The Stacks project

Lemma 61.26.6. Let $j : U \to X$ be a quasi-compact open immersion morphism of schemes. The functor $j_! : \textit{Ab}(U_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(X_{pro\text{-}\acute{e}tale})$ commutes with limits.

Proof. Since $j_!$ is exact it suffices to show that $j_!$ commutes with products. The question is local on $X$, hence we may assume $X$ affine. Let $\mathcal{G}$ be an abelian sheaf on $U_{pro\text{-}\acute{e}tale}$. We have $j^{-1}j_*\mathcal{G} = \mathcal{G}$. Hence applying the exact sequence of Lemma 61.26.5 we get

\[ 0 \to j_!\mathcal{G} \to j_*\mathcal{G} \to i_*i^{-1}j_*\mathcal{G} \to 0 \]

where $i : Z \to X$ is the inclusion of the reduced induced scheme structure on the complement $Z = X \setminus U$. The functors $j_*$ and $i_*$ commute with products as right adjoints. The functor $i^{-1}$ commutes with products by Lemma 61.25.3. Hence $j_!$ does because on the pro-├ętale site products are exact (Cohomology on Sites, Proposition 21.51.2). $\square$


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Your email address will not be published.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 09BP. Beware of the difference between the letter 'O' and the digit '0'.