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*}

Theorem 10.92.3. Let $M$ be an $R$-module. Then $M$ is projective if and only it satisfies:

  1. $M$ is flat,

  2. $M$ is Mittag-Leffler,

  3. $M$ is a direct sum of countably generated $R$-modules.

Proof. First suppose $M$ is projective. Then $M$ is a direct summand of a free module, so $M$ is flat and Mittag-Leffler since these properties pass to direct summands. By Kaplansky's theorem (Theorem 10.83.5), $M$ satisfies (3).

Conversely, suppose $M$ satisfies (1)-(3). Since being flat and Mittag-Leffler passes to direct summands, $M$ is a direct sum of flat, Mittag-Leffler, countably generated $R$-modules. Lemma 10.92.1 implies $M$ is a direct sum of projective modules. Hence $M$ is projective. $\square$


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