Lemma 61.17.4. Let $S$ be a scheme. For $K \in D(S_{pro\text{-}\acute{e}tale})$ the map

is an isomorphism.

Lemma 61.17.4. Let $S$ be a scheme. For $K \in D(S_{pro\text{-}\acute{e}tale})$ the map

\[ K \longrightarrow R\pi _{S, *}\pi _ S^{-1}K \]

is an isomorphism.

**Proof.**
This is true because both $\pi _ S^{-1}$ and $\pi _{S, *} = i_ S^{-1}$ are exact functors and the composition $\pi _{S, *} \circ \pi _ S^{-1}$ is the identity functor.
$\square$

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