Lemma 67.51.1 (0418)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.51: Applications
Lemma 67.51.1 (cite)

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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.

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. $\square$

Comments (2)

Comment #790 by Kestutis Cesnavicius on July 07, 2014 at 14:52

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 Johan on July 09, 2014 at 13:41

Yeah, this is something that is bothering me too. Maybe eventually we should just replace the statement in Proposition 67.50.2 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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