The Stacks Project


Tag 098W

Chapter 55: Pro-étale Cohomology > Section 55.11: The pro-étale site

Lemma 55.11.23. Let $S$ be a scheme. Let $S_{affine, {pro\text{-}\acute{e}tale}}$ denote the full subcategory of $S_{pro\text{-}\acute{e}tale}$ consisting of affine objects. A covering of $S_{affine, {pro\text{-}\acute{e}tale}}$ will be a standard étale covering, see Definition 55.11.6. Then restriction $$ \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, {\acute{e}tale}}} $$ defines an equivalence of topoi $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \cong \mathop{\textit{Sh}}\nolimits(S_{affine, {pro\text{-}\acute{e}tale}})$.

Proof. This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma 7.28.1 by checking that the inclusion functor $S_{affine, {pro\text{-}\acute{e}tale}} \to S_{pro\text{-}\acute{e}tale}$ is a special cocontinuous functor (see Sites, Definition 7.28.2). $\square$

    The code snippet corresponding to this tag is a part of the file proetale.tex and is located in lines 2534–2546 (see updates for more information).

    \begin{lemma}
    \label{lemma-alternative}
    Let $S$ be a scheme. Let $S_{affine, \proetale}$ denote the full subcategory
    of $S_\proetale$ consisting of affine objects. A covering of
    $S_{affine, \proetale}$ will be a standard \'etale covering, see
    Definition \ref{definition-standard-proetale}.
    Then restriction
    $$
    \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, \etale}}
    $$
    defines an equivalence of topoi
    $\Sh(S_\proetale) \cong \Sh(S_{affine, \proetale})$.
    \end{lemma}
    
    \begin{proof}
    This you can show directly from the definitions, and is a good exercise.
    But it also follows immediately from
    Sites, Lemma \ref{sites-lemma-equivalence}
    by checking that the inclusion functor
    $S_{affine, \proetale} \to S_\proetale$
    is a special cocontinuous functor (see
    Sites, Definition \ref{sites-definition-special-cocontinuous-functor}).
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    There are also 4 comments on Section 55.11: Pro-étale Cohomology.

    Add a comment on tag 098W

    Your email address will not be published.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 lower-right corner).

    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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?