The Stacks project

Lemma 15.59.10. Let $R$ be a ring. For any complex $M^\bullet $ there exists a K-flat complex $K^\bullet $ whose terms are flat $R$-modules and a quasi-isomorphism $K^\bullet \to M^\bullet $ which is termwise surjective.

Proof. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\text{Mod}_ R)$ be the class of flat $R$-modules. By Derived Categories, Lemma 13.29.1 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 15.59.7. The induced map $\mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet \to M^\bullet $ is a quasi-isomorphism and termwise surjective by construction. The complex $\mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet $ is K-flat by Lemma 15.59.8. The terms $\mathop{\mathrm{colim}}\nolimits K_ i^ n$ are flat because filtered colimits of flat modules are flat, see Algebra, Lemma 10.39.3. $\square$

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