Lemma 36.12.4 (09UG)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.12: Descent finiteness properties of complexes
Lemma 36.12.4 (cite)

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Lemma 36.12.4. Let $\{ f_ i : X_ i \to X\}$ be an fpqc covering of schemes. Let $E \in D(\mathcal{O}_ X)$ . Then $E$ is perfect if and only if each $Lf_ i^*E$ is perfect.

Proof. Pullback always preserves perfect complexes, see Cohomology, Lemma 20.49.6. Conversely, assume that $Lf_ i^*E$ is perfect for all $i$ . Then the cohomology sheaves of each $Lf_ i^*E$ are quasi-coherent, see Lemma 36.10.1 and Cohomology, Lemma 20.49.5. Since the morphisms $f_ i$ is flat we see that $H^ p(Lf_ i^*E) = f_ i^*H^ p(E)$ . Thus the cohomology sheaves of $E$ are quasi-coherent by Descent, Proposition 35.5.2. Having said this the lemma follows formally from Cohomology, Lemma 20.49.5 and Lemmas 36.12.1 and 36.12.2. $\square$

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