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 .
For any , we define to be the set of all lifts of to .
(Étale fundamental group)
Let be a scheme and a base point. The étale fundamental group is defined to be , the group of all automorphisms of the functor .
In the case of covering theory or Galois theory, this functor
is “representable”. For Galois groups, this is represented by the separable closure
, 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,
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.
A functor is pro-representable if there is a directed/pro-system of objects in such that
for all .
In this case, we have
and in particular, this is a profinite group (since each acts freely on the finite set ). In the case of Galois theory, the functor is pro-represented by the pro-system of all Galois extensions. In covering space theory, is pro-represented by the pro-system of all normal covers. In fact,
(Fundamental Theorem of Galois Theory)
is always pro-representable. Hence, is profinite, acting continuously on .
is equivalent to the category of finite continuous -sets.
In particular, the isomorphism class of does not depend on the choice of basepoint. Thus, we may write instead.
The proof of the theorem is the content of the next section. For now, we shall amuse ourselves by computing some fundamental groups.
Let be a field. Then .
Let be a projective variety over . Then is the pro-completion of the topological fundamental group, by the characterization of étale maps and the Riemann existence theorem.
By Riemann–Hurwitz, over any algebraically closed field of characteristic zero, any covering of is trivial. So . This agrees with what topology tells us if we work over .