The Stacks project

Lemma 61.31.2. Let $S$ be a scheme. Let $S_{pro\text{-}\acute{e}tale}\subset S_{pro\text{-}\acute{e}tale}'$ be two small pro-étale sites of $S$ as constructed in Definition 61.12.8. Then the inclusion functor satisfies the assumptions of Sites, Lemma 7.21.8. Hence there exist morphisms of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \ar[r]^ g & \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}') \ar[r]^ f & \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) } \]

whose composition is isomorphic to the identity and with $f_* = g^{-1}$. Moreover,

  1. for $\mathcal{F}' \in \textit{Ab}(S_{pro\text{-}\acute{e}tale}')$ we have $H^ p(S_{pro\text{-}\acute{e}tale}', \mathcal{F}') = H^ p(S_{pro\text{-}\acute{e}tale}, g^{-1}\mathcal{F}')$,

  2. for $\mathcal{F} \in \textit{Ab}(S_{pro\text{-}\acute{e}tale})$ we have

    \[ H^ p(S_{pro\text{-}\acute{e}tale}, \mathcal{F}) = H^ p(S_{pro\text{-}\acute{e}tale}', g_*\mathcal{F}) = H^ p(S_{pro\text{-}\acute{e}tale}', f^{-1}\mathcal{F}). \]

Proof. The inclusion functor is fully faithful and continuous. We have seen that $S_{pro\text{-}\acute{e}tale}$ and $S_{pro\text{-}\acute{e}tale}'$ have fibre products and final objects and that our functor commutes with these (Lemma 61.12.10). It follows from Lemma 61.31.1 that the inclusion functor is cocontinuous. Hence the existence of $f$ and $g$ follows from Sites, Lemma 7.21.8. The equality in (1) is Cohomology on Sites, Lemma 21.7.2. Part (2) follows from (1) as $\mathcal{F} = g^{-1}g_*\mathcal{F} = g^{-1}f^{-1}\mathcal{F}$. $\square$

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Your email address will not be published.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 098Y. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 61.31.2 as pdf