Lemma 20.26.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $0 \to \mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \mathcal{K}_3^\bullet \to 0$ be a short exact sequence of complexes such that the terms of $\mathcal{K}_3^\bullet$ are flat $\mathcal{O}_ X$-modules. If two out of three of $\mathcal{K}_ i^\bullet$ are K-flat, so is the third.

Proof. By Modules, Lemma 17.17.7 for every complex $\mathcal{L}^\bullet$ we obtain a short exact sequence

$0 \to \text{Tot}(\mathcal{L}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{K}_1^\bullet ) \to \text{Tot}(\mathcal{L}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{K}_1^\bullet ) \to \text{Tot}(\mathcal{L}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{K}_1^\bullet ) \to 0$

of complexes. Hence the lemma follows from the long exact sequence of cohomology sheaves and the definition of K-flat complexes. $\square$

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