The Stacks project

Lemma 106.13.5. 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 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 étale and the vertical arrows are the moduli spaces constructed in Lemma 106.13.4.

Proof. Observe that $h$ is representable by algebraic spaces by Morphisms of Stacks, Lemmas 101.45.3 and 101.45.1. 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 106.13.3 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 106.13.2. 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 101.45.3). Recall that $M$, resp. $M'$ was constructed in Lemma 106.13.4 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 83.3.6. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DUQ. Beware of the difference between the letter 'O' and the digit '0'.