Lemma 20.44.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. If $\mathcal{F}^\bullet $ is a strictly perfect complex of $\mathcal{O}_ Y$-modules, then $f^*\mathcal{F}^\bullet $ is a strictly perfect complex of $\mathcal{O}_ X$-modules.

**Proof.**
The pullback of a finite free module is finite free. The functor $f^*$ is additive functor hence preserves direct summands. The lemma follows.
$\square$

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