Lemma 21.23.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume

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

for every $V \in \mathcal{B}$ there exist an integer $d_ V \geq 0$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that

\[ H^ p(V_ i, H^ q(E)) = 0 \text{ for } \{ V_ i \to V\} \in \text{Cov}_ V,\ p > d_ V, \text{ and }q < 0 \]

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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