Lemma 70.14.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If there exists an $f$-ample invertible sheaf, then $f$ is representable, quasi-compact, and separated.

Proof. This is clear from the definitions and Morphisms, Lemma 29.37.3. (If in doubt, take a look at the principle of Algebraic Spaces, Lemma 64.5.8.) $\square$

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