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The Stacks project

Lemma 20.37.8. Let (X, \mathcal{O}_ X) be a ringed space. Let E \in D(\mathcal{O}_ X). Assume there exist a function p(-) : \mathbf{Z} \to \mathbf{Z} and a set \mathcal{B} of opens of X such that

  1. every open in X has a covering whose members are elements of \mathcal{B}, and

  2. H^ p(U, H^{m - p}(E)) = 0 for p > p(m) and U \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}_ X).

Proof. Apply Lemma 20.37.6 with p(x, m) = p(m) and \mathfrak {U}_ x = \{ U \in \mathcal{B} \mid x \in U\} . \square


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