The Stacks project

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:

  1. $f$ is an (open, resp. closed) immersion, and

  2. $V = W \times _\mathcal {Y} \mathcal{X}$ 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. $\square$

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