Lemma 100.27.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks with diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. If $f$ is locally of finite type then $\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta$ is of finite presentation.

Proof. Note that $\Delta$ is a morphism over $\mathcal{X}$ (via the second projection). If $f$ is locally of finite type, then $\mathcal{X}$ is of finite presentation over $\mathcal{X}$ and $\text{pr}_2 : \mathcal{X} \times _\mathcal {Y} \mathcal{X} \to \mathcal{X}$ is locally of finite type by Lemma 100.17.3. Thus the first statement holds by Lemma 100.27.6. The second statement follows from the first and the definitions (because $f$ being quasi-separated means by definition that $\Delta _ f$ is quasi-compact and quasi-separated). $\square$

There are also:

• 2 comment(s) on Section 100.27: Morphisms of finite presentation

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).