Lemma 105.3.9. Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks. Then

$\mathcal{X}$ is an algebraic space if and only if $\mathcal{X}'$ is an algebraic space,

$\mathcal{X}$ is a scheme if and only if $\mathcal{X}'$ is a scheme,

$\mathcal{X}$ is DM if and only if $\mathcal{X}'$ is DM,

$\mathcal{X}$ is quasi-DM if and only if $\mathcal{X}'$ is quasi-DM,

$\mathcal{X}$ is separated if and only if $\mathcal{X}'$ is separated,

$\mathcal{X}$ is quasi-separated if and only if $\mathcal{X}'$ is quasi-separated, and

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