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.
We check this on stalks. Suppose
is a Noetherian local ring and
is a flat
-module. We want to show
is free. By Nakayama, there is a surjection
whose quotient by
is an isomorphism. Then by the Tor long exact sequence, the quotient of the kernel by
is also trivial. By Nakayama, this implies
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 .
is an étale morphism over a separably closed field, hence a discrete number of copies of
. This number is the rank of
-module, and is locally constant, hence constant.
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 pull back
along itself, then we can split off the diagonal as a trivial component of
. So we are done by induction on the degree.
If we have a composition
where and are étale covers, then so is .
By fpqc descent, we may assume that
are both trivial covers, in which case the proposition is clear.