The Stacks project

[Theorem 2.2.5, Conrad-moduli]

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

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

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

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

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

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

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

  7. add more here.

Proof. In each case we reduce to a question about the diagonal and then we use Lemma 105.3.8 applied to the morphism of thickenings

\[ (\mathcal{X} \subset \mathcal{X}') \to \left(\mathop{\mathrm{Spec}}(\mathbf{Z}) \subset \mathop{\mathrm{Spec}}(\mathbf{Z})\right) \]

We do this after viewing $\mathcal{X} \subset \mathcal{X}'$ as a thickening of algebraic stacks over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ via Algebraic Stacks, Definition 93.19.2.

Case (1). An algebraic stack is an algebraic space if and only if its diagonal is a monomorphism, see Morphisms of Stacks, Lemma 100.6.3 (this also follows immediately from Algebraic Stacks, Proposition 93.13.3).

Case (2). By (1) we may assume that $\mathcal{X}$ and $\mathcal{X}'$ are algebraic spaces and then we can use More on Morphisms of Spaces, Lemma 75.9.5.

Case (3) – (6). Each of these cases corresponds to a condition on the diagonal, see Morphisms of Stacks, Definitions 100.4.1 and 100.4.2. $\square$

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