The Stacks project

Lemma 15.57.8. Let $R$ be a ring. Let $0 \to K_1^\bullet \to K_2^\bullet \to K_3^\bullet \to 0$ be a short exact sequence of complexes. If $K_3^ n$ is flat for all $n \in \mathbf{Z}$ and two out of three of $K_ i^\bullet $ are K-flat, so is the third.

Proof. Let $L^\bullet $ be a complex of $R$-modules. Then

\[ 0 \to \text{Tot}(L^\bullet \otimes _ R K_1^\bullet ) \to \text{Tot}(L^\bullet \otimes _ R K_2^\bullet ) \to \text{Tot}(L^\bullet \otimes _ R K_3^\bullet ) \to 0 \]

is a short exact sequence of complexes. Namely, for each $n, m$ the sequence of modules $0 \to L^ n \otimes _ R K_1^ m \to L^ n \otimes _ R K_2^ m \to L^ n \otimes _ R K_3^ m \to 0$ is exact by Algebra, Lemma 10.38.12 and the sequence of complexes is a direct sum of these. Thus the lemma follows from this and the fact that in a short exact sequence of complexes if two out of three are acyclic, so is the third. $\square$


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