Lemma 10.89.9. 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 10.89.5. 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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