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∈X.
Definition
For any Y∈FEtX, we define F(Y) to be the set of all lifts of x to Y.
By definition,
∣F(Y)∣=degY.
Definition
(Étale fundamental group)
Let X be a scheme and x a base point. The étale fundamental group π1(X,x) is defined to be AutF, the group of all automorphisms of the functor F:FEtX→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
kˉ, 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,
Speckˉ→Speck 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:C→Sets is pro-representable if there is a directed/pro-system (Xα)α of objects in C such that
F(Y)=αcolimHomC(Xα,Y)
for all Y∈C.
In this case, we have
Aut(F)=αlimAut(Xα)op,
and in particular, this is a profinite group (since each Aut(Xα) acts freely on the finite set F(Xα)). 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, π1(X,x) is profinite, acting continuously on F.
FEtX is equivalent to the category of finite continuous π1(X,x)-sets.
In particular, the isomorphism class of π1(X,x) does not depend on the choice of basepoint. Thus, we may write π1(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 π1(Speck)=Gal(kˉ/k).
Example
Let X be a projective variety over C. Then π1(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 is trivial. So π1(P1)=0. This agrees with what topology tells us if we work over C.