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 $\mathscr {FE}t_{X}$ for the category of étale covers of $X$.

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

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 $f: Y \to X$ is locally free, i.e. $f_* \mathcal{O}_ Y$ is a locally free $\mathcal{O}_ X$-module.

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

Corollary

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

Proof
$x^*Y \to \operatorname{Spec}\bar{k}$ is an étale morphism over a separably closed field, hence a discrete number of copies of $\operatorname{Spec}\bar{k}$. This number is the rank of $f_* \mathcal{O}_ Y$ as an $\mathcal{O}_ X$-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 $X$ as an “open set” of $X$.

Lemma

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

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

Corollary

If we have a composition

where $p$ and $p \circ q$ are étale covers, then so is $q$.

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