Lemma 20.42.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $, $\mathcal{F}^\bullet $ be complexes of $\mathcal{O}_ X$-modules with $\mathcal{E}^\bullet $ strictly perfect.

For any element $\alpha \in \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ there exists an open covering $X = \bigcup U_ i$ such that $\alpha |_{U_ i}$ is given by a morphism of complexes $\alpha _ i : \mathcal{E}^\bullet |_{U_ i} \to \mathcal{F}^\bullet |_{U_ i}$.

Given a morphism of complexes $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet $ whose image in the group $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is zero, there exists an open covering $X = \bigcup U_ i$ such that $\alpha |_{U_ i}$ is homotopic to zero.

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