The Stacks project

Lemma 21.44.4. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. If $\mathcal{F}^\bullet $ is a strictly perfect complex of $\mathcal{O}_\mathcal {D}$-modules, then $f^*\mathcal{F}^\bullet $ is a strictly perfect complex of $\mathcal{O}_\mathcal {C}$-modules.

Proof. We have seen in Modules on Sites, Lemma 18.17.2 that the pullback of a finite free module is finite free. The functor $f^*$ is additive functor hence preserves direct summands. The lemma follows. $\square$

Comments (2)

Comment #1259 by typo on

By the reference, the result seems to be true for a morphism of ringed topoi.

Maybe should extend the definition of a strictly perfect complex (Tag 08FL) to ringed topoi.

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