The Stacks project

Remark 10.88.8. Let $M$ be a flat $R$-module. By Lazard's theorem (Theorem 10.81.4) we can write $M = \mathop{\mathrm{colim}}\nolimits M_ i$ as the colimit of a directed system $(M_ i, f_{ij})$ where the $M_ i$ are free finite $R$-modules. For $M$ to be Mittag-Leffler, it is enough for the inverse system of duals $(\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, R), \mathop{\mathrm{Hom}}\nolimits _ R(f_{ij}, R))$ to be Mittag-Leffler. This follows from criterion (4) of Proposition 10.88.6 and the fact that for a free finite $R$-module $F$, there is a functorial isomorphism $\mathop{\mathrm{Hom}}\nolimits _ R(F, R) \otimes _ R N \cong \mathop{\mathrm{Hom}}\nolimits _ R(F, N)$ for any $R$-module $N$.