The Stacks project

Lemma 78.20.3. Assumptions and notation as in Lemma 78.20.2. There exists a canonical $2$-morphism $\alpha : \pi \circ s \to \pi \circ t$ making the diagram

\[ \xymatrix{ \mathcal{S}_ R \ar[r]_ s \ar[d]_ t & \mathcal{S}_ U \ar[d]^\pi \\ \mathcal{S}_ U \ar[r]^-\pi & [U/R] } \]

$2$-commutative.

Proof. Let $S'$ be a scheme over $S$. Let $r : S' \to R$ be a morphism over $S$. Then $r \in R(S')$ is an isomorphism between the objects $s \circ r, t \circ r \in U(S')$. Moreover, this construction is compatible with pullbacks. This gives a canonical $2$-morphism $\alpha _ p : \pi _ p \circ s \to \pi _ p \circ t$ where $\pi _ p : \mathcal{S}_ U \to [U/_{\! p}R]$ is as in the proof of Lemma 78.20.2. Thus even the diagram

\[ \xymatrix{ \mathcal{S}_ R \ar[r]_ s \ar[d]^ t & \mathcal{S}_ U \ar[d]^{\pi _ p} \\ \mathcal{S}_ U \ar[r]^-{\pi _ p} & [U/_{\! p}R] } \]

is $2$-commutative. Thus a fortiori the diagram of the lemma is $2$-commutative. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 78.20: Quotient stacks

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Your email address will not be published.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 044S. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 78.20.3 as pdf