The Stacks project

Lemma 20.36.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $E \in D(\mathcal{O}_ X)$. Assume there exist an integer $d \geq 0$ and a basis $\mathcal{B}$ for the topology of $X$ such that

\[ H^ p(U, H^ q(E)) = 0 \text{ for } U \in \mathcal{B},\ p > d, \text{ and }q < 0 \]

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

Proof. Apply Lemma 20.36.7 with $d_ x = d$ and $\mathfrak {U}_ x = \{ U \in \mathcal{B} \mid x \in U\} $. $\square$

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