Lemma 76.14.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:
$f$ is a closed immersion,
$f$ is universally closed, unramified, and a monomorphism,
$f$ is universally closed, unramified, and universally injective,
$f$ is universally closed, locally of finite type, and a monomorphism,
$f$ is universally closed, universally injective, locally of finite type, and formally unramified.
Proof. The equivalence of (2) – (5) follows immediately from Lemma 76.14.8. Moreover, if (2) – (5) are satisfied then $f$ is representable. Similarly, if (1) is satisfied then $f$ is representable. Hence the result follows from the case of schemes, see Étale Morphisms, Lemma 41.7.2. $\square$
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