The Stacks project

Definition 20.49.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. We say $\mathcal{E}^\bullet $ is perfect if there exists an open covering $X = \bigcup U_ i$ 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 $ a strictly perfect complex of $\mathcal{O}_{U_ i}$-modules. An object $E$ of $D(\mathcal{O}_ X)$ is perfect if it can be represented by a perfect complex of $\mathcal{O}_ X$-modules.

Comments (2)

Comment #8373 by Nicolás on

It might be useful to add a small remark like the one right after Definition 20.47.1, about perfect complexes being bounded. Something like

"If is quasi-compact, then a perfect object of is in . But this need not be the case if is not quasi-compact."

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  • 2 comment(s) on Section 20.49: Perfect complexes

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