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.