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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