Lemma 59.103.6. Let X be a scheme. For K \in D^+(X_{\acute{e}tale}) with torsion cohomology sheaves the map
K \longrightarrow Ra_{X, *}a_ X^{-1}K
is an isomorphism with a_ X : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_ h) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) 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 of torsion abelian sheaves. This is possible by Cohomology on Sites, Lemma 21.19.8. 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} where \mathcal{F} is a torsion abelian sheaf.
By Lemma 59.103.1 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 (Cohomology on Sites, Lemma 21.14.7). By Lemma 59.103.5 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 59.99.4 we have R^ j\pi _{X, *}(\pi _ X^{-1}\mathcal{F}) = 0 for j > 0. This concludes the proof.
\square
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