Lemma 61.12.18. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Consider a cartesian diagram
in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. Then $i_ g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.
Proof. Since the diagram is cartesian, we have for $U'/S'$ that $U' \times _{S'} T' = U' \times _ S T$. Hence both $i_ g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\mathit{Sch}/T)_{pro\text{-}\acute{e}tale}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times _{S'} T')$ on $S'_{pro\text{-}\acute{e}tale}$ (use Lemmas 61.12.12 and 61.12.15). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 7.28.1. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: