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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