The Stacks project

Lemma 21.17.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. 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 $\mathcal{O}$-modules. If two out of three of $\mathcal{K}_ i^\bullet $ are K-flat, so is the third.

Proof. By Modules on Sites, Lemma 18.28.9 for every complex $\mathcal{L}^\bullet $ we obtain a short exact sequence

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

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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