The Stacks project

Lemma 105.14.2. 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 105.13.9.

  1. If $\mathcal{X}$ is quasi-separated, then $M$ is quasi-separated.

  2. If $\mathcal{X}$ is separated, then $M$ is separated.

  3. 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 66.40.9.

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

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