The Stacks project

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. $\square$

Proof. To prove this consider the following diagram

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. $\square$

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. $\square$