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 20.47.6. 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 20.47.5. 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 20.47.5 and Lemmas 36.12.1 and 36.12.2. $\square$


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