Lemma 21.31.11. Let $X$ be an object of $\textit{LC}_{qc}$. For $K \in D^+(X)$ the map

$K \longrightarrow Ra_{X, *}a_ X^{-1}K$

is an isomorphism with $a_ X : \mathop{\mathit{Sh}}\nolimits (\textit{LC}_{qc}/X) \to \mathop{\mathit{Sh}}\nolimits (X)$ as above.

Proof. We first reduce the statement to the case where $K$ is given by a single abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet$. By the case of a sheaf we see that $\mathcal{F}^ n = a_{X, *} a_ X^{-1} \mathcal{F}^ n$ and that the sheaves $R^ qa_{X, *}a_ X^{-1}\mathcal{F}^ n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) applied to $a_ X^{-1}\mathcal{F}^\bullet$ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$.

By Lemma 21.31.6 we have $a_{X, *}a_ X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^ qa_{X, *}a_ X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_ X = \epsilon _ X \circ \pi _ X$ and the Leray spectral sequence Lemma 21.14.7. By Lemma 21.31.10 we have $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$ and $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$. By Lemma 21.31.7 we have $R^ j\pi _{X, *}(\pi _ X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof. $\square$

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