Lemma 10.95.1. Let R \to S be a faithfully flat ring map. Let M be an R-module. If the S-module M \otimes _ R S is Mittag-Leffler, then M is Mittag-Leffler.
Email from Juan Pablo Acosta Lopez dated 12/20/14.
Proof. Write M = \mathop{\mathrm{colim}}\nolimits _{i\in I} M_ i as a directed colimit of finitely presented R-modules M_ i. Using Proposition 10.88.6, we see that we have to prove that for each i \in I there exists i \leq j, j\in I such that M_ i\rightarrow M_ j dominates M_ i\rightarrow M.
Take N the pushout
\xymatrix{ M_ i \ar[r] \ar[d] & M_ j \ar[d] \\ M \ar[r] & N }
Then the lemma is equivalent to the existence of j such that M_ j\rightarrow N is universally injective, see Lemma 10.88.4. Observe that the tensorization by S
\xymatrix{ M_ i\otimes _ R S \ar[r] \ar[d] & M_ j\otimes _ R S \ar[d] \\ M\otimes _ R S \ar[r] & N\otimes _ R S }
Is a pushout diagram. So because M \otimes _ R S = \mathop{\mathrm{colim}}\nolimits _{i\in I} M_ i \otimes _ R S expresses M\otimes _ R S as a colimit of S-modules of finite presentation, and M\otimes _ R S is Mittag-Leffler, there exists j \geq i such that M_ j\otimes _ R S\rightarrow N\otimes _ R S is universally injective. So using that R\rightarrow S is faithfully flat we conclude that M_ j\rightarrow N is universally injective too. \square
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