The Stacks project

Lemma 36.10.7. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M^\bullet $ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_ X)$. Then $E$ is a perfect object of $D(\mathcal{O}_ X)$ if and only if $M^\bullet $ is perfect as an object of $D(A)$.

Proof. This is a logical consequence of Lemmas 36.10.2 and 36.10.4, Cohomology, Lemma 20.49.5, and More on Algebra, Lemma 15.74.2. $\square$

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