3 Étale Covers
An (finite) étale cover is a morphism that is finite and étale. We write for the category of étale covers of .
A trivial cover of is one that is a finite disjoint union of copies of .
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 is locally free, i.e. is a locally free -module.
For any and geometric point , the pullback is finite and the cardinality does not depend on . In particular, is surjective, hence faithfully flat. We call this cardinality the degree of .
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 as an “open set” of .
If is an étale cover, then there is an étale cover such that is trivial.
If we have a composition
where and are étale covers, then so is .