# 3 Étale Covers

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 $X$.

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

Affine, and in particular étale morphisms are separated.

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

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.

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 $x$.

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 $X$.

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.

If we have a composition

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