Lemma 105.13.4. Let the algebraic stack $\mathcal{X}$ be well-nigh affine. There exists a uniform categorical moduli space

$f : \mathcal{X} \longrightarrow M$

in the category of affine schemes. Moreover $f$ is separated, quasi-compact, and a universal homeomorphism.

Proof. Write $\mathcal{X} = [U/R]$ with $(U, R, s, t, c)$ as in Lemma 105.13.2. 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 105.12.2. 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 105.12.3).

Since $\mathcal{X}$ is separated by Lemma 105.13.2 we find that $f$ is separated by Morphisms of Stacks, Lemma 100.4.12.

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 100.7.6.

By Groupoids, Lemma 39.23.4 the composition

$U \to \mathcal{X} \to M$

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 100.37.6). 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 100.20.2. 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 82.3.6. Let $C^1$ be the ring of $R'$-invariant functions on $U'$. Set $M^1 = \mathop{\mathrm{Spec}}(C^1)$ and consider the diagram

$\xymatrix{ \mathcal{X}' \ar[d]^{f'} \ar[r] & \mathcal{X} \ar[dd]^ f \\ M^1 \ar[d] \\ M' \ar[r] & M }$

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 100.14.7.

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. $\square$

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