Lemma 21.47.5 (08H6)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.47: Perfect complexes
Lemma 21.47.5 (cite)

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Lemma 21.47.5. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal {D})$ . If $E$ is perfect in $D(\mathcal{O}_\mathcal {D})$ , then $Lf^*E$ is perfect in $D(\mathcal{O}_\mathcal {C})$ .

Proof. This follows from Lemma 21.47.4, 21.46.5, and 21.45.3. $\square$

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