Lemma 67.51.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally quasi-finite and separated, then $f$ is representable.
67.51 Applications
An alternative proof of the following lemma is to see it as a consequence of Zariski's main theorem for (nonrepresentable) morphisms of algebraic spaces as discussed in More on Morphisms of Spaces, Section 76.34. Namely, More on Morphisms of Spaces, Lemma 76.34.2 implies that a quasi-finite and separated morphism of algebraic spaces is quasi-affine and therefore representable.
Proof. This is immediate from Proposition 67.50.2 and the fact that being locally quasi-finite and separated is preserved under any base change, see Lemmas 67.27.4 and 67.4.4. $\square$
Lemma 67.51.2. Let $S$ be a scheme. Let $f : X \to Y$ be an étale and universally injective morphism of algebraic spaces over $S$. Then $f$ is an open immersion.
Proof. Let $T \to Y$ be a morphism from a scheme into $Y$. If we can show that $X \times _ Y T \to T$ is an open immersion, then we are done. Since being étale and being universally injective are properties of morphisms stable under base change (see Lemmas 67.39.4 and 67.19.5) we may assume that $Y$ is a scheme. Note that the diagonal $\Delta _{X/Y} : X \to X \times _ Y X$ is étale, a monomorphism, and surjective by Lemma 67.19.2. Hence we see that $\Delta _{X/Y}$ is an isomorphism (see Spaces, Lemma 65.5.9), in particular we see that $X$ is separated over $Y$. It follows that $X$ is a scheme too, by Proposition 67.50.2. Finally, $X \to Y$ is an open immersion by the fundamental theorem for étale morphisms of schemes, see Étale Morphisms, Theorem 41.14.1. $\square$
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