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10.92 Countably generated Mittag-Leffler modules

It turns out that countably generated Mittag-Leffler modules have a particularly simple structure.

Lemma 10.92.1. Let M be an R-module. Write M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i where (M_ i, f_{ij}) is a directed system of finitely presented R-modules. If M is Mittag-Leffler and countably generated, then there is a directed countable subset I' \subset I such that M \cong \mathop{\mathrm{colim}}\nolimits _{i \in I'} M_ i.

Proof. Let x_1, x_2, \ldots be a countable set of generators for M. For each x_ n choose i \in I such that x_ n is in the image of the canonical map f_ i: M_ i \to M; let I'_{0} \subset I be the set of all these i. Now since M is Mittag-Leffler, for each i \in I'_{0} we can choose j \in I such that j \geq i and f_{ij}: M_ i \to M_ j factors through f_{ik}: M_ i \to M_ k for all k \geq i (condition (3) of Proposition 10.88.6); let I'_1 be the union of I'_0 with all of these j. Since I'_1 is a countable, we can enlarge it to a countable directed set I'_{2} \subset I. Now we can apply the same procedure to I'_{2} as we did to I'_{0} to get a new countable set I'_{3} \subset I. Then we enlarge I'_{3} to a countable directed set I'_{4}. Continuing in this way—adding in a j as in Proposition 10.88.6 (3) for each i \in I'_{\ell } if \ell is odd and enlarging I'_{\ell } to a directed set if \ell is even—we get a sequence of subsets I'_{\ell } \subset I for \ell \geq 0. The union I' = \bigcup I'_{\ell } satisfies:

  1. I' is countable and directed;

  2. each x_ n is in the image of f_ i: M_ i \to M for some i \in I';

  3. if i \in I', then there is j \in I' such that j \geq i and f_{ij}: M_ i \to M_ j factors through f_{ik}: M_ i \to M_ k for all k \in I with k \geq i. In particular \mathop{\mathrm{Ker}}(f_{ik}) \subset \mathop{\mathrm{Ker}}(f_{ij}) for k \geq i.

We claim that the canonical map \mathop{\mathrm{colim}}\nolimits _{i \in I'} M_ i \to \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i = M is an isomorphism. By (2) it is surjective. For injectivity, suppose x \in \mathop{\mathrm{colim}}\nolimits _{i \in I'} M_ i maps to 0 in \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i. Representing x by an element \tilde{x} \in M_ i for some i \in I', this means that f_{ik}(\tilde{x}) = 0 for some k \in I, k \geq i. But then by (3) there is j \in I', j \geq i, such that f_{ij}(\tilde{x}) = 0. Hence x = 0 in \mathop{\mathrm{colim}}\nolimits _{i \in I'} M_ i. \square

Lemma 10.92.1 implies that a countably generated Mittag-Leffler module M over R is the colimit of a system

M_1 \to M_2 \to M_3 \to M_4 \to \ldots

with each M_ n a finitely presented R-module. To see this argue as in the proof of Lemma 10.86.3 to see that a countable directed set has a cofinal subset isomorphic to (\mathbf{N}, \geq ). Suppose R = k[x_1, x_2, x_3, \ldots ] and M = R/(x_ i). Then M is finitely generated but not finitely presented, hence not Mittag-Leffler (see Example 10.91.1 part (1)). But of course you can write M = \mathop{\mathrm{colim}}\nolimits _ n M_ n by taking M_ n = R/(x_1, \ldots , x_ n), hence the condition that you can write M as such a limit does not imply that M is Mittag-Leffler.

Lemma 10.92.2. Let R be a ring. Let M be an R-module. Assume M is Mittag-Leffler and countably generated. For any R-module map f : P \to M with P finitely generated there exists an endomorphism \alpha : M \to M such that

  1. \alpha : M \to M factors through a finitely presented R-module, and

  2. \alpha \circ f = f.

Proof. Write M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i as a directed colimit of finitely presented R-modules with I countable, see Lemma 10.92.1. The transition maps are denoted f_{ij} and we use f_ i : M_ i \to M to denote the canonical maps into M. Set N = \prod _{s \in I} M_ s. Denote

M_ i^* = \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N) = \prod \nolimits _{s \in I} \mathop{\mathrm{Hom}}\nolimits _ R(M_ i, M_ s)

so that (M_ i^*) is an inverse system of R-modules over I. Note that \mathop{\mathrm{Hom}}\nolimits _ R(M, N) = \mathop{\mathrm{lim}}\nolimits M_ i^*. As M is Mittag-Leffler, we find for every i \in I an index k(i) \geq i such that

E_ i := \bigcap \nolimits _{i' \geq i} \mathop{\mathrm{Im}}(M_{i'}^* \to M_ i^*) = \mathop{\mathrm{Im}}(M_{k(i)}^* \to M_ i^*)

Choose and fix j \in I such that \mathop{\mathrm{Im}}(P \to M) \subset \mathop{\mathrm{Im}}(M_ j \to M). This is possible as P is finitely generated. Set k = k(j). Let x = (0, \ldots , 0, \text{id}_{M_ k}, 0, \ldots , 0) \in M_ k^* and note that this maps to y = (0, \ldots , 0, f_{jk}, 0, \ldots , 0) \in M_ j^*. By our choice of k we see that y \in E_ j. By Example 10.86.2 the transition maps E_ i \to E_ j are surjective for each i \geq j and \mathop{\mathrm{lim}}\nolimits E_ i = \mathop{\mathrm{lim}}\nolimits M_ i^* = \mathop{\mathrm{Hom}}\nolimits _ R(M, N). Hence Lemma 10.86.3 guarantees there exists an element z \in \mathop{\mathrm{Hom}}\nolimits _ R(M, N) which maps to y in E_ j \subset M_ j^*. Let z_ k be the kth component of z. Then z_ k : M \to M_ k is a homomorphism such that

\xymatrix{ M \ar[r]_{z_ k} & M_ k \\ M_ j \ar[ru]_{f_{jk}} \ar[u]^{f_ j} }

commutes. Let \alpha : M \to M be the composition f_ k \circ z_ k : M \to M_ k \to M. Then \alpha factors through a finitely presented module by construction and \alpha \circ f_ j = f_ j. Since the image of f is contained in the image of f_ j this also implies that \alpha \circ f = f. \square

We will see later (see Lemma 10.153.13) that Lemma 10.92.2 means that a countably generated Mittag-Leffler module over a henselian local ring is a direct sum of finitely presented modules.


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