Definition 21.47.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet $ be a complex of $\mathcal{O}$-modules. We say $\mathcal{E}^\bullet $ is perfect if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that for each $i$ there exists a morphism of complexes $\mathcal{E}_ i^\bullet \to \mathcal{E}^\bullet |_{U_ i}$ which is a quasi-isomorphism with $\mathcal{E}_ i^\bullet $ strictly perfect. An object $E$ of $D(\mathcal{O})$ is perfect if it can be represented by a perfect complex of $\mathcal{O}$-modules.
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