The Stacks project

Lemma 99.4.8. Let $\mathcal{T}$ be an algebraic stack. Let $g : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over $\mathcal{T}$. Consider the graph $i : \mathcal{X} \to \mathcal{X} \times _\mathcal {T} \mathcal{Y}$ of $g$. Then

  1. $i$ is representable by algebraic spaces and locally of finite type,

  2. if $\mathcal{Y} \to \mathcal{T}$ is DM, then $i$ is unramified,

  3. if $\mathcal{Y} \to \mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite,

  4. if $\mathcal{Y} \to \mathcal{T}$ is separated, then $i$ is proper, and

  5. if $\mathcal{Y} \to \mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated.

Proof. This is a special case of Lemma 99.4.7 applied to the morphism $\mathcal{X} = \mathcal{X} \times _\mathcal {Y} \mathcal{Y} \to \mathcal{X} \times _\mathcal {T} \mathcal{Y}$. $\square$


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