This is the second in a three-part post about a proof that I contributed to the Stacks project. The result was already there, but I found a slightly easier proof. This post contains the actual result, and the next post will contain a fun application.
Remark. Recall that a morphism of schemes is étale if
is flat and locally of finite presentation, and
.
Lemma. Suppose is étale and universally injective. Then
is an open immersion.
Proof. Since étale morphisms are open and surjectivity is stable under base change, we may assume that is universally bijective; then we have to prove that
is an isomorphism. Since étale morphisms are open,
is in fact a universal homeomorphism. By Tag 04DE,
is affine.
The question is local on , so we may assume
is affine, and hence so is
. Say
is induced by
. Now
is proper and affine, hence finite. Moreover, since
is finitely presented and finite as
-algebra, and
is a finitely presented
-module, it is also a finitely presented
-module (Tag 0564).
Now is flat of finite presentation over
, hence locally free (actually, we need the slightly stronger result that I mention in the first remark; see Tag 00NX for statement and proof). Since the question is local, we may assume
is free of rank
.
Now let be a geometric point; that is, let
be a map to an algebraically closed field. Then the tensor product
is étale of dimension
over
. Hence,
is a union of
points. Since
is universally bijective, we have
. Then the result follows from my previous post.