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. $\square$

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