The Stacks project

Lemma 61.19.10. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. Let $D_{flc}(X_{\acute{e}tale}, \Lambda )$, resp. $D_{flc}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ be the full subcategory of $D(X_{\acute{e}tale}, \Lambda )$, resp. $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ consisting of those complexes whose cohomology sheaves are locally constant sheaves of $\Lambda $-modules of finite type. Then

\[ \epsilon ^{-1} : D_{flc}^+(X_{\acute{e}tale}, \Lambda ) \longrightarrow D_{flc}^+(X_{pro\text{-}\acute{e}tale}, \Lambda ) \]

is an equivalence of categories.

Proof. The categories $D_{flc}(X_{\acute{e}tale}, \Lambda )$ and $D_{flc}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ are strictly full, saturated, triangulated subcategories of $D(X_{\acute{e}tale}, \Lambda )$ and $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ by Modules on Sites, Lemma 18.43.5 and Derived Categories, Section 13.17. The statement of the lemma follows by combining Lemmas 61.19.8 and 61.19.9. $\square$

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Your email address will not be published.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 099Z. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 61.19.10 as pdf