## 10.88 Mittag-Leffler modules

A Mittag-Leffler module is (very roughly) a module which can be written as a directed limit whose dual is a Mittag-Leffler system. To be able to give a precise definition we need to do a bit of work.

Definition 10.88.1. Let $(M_ i, f_{ij})$ be a directed system of $R$-modules. We say that $(M_ i, f_{ij})$ is a Mittag-Leffler directed system of modules if each $M_ i$ is an $R$-module of finite presentation and if for every $R$-module $N$, the inverse system

$(\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N), \mathop{\mathrm{Hom}}\nolimits _ R(f_{ij}, N))$

is Mittag-Leffler.

We are going to characterize those $R$-modules that are colimits of Mittag-Leffler directed systems of modules.

Definition 10.88.2. Let $f: M \to N$ and $g: M \to M'$ be maps of $R$-modules. Then we say $g$ dominates $f$ if for any $R$-module $Q$, we have $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ Q) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ Q)$.

It is enough to check this condition for finitely presented modules.

Lemma 10.88.3. Let $f: M \to N$ and $g: M \to M'$ be maps of $R$-modules. Then $g$ dominates $f$ if and only if for any finitely presented $R$-module $Q$, we have $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ Q) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ Q)$.

Proof. Suppose $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ Q) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ Q)$ for all finitely presented modules $Q$. If $Q$ is an arbitrary module, write $Q = \mathop{\mathrm{colim}}\nolimits _{i \in I} Q_ i$ as a colimit of a directed system of finitely presented modules $Q_ i$. Then $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_{Q_ i}) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_{Q_ i})$ for all $i$. Since taking directed colimits is exact and commutes with tensor product, it follows that $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ Q) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ Q)$. $\square$

Lemma 10.88.4. Let $f : M \to N$ and $g : M \to M'$ be maps of $R$-modules. Consider the pushout of $f$ and $g$,

$\xymatrix{ M \ar[r]_ f \ar[d]_ g & N \ar[d]^{g'} \\ M' \ar[r]^{f'} & N' }$

Then $g$ dominates $f$ if and only if $f'$ is universally injective.

Proof. Recall that $N'$ is $M' \oplus N$ modulo the submodule consisting of elements $(g(x), -f(x))$ for $x \in M$. From the construction of $N'$ we have a short exact sequence

$0 \to \mathop{\mathrm{Ker}}(f) \cap \mathop{\mathrm{Ker}}(g) \to \mathop{\mathrm{Ker}}(f) \to \mathop{\mathrm{Ker}}(f') \to 0.$

Since tensoring commutes with taking pushouts, we have such a short exact sequence

$0 \to \mathop{\mathrm{Ker}}(f \otimes \text{id}_ Q ) \cap \mathop{\mathrm{Ker}}(g \otimes \text{id}_ Q) \to \mathop{\mathrm{Ker}}(f \otimes \text{id}_ Q) \to \mathop{\mathrm{Ker}}(f' \otimes \text{id}_ Q) \to 0$

for every $R$-module $Q$. So $f'$ is universally injective if and only if $\mathop{\mathrm{Ker}}(f \otimes \text{id}_ Q ) \subset \mathop{\mathrm{Ker}}(g \otimes \text{id}_ Q)$ for every $Q$, if and only if $g$ dominates $f$. $\square$

The above definition of domination is sometimes related to the usual notion of domination of maps as the following lemma shows.

Lemma 10.88.5. Let $f: M \to N$ and $g: M \to M'$ be maps of $R$-modules. Suppose $\mathop{\mathrm{Coker}}(f)$ is of finite presentation. Then $g$ dominates $f$ if and only if $g$ factors through $f$, i.e. there exists a module map $h: N \to M'$ such that $g = h \circ f$.

Proof. Consider the pushout of $f$ and $g$ as in the statement of Lemma 10.88.4. From the construction of the pushout it follows that $\mathop{\mathrm{Coker}}(f') = \mathop{\mathrm{Coker}}(f)$, so $\mathop{\mathrm{Coker}}(f')$ is of finite presentation. Then by Lemma 10.82.4, $f'$ is universally injective if and only if

$0 \to M' \xrightarrow {f'} N' \to \mathop{\mathrm{Coker}}(f') \to 0$

splits. This is the case if and only if there is a map $h' : N' \to M'$ such that $h' \circ f' = \text{id}_{M'}$. From the universal property of the pushout, the existence of such an $h'$ is equivalent to $g$ factoring through $f$. $\square$

Proposition 10.88.6. Let $M$ be an $R$-module. Let $(M_ i, f_{ij})$ be a directed system of finitely presented $R$-modules, indexed by $I$, such that $M = \mathop{\mathrm{colim}}\nolimits M_ i$. Let $f_ i: M_ i \to M$ be the canonical map. The following are equivalent:

1. For every finitely presented $R$-module $P$ and module map $f: P \to M$, there exists a finitely presented $R$-module $Q$ and a module map $g: P \to Q$ such that $g$ and $f$ dominate each other, i.e., $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ N) = \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ N)$ for every $R$-module $N$.

2. For each $i \in I$, there exists $j \geq i$ such that $f_{ij}: M_ i \to M_ j$ dominates $f_ i: M_ i \to M$.

3. For each $i \in I$, there exists $j \geq i$ such that $f_{ij}: M_ i \to M_ j$ factors through $f_{ik}: M_ i \to M_ k$ for all $k \geq i$.

4. For every $R$-module $N$, the inverse system $(\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N), \mathop{\mathrm{Hom}}\nolimits _ R(f_{ij}, N))$ is Mittag-Leffler.

5. For $N = \prod _{s \in I} M_ s$, the inverse system $(\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N), \mathop{\mathrm{Hom}}\nolimits _ R(f_{ij}, N))$ is Mittag-Leffler.

Proof. First we prove the equivalence of (1) and (2). Suppose (1) holds and let $i \in I$. Corresponding to the map $f_ i: M_ i \to M$, we can choose $g: M_ i \to Q$ as in (1). Since $M_ i$ and $Q$ are of finite presentation, so is $\mathop{\mathrm{Coker}}(g)$. Then by Lemma 10.88.5, $f_ i : M_ i \to M$ factors through $g: M_ i \to Q$, say $f_ i = h \circ g$ for some $h: Q \to M$. Then since $Q$ is finitely presented, $h$ factors through $M_ j \to M$ for some $j \geq i$, say $h = f_ j \circ h'$ for some $h': Q \to M_ j$. In total we have a commutative diagram

$\xymatrix{ & M & \\ M_ i \ar[dr]_ g \ar[ur]^{f_ i} \ar[rr]^{f_{ij}} & & M_ j \ar[ul]_{f_ j} \\ & Q \ar[ur]_{h'} & }$

Thus $f_{ij}$ dominates $g$. But $g$ dominates $f_ i$, so $f_{ij}$ dominates $f_ i$.

Conversely, suppose (2) holds. Let $P$ be of finite presentation and $f: P \to M$ a module map. Then $f$ factors through $f_ i: M_ i \to M$ for some $i \in I$, say $f = f_ i \circ g'$ for some $g': P \to M_ i$. Choose by (2) a $j \geq i$ such that $f_{ij}$ dominates $f_ i$. We have a commutative diagram

$\xymatrix{ P \ar[d]_{g'} \ar[r]^{f} & M \\ M_ i \ar[ur]^{f_ i} \ar[r]_{f_{ij}} & M_ j \ar[u]_{f_ j} }$

From the diagram and the fact that $f_{ij}$ dominates $f_ i$, we find that $f$ and $f_{ij} \circ g'$ dominate each other. Hence taking $g = f_{ij} \circ g' : P \to M_ j$ works.

Next we prove (2) is equivalent to (3). Let $i \in I$. It is always true that $f_ i$ dominates $f_{ik}$ for $k \geq i$, since $f_ i$ factors through $f_{ik}$. If (2) holds, choose $j \geq i$ such that $f_{ij}$ dominates $f_ i$. Then since domination is a transitive relation, $f_{ij}$ dominates $f_{ik}$ for $k \geq i$. All $M_ i$ are of finite presentation, so $\mathop{\mathrm{Coker}}(f_{ik})$ is of finite presentation for $k \geq i$. By Lemma 10.88.5, $f_{ij}$ factors through $f_{ik}$ for all $k \geq i$. Thus (2) implies (3). On the other hand, if (3) holds then for any $R$-module $N$, $f_{ij} \otimes _ R \text{id}_ N$ factors through $f_{ik} \otimes _ R \text{id}_ N$ for $k \geq i$. So $\mathop{\mathrm{Ker}}(f_{ik} \otimes _ R \text{id}_ N) \subset \mathop{\mathrm{Ker}}(f_{ij} \otimes _ R \text{id}_ N)$ for $k \geq i$. But $\mathop{\mathrm{Ker}}(f_ i \otimes _ R \text{id}_ N: M_ i \otimes _ R N \to M \otimes _ R N)$ is the union of $\mathop{\mathrm{Ker}}(f_{ik} \otimes _ R \text{id}_ N)$ for $k \geq i$. Thus $\mathop{\mathrm{Ker}}(f_ i \otimes _ R \text{id}_ N) \subset \mathop{\mathrm{Ker}}(f_{ij} \otimes _ R \text{id}_ N)$ for any $R$-module $N$, which by definition means $f_{ij}$ dominates $f_ i$.

It is trivial that (3) implies (4) implies (5). We show (5) implies (3). Let $N = \prod _{s \in I} M_ s$. If (5) holds, then given $i \in I$ choose $j \geq i$ such that

$\mathop{\mathrm{Im}}( \mathop{\mathrm{Hom}}\nolimits (M_ j, N) \to \mathop{\mathrm{Hom}}\nolimits (M_ i, N)) = \mathop{\mathrm{Im}}( \mathop{\mathrm{Hom}}\nolimits (M_ k, N) \to \mathop{\mathrm{Hom}}\nolimits (M_ i, N))$

for all $k \geq j$. Passing the product over $s \in I$ outside of the $\mathop{\mathrm{Hom}}\nolimits$'s and looking at the maps on each component of the product, this says

$\mathop{\mathrm{Im}}( \mathop{\mathrm{Hom}}\nolimits (M_ j, M_ s) \to \mathop{\mathrm{Hom}}\nolimits (M_ i, M_ s)) = \mathop{\mathrm{Im}}( \mathop{\mathrm{Hom}}\nolimits (M_ k, M_ s) \to \mathop{\mathrm{Hom}}\nolimits (M_ i, M_ s))$

for all $k \geq j$ and $s \in I$. Taking $s = j$ we have

$\mathop{\mathrm{Im}}( \mathop{\mathrm{Hom}}\nolimits (M_ j, M_ j) \to \mathop{\mathrm{Hom}}\nolimits (M_ i, M_ j)) = \mathop{\mathrm{Im}}( \mathop{\mathrm{Hom}}\nolimits (M_ k, M_ j) \to \mathop{\mathrm{Hom}}\nolimits (M_ i, M_ j))$

for all $k \geq j$. Since $f_{ij}$ is the image of $\text{id} \in \mathop{\mathrm{Hom}}\nolimits (M_ j, M_ j)$ under $\mathop{\mathrm{Hom}}\nolimits (M_ j, M_ j) \to \mathop{\mathrm{Hom}}\nolimits (M_ i, M_ j)$, this shows that for any $k \geq j$ there is $h \in \mathop{\mathrm{Hom}}\nolimits (M_ k, M_ j)$ such that $f_{ij} = h \circ f_{ik}$. If $j \geq k$ then we can take $h = f_{kj}$. Hence (3) holds. $\square$

Definition 10.88.7. Let $M$ be an $R$-module. We say that $M$ is Mittag-Leffler if the equivalent conditions of Proposition 10.88.6 hold.

In particular a finitely presented module is Mittag-Leffler.

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 $N$.

Lemma 10.88.9. If $R$ is a ring and $M$, $N$ are Mittag-Leffler modules over $R$, then $M \otimes _ R N$ is a Mittag-Leffler module.

Proof. Write $M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i$ and $N = \mathop{\mathrm{colim}}\nolimits _{j \in J} N_ j$ as directed colimits of finitely presented $R$-modules. Denote $f_{ii'} : M_ i \to M_{i'}$ and $g_{jj'} : N_ j \to N_{j'}$ the transition maps. Then $M_ i \otimes _ R N_ j$ is a finitely presented $R$-module (see Lemma 10.12.14), and $M \otimes _ R N = \mathop{\mathrm{colim}}\nolimits _{(i, j) \in I \times J} M_ i \otimes _ R M_ j$. Pick $(i, j) \in I \times J$. By the definition of a Mittag-Leffler module we have Proposition 10.88.6 (3) for both systems. In other words there exist $i' \geq i$ and $j' \geq j$ such that for every choice of $i'' \geq i$ and $j'' \geq j$ there exist maps $a : M_{i''} \to M_{i'}$ and $b : M_{j''} \to M_{j'}$ such that $f_{ii'} = a \circ f_{ii''}$ and $g_{jj'} = b \circ g_{jj''}$. Then it is clear that $a \otimes b : M_{i''} \otimes _ R N_{j''} \to M_{i'} \otimes _ R N_{j'}$ serves the same purpose for the system $(M_ i \otimes _ R N_ j, f_{ii'} \otimes g_{jj'})$. Thus by the characterization Proposition 10.88.6 (3) we conclude that $M \otimes _ R N$ is Mittag-Leffler. $\square$

Lemma 10.88.10. Let $R$ be a ring and $M$ an $R$-module. Then $M$ is Mittag-Leffler if and only if for every finite free $R$-module $F$ and module map $f: F \to M$, there exists a finitely presented $R$-module $Q$ and a module map $g : F \to Q$ such that $g$ and $f$ dominate each other, i.e., $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ N) = \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ N)$ for every $R$-module $N$.

Proof. Since the condition is clear weaker than condition (1) of Proposition 10.88.6 we see that a Mittag-Leffler module satisfies the condition. Conversely, suppose that $M$ satisfies the condition and that $f : P \to M$ is an $R$-module map from a finitely presented $R$-module $P$ into $M$. Choose a surjection $F \to P$ where $F$ is a finite free $R$-module. By assumption we can find a map $F \to Q$ where $Q$ is a finitely presented $R$-module such that $F \to Q$ and $F \to M$ dominate each other. In particular, the kernel of $F \to Q$ contains the kernel of $F \to P$, hence we obtain an $R$-module map $g : P \to Q$ such that $F \to Q$ is equal to the composition $F \to P \to Q$. Let $N$ be any $R$-module and consider the commutative diagram

$\xymatrix{ F \otimes _ R N \ar[d] \ar[r] & Q \otimes _ R N \\ P \otimes _ R N \ar[ru] \ar[r] & M \otimes _ R N }$

By assumption the kernels of $F \otimes _ R N \to Q \otimes _ R N$ and $F \otimes _ R N \to M \otimes _ R N$ are equal. Hence, as $F \otimes _ R N \to P \otimes _ R N$ is surjective, also the kernels of $P \otimes _ R N \to Q \otimes _ R N$ and $P \otimes _ R N \to M \otimes _ R N$ are equal. $\square$

Lemma 10.88.11. Let $R \to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module. If $M$ is a Mittag-Leffler module over $S$ then $M$ is a Mittag-Leffler module over $R$.

Proof. Assume $M$ is a Mittag-Leffler module over $S$. Write $M = \mathop{\mathrm{colim}}\nolimits M_ i$ as a directed colimit of finitely presented $S$-modules $M_ i$. As $M$ is Mittag-Leffler over $S$ there exists for each $i$ an index $j \geq i$ such that for all $k \geq j$ there is a factorization $f_{ij} = h \circ f_{ik}$ (where $h$ depends on $i$, the choice of $j$ and $k$). Note that by Lemma 10.36.23 the modules $M_ i$ are also finitely presented as $R$-modules. Moreover, all the maps $f_{ij}, f_{ik}, h$ are maps of $R$-modules. Thus we see that the system $(M_ i, f_{ij})$ satisfies the same condition when viewed as a system of $R$-modules. Thus $M$ is Mittag-Leffler as an $R$-module. $\square$

Lemma 10.88.12. Let $R$ be a ring. Let $S = R/I$ for some finitely generated ideal $I$. Let $M$ be an $S$-module. Then $M$ is a Mittag-Leffler module over $R$ if and only if $M$ is a Mittag-Leffler module over $S$.

Proof. One implication follows from Lemma 10.88.11. To prove the other, assume $M$ is Mittag-Leffler as an $R$-module. Write $M = \mathop{\mathrm{colim}}\nolimits M_ i$ as a directed colimit of finitely presented $S$-modules. As $I$ is finitely generated, the ring $S$ is finite and finitely presented as an $R$-algebra, hence the modules $M_ i$ are finitely presented as $R$-modules, see Lemma 10.36.23. Next, let $N$ be any $S$-module. Note that for each $i$ we have $\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N) = \mathop{\mathrm{Hom}}\nolimits _ S(M_ i, N)$ as $R \to S$ is surjective. Hence the condition that the inverse system $(\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N))_ i$ satisfies Mittag-Leffler, implies that the system $(\mathop{\mathrm{Hom}}\nolimits _ S(M_ i, N))_ i$ satisfies Mittag-Leffler. Thus $M$ is Mittag-Leffler over $S$ by definition. $\square$

Remark 10.88.13. Let $R \to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module which is Mittag-Leffler as an $R$-module. Then it is in general not the case that $M$ is Mittag-Leffler as an $S$-module. For example suppose that $S$ is the ring of dual numbers over $R$, i.e., $S = R \oplus R\epsilon$ with $\epsilon ^2 = 0$. Then an $S$-module consists of an $R$-module $M$ endowed with a square zero $R$-linear endomorphism $\epsilon : M \to M$. Now suppose that $M_0$ is an $R$-module which is not Mittag-Leffler. Choose a presentation $F_1 \xrightarrow {u} F_0 \to M_0 \to 0$ with $F_1$ and $F_0$ free $R$-modules. Set $M = F_1 \oplus F_0$ with

$\epsilon = \left( \begin{matrix} 0 & 0 \\ u & 0 \end{matrix} \right) : M \longrightarrow M.$

Then $M/\epsilon M \cong F_1 \oplus M_0$ is not Mittag-Leffler over $R = S/\epsilon S$, hence not Mittag-Leffler over $S$ (see Lemma 10.88.12). On the other hand, $M/\epsilon M = M \otimes _ S S/\epsilon S$ which would be Mittag-Leffler over $S$ if $M$ was, see Lemma 10.88.9.

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