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The Stacks project

[Proposition 3.13, Spaltenstein]

Lemma 20.37.7. Let (X, \mathcal{O}_ X) be a ringed space. Let E \in D(\mathcal{O}_ X). Assume that for every x \in X there exist an integer d_ x \geq 0 and a fundamental system \mathfrak {U}_ x of open neighbourhoods of x such that

H^ p(U, H^ q(E)) = 0 \text{ for } U \in \mathfrak {U}_ x,\ p > d_ x, \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}_ X).

Proof. This follows from Lemma 20.37.6 with p(x, m) = d_ x + \max (0, m). \square


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