The Stacks project

Lemma 36.37.2. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then every perfect object of $D(\mathcal{O}_ X)$ can be represented by a bounded complex of finite locally free $\mathcal{O}_ X$-modules.

Proof. Let $E$ be a perfect object of $D(\mathcal{O}_ X)$. By Lemma 36.36.10 we see that $X$ has affine diagonal. Hence by Proposition 36.7.5 we can represent $E$ by a complex $\mathcal{F}^\bullet $ of quasi-coherent $\mathcal{O}_ X$-modules. Observe that $E$ is in $D^ b(\mathcal{O}_ X)$ because $X$ is quasi-compact. Hence $\tau _{\geq n}\mathcal{F}^\bullet $ is a bounded below complex of quasi-coherent $\mathcal{O}_ X$-modules which represents $E$ if $n \ll 0$. Thus we may apply Lemma 36.37.1 to conclude. $\square$

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