Lemma 99.9.16. Let $\mathcal X$ be an algebraic stack. Let $\mathcal{X}_ i$, $i \in I$ be a set of open substacks of $\mathcal{X}$. Assume

1. $\mathcal{X} = \bigcup _{i \in I} \mathcal{X}_ i$, and

2. each $\mathcal{X}_ i$ is an algebraic space.

Then $\mathcal{X}$ is an algebraic space.

Proof. Apply Stacks, Lemma 8.6.10 to the morphism $\coprod _{i \in I} \mathcal{X}_ i \to \mathcal{X}$ and the morphism $\text{id} : \mathcal{X} \to \mathcal{X}$ to see that $\mathcal{X}$ is a stack in setoids. Hence $\mathcal{X}$ is an algebraic space, see Algebraic Stacks, Proposition 93.13.3. $\square$

There are also:

• 2 comment(s) on Section 99.9: Immersions of algebraic stacks

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).