The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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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