Lemma 21.17.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. 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

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

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

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