The Stacks project

Lemma 105.13.5. Let $h : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks. Assume $\mathcal{X}'$ and $\mathcal{X}$ are well-nigh affine, $h$ is étale, and $h$ induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark 100.19.5). 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 étale and the vertical arrows are the moduli spaces constructed in Lemma 105.13.4.

Proof. Observe that $h$ is representable by algebraic spaces by Morphisms of Stacks, Lemmas 100.45.3 and 100.45.1. Choose an affine scheme $U$ and a surjective, flat, finite, and finitely presented morphism $U \to \mathcal{X}$. Then $U' = \mathcal{X}' \times _\mathcal {X} U$ is an algebraic space with a finite (in particular affine) morphism $U' \to \mathcal{X}'$. By Lemma 105.13.3 we conclude that $U'$ is affine. Setting $R = U \times _\mathcal {X} U$ and $R' = U' \times _{\mathcal{X}'} U'$ we obtain groupoids $(U, R, s, t, c)$ and $(U', R', s', t', c')$ such that $\mathcal{X} = [U/R]$ and $\mathcal{X}' = [U'/R']$, see proof of Lemma 105.13.2. we see that the diagrams

\[ \xymatrix{ R' \ar[d]_{s'} \ar[r]_ f & R \ar[d]^ s \\ U' \ar[r]^ f & U } \quad \quad \xymatrix{ R' \ar[d]_{t'} \ar[r]_ f & R \ar[d]^ t \\ U' \ar[r]^ f & U } \quad \quad \xymatrix{ G' \ar[d] \ar[r]_ f & G \ar[d] \\ U' \ar[r]^ f & U } \]

are cartesian where $G$ and $G'$ are the stabilizer group schemes. This follows for the first two by transitivity of fibre products and for the last one this follows because it is the pullback of the isomorphism $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}' \times _\mathcal {X} \mathcal{I}_\mathcal {X}$ (by the already used Morphisms of Stacks, Lemma 100.45.3). Recall that $M$, resp. $M'$ was constructed in Lemma 105.13.4 as the spectrum of the ring of $R$-invariant functions on $U$, resp. the ring of $R'$-invariant functions on $U'$. Thus $M' \to M$ is étale and $U' = M' \times _ M U$ by Groupoids, Lemma 39.23.7. It follows that $R' = M' \times _ M U$, in other words the groupoid $(U', R', s', t', c')$ is the base change of $(U, R, s, t, c)$ by $M' \to M$. This implies that the diagram in the lemma is cartesian by Quotients of Groupoids, Lemma 82.3.6. $\square$

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