The Stacks project

Lemma 100.3.6. Let $g : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Let $[U/R] \to \mathcal{X}$ be a presentation. Set $U' = U \times _\mathcal {X} \mathcal{X}'$, and $R' = R \times _\mathcal {X} \mathcal{X}'$. Then there exists a groupoid in algebraic spaces of the form $(U', R', s', t', c')$, a presentation $[U'/R'] \to \mathcal{X}'$, and the diagram

\[ \xymatrix{ [U'/R'] \ar[d]_{[\text{pr}]} \ar[r] & \mathcal{X}' \ar[d]^ g \\ [U/R] \ar[r] & \mathcal{X} } \]

is $2$-commutative where the morphism $[\text{pr}]$ comes from a morphism of groupoids $\text{pr} : (U', R', s', t', c') \to (U, R, s, t, c)$.

Proof. Since $U \to \mathcal{Y}$ is surjective and smooth, see Algebraic Stacks, Lemma 94.17.2 the base change $U' \to \mathcal{X}'$ is also surjective and smooth. Hence, by Algebraic Stacks, Lemma 94.16.2 it suffices to show that $R' = U' \times _{\mathcal{X}'} U'$ in order to get a smooth groupoid $(U', R', s', t', c')$ and a presentation $[U'/R'] \to \mathcal{X}'$. Using that $R = V \times _\mathcal {Y} V$ (see Groupoids in Spaces, Lemma 78.22.2) this follows from

\[ R' = U \times _\mathcal {X} U \times _\mathcal {X} \mathcal{X}' = (U \times _\mathcal {X} \mathcal{X}') \times _{\mathcal{X}'} (U \times _\mathcal {X} \mathcal{X}') \]

see Categories, Lemmas 4.31.8 and 4.31.10. Clearly the projection morphisms $U' \to U$ and $R' \to R$ give the desired morphism of groupoids $\text{pr} : (U', R', s', t', c') \to (U, R, s, t, c)$. Hence the morphism $[\text{pr}]$ of quotient stacks by Groupoids in Spaces, Lemma 78.21.1.

We still have to show that the diagram $2$-commutes. It is clear that the diagram

\[ \xymatrix{ U' \ar[d]_{\text{pr}_ U} \ar[r]_{f'} & \mathcal{X}' \ar[d]^ g \\ U \ar[r]^ f & \mathcal{X} } \]

$2$-commutes where $\text{pr}_ U : U' \to U$ is the projection. There is a canonical $2$-arrow $\tau : f \circ t \to f \circ s$ in $\mathop{\mathrm{Mor}}\nolimits (R, \mathcal{X})$ coming from $R = U \times _\mathcal {X} U$, $t = \text{pr}_0$, and $s = \text{pr}_1$. Using the isomorphism $R' \to U' \times _{\mathcal{X}'} U'$ we get similarly an isomorphism $\tau ' : f' \circ t' \to f' \circ s'$. Note that $g \circ f' \circ t' = f \circ t \circ \text{pr}_ R$ and $g \circ f' \circ s' = f \circ s \circ \text{pr}_ R$, where $\text{pr}_ R : R' \to R$ is the projection. Thus it makes sense to ask if
\begin{equation} \label{stacks-properties-equation-verify} \tau \star \text{id}_{\text{pr}_ R} = \text{id}_ g \star \tau '. \end{equation}

Now we make two claims: (1) if Equation ( holds, then the diagram $2$-commutes, and (2) Equation ( holds. We omit the proof of both claims. Hints: part (1) follows from the construction of $f = f_{can}$ and $f' = f'_{can}$ in Algebraic Stacks, Lemma 94.16.1. Part (2) follows by carefuly working through the definitions. $\square$

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 04Y6. Beware of the difference between the letter 'O' and the digit '0'.