Lemma 85.36.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is a proper hypercovering of $X$, then for $K \in D^+(X_{\acute{e}tale})$
is an isomorphism. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 85.32.
Proof. Consider the diagram of Lemma 85.36.1. Observe that $Rh_{n, *}h_ n^{-1}$ is the identity functor on $D^+(U_{n, {\acute{e}tale}})$ by More on Cohomology of Spaces, Lemma 84.8.2. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_{\acute{e}tale})$ by Lemma 85.5.3. We have
The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 85.25.1, the third equality by Lemma 85.21.2 as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{ph}$ by Topologies on Spaces, Lemma 73.8.3, and the last equality by the already used More on Cohomology of Spaces, Lemma 84.8.2. $\square$
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