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The Stacks project

Lemma 101.4.13. Let \mathcal{X} be an algebraic stack over the base scheme S.

  1. \mathcal{X} is DM \Leftrightarrow \mathcal{X} is DM over S.

  2. \mathcal{X} is quasi-DM \Leftrightarrow \mathcal{X} is quasi-DM over S.

  3. If \mathcal{X} is separated, then \mathcal{X} is separated over S.

  4. If \mathcal{X} is quasi-separated, then \mathcal{X} is quasi-separated over S.

Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks over the base scheme S.

  1. If \mathcal{X} is DM over S, then f is DM.

  2. If \mathcal{X} is quasi-DM over S, then f is quasi-DM.

  3. If \mathcal{X} is separated over S and \Delta _{\mathcal{Y}/S} is separated, then f is separated.

  4. If \mathcal{X} is quasi-separated over S and \Delta _{\mathcal{Y}/S} is quasi-separated, then f is quasi-separated.

Proof. Parts (5), (6), (7), and (8) follow immediately from Lemma 101.4.12 and Spaces, Definition 65.13.2. To prove (3) and (4) think of X and Y as algebraic stacks over \mathop{\mathrm{Spec}}(\mathbf{Z}) and apply Lemma 101.4.12. Similarly, to prove (1) and (2), think of \mathcal{X} as an algebraic stack over \mathop{\mathrm{Spec}}(\mathbf{Z}) consider the morphisms

\mathcal{X} \longrightarrow \mathcal{X} \times _ S \mathcal{X} \longrightarrow \mathcal{X} \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} \mathcal{X}

Both arrows are representable by algebraic spaces. The second arrow is unramified and locally quasi-finite as the base change of the immersion \Delta _{S/\mathbf{Z}}. Hence the composition is unramified (resp. locally quasi-finite) if and only if the first arrow is unramified (resp. locally quasi-finite), see Morphisms of Spaces, Lemmas 67.38.3 and 67.38.11 (resp. Morphisms of Spaces, Lemmas 67.27.3 and 67.27.8). \square


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