The Stacks project

Lemma 20.26.11. Let $(X, \mathcal{O}_ X)$ be a ringed space. For any complex $\mathcal{G}^\bullet $ there exists a $K$-flat complex $\mathcal{K}^\bullet $ and a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet $. Moreover, each $\mathcal{K}^ n$ is a flat $\mathcal{O}_ X$-module.

Proof. Choose a diagram as in Lemma 20.26.10. Each complex $\mathcal{K}_ n^\bullet $ is a bounded above complex of flat modules, see Modules, Lemma 17.16.5. Hence $\mathcal{K}_ n^\bullet $ is K-flat by Lemma 20.26.8. The induced map $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism by construction. Thus $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet $ is K-flat by Lemma 20.26.9. Property (3) of Lemma 20.26.10 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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