The Stacks project

106.14 Properties of moduli spaces

Once the existence of a moduli space has been proven, it becomes possible (and is usually straightforward) to esthablish properties of these moduli spaces.

Lemma 106.14.1. Let $p : \mathcal{X} \to Y$ be a morphism of an algebraic stack to an algebraic space. Assume

  1. $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is finite,

  2. $Y$ is locally Noetherian, and

  3. $p$ is locally of finite type.

Let $f : \mathcal{X} \to M$ be the moduli space constructed in Theorem 106.13.9. Then $M \to Y$ is locally of finite type.

Proof. Since $f$ is a uniform categorical moduli space we obtain the morphism $M \to Y$. It suffices to check that $M \to Y$ is locally of finite type étale locally on $M$ and $Y$. Since $f$ is a uniform categorical moduli space, we may first replace $Y$ by an affine scheme étale over $Y$. Next, we may choose $I$ and $g_ i : \mathcal{X}_ i \to \mathcal{X}$ as in Lemma 106.13.8. Then by Lemma 106.13.10 we reduce to the case $\mathcal{X} = \mathcal{X}_ i$. In other words, we may assume $\mathcal{X}$ is well-nigh affine. In this case we have $Y = \mathop{\mathrm{Spec}}(A_0)$, we have $\mathcal{X} = [U/R]$ with $U = \mathop{\mathrm{Spec}}(A)$ and $M = \mathop{\mathrm{Spec}}(C)$ where $C \subset A$ is the set of $R$-invariant functions on $U$. See Lemmas 106.13.2 and 106.13.4. Then $A_0$ is Noetherian and $A_0 \to A$ is of finite type. Moreover $A$ is integral over $C$ by Groupoids, Lemma 39.23.4, hence finite over $C$ (being of finite type over $A_0$). Thus we may finally apply Algebra, Lemma 10.51.7 to conclude. $\square$

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.

  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 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. $\square$

Lemma 106.14.3. Let $p : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. Assume

  1. $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is finite,

  2. $p$ is proper, and

  3. $Y$ is locally Noetherian.

Let $f : \mathcal{X} \to M$ be the moduli space constructed in Theorem 106.13.9. Then $M \to Y$ is proper.

Proof. By Lemma 106.14.1 we see that $M \to Y$ is locally of finite type. By Lemma 106.14.2 we see that $M \to Y$ is separated. Of course $M \to Y$ is quasi-compact and universally closed as these are topological properties and $\mathcal{X} \to Y$ has these properties and $\mathcal{X} \to M$ is a universal homeomorphism. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DUW. Beware of the difference between the letter 'O' and the digit '0'.