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 10.87.9. If $R$ is a ring and $M$, $N$ are Mittag-Leffler modules over $R$, then $M \otimes _ R N$ is a Mittag-Leffler module.

Proof. Write $M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i$ and $N = \mathop{\mathrm{colim}}\nolimits _{j \in J} N_ j$ as directed colimits of finitely presented $R$-modules. Denote $f_{ii'} : M_ i \to M_{i'}$ and $g_{jj'} : N_ j \to N_{j'}$ the transition maps. Then $M_ i \otimes _ R N_ j$ is a finitely presented $R$-module (see Lemma 10.11.14), and $M \otimes _ R N = \mathop{\mathrm{colim}}\nolimits _{(i, j) \in I \times J} M_ i \otimes _ R M_ j$. Pick $(i, j) \in I \times J$. By the definition of a Mittag-Leffler module we have Proposition 10.87.6 (3) for both systems. In other words there exist $i' \geq i$ and $j' \geq j$ such that for every choice of $i'' \geq i$ and $j'' \geq j$ there exist maps $a : M_{i''} \to M_{i'}$ and $b : M_{j''} \to M_{j'}$ such that $f_{ii'} = a \circ f_{ii''}$ and $g_{jj'} = b \circ g_{jj''}$. Then it is clear that $a \otimes b : M_{i''} \otimes _ R N_{j''} \to M_{i'} \otimes _ R N_{j'}$ serves the same purpose for the system $(M_ i \otimes _ R N_ j, f_{ii'} \otimes g_{jj'})$. Thus by the characterization Proposition 10.87.6 (3) we conclude that $M \otimes _ R N$ is Mittag-Leffler. $\square$

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