Lemma 15.57.12. Let $R$ be a ring. For any complex $M^\bullet$ there exists a K-flat complex $K^\bullet$ and a quasi-isomorphism $K^\bullet \to M^\bullet$. Moreover each $K^ n$ is a flat $R$-module.

Proof. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\text{Mod}_ R)$ be the class of flat $R$-modules. By Derived Categories, Lemma 13.28.1 there exists a system $K_1^\bullet \to K_2^\bullet \to \ldots$ and a diagram

$\xymatrix{ K_1^\bullet \ar[d] \ar[r] & K_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}M^\bullet \ar[r] & \tau _{\leq 2}M^\bullet \ar[r] & \ldots }$

with the properties (1), (2), (3) listed in that lemma. These properties imply each complex $K_ i^\bullet$ is a bounded above complex of flat modules. Hence $K_ i^\bullet$ is K-flat by Lemma 15.57.9. The induced map $\mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet \to M^\bullet$ is a quasi-isomorphism by construction. The complex $\mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet$ is K-flat by Lemma 15.57.10. The final assertion of the lemma is true because the colimit of a system of flat modules is flat, see Algebra, Lemma 10.38.3. $\square$

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