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Étale morphisms and the fundamental group

In this post I’m going to give the definition and basic properties of the étale fundamental group, whose abelianization appears on one side of class field theory. This is the object which is related a priori to our goal of understanding (abelian) covers of a curve: it precisely encodes the symmetries of all such covers. My initial goal of proving unramified global class field theory in the next post was too ambitious: I’ll slow down the pace a bit and spend the next four or five posts covering foundational material, then move on to the main theorems. This post is quite long, and to prevent it from being even longer I have not attempted to give any proofs or even the most general statements. Instead I refer the reader to SGA1, which is the standard reference, or chapters 4 and 5 of Szamuely’s excellent book Galois Groups and Fundamental Groups.

The idea is to introduce an algebro-geometric analogue of the topological fundamental group, but the conventional approach to the latter using loops is inappropriate in this algebraic setting. Instead, we find our inspiration in the following observation: if $X$ is a connected topological space and $x \in X$, then for any cover $f : Y \to X$ the topological fundamental group $\pi_1(X,x)$ acts on the fiber $f^{-1}(x)$. This construction extends to a functor from the category of covers of $X$ to $\pi_1(X,x)$-sets, which is an equivalence provided that $X$ admits a universal cover, e.g. if $X$ is a manifold.

Let’s try to imitate this idea in the world of algebraic geometry. For most of today we can work with any connected smooth variety $X$ over a field $k$ (all morphisms, products, etc. will be over $k$). The analogue of a topological finite cover is a finite étale morphism $f : Y \to X$, where finite means $f_*\mathcal{O}_Y$ is a finitely generated $\mathcal{O}_X$-module, and étale means the following. If $k$ is algebraically closed, we call $f$ étale provided that the differential $df_y : T_yY \to T_{f(y)}X$ is an isomorphism for every $y \in Y$. For arbitrary $k$, we say $f$ is étale provided that the map $\overline{f} : \overline{Y} \to \overline{X}$ obtained by extending scalars is étale (here $\overline{X} = X \times_k \overline{k}$, where $\overline{k}$ is an algebraic closure of $k$ and we are systematically confusing fields with their spectra). We should warn the reader here that this definition of étale is only appropriate when $X$ is smooth, but since this is the only case which concerns us we will not attempt to give more general definitions.

Why étale morphisms? We want a notion which behaves like a local diffeomorphism of smooth manifolds, and that condition is equivalent to inducing an isomorphism on tangent spaces at each point by the inverse function theorem. In algebraic geometry the Zariski topology is too coarse for actual local isomorphisms to be of any use, but étale morphisms are the appropriate replacement. To see why, let $k = \mathbb{C}$ and consider the map $f : \mathbb{C} \setminus \{ 0 \} \to \mathbb{C} \setminus \{ 0 \}$ which sends $z \mapsto z^n$. This map is a cover in the topological sense with respect to the analytic (i.e. metric) topology, but is not even locally injective in the Zariski topology, since Zariski open sets are cofinite and $f$ is generically $n$-to-$1$ on any such set. However, since the derivative of $f$ is everywhere nonzero, this map is étale.

The finiteness hypothesis is easier to explain: an étale morphism automatically has finite fibers, so for example, there is no algebraic version of $\text{exp} : \mathbb{C} \to \mathbb{C} \setminus \{ 0 \}$. But (Zariski) open embeddings are étale, and we do not want to allow any of these other than isomorphisms. Note that since finite morphisms are closed and étale morphisms are open, any finite étale morphism into $X$ is surjective (recall that we are assuming that $X$ is connected).

So far our examples have been over $\mathbb{C}$, but there is another crucial feature which one does not notice when $k$ is algebraically closed: if $\ell / k$ is a finite separable extension of the ground field, then $\text{Spec } \ell \to \text{Spec } k$ is a finite étale morphism, and conversely any connected finite étale cover of $\text{Spec } k$ has this form. Since finite étale morphisms are stable under base change, any map of the form $X \times_k \ell \to X$ is finite étale.

Denote the category of all finite étale morphisms $Y \to X$ by $\text{fEt}_X$, with arrows in this category being given by commutative triangles in the usual way. Let $\Omega$ be a separably closed field containing $k$ and pick a morphism $\overline{x} : \text{Spec }\Omega \to X$, which we call a geometric point of $X$. Now if $Y \to X$ is a finite étale morphism, the fiber $Y_{\overline{x}} = (Y \times_X \Omega)(\Omega)$ is a finite set, and $Y \to Y_{\overline{x}}$ evidently extends to a functor $F_{\overline{x}} : \text{fEt}_X \to \text{fSet}$.

Definition The étale fundamental group of $X$ based at $\overline{x}$, denoted by $\pi_1(X,\overline{x})$, is the automorphism group of $F_{\overline{x}}$.

The group $\pi_1(X,\overline{x})$ has a natural profinite topology. Namely, a basis of open subgroups at the identity is given by the stabilizers of elements of $F_{\overline{x}}(Y) = Y_{\overline{x}}$ as $Y \to X$ varies through all of $\text{fEt}_X$, so that the action of $\pi_1(X,\overline{x})$ on each $F_{\overline{x}}(Y)$ is continuous. To summarize, $F_{\overline{x}}$ factors through the forgetful functor $\pi_1(X,\overline{x})\text{-fSet} \to \text{fSet}$, where $\pi_1(X,\overline{x})\text{-fSet}$ is the category of finite sets equipped with a continuous $\pi_1(X,\overline{x})$-action. Moreover, we have the following theorem, which confirms that we have found the correct analogue of the topological fundamental group.

Theorem The functor $F_{\overline{x}} : \text{fEt}_X \to \pi_1(X,\overline{x})\text{-fSet}$ is an equivalence.

From this one can deduce that if $k = \mathbb{C}$, the étale fundamental group is the profinite completion of the topological fundamental group based at the same point.

It is not hard to see that the étale fundamental group is functorial in the sense that if $Y \to X$ is a morphism and $\overline{y} \in Y_{\overline{x}}$, there is an induced continuous homomorphism $\pi_1(Y,\overline{y}) \to \pi_1(X,\overline{x})$. Also, if $\overline{x}' \in X(\Omega)$ is another geometric point, the groups $\pi_1(X,\overline{x})$ and $\pi_1(X,\overline{x}')$ are isomorphic, but noncanonically: the isomorphism is only determined up to conjugation. In particular, there is a canonical isomorphism $\pi_1(X,\overline{x})^{\text{ab}} \cong \pi_1(X,\overline{x}')^{\text{ab}}$, so after abelianization we may omit the basepoint from our notation and simply write $\pi_1(X)^{\text{ab}}$.

Finally, we explain the relationship between the étale fundamental group and Galois groups in the cases which interest us. This will establish the connection to the arithmetic language in which class field theory was originally developed. First, consider the case $X = \text{Spec } k$. Once we have chosen a separable closure $k^{\text{sep}}$ of $k$ there is a unique geometric point $\text{Spec } k^{\text{sep}} \to \text{Spec } k$, and we will simply write $\pi_1(k)$ for the fundamental group of $\text{Spec } k$ based there. It follows from the definitions and basic Galois theory that there is a canonical isomorphism $\pi_1(k) \cong \text{Gal}(k^{\text{sep}}/k)$.

Let’s return to the situation of the last post: let $X$ be a smooth, projective, and geometrically connected curve over $k$. Denote by $K$ its field of rational functions, $K^{\text{sep}}$ a fixed separable closure, and $K^{\text{nr}}$ the maximal unramified extension contained in $K^{\text{sep}}$. There is a natural choice of basepoint for $X$, namely the geometric generic point $\overline{\eta} : \text{Spec } K^{\text{sep}} \to X$.

Proposition There is a canonical isomorphism $\pi_1(X,\overline{\eta}) \cong \text{Gal}(K^{\text{nr}}/K)$.

Next time, we’ll discuss the Picard scheme and how it relates to the adèles in number theory.