The Stacks project

Lemma 21.17.11. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. For any complex $\mathcal{G}^\bullet $ there exists a $K$-flat complex $\mathcal{K}^\bullet $ whose terms are flat $\mathcal{O}$-modules and a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet $ which is termwise surjective.

Proof. Choose a diagram as in Lemma 21.17.10. Each complex $\mathcal{K}_ n^\bullet $ is a bounded above complex of flat modules, see Modules on Sites, Lemma 18.28.7. Hence $\mathcal{K}_ n^\bullet $ is K-flat by Lemma 21.17.8. Thus $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet $ is K-flat by Lemma 21.17.9. The induced map $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism and termwise surjective by construction. Property (3) of Lemma 21.17.10 shows that $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^ m$ is a direct sum of flat modules and hence flat which proves the final assertion. $\square$

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