5 Galois Theory
The usual proof of the usual Fundamental Theorem of Galois Theory can pretty much be carried over to the general case if we can say the word “Galois”. Fortunately, the word is not too difficult to utter.
A Galois cover of is an element such that acts transitively on .
The following proposition should be reassuring, but will not be used.
Let be Galois. Then splits as finitely many disjoint copies of , and acts freely and transitively over the copies, where acts as an automorphism of by pullback.
The diagonal map
gives an isomorphism between
and a component of
. We fix a geometric point
, and then we have a diagram
By definition of the pullback, the lifts of to bijects with the lifts of to , and this preserves the action of . Since acts freely on the lifts of , it must act freely on . So splits as copies of , and there is nothing left since has degree .
In Galois theory, we can construct Galois closures. We can do the same here.
Let be connected. Then there is a Galois cover with a map such that if is any Galois cover, then any map factors through .
In Galois theory, the Galois closure is constructed by taking the field generated by all field embeddings into the algebraic closure. The construction here is similar.
, which gives rise to a geometric point
be the component of
, mapping to
by first projection. This is an étale cover since it is a connected component of one. We claim that this works.
If , then for all and .
is the projection onto the
coordinates, then this is the same as saying
. But since
is open and closed and
is continuous, we know
or empty. But it does not contain
. So it is empty.
Any element in
is of the form
, and by the above, if it is in
, then it is of the form
for some permutation
also acts on
by permuting the coordinates, and
is non-empty since it contains
, and hence
acts transitively on
If is Galois and , then it factors through .
Fix a lift . Then by composing with automorphisms of , we can pick maps such that . Then the image of includes a point in , namely , and hence maps into by connectedness of .
Let be the poset of Galois covers of , ordered under if there is a morphism . For each , pick a point . Then if , we pick the morphism that sends to for use in the pro-system. Then this pro-system pro-represents .
This is indeed a pro-system since the pullback of two Galois covers is yet another Galois cover of
. There is a natural transformation
that sends a map to . Conversely, given any , we may restrict to the component of and assume is connected. Then since is surjective and acts transitively on , precomposing with an automorphism gives a morphism that sends to , and such a map is unique since acts freely. This gives the desired inverse map.
is naturally equivalent to the category of continuous finite -sets.
The functor is given by
. Conversely, given a continuous finite
acts transitively with stabilizer
. Functions can be constructed similarly.