Lemma 21.17.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A bounded above complex of flat $\mathcal{O}$-modules is K-flat.

**Proof.**
Let $\mathcal{K}^\bullet $ be a bounded above complex of flat $\mathcal{O}$-modules. Let $\mathcal{L}^\bullet $ be an acyclic complex of $\mathcal{O}$-modules. Note that $\mathcal{L}^\bullet = \mathop{\mathrm{colim}}\nolimits _ m \tau _{\leq m}\mathcal{L}^\bullet $ where we take termwise colimits. Hence also

termwise. Hence to prove the complex on the left is acyclic it suffices to show each of the complexes on the right is acyclic. Since $\tau _{\leq m}\mathcal{L}^\bullet $ is acyclic this reduces us to the case where $\mathcal{L}^\bullet $ is bounded above. In this case the spectral sequence of Homology, Lemma 12.25.3 has

which is zero as $\mathcal{K}^ q$ is flat and $\mathcal{L}^\bullet $ acyclic. Hence we win. $\square$

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