Lemma 77.22.2. Assume $B \to S$, $(U, R, s, t, c)$, and $\pi : \mathcal{S}_ U \to [U/R]$ are as in Lemma 77.20.2. The $2$-commutative square

of Lemma 77.20.3 is a $2$-fibre product of stacks in groupoids of $(\mathit{Sch}/S)_{fppf}$.

Lemma 77.22.2. Assume $B \to S$, $(U, R, s, t, c)$, and $\pi : \mathcal{S}_ U \to [U/R]$ are as in Lemma 77.20.2. The $2$-commutative square

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

of Lemma 77.20.3 is a $2$-fibre product of stacks in groupoids of $(\mathit{Sch}/S)_{fppf}$.

**Proof.**
According to Stacks, Lemma 8.5.6 the lemma makes sense. It also tells us that we have to show that the functor

\[ \mathcal{S}_ R \longrightarrow \mathcal{S}_ U \times _{[U/R]} \mathcal{S}_ U \]

which maps $r : T \to R$ to $(T, t(r), s(r), \alpha (r))$ is an equivalence, where the right hand side is the $2$-fibre product as described in Categories, Lemma 4.32.3. This is, after spelling out the definitions, exactly the content of Lemma 77.22.1. (Alternative proof: Work out the meaning of Lemma 77.21.2 in this situation will give you the result also.) $\square$

Your email address will not be published.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.

## Comments (0)