The Stacks project

Lemma 36.11.8. Let $X$ be a Noetherian regular scheme of finite dimension. Then every object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ is perfect and conversely every perfect object of $D(\mathcal{O}_ X)$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$.

Proof. Combine More on Algebra, Lemma 15.73.14 with Lemma 36.10.7. $\square$

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