Lemma 100.9.4 (0503)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 100: Properties of Algebraic Stacks
Section 100.9: Immersions of algebraic stacks
Lemma 100.9.4 (cite)

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Lemma 100.9.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W$ be an algebraic space and let $W \to \mathcal{Y}$ be a surjective, flat morphism which is locally of finite presentation. The following are equivalent:

$f$ is an (open, resp. closed) immersion, and
$V = W \times _\mathcal {Y} \mathcal{X}$ is an algebraic space, and $V \to W$ is an (open, resp. closed) immersion.

Proof. This follows from the general discussion in Section 100.3 and in particular Lemmas 100.3.1 and 100.3.3. $\square$

Comments (2)

Comment #9036 by Pat on May 02, 2024 at 10:52

There seems to be a grammar typo at the start of the second sentence.

Comment #9183 by Stacks project on May 14, 2024 at 15:22

Fixed here.

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