Lemma 21.17.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $\mathcal{K}^\bullet $, $\mathcal{L}^\bullet $ are K-flat complexes of $\mathcal{O}$-modules, then $\text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet )$ is a K-flat complex of $\mathcal{O}$-modules.

**Proof.**
Follows from the isomorphism

\[ \text{Tot}(\mathcal{M}^\bullet \otimes _\mathcal {O} \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet )) = \text{Tot}(\text{Tot}(\mathcal{M}^\bullet \otimes _\mathcal {O} \mathcal{K}^\bullet ) \otimes _\mathcal {O} \mathcal{L}^\bullet ) \]

and the definition. $\square$

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