Theorem 106.13.9 (0DUT): Keel-Mori

Theorem 106.13.9 (Keel-Mori). Let $\mathcal{X}$ be an algebraic stack. Assume $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is finite. Then there exists a uniform categorical moduli space

\[ f : \mathcal{X} \longrightarrow M \]

and $f$ is separated, quasi-compact, and a universal homeomorphism.

Proof. We choose a set $I$¹ and for $i \in I$ a morphism of algebraic stacks $g_ i : \mathcal{X}_ i \to \mathcal{X}$ as in Lemma 106.13.8; we will use all of the properties listed in this lemma without further mention. Let

\[ f_ i : \mathcal{X}_ i \to M_ i \]

be as in Lemma 106.13.4. Consider the stacks

\[ \mathcal{X}_{ij} = \mathcal{X}_ i \times _{g_ i, \mathcal{X}, g_ j} \mathcal{X}_ j \]

for $i, j \in I$. The projections $\mathcal{X}_{ij} \to \mathcal{X}_ i$ and $\mathcal{X}_{ij} \to \mathcal{X}_ j$ are separated by Morphisms of Stacks, Lemma 101.4.4, étale by Morphisms of Stacks, Lemma 101.35.3, and induce isomorphisms on automorphism groups (as in Morphisms of Stacks, Remark 101.19.5) by Morphisms of Stacks, Lemma 101.45.5. Thus we may apply Lemma 106.13.7 to find a commutative diagram

\[ \xymatrix{ \mathcal{X}_ i \ar[d]_{f_ i} & \mathcal{X}_{ij} \ar[d]_{f_{ij}} \ar[l] \ar[r] & \mathcal{X}_ j \ar[d]_{f_ j} \\ M_ i & M_{ij} \ar[l] \ar[r] & M_ j } \]

with cartesian squares where $M_{ij} \to M_ i$ and $M_{ij} \to M_ j$ are separated étale morphisms of schemes; here we also use that $f_ i$ is a uniform categorical quotient by Lemma 106.13.6. Claim:

\[ \coprod M_{ij} \longrightarrow \coprod M_ i \times \coprod M_ i \]

is an étale equivalence relation.

Proof of the claim. Set $R = \coprod M_{ij}$ and $U = \coprod M_ i$. We have already seen that $t : R \to U$ and $s : R \to U$ are étale. Let us construct a morphism $c : R \times _{s, U, t} R \to R$ compatible with $\text{pr}_{13} : U \times U \times U \to U \times U$. Namely, for $i, j, k \in I$ we consider

\[ \mathcal{X}_{ijk} = \mathcal{X}_ i \times _{g_ i, \mathcal{X}, g_ j} \mathcal{X}_ j \times _{g_ j, \mathcal{X}, g_ k} \mathcal{X}_ k = \mathcal{X}_{ij} \times _{\mathcal{X}_ j} \mathcal{X}_{jk} \]

Arguing exactly as in the previous paragraph, we find that $M_{ijk} = M_{ij} \times _{M_ j} M_{jk}$ is a categorical moduli space for $\mathcal{X}_{ijk}$. In particular, there is a canonical morphism $M_{ijk} = M_{ij} \times _{M_ j} M_{jk} \to M_{ik}$ coming from the projection $\mathcal{X}_{ijk} \to \mathcal{X}_{ik}$. Putting these morphisms together we obtain the morphism $c$. In a similar fashion we construct a morphism $e : U \to R$ compatible with $\Delta : U \to U \times U$ and $i : R \to R$ compatible with the flip $U \times U \to U \times U$. Let $k$ be an algebraically closed field. Then

\[ \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X}_ i) \to \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), M_ i) = M_ i(k) \]

is bijective on isomorphism classes and the same remains true after any base change by a morphism $M' \to M$. This follows from our choice of $f_ i$ and Morphisms of Stacks, Lemmas 101.14.5 and 101.14.6. By construction of $2$-fibred products the diagram

\[ \xymatrix{ \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X}_{ij}) \ar[d] \ar[r] & \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X}_ j) \ar[d] \\ \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X}_ i) \ar[r] & \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X}) } \]

is a fibre product of categories. By our choice of $g_ i$ the functors in this diagram induce bijections on automorphism groups. It follows that this diagram induces a fibre product diagram on sets of isomorphism classes! Thus we see that

\[ R(k) = U(k) \times _{|\mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X})|} U(k) \]

where $|\mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X})|$ denotes the set of isomorphism classes. In particular, for any algebraically closed field $k$ the map on $k$-valued point is an equivalence relation. We conclude the claim holds by Groupoids, Lemma 39.3.5.

Let $M = U/R$ be the algebraic space which is the quotient of the above étale equivalence relation, see Spaces, Theorem 65.10.5. There is a canonical morphism $f : \mathcal{X} \to M$ fitting into commutative diagrams

106.13.9.1

\begin{equation} \label{stacks-more-morphisms-equation-fundamental-diagram} \xymatrix{ \mathcal{X}_ i \ar[r]_{g_ i} \ar[d]_{f_ i} & \mathcal{X} \ar[d]^ f \\ M_ i \ar[r] & M } \end{equation}

Namely, such a morphism $f$ is given by a functor

\[ f : \mathop{\mathrm{Mor}}\nolimits (T, \mathcal{X}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits (T, M) \]

for any scheme $T$ compatible with base change. Let $a : T \to \mathcal{X}$ be an object of the left hand side. We obtain an étale covering $\{ T_ i \to T\} $ with $T_ i = \mathcal{X}_ i \times _\mathcal {X} T$ and morphisms $a_ i : T_ i \to \mathcal{X}_ i$. Then we get $b_ i = f_ i \circ a_ i : T_ i \to M_ i$. Since $T_ i \times _ T T_ j = \mathcal{X}_{ij} \times _\mathcal {X} T$ we moreover get a morphism $a_{ij} : T_ i \times _ T T_ j \to \mathcal{X}_{ij}$. Setting $b_{ij} = f_{ij} \circ a_{ij}$ we find that $b_ i \times b_ j$ factors through the monomorphism $M_{ij} \to M_ i \times M_ j$. Hence the morphisms

\[ T_ i \xrightarrow {b_ i} M_ i \to M \]

agree on $T_ i \times _ T T_ j$. As $M$ is a sheaf for the étale topology, we see that these morphisms glue to a unique morphism $b = f(a) : T \to M$. We omit the verification that this construction is compatible with base change and we omit the verification that the diagrams (106.13.9.1) commute.

Claim: the diagrams (106.13.9.1) are cartesian. To see this we study the induced morphism

\[ h_ i : \mathcal{X}_ i \longrightarrow M_ i \times _ M \mathcal{X} \]

This is a morphism of stacks étale over $\mathcal{X}$ and hence $h_ i$ is étale (Morphisms of Stacks, Lemma 101.35.6). Since $g_ i$ is separated, we see $h_ i$ is separated (use Morphisms of Stacks, Lemma 101.4.12 and the fact seen above that the diagonal of $\mathcal{X}$ is separated). The morphism $h_ i$ induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark 101.19.5) as this is true for $g_ i$. For an algebraically closed field $k$ the diagram

\[ \xymatrix{ \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), M_ i \times _ M \mathcal{X}) \ar[r] \ar[d] & \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X}) \ar[d] \\ M_ i(k) \ar[r] & M(k) } \]

is a catesian diagram of categories and the top arrow induces bijections on automorphism groups. On the other hand, we have

\[ M(k) = U(k)/R(k) = U(k)/ U(k) \times _{|\mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X})|} U(k) = |\mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X})| \]

by what we said above. Thus the right vertical arrow in the cartesian diagram above is a bijection on isomorphism classes. We conclude that $|\mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), M_ i \times _ M \mathcal{X})| \to M_ i(k)$ is bijective. Review: $h_ i$ is a separated, étale, induces isomorphisms on automorphism groups (as in Morphisms of Stacks, Remark 101.19.5), and induces an equivalence on fibre categories over algebraically closed fields. Hence it is an isomorphism by Morphisms of Stacks, Lemma 101.45.7.

From the claim we get in particular the following: we have a surjective étale morphism $U \to M$ such that the base change of $f$ is separated, quasi-compact, and a universal homeomorphism. It follows that $f$ is separated, quasi-compact, and a universal homeomorphism. See Morphisms of Stacks, Lemma 101.4.5, 101.7.10, and 101.15.5

To finish the proof we have to show that $f : \mathcal{X} \to M$ is a uniform categorical moduli space. To prove this it suffices to show that given a flat morphism $M' \to M$ of algebraic spaces, the base change

\[ M' \times _ M \mathcal{X} \longrightarrow M' \]

is a categorical moduli space. Thus we consider a morphism

\[ \theta : M' \times _ M \mathcal{X} \longrightarrow E \]

where $E$ is an algebraic space. For each $i$ we know that $f_ i$ is a uniform categorical moduli space. Hence we obtain

\[ \xymatrix{ M' \times _ M \mathcal{X}_ i \ar[d] \ar[r] & M' \times _ M \mathcal{X} \ar[d]^\theta \\ M' \times _ M M_ i \ar[r]^{\psi _ i} & E } \]

Since $\{ M' \times _ M M_ i \to M'\} $ is an étale covering, to obtain the desired morphism $\psi : M' \to E$ it suffices to show that $\psi _ i$ and $\psi _ j$ agree over $M' \times _ M M_ i \times _ M M_ j = M' \times _ M M_{ij}$. This follows easily from the fact that $f_{ij} : \mathcal{X}_{ij} = \mathcal{X}_ i \times _\mathcal {X} \mathcal{X}_ j \to M_{ij}$ is a uniform categorical quotient; details omitted. Then finally one shows that $\psi $ fits into the commutative diagram

\[ \xymatrix{ M' \times _ M \mathcal{X} \ar[d] \ar[rd]^\theta \\ M' \ar[r]^\psi & E } \]

because “$\{ M' \times _ M \mathcal{X}_ i \to M' \times _ M \mathcal{X}\} $ is an étale covering” and the morphisms $\psi _ i$ fit into the corresponding commutative diagrams by construction. This finishes the proof of the Keel-Mori theorem. $\square$

[1] The reader who is still keeping track of set theoretic issues should make sure $I$ is not too large.

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