Lemma 10.93.1. Let $M$ be an $R$-module. If $M$ is flat, Mittag-Leffler, and countably generated, then $M$ is projective.
Proof. By Lazard's theorem (Theorem 10.81.4), we can write $M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i$ for a directed system of finite free $R$-modules $(M_ i, f_{ij})$ indexed by a set $I$. By Lemma 10.92.1, we may assume $I$ is countable. Now let
\[ 0 \to N_1 \to N_2 \to N_3 \to 0 \]
be an exact sequence of $R$-modules. We must show that applying $\mathop{\mathrm{Hom}}\nolimits _ R(M, -)$ preserves exactness. Since $M_ i$ is finite free,
\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N_1) \to \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N_2) \to \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N_3) \to 0 \]
is exact for each $i$. Since $M$ is Mittag-Leffler, $(\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N_{1}))$ is a Mittag-Leffler inverse system. So by Lemma 10.86.4,
\[ 0 \to \mathop{\mathrm{lim}}\nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N_1) \to \mathop{\mathrm{lim}}\nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N_2) \to \mathop{\mathrm{lim}}\nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N_3) \to 0 \]
is exact. But for any $R$-module $N$ there is a functorial isomorphism $\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \cong \mathop{\mathrm{lim}}\nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N)$, so
\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N_1) \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N_2) \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N_3) \to 0 \]
is exact. $\square$
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