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. $\square$

Comment #9529 by nkym on

In the proof the last two $\mathcal{K}_1$ should be $\mathcal{K}_2$ and $\mathcal{K}_3$

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