Lemma 20.26.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \ldots$ be a system of K-flat complexes. Then $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{K}_ i^\bullet$ is K-flat.

Proof. Because we are taking termwise colimits it is clear that

$\mathop{\mathrm{colim}}\nolimits _ i \text{Tot}( \mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{K}_ i^\bullet ) = \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathop{\mathrm{colim}}\nolimits _ i \mathcal{K}_ i^\bullet )$

Hence the lemma follows from the fact that filtered colimits are exact. $\square$

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