The Stacks project

Lemma 36.11.8. Let $X$ be a Noetherian regular scheme. 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. Since $X$ is Noetherian, it is in particular quasi-compact. Hence being bounded can be checked on the members of a finite affine open covering of $X$. This remark, plus similar remarks on having coherent cohomology sheaves and being perfect, shows that it suffices to prove the lemma when $X$ is affine. This case translated via Lemma 36.10.7 into More on Algebra, Lemma 15.74.14. $\square$

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