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The Stacks project

Lemma 21.44.8. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let U be an object of \mathcal{C}. Let \mathcal{E}^\bullet , \mathcal{F}^\bullet be complexes of \mathcal{O}_ U-modules with \mathcal{E}^\bullet strictly perfect.

  1. For any element \alpha \in \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet ) there exists a covering \{ U_ i \to U\} 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}.

  2. 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}_ U)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet ) is zero, there exists a covering \{ U_ i \to U\} such that \alpha |_{U_ i} is homotopic to zero.

Proof. Proof of (1). By the construction of the derived category we can find a quasi-isomorphism f : \mathcal{F}^\bullet \to \mathcal{G}^\bullet and a map of complexes \beta : \mathcal{E}^\bullet \to \mathcal{G}^\bullet such that \alpha = f^{-1}\beta . Thus the result follows from Lemma 21.44.7. We omit the proof of (2). \square


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