The Stacks project

Lemma 61.19.8. Let $X$ be a scheme. Let $\Lambda $ be a ring.

  1. The essential image of the fully faithful functor $\epsilon ^{-1} : \textit{Mod}(X_{\acute{e}tale}, \Lambda ) \to \textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ is a weak Serre subcategory $\mathcal{C}$.

  2. The functor $\epsilon ^{-1}$ defines an equivalence of categories of $D^+(X_{\acute{e}tale}, \Lambda )$ with $D^+_\mathcal {C}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ with question inverse given by $R\epsilon _*$.

Proof. To prove (1) we will prove conditions (1) – (4) of Homology, Lemma 12.10.3. Since $\epsilon ^{-1}$ is fully faithful (Lemma 61.19.2) and exact, everything is clear except for condition (4). However, if

\[ 0 \to \epsilon ^{-1}\mathcal{F}_1 \to \mathcal{G} \to \epsilon ^{-1}\mathcal{F}_2 \to 0 \]

is a short exact sequence of sheaves of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$, then we get

\[ 0 \to \epsilon _*\epsilon ^{-1}\mathcal{F}_1 \to \epsilon _*\mathcal{G} \to \epsilon _*\epsilon ^{-1}\mathcal{F}_2 \to R^1\epsilon _*\epsilon ^{-1}\mathcal{F}_1 \]

which by Lemma 61.19.5 is the same as a short exact sequence

\[ 0 \to \mathcal{F}_1 \to \epsilon _*\mathcal{G} \to \mathcal{F}_2 \to 0 \]

Pulling pack we find that $\mathcal{G} = \epsilon ^{-1}\epsilon _*\mathcal{G}$. This proves (1).

Part (2) follows from part (1) and Cohomology on Sites, Lemma 21.28.5. $\square$


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Your email address will not be published.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 09B1. Beware of the difference between the letter 'O' and the digit '0'.