Lemma 20.37.1. In the situation described above. Denote $\mathcal{H}^ m = H^ m(\mathcal{F}^\bullet )$ the $m$th cohomology sheaf. Let $\mathcal{B}$ be a set of open subsets of $X$. Let $d \in \mathbf{N}$. Assume

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

for every $U \in \mathcal{B}$ we have $H^ p(U, \mathcal{H}^ q) = 0$ for $p > d$ and $q < 0$

^{1}.

Then (20.37.0.1) is a quasi-isomorphism.

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