Lemma 21.17.17. Let $(\mathcal{C}, \mathcal{O})$ be a ringed space. Let $a : \mathcal{K}^\bullet \to \mathcal{L}^\bullet $ be a map of complexes of $\mathcal{O}$-modules. If $\mathcal{K}^\bullet $ is K-flat, then there exist a complex $\mathcal{N}^\bullet $ and maps of complexes $b : \mathcal{K}^\bullet \to \mathcal{N}^\bullet $ and $c : \mathcal{N}^\bullet \to \mathcal{L}^\bullet $ such that

$\mathcal{N}^\bullet $ is K-flat,

$c$ is a quasi-isomorphism,

$a$ is homotopic to $c \circ b$.

If the terms of $\mathcal{K}^\bullet $ are flat, then we may choose $\mathcal{N}^\bullet $, $b$, and $c$ such that the same is true for $\mathcal{N}^\bullet $.

**Proof.**
We will use that the homotopy category $K(\textit{Mod}(\mathcal{O}))$ is a triangulated category, see Derived Categories, Proposition 13.10.3. Choose a distinguished triangle $\mathcal{K}^\bullet \to \mathcal{L}^\bullet \to \mathcal{C}^\bullet \to \mathcal{K}^\bullet [1]$. Choose a quasi-isomorphism $\mathcal{M}^\bullet \to \mathcal{C}^\bullet $ with $\mathcal{M}^\bullet $ K-flat with flat terms, see Lemma 21.17.11. By the axioms of triangulated categories, we may fit the composition $\mathcal{M}^\bullet \to \mathcal{C}^\bullet \to \mathcal{K}^\bullet [1]$ into a distinguished triangle $\mathcal{K}^\bullet \to \mathcal{N}^\bullet \to \mathcal{M}^\bullet \to \mathcal{K}^\bullet [1]$. By Lemma 21.17.6 we see that $\mathcal{N}^\bullet $ is K-flat. Again using the axioms of triangulated categories, we can choose a map $\mathcal{N}^\bullet \to \mathcal{L}^\bullet $ fitting into the following morphism of distinghuised triangles

\[ \xymatrix{ \mathcal{K}^\bullet \ar[r] \ar[d] & \mathcal{N}^\bullet \ar[r] \ar[d] & \mathcal{M}^\bullet \ar[r] \ar[d] & \mathcal{K}^\bullet [1] \ar[d] \\ \mathcal{K}^\bullet \ar[r] & \mathcal{L}^\bullet \ar[r] & \mathcal{C}^\bullet \ar[r] & \mathcal{K}^\bullet [1] } \]

Since two out of three of the arrows are quasi-isomorphisms, so is the third arrow $\mathcal{N}^\bullet \to \mathcal{L}^\bullet $ by the long exact sequences of cohomology associated to these distinguished triangles (or you can look at the image of this diagram in $D(\mathcal{O})$ and use Derived Categories, Lemma 13.4.3 if you like). This finishes the proof of (1), (2), and (3). To prove the final assertion, we may choose $\mathcal{N}^\bullet $ such that $\mathcal{N}^ n \cong \mathcal{M}^ n \oplus \mathcal{K}^ n$, see Derived Categories, Lemma 13.10.7. Hence we get the desired flatness if the terms of $\mathcal{K}^\bullet $ are flat.
$\square$

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