The Stacks project

Lemma 36.11.8. Let $X$ be a locally Noetherian regular scheme. Then every object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ is perfect. If $X$ is quasi-compact, i.e., Noetherian regular, then conversely every perfect object of $D(\mathcal{O}_ X)$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$.

Proof. Let $K$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. To check that $K$ is perfect, we may work affine locally on $X$ (see Cohomology, Section 20.47). Then $K$ is perfect by Lemma 36.10.7 and More on Algebra, Lemma 15.74.14. The converse is Lemma 36.11.6. $\square$

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