   This is the final preprint version of a paper which appeared at
            Proc. Amer. Math. Soc. 141 (2013) 1177-1183.
                The published version is readable at
         http://dx.doi.org/10.1090/S0002-9939-2012-11453-4

EVERY MODULE IS AN INVERSE LIMIT OF INJECTIVES

GEORGE M. BERGMAN

Abstract. It is shown that any left module A over a ring R can be written as the intersection of a downward directed system of injective submodules of an injective module; equivalently, as an inverse limit of one-to-one homomorphisms of injectives. If R is left Noetherian, A can also be written as the inverse limit of a system of surjective homomorphisms of injectives.

Some questions are raised.

The flat modules over a ring are precisely the direct limits of projective modules [11] [6] [10, Theorem 2.4.34]. Which modules are, dually, inverse limits of injectives?

I sketched the answer in [1], but in view of the limited distribution of that item, it seems worthwhile to make the result more widely available. The construction from [1] is Theorem 2 below; the connecting maps there are inclusions. In Theorem 4, we shall see that the connecting maps can, alternatively, be taken to be onto if R is Noetherian on the appropriate side.

In §2 we ask some questions, in §3 we take some steps toward answering one of them, and in §4 we note what the proofs of our results tell us when applied to not necessarily injective modules.

Throughout, “ring” means associative ring with unit, and modules are unital. I am indebted to Pace Nielsen for pointing out the need to assume κ regular in

Lemma 1, and to the referee for some useful suggestions.

1. Main results

We will need the following generalization of the familiar observation ([4, Proposition I.3.1], [9, Proposition IV.3.7]) that a direct product of injective modules is injective. (It is a generalization because on taking κ > card(I), it yields that result.)

Lemma 1. Let R be a ring, κ an infinite regular cardinal such that every left ideal of R can be generated by < κ elements, and (M_{i})_{i∈I} a family of left R-modules. Let

(1) ∏_{I}^{κ} M_{i} = {x ∈ ∏_{I} M_{i} | x has support of cardinality < κ in I }.

Then if all M_{i} are injective, so is ∏_{I}^{κ} M_{i}.

2010 Mathematics Subject Classification. Primary: 16D50, 18A30. Secondary: 13C11, 16D90.

Key words and phrases. inverse limit of injective modules.

http://arXiv.org/abs/arXiv:1104.3173 . After publication of this note, updates, errata, related references etc., if found, will be recorded at /~gbergman/papers/.

Proof. To show ∏_{I}^{κ} M_{i} injective, it suffices by [4, Theorem I.3.2] [9, Lemma IV.3.8] to show that for every left ideal L of R, every module homomorphism h : L → ∏_{I}^{κ} M_{i} can be extended to all of R. By choice of κ, L has a generating set X of cardinality < κ, and by definition of ∏_{I}^{κ} M_{i}, the image under h of each member of X has support of cardinality < κ in I. Hence by regularity of κ, the union I_{0} ⊆ I of these supports has cardinality < κ. Clearly h(L) has support in I_{0}, hence h can be looked at as a homomorphism L → ∏_{I_{0}} M_{i}. As each M_{i} is injective, we can now lift h componentwise to a homomorphism R → ∏_{I_{0}} M_{i} ⊆ ∏_{I}^{κ} M_{i}, as desired. □

Theorem 2. Let R be a ring. Then every left R-module A can be written as the intersection of a downward directed system of injective submodules of an injective module; in other words, as the inverse limit of a system of injective modules and one-to-one homomorphisms.

Proof. Given A, choose an exact sequence of modules

(2) 0 → A → M → N

with M and N injective, as we may by [4, Theorem I.3.3], and call the second map f : M → N. Taking a cardinal κ as in the preceding lemma (for example, any infinite regular cardinal > |R|), and a set I of cardinality ≥ κ, one element of which we will denote 0, we define R-modules M_{i} (i ∈ I) by letting M_{0} = M, and M_{i} = N for i ≠ 0.

Now let P = ∏_{I}^{κ} M_{i}, and for each finite subset D ⊆ I − {0}, let P_{D} ⊆ P be the submodule of elements (x_{i})_{i∈I} such that for all i ∈ D, x_{i} = f(x_{0}). Clearly, each element of P_{D} is determined by its components at the indices in I − D, from which we see that P_{D} ≅ ∏_{I−D}^{κ} M_{i}; so by Lemma 1, P_{D} is injective. The family of submodules P_{D} is downward directed, since P_{D_{1}} ∩ P_{D_{2}} = P_{D_{1}∪D_{2}}.

Now ⋂_{D} P_{D} ⊆ P consists of the elements x ∈ P such that for all i ∈ I − {0}, x_{i} = f(x_{0}). Each such x is determined by its coordinate x_{0} ∈ M; but to lie in P, such an element must have support of cardinality < κ, which only happens if x_{0} ∈ ker f. Thus, ⋂_{D} P_{D} ≅ ker f = A. □

Note that in the construction of the above proof, if R is left Noetherian then κ can be taken to be ℵ_{0}, and I countable; so the intersection is over the finite subsets of a countable set, giving a countably indexed inverse system. In that situation, ∏_{I}^{κ} M_{i} is simply ⨁_{I} M_{i}, and Lemma 1 then says that the class of injective Rmodules is closed under direct sums (a known result, [12, Proposition 2.1]. In fact, that condition is necessary and sufficient for R to be left Noetherian [13, Theorem 1] [3, Theorem 1.1] [5, Theorem 20.1], a result variously called the Matlis-Papp Theorem, the Cartan-Eilenberg-Bass Theorem, and by other combinations of these names.) We shall use this closure under direct sums in the proof of our next theorem, along with the following fact.

(3)

There exists an inverse system, indexed by the first uncountable ordinal ω_{1}, of nonempty sets S_{α} and surjective maps f_{αβ} : S_{β} → S_{α} (α ≤ β ∈ ω_{1}), which has empty inverse limit [7] [8, §2] [2].

Again, we begin with a general lemma.

Lemma 3 (after [8, §3]; cf. [2, Corollary 8]). Suppose (S_{α}, f_{αβ})_{α≤β∈ω_{1}} is an inverse system of sets with the properties stated in (3), and N a left module over a ring R. To each α ∈ ω_{1}, let us associate the direct sum ⨁_{S_{α}} N of an S_{α}-tuple of copies of N; and for α ≤ β, let φ_{αβ} : ⨁_{S_{β}} N → ⨁_{S_{α}} N be the map sending (x_{j})_{j∈S_{β}} to the element (y_{i})_{i∈S_{α}} with components y_{i} = ∑_{f_{αβ}(j) = i} x_{j}.

Then each φ_{αβ} is surjective, but the inverse limit of the above system is the zero module.

Sketch of proof. We imitate the argument of [8] (where R was a field and N was R). Surjectivity of the φ_{αβ} is clear. Now suppose x belongs to the inverse limit, and let us write its components x^{(α)} ∈ ⨁_{S_{α}} N (α ∈ ω_{1}). For each α ∈ ω_{1}, let T_{α} ⊆ S_{α} be the (finite) support of x^{(α)}. We see that the cardinalities of the T_{α} are monotonically nondecreasing in α; hence, since ω_{1} has uncountable cofinality, the supremum of those cardinalities must be finite. (Indeed, for each n such that some |T_{α}| equals n, let us choose an α_{n} realizing this value. Then the at most countably many indices α_{n} have a supremum, α_{sup} ∈ ω_{1}, and the finite value |T_{α_{sup}}| will bound all |T_{α}|.)

Calling this supremum n, we see that the set of α ∈ ω_{1} such that |T_{α}| = n is an up-set in ω_{1}, and that whenever α ≤ β are both in this up-set, the connecting map f_{αβ} gives a bijection T_{β} → T_{α}. These n-element sets T_{α} thus lead to an n-tuple of elements of \underleftarrow{lim} S_{α}. But by assumption, that limit set is empty. Hence n = 0, so all x^{(α)} are 0, so x = 0. □

We can now prove

Theorem 4. Let R be a left Noetherian ring. Then every left R-module A can be written as the inverse limit of a system, indexed by ω_{1}, of surjective homomorphisms of injective modules.

Proof. Again let f : M → N be a homomorphism of injective left R-modules with kernel A. Let us take the inverse system of direct sums of copies of N described in Lemma 3, and append to each of these direct sums a copy of M, getting modules

(4) M ⊕ ⨁_{S_{α}} N (α ∈ ω_{1}),

which we connect using maps that act on M as the identity, and act on the direct sums of copies of N by the connecting morphisms of Lemma 3. Assuming for notational convenience that none of the S_{α} contains an element named 0, let us write the general element of (4) as (x_{i})_{i∈{0}∪S_{α}}, where x_{0} ∈ M and the other components are in N.

We now define, for each α ∈ ω_{1},

(5) P_{α} = {x = (x_{i})_{i∈{0}∪S_{α}} ∈ M ⊕ ⨁_{S_{α}} N | ∑_{i∈S_{α}} x_{i} = f(x_{0})}.

Note that for each α, if we choose any i_{0} ∈ S_{α}, then we can specify an element x ∈ P_{α} by choosing its components other than x_{i_{0}} to comprise an arbitrary member of M ⊕ ⨁_{S_{α}−{i_{0}}}N. The value of x_{i_{0}} will then be determined by the relation ∑_{i∈S_{α}} x_{i} = f(x_{0}). Thus, P_{α} ≅ M ⊕ ⨁_{S_{α}−{i_{0}}}N, a direct sum of injectives, so since R is left Noetherian, each P_{α} is injective. Clearly, the inverse system of surjective maps among the modules (4) induces an inverse system of surjective maps among the submodules (5).

In a member of \underleftarrow{lim}_{ω_{1}} P_{α}, the ⨁_{S_{α}}N-components, as α ranges over ω_{1}, will form a member of the inverse limit of the system of Lemma 3; hence these components must all be zero. Thus, the corresponding M-components must belong to ker f = A. Since the connecting maps on these components are the identity map of M, the inverse limit is A ⊆ M. □

(Incidentally, Theorem 2 or 4 yields a correct proof of [14, Lemma 3], the statement that ℤ is an inverse limit of injective abelian groups. The construction of [14] is similar to our proof of Theorem 2, but since the groups H_{j} used there are uniquely p-divisible for all odd primes p, their intersection is p-divisible, and so is not ℤ.)

For further examples of unexpectedly small inverse limits, see [2], [7], [8], [15]. Some questions about these are noted in [2, §§4-5].

2. Questions

Theorem 4 leaves open

Question 5. For non-left-Noetherian R, which left R-modules are inverse limits of systems of surjective maps of injective R-modules? (All?) Does the answer change if we restrict ourselves to systems indexed, as in Theorem 4, by ω_{1} ?

We noted following Theorem 2 that for R Noetherian, the construction used there involved a countable inverse system. This suggests

Question 6. For non-left-Noetherian R, which left R-modules are inverse limits of countable systems of one-to-one maps of injective R-modules? (All?)

On the other hand, the construction of Theorem 4 used uncountable inverse systems in all cases, and so leaves open

Question 7. For a (left Noetherian or general) ring R, which left R-modules are inverse limits of countable systems of surjective maps of injective left R-modules?

3. Partial results on Question 7

The answer to Question 7 cannot be either “all modules” or “only the injectives”, even for R = ℤ, as will be shown by Corollary 9 and Example 10, respectively.

In describing inverse limits, we have indexed our inverse systems so that the connecting maps go from higher- to lower-indexed objects. In direct limits, which appear beside inverse limits in the following preparatory lemma, we shall take the connecting maps to go from lower- to higher-indexed objects. (Thus, in each kind of limit, our index-sets are upward directed.)

Lemma 8. Let R be a ring. Let M be the inverse limit of a countable system of injective left R-modules M_{α} and surjective homomorphisms φ_{αβ} : M_{β} → M_{α} (α ≤ β, α, β ∈ I), and let N be the direct limit of a countable system of projective left R-modules N_{γ} and one-to-one homomorphisms ψ_{δγ} : N_{γ} → N_{δ} (γ ≤ δ, γ, δ ∈ J).

Then any homomorphism

(6) f : N_{γ} → M_{α}, where γ ∈ J, α ∈ I

can be factored

(7) N_{γ} → N → M → M_{α},

where the first and last maps are the canonical ones associated with the given direct and inverse limits (and the indices γ and α), while the middle map is an arbitrary module homomorphism.

Proof. Let us be given a homomorphism (6).

Recall that every countable directed partially ordered set (or more generally, any directed partially ordered set of countable cofinality) has a cofinal chain isomorphic to ω, and that a direct or inverse limit over the original set is isomorphic to the corresponding construction over any such chain. In our present situation, we can clearly take such a chain in I which begins with the index α of (6), and such a chain in J beginning with the index γ. Hence, replacing the two given systems with the systems determined by these chains, we may assume that our direct and inverse system are both indexed by ω, and name the map we wish to extend f_{0} : N_{0} → M_{0} (see (8) below).

Using the projectivity of N_{0} and the surjectivity of φ_{01} : M_{1} → M_{0}, we can now factor f_{0} as φ_{01} g_{0} for some homomorphism g_{0} : N_{0} → M_{1}, and then, similarly using the injectivity of M_{1} and one-one-ness of ψ_{10} : N_{0} → N_{1}, factor g_{0} as f_{1} ψ_{10} for some f_{1} : N_{1} → M_{1}. Thus, we get f_{0} = φ_{01} f_{1} ψ_{10}.

We now iterate this process, getting f_{2} : N_{2} → M_{2}, etc., where each composite N_{i−1} → N_{i} → M_{i} → M_{i−1} is the preceding map f_{i−1} :

(8)  [figure: ⋯; ⋯; N_{i}; M_{i}; f_{i}; ⋯; ⋯; N_{2}  N_{1} ψ_{21}  N_{0} ψ_{10} M_{2}  M_{1} φ_{12}  M_{0} . φ_{01}^{f_{2}  f_{1}  f_{0}}]

In particular, each composite N_{0} → N_{i} → M_{i} → M_{0} is our original map f_{0}. Using the universal properties of direct and inverse limits, we see that these maps induce a map N → M such that the composite N_{0} → N → M → M_{0} is f_{0}, as required. □

Now suppose that R is a commutative principal ideal domain. Then it is easy to verify that an R-module M is injective if and only if it is divisible, i.e., if and only if it is a homomorphic image, as an R-module, of some K-module, where K is the field of fractions of R. If, further, R ≠ K, and R has at most countably many primes, say p_{1}, p_{2}, … (where we allow repetitions in this list, in case R has only finitely many), then K is, as an R-module, the direct limit of a chain of inclusions of free R-modules of rank 1

(9) R ⊆ p_{1}^{−1}R ⊆ p_{1}^{−2} p_{2}^{−2}R ⊆ ⋯ ⊆ p_{1}^{−i} p_{2}^{−i}… p_{i}^{−i}R ⊆ · · · .

Hence we can apply Lemma 8 with K as N, calling the modules of (9) N_{0} ⊆ N_{1} ⊆ ⋯; but still letting (M_{α})_{α∈I} be an arbitary countable inverse system of injectives. For any α ∈ I, every x ∈ M_{α} is, of course, the image of the generator 1 ∈ R = N_{0} under some homomorphism f : N_{0} → M_{α}. Hence Lemma 8 tells us that x lies in the image of a homomorphism K = N → M → M_{α}; so the span in M of the images of all homomorphisms K → M maps surjectively to each M_{α}. For brevity and concreteness, we state this result below for R = ℤ.

Corollary 9. Let M be the inverse limit of a countable system of injective ℤmodules M_{α} and surjective homomorphisms φ_{αβ} : M_{β} → M_{α}. Let M_{div} be the largest divisible (equivalently, injective) submodule of M, namely, the sum of the images of all ℤ-module homomorphisms ℚ → M. Then M_{div} projects surjectively to each M_{α}; i.e., the composite maps M_{div} ↪ M → M_{α} are surjective. □

This shows that if M is nontrivial, it must have a sizable injective submodule. (In particular, M cannot be a nonzero finitely generated ℤ-module.) However, the following example shows that that submodule need not be all of M.

Example 10. A countable inverse system ⋯ → M_{2} → M_{1} → M_{0} of injective ℤ-modules and surjective homomorphisms, whose inverse limit M is not injective.

Construction and proof. For each n ≥ 0, let

(10)  M_{n} = ℚ ⊕ … ⊕ ℚ ⊕ (ℚ/ℤ) ⊕ … ⊕ (ℚ/ℤ) ⊕ …,

where the summands ℚ are indexed by i = 0, …, n − 1, and the ℚ/ℤ by the i ≥ n. Define connecting maps φ_{mn} : M_{n} → M_{m} (m ≤ n) to act componentwise; namely, as the identity map of ℚ, respectively, of ℚ/ℤ, on the components with indices i < m or i ≥ n, and as the quotient map ℚ → ℚ/ℤ on the n − m intermediate components.

It is not hard to verify that the inverse limit M of these modules can be identified with the submodule of ℚ^{ω} consisting of those elements all but finitely many of whose components lie in ℤ. (Given x ∈ M, its image in M_{0} will have all but finitely many components 0 ∈ ℚ/ℤ, and these zero components will correspond to the components of x which lie in ℤ.)

If we take an element x ∈ M and a positive integer k such that the entries of x in ℤ are not almost all divisible by k, then x is not divisible by k in M. Hence M is not a divisible group, i.e., is not injective. □

Returning to Corollary 9, we remark that its method of proof, applied to a countable inverse limit M of injective modules and surjective homomorphisms over any integral domain R, shows that M contains many “highly divisible” elements. For most R, this shows that not all R-modules can occur as such inverse limits.

4. Not necessarily injective modules

None of the constructions we have used to get an inverse system of modules from an exact sequence 0 → A → M → N are limited to the case where M and N are injective. Let us record what they give us in general.

Corollary 11 (to proofs of Theorems 2 and 4, and Example 10). Let R be a ring, M a class of left R-modules, κ an infinite regular cardinal such that M is closed under κ-restricted direct products ∏_{I}^{κ} M_{α}, and 0 → A → M → N any exact sequence of left R-modules with M, N ∈ M. Then

(a) A can be written as the inverse limit of a system of modules in M and one-to-one homomorphisms.

(b) If κ = ℵ_{0} (so that the hypothesis on M is that it is closed under direct sums), then A can be written as the inverse limit of an ω_{1}-indexed system of modules in M and surjective homomorphisms.

(c) If, again, κ = ℵ_{0}, then the submodule of M^{ω} consisting of those elements with all but finitely many components in A can be written as the inverse limit of a countable system of modules in M and surjective homomorphisms. □

So, for instance, by (b), for any ring R, any R-module which can be written as the kernel of a homomorphism of projective modules can also be written as the inverse limit of a system of projective modules and surjective homomorphisms.

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University of California, Berkeley, CA 94720-3840, USA

E-mail address: gbergman@math.berkeley.edu