The Étale Fundamental Group — Étale Covers

3 Étale Covers

Definition (Étale cover)

An (finite) étale cover is a morphism that is finite and étale. We write FEtX\mathscr {FE}t_{X} for the category of étale covers of XX.

A trivial cover of XX is one that is a finite disjoint union of copies of XX.

Lemma

Étale covers are closed under pullback and composition, and satisfy fpqc descent.

Lemma

Affine, and in particular étale morphisms are separated.

It turns out being an étale cover imposes strong conditions on the map.

Lemma

A finite and flat morphism f:YXf: Y \to X is locally free, i.e. fOYf_* \mathcal{O}_ Y is a locally free OX\mathcal{O}_ X-module.

Proof
We check this on stalks. Suppose AA is a Noetherian local ring and MM is a flat AA-module. We want to show AA is free. By Nakayama, there is a surjection AkMA^ k \to M whose quotient by m\mathfrak {m} is an isomorphism. Then by the Tor long exact sequence, the quotient of the kernel by m\mathfrak {m} is also trivial. By Nakayama, this implies AkMA^ k \to M is injective.

Corollary

For any YFEtXY \in \mathscr {FE}t_{X} and geometric point x:SpeckˉXx: \operatorname{Spec}\bar{k} \to X, the pullback xYx^*Y is finite and the cardinality does not depend on xx. In particular, YXY \to X is surjective, hence faithfully flat. We call this cardinality the degree of xx.

Proof
xYSpeckˉx^*Y \to \operatorname{Spec}\bar{k} is an étale morphism over a separably closed field, hence a discrete number of copies of Speckˉ\operatorname{Spec}\bar{k}. This number is the rank of fOYf_* \mathcal{O}_ Y as an OX\mathcal{O}_ X-module, and is locally constant, hence constant.

Proposition

The degree is invariant under pullback, and non-empty covers have non-empty degree.

In algebraic topology, a map is a covering space if it is locally trivial. The same is true for étale covers, if we allow ourselves to view any étale morphism to XX as an “open set” of XX.

Lemma

If p:YXp: Y \to X is an étale cover, then there is an étale cover f:XXf: X' \to X such that fpf^* p is trivial.

Proof
If we pull back pp along itself, then we can split off the diagonal as a trivial component of Y×XYY \times _ X Y. So we are done by induction on the degree.

Corollary

If we have a composition

\begin{useimager} 
    \[
      \begin{tikzcd}
        Z \ar[r, "q"] & Y \ar[r, "p"] & X
      \end{tikzcd}
    \]
  \end{useimager}

where pp and pqp \circ q are étale covers, then so is qq.

Proof
By fpqc descent, we may assume that ZZ and YY are both trivial covers, in which case the proposition is clear.