Lemma 76.14.9 (05W8)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.14: Formally unramified morphisms
Lemma 76.14.9 (cite)

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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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