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The Stacks project

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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