The Stacks project

Proof. Write $\mathcal{X} = [U/R]$ with $(U, R, s, t, c)$ as in Lemma 106.13.2. Let $C$ be the ring of $R$-invariant functions on $U$, see Groupoids, Section 39.23. We set $M = \mathop{\mathrm{Spec}}(C)$. The $R$-invariant morphism $U \to M$ corresponds to a morphism $f : \mathcal{X} \to M$ by Lemma 106.12.2. The characterization of morphisms into affine schemes given in Schemes, Lemma 26.6.4 immediately guarantees that $\phi : U \to M$ is a categorical quotient in the category of affine schemes. Hence $f$ is a categorical moduli space in the category of affine schemes (Lemma 106.12.3).

Since $\mathcal{X}$ is separated by Lemma 106.13.2 we find that $f$ is separated by Morphisms of Stacks, Lemma 101.4.12.

Since $U \to \mathcal{X}$ is surjective and since $U \to M$ is quasi-compact, we see that $f$ is quasi-compact by Morphisms of Stacks, Lemma 101.7.6.

By Groupoids, Lemma 39.23.4 the composition

is an integral morphism of affine schemes. In particular, it is universally closed (Morphisms, Lemma 29.44.7). Since $U \to \mathcal{X}$ is surjective, it follows that $\mathcal{X} \to M$ is universally closed (Morphisms of Stacks, Lemma 101.37.6). To conclude that $\mathcal{X} \to M$ is a universal homeomorphism, it is enough to show that it is universally bijective, i.e., surjective and universally injective.

We have $|\mathcal{X}| = |U|/|R|$ by Morphisms of Stacks, Lemma 101.20.2. Thus $|f|$ is surjective and even bijective by Groupoids, Lemma 39.23.6.

Let $C \to C'$ be a ring map. Let $(U', R', s', t', c')$ be the base change of $(U, R, s, t, c)$ by $M' = \mathop{\mathrm{Spec}}(C') \to M$. Setting $\mathcal{X}' = [U'/R']$, we observe that $M' \times _ M \mathcal{X} = \mathcal{X}'$ by Quotients of Groupoids, Lemma 83.3.6. Let $C^1$ be the ring of $R'$-invariant functions on $U'$. Set $M^1 = \mathop{\mathrm{Spec}}(C^1)$ and consider the diagram

By Groupoids, Lemma 39.23.5 and Algebra, Lemma 10.46.11 the morphism $M^1 \to M'$ is a homeomorphism. On the other hand, the previous paragraph applied to $(U', R', s', t', c')$ shows that $|f'|$ is bijective. We conclude that $f$ induces a bijection on points after any base change by an affine scheme. Thus $f$ is universally injective by Morphisms of Stacks, Lemma 101.14.7.

Finally, we still have to show that $f$ is a uniform moduli space in the category of affine schemes. This follows from the discussion above and the fact that if the ring map $C \to C'$ is flat, then $C' \to C^1$ is an isomorphism by Groupoids, Lemma 39.23.5. $\square$