The Étale Fundamental Group — The Étale Fundamental Group

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 xXx \in X.

Definition

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

By definition, F(Y)=degY|F(Y)| = \deg Y.

Definition (Étale fundamental group)

Let XX be a scheme and xx a base point. The étale fundamental group π1(X,x)\pi _1(X, x) is defined to be AutF\operatorname{Aut}F, the group of all automorphisms of the functor F:FEtXSetsF: \mathscr {FE}t_{X} \to \mathrm{Sets}.

In the case of covering theory or Galois theory, this functor FF is “representable”. For Galois groups, this is represented by the separable closure kˉ\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, SpeckˉSpeck\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:CSetsF: \mathcal{C} \to \mathrm{Sets} is pro-representable if there is a directed/pro-system (Xα)α(X_\alpha )_\alpha of objects in C\mathcal{C} such that

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

for all YCY \in \mathcal{C}.

In this case, we have

Aut(F)=limαAut(Xα)op, \operatorname{Aut}(F) = \lim _\alpha \operatorname{Aut}(X_\alpha )^\mathrm{op},

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

Theorem (Fundamental Theorem of Galois Theory)
  1. FF is always pro-representable. Hence, π1(X,x)\pi _1(X, x) is profinite, acting continuously on FF.

  2. FEtX\mathscr {FE}t_{X} is equivalent to the category of finite continuous π1(X,x)\pi _1(X, x)-sets.

  3. In particular, the isomorphism class of π1(X,x)\pi _1(X, x) does not depend on the choice of basepoint. Thus, we may write π1(X)\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 kk be a field. Then π1(Speck)=Gal(kˉ/k)\pi _1(\operatorname{Spec}k) = \operatorname{Gal}(\bar{k}/k).

Example

Let XX be a projective variety over C\mathbb {C}. Then π1(X)\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 P1\mathbb {P}^1 is trivial. So π1(P1)=0\pi _1(\mathbb {P}^1) = 0. This agrees with what topology tells us if we work over C\mathbb {C}.