The Stacks project

Lemma 106.13.7. Let $h : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks. Assume $\mathcal{X}$ is well-nigh affine, $h$ is étale, $h$ is separated, and $h$ induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark 101.19.5). Then there exists a cartesian diagram

\[ \xymatrix{ \mathcal{X}' \ar[d] \ar[r] & \mathcal{X} \ar[d] \\ M' \ar[r] & M } \]

where $M' \to M$ is a separated étale morphism of schemes and $\mathcal{X} \to M$ is the moduli space constructed in Lemma 106.13.4.

Proof. Choose an affine scheme $U$ and a surjective, flat, finite, and locally finitely presented morphism $U \to \mathcal{X}$. Since $h$ is representable by algebraic spaces (Morphisms of Stacks, Lemmas 101.45.3 and 101.45.1) we see that $U' = \mathcal{X}' \times _\mathcal {X} U$ is an algebraic space. Since $U' \to U$ is separated and étale, we see that $U'$ is a scheme and that every finite set of points of $U'$ is contained in an affine open, see Morphisms of Spaces, Lemma 67.51.1 and More on Morphisms, Lemma 37.45.1. Setting $R' = U' \times _{\mathcal{X}'} U'$ we see that $s', t' : R' \to U'$ are finite locally free. By Groupoids, Lemma 39.24.1 there exists an open covering $U' = \bigcup U'_ i$ by $R'$-invariant affine open subschemes $U'_ i \subset U'$. Let $\mathcal{X}'_ i \subset \mathcal{X}'$ be the corresponding open substacks. These are well-nigh affine as $U'_ i \to \mathcal{X}'_ i$ is surjective, flat, finite and of finite presentation. By Lemma 106.13.5 we obtain cartesian diagrams

\[ \xymatrix{ \mathcal{X}'_ i \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ M'_ i \ar[r] & M } \]

with $M'_ i \to M$ an étale morphism of affine schemes and vertical arrows as in Lemma 106.13.4. Observe that $\mathcal{X}'_{ij} = \mathcal{X}'_ i \cap \mathcal{X}'_ j$ is an open subspace of $\mathcal{X}'_ i$ and $\mathcal{X}'_ j$. Hence we get corresponding open subschemes $V_{ij} \subset M'_ i$ and $V_{ji} \subset M'_ j$. By the result of Lemma 106.13.6 we see that both $\mathcal{X}'_{ij} \to V_{ij}$ and $\mathcal{X}'_{ji} \to V_{ji}$ are categorical moduli spaces! Thus we get a unique isomorphism $\varphi _{ij} : V_{ij} \to V_{ji}$ such that

\[ \xymatrix{ \mathcal{X}'_ i \ar[d] & & \mathcal{X}'_ i \cap \mathcal{X}'_ j \ar[rr] \ar[ll] \ar[ld] \ar[rd] & & \mathcal{X}'_ j \ar[d] \\ M'_ i & V_{ij} \ar[l] \ar[rr]^{\varphi _{ij}} & & V_{ji} \ar[r] & M'_ j } \]

is commutative. These isomorphisms satisfy the cocyclce condition of Schemes, Section 26.14 by a computation (and another application of the previous lemma) which we omit. Thus we can glue the affine schemes in to scheme $M'$, see Schemes, Lemma 26.14.1. Let us identify the $M'_ i$ with their image in $M'$. We claim there is a morphism $\mathcal{X}' \to M'$ fitting into cartesian diagrams

\[ \xymatrix{ \mathcal{X}'_ i \ar[r] \ar[d] & \mathcal{X}' \ar[d] \\ M'_ i \ar[r] & M' } \]

This is clear from the description of the morphisms into the glued scheme $M'$ in Schemes, Lemma 26.14.1 and the fact that to give a morphism $\mathcal{X}' \to M'$ is the same thing as given a morphism $T \to M'$ for any morphism $T \to \mathcal{X}'$. Similarly, there is a morphism $M' \to M$ restricting to the given morphisms $M'_ i \to M$ on $M'_ i$. The morphism $M' \to M$ is étale (being étale on the members of an étale covering) and the fibre product property holds as it can be checked on members of the (affine) open covering $M' = \bigcup M'_ i$. Finally, $M' \to M$ is separated because the composition $U' \to \mathcal{X}' \to M'$ is surjective and universally closed and we can apply Morphisms, Lemma 29.41.11. $\square$

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