Lemma 20.26.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. If $\mathcal{K}^\bullet $, $\mathcal{L}^\bullet $ are K-flat complexes of $\mathcal{O}_ X$-modules, then $\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet )$ is a K-flat complex of $\mathcal{O}_ X$-modules.

**Proof.**
Follows from the isomorphism

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

and the definition. $\square$

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