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. $\square$

Comment #790 by Kestutis Cesnavicius on

Wouldn't it be better to move this lemma to the preceding section? Once this is done, one could change the section title "Applications" to something more descriptive, e.g., to "Etale universally injective morphisms".

Comment #801 by on

Yeah, this is something that is bothering me too. Maybe eventually we should just replace the statement in Proposition 66.50.2 to the statement of this lemma (because we are always allowed to strengthen statements of lemmas, propositions, and theorems) and then move this lemma into the obsolete chapter. Leaving this for now.

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