The Stacks project

Lemma 21.23.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Assume there exists a function $p(-) : \mathbf{Z} \to \mathbf{Z}$ and a subset $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that

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

  2. $H^ p(V, H^{m - p}(E)) = 0$ for $p > p(m)$ and $V \in \mathcal{B}$.

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 $D(\mathcal{O})$.

Proof. Apply Lemma 21.23.7 with $p(V, m) = p(m)$ and $\text{Cov}_ V$ equal to the set of coverings $\{ V_ i \to V\} $ with $V_ i \in \mathcal{B}$ for all $i$. $\square$

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