The Stacks project

Lemma 21.47.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$.

  1. If $\mathcal{C}$ has a final object $X$ and there exist a covering $\{ U_ i \to X\} $, strictly perfect complexes $\mathcal{E}_ i^\bullet $ of $\mathcal{O}_{U_ i}$-modules, and isomorphisms $\alpha _ i : \mathcal{E}_ i^\bullet \to E|_{U_ i}$ in $D(\mathcal{O}_{U_ i})$, then $E$ is perfect.

  2. If $E$ is perfect, then any complex representing $E$ is perfect.

Proof. Identical to the proof of Lemma 21.45.2. $\square$

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