The Stacks project

Lemma 85.25.3. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $a : U \to X$ gives a proper hypercovering of $X$, then for $K \in D^+(X)$

\[ K \to Ra_*(a^{-1}K) \]

is an isomorphism where $a : \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ is as in Lemma 85.2.8.

Proof. Consider the diagram of Lemma 85.25.1. Observe that $Rh_{n, *}h_ n^{-1}$ is the identity functor on $D^+(U_ n)$ by Cohomology on Sites, Lemma 21.31.11. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_{Zar})$ by Lemma 85.5.3. We have

\begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{qc, *}a_{qc}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*}

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 ($U$ is a hypercovering of $X$ in $\textit{LC}_{qc}$ by Cohomology on Sites, Lemma 21.31.4), and the last equality by the already used Cohomology on Sites, Lemma 21.31.11. $\square$

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