Lemma 21.22.10. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Assume there exists an integer $d \geq 0$ and a subset $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that

every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,

$H^ p(V, H^ q(E)) = 0$ for $p > d$, $q < 0$, and $V \in \mathcal{B}$.

Then the canonical map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.

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