The Stacks project

Lemma 73.13.5. Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_ X)$. Then $E$ is a perfect object of $D(\mathcal{O}_ X)$ if and only if $\epsilon ^*E$ is a perfect object of $D(\mathcal{O}_{\acute{e}tale})$. Here $\epsilon $ is as in (73.4.0.1).

Proof. The easy implication follows from the general result contained in Cohomology on Sites, Lemma 21.45.5. For the converse, we can use the equivalence of Cohomology on Sites, Lemma 21.45.4 and the corresponding results for pseudo-coherent and complexes of finite tor dimension, namely Lemmas 73.13.2 and 73.13.3. Some details omitted. $\square$


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