Lemma 20.49.6. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\mathcal{O}_ Y)$. If $E$ is perfect in $D(\mathcal{O}_ Y)$, then $Lf^*E$ is perfect in $D(\mathcal{O}_ X)$.

**Proof.**
This follows from Lemma 20.49.5, 20.48.4, and 20.47.3. (An alternative proof is to copy the proof of Lemma 20.47.3.)
$\square$

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