The Stacks project

Lemma 101.6.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then

  1. $f$ is separated if and only if $\Delta _{f, 1}$ and $\Delta _{f, 2}$ are universally closed, and

  2. $f$ is quasi-separated if and only if $\Delta _{f, 1}$ and $\Delta _{f, 2}$ are quasi-compact.

  3. $f$ is quasi-DM if and only if $\Delta _{f, 1}$ and $\Delta _{f, 2}$ are locally quasi-finite.

  4. $f$ is DM if and only if $\Delta _{f, 1}$ and $\Delta _{f, 2}$ are unramified.

Proof. Proof of (1). Assume that $\Delta _{f, 2}$ and $\Delta _{f, 1}$ are universally closed. Then $\Delta _{f, 1}$ is separated and universally closed by Lemma 101.6.4. By Morphisms of Spaces, Lemma 67.9.7 and Algebraic Stacks, Lemma 94.10.9 we see that $\Delta _{f, 1}$ is quasi-compact. Hence it is quasi-compact, separated, universally closed and locally of finite type (by Lemma 101.3.3) so proper. This proves “$\Leftarrow $” of (1). The proof of the implication in the other direction is omitted.

Proof of (2). This follows immediately from Lemma 101.6.4.

Proof of (3). This follows from the fact that $\Delta _{f, 2}$ is always locally quasi-finite by Lemma 101.3.4 applied to $\Delta _ f = \Delta _{f, 1}$.

Proof of (4). This follows from the fact that $\Delta _{f, 2}$ is always unramified as Lemma 101.3.4 applied to $\Delta _ f = \Delta _{f, 1}$ shows that $\Delta _{f, 2}$ is locally of finite type and a monomorphism. See More on Morphisms of Spaces, Lemma 76.14.8. $\square$

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