Lemma 106.14.2 (0DUY)—The Stacks project

Loading web-font TeX/Caligraphic/Regular

Table of contents
Part 7: Algebraic Stacks
Chapter 106: More on Morphisms of Stacks
Section 106.14: Properties of moduli spaces
Lemma 106.14.2 (cite)

numberstags

Lemma 106.14.2. Let $\mathcal{X}$ be an algebraic stack. Assume $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is finite. Let $f : \mathcal{X} \to M$ be the moduli space constructed in Theorem 106.13.9.

If $\mathcal{X}$ is quasi-separated, then $M$ is quasi-separated.
If $\mathcal{X}$ is separated, then $M$ is separated.
Add more here, for example relative versions of the above.

Proof. To prove this consider the following diagram

$\xymatrix{ \mathcal{X} \ar[d]_ f \ar[r]_{\Delta _\mathcal {X}} & \mathcal{X} \times \mathcal{X} \ar[d]^{f \times f} \\ M \ar[r]^{\Delta _ M} & M \times M }$

Since $f$ is a universal homeomorphism, we see that $f \times f$ is a universal homeomorphism.

If $\mathcal{X}$ is separated, then $\Delta _\mathcal {X}$ is proper, hence $\Delta _\mathcal {X}$ is universally closed, hence $\Delta _ M$ is universally closed, hence $M$ is separated by Morphisms of Spaces, Lemma 67.40.9.

If $\mathcal{X}$ is quasi-separated, then $\Delta _\mathcal {X}$ is quasi-compact, hence $\Delta _ M$ is quasi-compact, hence $M$ is quasi-separated. $\square$

Comments (0)

There are also:

1 comment(s) on Section 106.14: Properties of moduli spaces

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).

Comments (0)

Post a comment