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 quasi-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 (2)

Comment #8865 by Wataru on

"question inverse" in the statement of Lemma 09B1 is probably a typo for "quasi-inverse".

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