Lemma 21.31.10. With notation as above.

For $X \in \mathop{\mathrm{Ob}}\nolimits (\textit{LC}_{qc})$ and an abelian sheaf $\mathcal{F}$ on $X$ we have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$.

For a proper morphism $f : X \to Y$ in $\textit{LC}_{qc}$ and abelian sheaf $\mathcal{F}$ on $X$ we have $a_ Y^{-1}(R^ if_*\mathcal{F}) = R^ if_{qc, *}(a_ X^{-1}\mathcal{F})$ for all $i$.

For $X \in \mathop{\mathrm{Ob}}\nolimits (\textit{LC}_{qc})$ and $K$ in $D^+(X)$ the map $\pi _ X^{-1}K \to R\epsilon _{X, *}(a_ X^{-1}K)$ is an isomorphism.

For a proper morphism $f : X \to Y$ in $\textit{LC}_{qc}$ and $K$ in $D^+(X)$ we have $a_ Y^{-1}(Rf_*K) = Rf_{qc, *}(a_ X^{-1}K)$.

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