# 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 $q$.