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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