Lemma 10.153.13. Let R be a henselian local ring. Any countably generated Mittag-Leffler module over R is a direct sum of finitely presented R-modules.
Proof. Let M be a countably generated and Mittag-Leffler R-module. We claim that for any element x \in M there exists a direct sum decomposition M = N \oplus K with x \in N, the module N finitely presented, and K Mittag-Leffler.
Suppose the claim is true. Choose generators x_1, x_2, x_3, \ldots of M. By the claim we can inductively find direct sum decompositions
with N_ i finitely presented, x_1, \ldots , x_ n \in N_1 \oplus \ldots \oplus N_ n, and K_ n Mittag-Leffler. Repeating ad infinitum we see that M = \bigoplus N_ i.
We still have to prove the claim. Let x \in M. By Lemma 10.92.2 there exists an endomorphism \alpha : M \to M such that \alpha factors through a finitely presented module, and \alpha (x) = x. Say \alpha factors as
Set a = \pi \circ \alpha \circ i : P \to P, so i \circ a \circ \pi = \alpha ^3. By Lemma 10.16.2 there exists a monic polynomial P \in R[T] such that P(a) = 0. Note that this implies formally that \alpha ^2 P(\alpha ) = 0. Hence we may think of M as a module over R[T]/(T^2P). Assume that x \not= 0. Then \alpha (x) = x implies that 0 = \alpha ^2P(\alpha )x = P(1)x hence P(1) = 0 in R/I where I = \{ r \in R \mid rx = 0\} is the annihilator of x. As x \not= 0 we see I \subset \mathfrak m_ R, hence 1 is a root of \overline{P} = P \bmod \mathfrak m_ R \in R/\mathfrak m_ R[T]. As R is henselian we can find a factorization
for some Q_1, Q_2 \in R[T] with Q_2 = (T - 1)^ e \bmod \mathfrak m_ R R[T] and Q_1(1) \not= 0 \bmod \mathfrak m_ R, see Lemma 10.153.3. Let N = \mathop{\mathrm{Im}}(\alpha ^2Q_1(\alpha ) : M \to M) and K = \mathop{\mathrm{Im}}(Q_2(\alpha ) : M \to M). As T^2Q_1 and Q_2 generate the unit ideal of R[T] we get a direct sum decomposition M = N \oplus K. Moreover, Q_2 acts as zero on N and T^2Q_1 acts as zero on K. Note that N is a quotient of P hence is finitely generated. Also x \in N because \alpha ^2Q_1(\alpha )x = Q_1(1)x and Q_1(1) is a unit in R. By Lemma 10.89.10 the modules N and K are Mittag-Leffler. Finally, the finitely generated module N is finitely presented as a finitely generated Mittag-Leffler module is finitely presented, see Example 10.91.1 part (1). \square
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