Lemma 61.12.18 (0F61)—The Stacks project

Processing math: 100%

Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.12: The pro-étale site
Lemma 61.12.18 (cite)

numberstags

Lemma 61.12.18. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site. Consider a cartesian diagram

$\xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^ f \\ S' \ar[r]^ g & S }$

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. $\square$

Comments (0)

There are also:

6 comment(s) on Section 61.12: The pro-étale site

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).

Comments (0)

Post a comment