The Étale Fundamental Group — Galois Theory

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.

Definition (Galois cover)

A Galois cover of XX is an element YFEtXY \in \mathscr {FE}t_{X} such that Aut(Y)\operatorname{Aut}(Y) acts transitively on F(Y)F(Y).

The following proposition should be reassuring, but will not be used.
Proposition

Let YFEtXY \in \mathscr {FE}t_{X} be Galois. Then Y×XYY \times _ X Y splits as finitely many disjoint copies of YY, and Aut(Y)\operatorname{Aut}(Y) acts freely and transitively over the copies, where Aut(Y)\operatorname{Aut}(Y) acts as an automorphism of π1:Y×XYY\pi _1: Y \times _ X Y \to Y by pullback.

Proof
The diagonal map YY×XYY \to Y \times _ X Y gives an isomorphism between YY and a component of Y×XYY \times _ X Y. We fix a geometric point zF(Y)z \in F(Y), and then we have a diagram
\begin{useimager} 
    \[
      \begin{tikzcd}
        (\Spec k)^n \ar[d] \ar[r] & Y \times_X Y \ar[d, "\pi_1"] \ar[r] \ar[d] & Y \ar[d]\\
        \Spec k \ar[r, "z"] & Y \ar[r] & X
      \end{tikzcd}
    \]
  \end{useimager}
By definition of the pullback, the lifts of zz to Y×XYY \times _ X Y bijects with the lifts of xx to YY, and this preserves the action of Aut(Y)\operatorname{Aut}(Y). Since Aut(Y)\operatorname{Aut}(Y) acts freely on the lifts of zz, it must act freely on YY×XYY \subseteq Y \times _ X Y. So Y×XYY \times _ X Y splits as nn copies of YY, and there is nothing left since Y×XYYY \times _ X Y \to Y has degree nn.

In Galois theory, we can construct Galois closures. We can do the same here.

Lemma

Let YFEtXY \in \mathscr {FE}t_{X} be connected. Then there is a Galois cover YY' with a map YYY' \to Y such that if ZFEtXZ \in \mathscr {FE}t_{X} is any Galois cover, then any map ZYZ \to Y factors through YY'.

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.

Proof

Let y1,,ynF(Y)y_1, \ldots , y_ n \in F(Y), which gives rise to a geometric point y=(y1,,yn)Yny = (y_1, \ldots , y_ n) \in Y^ n. Let YY' be the component of yYny \in Y^ n, mapping to YY by first projection. This is an étale cover since it is a connected component of one. We claim that this works.

Claim

If (x1,,xn)Y(x_1, \ldots , x_ n) \in Y', then xi̸=xjx_ i \not= x_ j for all ii and jj.

If πi,j:YnY2\pi _{i, j}: Y^ n \to Y^2 is the projection onto the i,ji, j coordinates, then this is the same as saying πij1(Δ(Y))Y=\pi _{ij}^{-1}(\Delta (Y)) \cap Y' = \emptyset . But since Δ(Y)\Delta (Y) is open and closed and πij1\pi _{ij}^{-1} is continuous, we know πij1(Δ(Y))Y\pi _{ij}^{-1}(\Delta (Y)) \cap Y' is either YY' or empty. But it does not contain (y1,,yn)Y(y_1, \ldots , y_ n) \in Y'. So it is empty.

Claim

YXY' \to X is Galois.

Any element in F(Yn)F(Y^ n) is of the form (xi1,,xin)(x_{i_1}, \ldots , x_{i_ n}), and by the above, if it is in YY', then it is of the form (xσ(1),,xσ(n))(x_{\sigma (1)}, \ldots , x_{\sigma (n)}) for some permutation σSn\sigma \in S_ n. This σ\sigma also acts on XnX^ n by permuting the coordinates, and σ(Y)Y\sigma (Y') \cap Y' is non-empty since it contains (xσ(1),,xσ(n))(x_{\sigma (1)}, \ldots , x_{\sigma (n)}). So σ(Y)=Y\sigma (Y') = Y', and hence σAut(Y)\sigma \in \operatorname{Aut}(Y'). So Aut(Y)\operatorname{Aut}(Y') acts transitively on F(Y)F(Y').

Claim

If ZZ is Galois and q:ZYq: Z \to Y, then it factors through YY'.

Fix a lift zF(Z)z \in F(Z). Then by composing with automorphisms of Aut(Z)\operatorname{Aut}(Z), we can pick maps q1,,qn:ZYq_1, \ldots , q_ n: Z \to Y such that qi(z)=yiq_ i(z) = y_ i. Then the image of (q1,,qn):ZYn(q_1, \ldots , q_ n): Z \to Y^ n includes a point in YY', namely yy, and hence maps into YY' by connectedness of ZZ.

Theorem

Let (Xα)αFEtX(X_\alpha )_\alpha \subseteq \mathscr {FE}t_{X} be the poset of Galois covers of XX, ordered under XαXβX_\alpha \leq X_\beta if there is a morphism XβXαX_\beta \to X_\alpha . For each XαX_\alpha , pick a point xαF(Xα)x_\alpha \in F(X_\alpha ). Then if XαXβX_\alpha \leq X_\beta , we pick the morphism that sends xαx_\alpha to xβx_\beta for use in the pro-system. Then this pro-system pro-represents FF.

Proof
This is indeed a pro-system since the pullback of two Galois covers is yet another Galois cover of XX. There is a natural transformation colimHom(Xα,Y)F(Y) \operatorname{colim}\operatorname{Hom}(X_\alpha , Y) \to F(Y) that sends a map q:XαYq: X_\alpha \to Y to q(xα)q(x_\alpha ). Conversely, given any yF(Y)y \in F(Y), we may restrict to the component of yy and assume YY is connected. Then since XαYX_\alpha \to Y is surjective and Aut(Xα)\operatorname{Aut}(X_\alpha ) acts transitively on F(Xα)F(X_\alpha ), precomposing with an automorphism gives a morphism XαYX_\alpha \to Y that sends xαx_\alpha to yy, and such a map is unique since Aut(Xα)\operatorname{Aut}(X_\alpha ) acts freely. This gives the desired inverse map.

Theorem

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

Proof
The functor is given by FF. Conversely, given a continuous finite π1(X,x)\pi _1(X, x) set SS, pick XαX_\alpha such that Aut(Xα)\operatorname{Aut}(X_\alpha ) acts transitively with stabilizer HH. Then XF(Xα)/HF(Xα/H)X \cong F(X_\alpha )/H \cong F(X_\alpha /H) 1 . Functions can be constructed similarly.