Lemma 21.25.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume there is an integer $N$ such that

$\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation 21.25.1,

$\mathcal{C}', \mathcal{O}', \mathcal{A}'$ satisfy the assumption of Situation 21.25.1,

$R^ pf_*\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ for $p \geq 0$ and $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$,

$R^ pf_*\mathcal{F} = 0$ for $p > N$ and $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$,

Then for $K$ in $D_\mathcal {A}(\mathcal{O})$ we have

$Rf_*K$ is in $D_{\mathcal{A}'}(\mathcal{O}')$,

the map $H^ j(Rf_*K) \to H^ j(Rf_*(\tau _{\geq -n}K))$ is an isomorphism for $j \geq N - n$.

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