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 map E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} E of Derived Categories, Remark 13.34.4 is an isomorphism in D(\mathcal{O}).
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