The Stacks project

Lemma 21.19.1. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be a ringed topos. For any complex of $\mathcal{O}_\mathcal {C}$-modules $\mathcal{G}^\bullet $ there exists a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet $ such that $f^*\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}_\mathcal {D}$-modules for any morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ of ringed topoi.

Proof. In the proof of Lemma 21.18.10 we find a quasi-isomorphism $\mathcal{K}^\bullet = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{K}_ i^\bullet \to \mathcal{G}^\bullet $ where each $\mathcal{K}_ i^\bullet $ is a bounded above complex of flat $\mathcal{O}_\mathcal {C}$-modules. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be a morphism of ringed topoi. By Modules on Sites, Lemma 18.38.1 we see that $f^*\mathcal{F}_ i^\bullet $ is a bounded above complex of flat $\mathcal{O}_\mathcal {D}$-modules. Hence $f^*\mathcal{K}^\bullet = \mathop{\mathrm{colim}}\nolimits _ i f^*\mathcal{K}_ i^\bullet $ is K-flat by Lemmas 21.18.7 and 21.18.8. $\square$

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