Lemma 84.5.5 (0DGC)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 84: More on Cohomology of Spaces
Section 84.5: Comparing big and small topoi
Lemma 84.5.5 (cite)

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Lemma 84.5.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . For $K \in D(X_{\acute{e}tale})$ the map

$K \longrightarrow R\pi _{X, *}\pi _ X^{-1}K$

is an isomorphism where $\pi _ X : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as above.

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

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