objects in this category.

Moreover, it turns out that, when one is working over the complex numbers, the étale fundamental group turns out to be the profinite completion of the usual fundamental group. This is one way of stating the so-called Riemann existence theorem:

Theorem 3 Let ${X}$ be a scheme of finite type over ${\mathrm{Spec} \mathbb{C}}$. Then the analytification functor establishes an equivalence of categories between the category of finite étale covers of ${X}$ and the category of finite covering spaces of the analytification.

In other words, we can use familiar transcendental techniques to compute the étale fundamental group of something in characterstic zero. The more interesting case is characteristic ${p}$.

Anyway, let’s accept this, and try to sketch the idea behind Grothendieck’s proof. Fix a smooth curve ${X_0}$ over a field of characteristic ${p}$. The first step is to lift ${X_0}$ to characteristic zero. The existence of the lifting is not trivial, but let’s assume it exists for now. In other words, choose a complete discrete valuation ring ${A}$ of characteristic zero whose residue field is the field of characteristic ${p}$ and a proper, smooth scheme ${X \rightarrow \mathrm{Spec} A}$ such that the special fiber is ${X_0}$.

Then, one can, using some of Grothendieck’s descent machinery, find relations between the fundamental groups of the special and general fiber. In particular, one can define a surjective “cospecialization” map between the general and specific fiber. But the general fiber is a curve in characteristic zero, and it has the same genus as ${X_0}$ by flatness of ${X \rightarrow \mathrm{Spec} A}$, so the usual transcendental methods allow one to compute ${\pi_1}$ of the general fiber. This will be the strategy.

This has been a short post, but it seems futile to actually start going into the technical details after a post intended to be an overview. Next time, I shall begin discussing the details, by discussing the process of lifting curves to characteristic zero.