The Stacks project

66.51 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 75.34. Namely, More on Morphisms of Spaces, Lemma 75.34.2 implies that a quasi-finite and separated morphism of algebraic spaces is quasi-affine and therefore representable.

Lemma 66.51.1. 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.

Proof. This is immediate from Proposition 66.50.2 and the fact that being locally quasi-finite and separated is preserved under any base change, see Lemmas 66.27.4 and 66.4.4. $\square$


Lemma 66.51.2. 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 66.39.4 and 66.19.5) 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 66.19.2. Hence we see that $\Delta _{X/Y}$ is an isomorphism (see Spaces, Lemma 64.5.9), in particular we see that $X$ is separated over $Y$. It follows that $X$ is a scheme too, by Proposition 66.50.2. 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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