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 10.88.9. If $M = \mathop{\mathrm{colim}}\nolimits M_ i$ is the colimit of a directed system of Mittag-Leffler $R$-modules $M_ i$ with universally injective transition maps, then $M$ is Mittag-Leffler.

Proof. Let $(Q_{\alpha })_{\alpha \in A}$ be a family of $R$-modules. We have to show that $M \otimes _ R (\prod Q_\alpha ) \to \prod M \otimes _ R Q_\alpha $ is injective and we know that $M_ i \otimes _ R (\prod Q_\alpha ) \to \prod M_ i \otimes _ R Q_\alpha $ is injective for each $i$, see Proposition 10.88.5. Since $\otimes $ commutes with filtered colimits, it suffices to show that $\prod M_ i \otimes _ R Q_\alpha \to \prod M \otimes _ R Q_\alpha $ is injective. This is clear as each of the maps $M_ i \otimes _ R Q_\alpha \to M \otimes _ R Q_\alpha $ is injective by our assumption that the transition maps are universally injective. $\square$


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