Contemporary Mathematics Volume 00, 0000

On the Abundance of ℵ_{1}-Separable Modules

BIRGE ZIMMERMANN HUISGEN

Dedicated to the memory of Bob Warfield

Abstract. It is shown that ℵ_{1}-separable modules are intimately linked with a class of short exact sequences which, next-to-best after split sequences, admit pointwise sections and retractions. Moreover, a rich supply of ℵ_{1}-separable modules which are not direct sums of countably generated components is made accessible. In fact, given any module M which fails to be Σ-pure injective, the direct product M^{ℵ_{1}} is proved to contain modules of the above ilk. The pertinent construction yields the equivalence of the following statements for a ring R: (i) The pure global dimension of R vanishes; (ii) R has ‘nontrivial’ ℵ_{1}-separable modules; (iii) R has nonsplit short exact sequences with pointwise sections and retractions.

1. Introduction.

Call a module ℵ_{1}-separable if each of its countable subsets belongs to a countably generated direct summand; moreover, label an ℵ_{1}-separable module trivial if it is a direct sum of countably generated components (here ℵ_{1} denotes the first uncountable cardinal number). Responding to an existence question for abelian groups which had been open for an extended period, Hill [13] was the first to construct an ℵ_{1}-separable abelian group which is not trivial. In [7], Griffith modified and generalized Hill’s construction technique to show that, given a ring R, all flat, torsionless ℵ_{1}-separable left R-modules are trivial precisely if R is left perfect. One of the goals of this paper is to demonstrate how ubiquitous nontrivial ℵ_{1}-separable modules really are. In fact, we will prove the equivalence of the following two statements:

(1) R has left pure global dimension zero (equivalently, all left R-modules are direct sums of finitely generated submodules);

(2) All ℵ_{1}-separable left R-modules are trivial.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page

1991 Mathematics Subject Classification. 16D70, 16D80, 16E10, 16G10.

This research was partially supported by a grant from the National Science Foundation.

This paper is in final form, and no version of it will be submitted for publication elsewhere.

Furthermore, we will see that, in case the left pure global dimension of R exceeds zero, the category R-Mod has an extremely rich supply of non-trivial ℵ_{1}- separable left modules; namely, given any left R-module M which fails to be Σ-pure injective, the direct product of ℵ_{1} copies of M contains an ℵ_{1}-separable module which is not trivial. Let us return to the classical framework for a moment: As a consequence of our results, we will see that, given a module M over a Dedekind domain, the product M^{ℵ_{1}} contains a non-trivial ℵ_{1}-separable module precisely when M is unbounded.

Our second objective is to illustrate the significance of non-trivial ℵ_{1}-separable modules in a wide context. In fact, the ℵ_{1}-separable modules are closely tied to ‘locally split’ exact sequences, and such sequences, in turn, abound as tools of proof in the literature (if not under this name); see Section 3 for specific instances. We call a short exact sequence

0 ⟶ A ⟶^{f} B ⟶^{g} C ⟶ 0

locally split if for every finite subset A′ of A and every finite subset C′ of C, there exist homomorphisms f′ : B ⟶ A, resp., g′ : C ⟶ B, such that f′f|_{A′} = id_{A′}, resp., gg′|_{C′} = id_{C′} (this terminology goes back to Azumaya [1]). To demonstrate the strength of the tie between locally split exact sequences and ℵ_{1}-separable modules, we will add another equivalent statement to (1) and (2) above, namely:

(3) All locally split exact sequences of left R-modules split.

Acknowledgement. This paper originated with a question of K. M. Rangaswamy who conjectured that the statements (1) and (3) of the introduction are equivalent; the author would like to express her gratitude for a stimulating correspondence and conversations with him and G. D. Abrams on this topic. Moreover, this paper owes much to ideas of P. A. Griffith [7], L. Gruson and C. U. Jensen [10], W. Zimmermann [22], and, by extension, to P. Hill’s pioneering example [13].

More terminology and some supporting machinery.

Let M be a (unitary) left module over an associative ring R with identity. Recall from [9] and [22] that a matrix subgroup of M is an additive subgroup of the form

[A, α]M = {m = m_{α} ∈ M | ∃ (m_{j})_{j∈J\{α}} such that ∑_{j∈J} a_{ij}m_{j} = 0 ∀ i ∈ I},

where A = (a_{ij})_{i∈I,j∈J} is a row-finite matrix with coefficients in R and α ∈ J; in other words, [A, α]M is the α-th projection of the solution set in M^{J} of the homogeneous system ∑_{j∈J} a_{ij}X_{j} = 0 for i ∈ I. Call [A, α]M a finite matrix subgroup whenever I and J are finite. For prototypical examples of (finite) matrix subgroups we refer to [22] and [15].

The (for us) crucial connection between matrix subgroups and Σ-pure injectivity (= Σ-pure algebraic compactness [20]) comes from [10] and [22].

ABUNDANCE OF ℵ_{1}–SEPARABLE MODULES 3

Theorem A. ([10, Théorème] and [22, Folgerung 3.4]) For a left R-module M the following statements are equivalent:

(i) M is Σ-pure injective.

(ii) M has the descending chain condition for matrix subgroups.

(iii) M has the descending chain condition for finite matrix subgroups. □

The following harmless lemma will facilitate our exposition.

Lemma B. Suppose that M is a left R-module and that U, V are finite matrix subgroups of M with U ⊇ V. Then there exist finite square matrices A, B over R (usually of different sizes) such that U = [A, 1]M, V = [B, 1]M, and

B = ( \underline{A  0} ; \begin{matrix}⋆ & ⋯ & ⋯ & ⋯ & ⋆ \\ ⋮ & & & & ⋮ \\ ⋆ & ⋯ & ⋯ & ⋯ & ⋆ \end{matrix} )

In other words, B is obtained from A by possibly adding rows and filling in zeros to reestablish a square format.

Proof. Let U = [Ã, α]M and V = [B̃, β]M. We may clearly assume that the original matrices Ã and B̃ are square − just fill in zeros if not − and that α = β = 1. If Ã = (ã_{ij}) ∈ R^{m×m} and B̃ = (b̃_{ij}) ∈ R^{n×n}, then the choices A = Ã and

B = ( \begin{matrix}A & 0 \\ \begin{matrix}b̃_{11} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ b̃_{n1} & 0 & ⋯ & 0 \end{matrix} & \begin{matrix}b̃_{12} & ⋯ & b̃_{1n} & 0 \\ ⋮ & & ⋮ & ⋮ \\ b̃_{n2} & ⋯ & b̃_{nn} & 0 \end{matrix} \end{matrix} )

are as desired. (The final column of zeros was inserted to make B an (n + m) × (n + m) matrix.) □

2. A technique for constructing non-trivial ℵ_{1}-separable modules. The very general construction principle which we describe in this section was

inspired by related techniques due to Hill [13], Griffith [7], Gruson/Jensen [10], and W. Zimmermann [22]. Our technique covers the ones listed as special cases, making them more transparent from a common angle of observation. The underlying idea is quite simple, which should become clear once the reader is familiar with the notational ballast which is − unfortunately − necessary to pin it down.

In the following we will denote the first uncountable ordinal number by Ω. Start with a left R-module M which is not Σ-pure injective. Our goal is to

locate a non-trivial ℵ_{1}-separable module inside the direct product M^{Ω}.

By Theorem A there is a strictly descending chain

[A_{1}, 1]M ⊋ [A_{2}, 1]M ⊋ [A_{3}, 1]M ⊋ ⋯

of finite matrix subgroups of M, and by Lemma B we may assume that all the matrices A_{n} are square, say A_{n} = (a_{ij})_{1≤i,j≤k(n)}, and that for each n, the matrix A_{n+1} is obtained from A_{n} by adding on rows and ‘filling up’ with zeros, i.e.,

A_{n+1} = ( \underline{A_{n}  0} ; \begin{matrix}⋆ & ⋯ & ⋯ & ⋯ & ⋆ \\ ⋮ & & & & ⋮ \\ ⋆ & ⋯ & ⋯ & ⋯ & ⋆ \end{matrix} ) .

In particular, a_{ij} = 0 whenever i ≤ k(n) and j > k(n). For each n pick an element

m^{(n,1)} ∈ [A_{n}, 1]M \ [A_{n+1}, 1]M;

by definition of [A_{n}, 1]M, we can supplement m^{(n,1)} by elements m^{(n,j)}, 2 ≤ j ≤ k(n), such that

∑_{j=1}^{k(n)} a_{ij}m^{(n,j)} = 0 for all i = 1, …, k(n).

Let \overline{M} be the submodule of M generated by all the elements m^{(n,j)}, n ∈ ℕ, 1 ≤ j ≤ k(n). Then clearly m^{(n,1)} ∈ [A_{n}, 1]\overline{M} \ [A_{n+1}, 1]\overline{M}; in particular, \overline{M} in turn fails to be Σ-pure injective. Since \overline{M}^{Ω} embeds into M^{Ω}, it is clearly enough to show that \overline{M}^{Ω} contains a non-trivial ℵ_{1}-separable submodule. Therefore we may assume that M = \overline{M} is countably generated.

Next we string up the selected solutions of the systems ∑_{j=1}^{k(n)} a_{ij}X_{j} = 0 for i = 1, …, k(n), n ≥ 1, to elements of the direct product P = M^{ℕ} = ∏_{n≥1} M_{n}, where M_{n} = M for n ≥ 1, as follows: For j ≥ 1 define m^{(j)} = (m^{(n,j)})_{n≥1} ∈ P, where we set m^{(n,j)} = 0 for j > k(n). Our construction then yields the following key property of the elements m^{(j)}.

Lemma 1. For each l ≥ 1, all entries of the sums

∑_{j=1}^{k(l)} a_{ij}m^{(j)} ∈ P, i = 1, …, k(l),

are zero with the possible exception of the entries in positions 1, …, l − 1, that is,

∑_{j=1}^{k(l)} a_{ij}m^{(j)} = \binom{-l-}{⋆, ⋯, ⋆, 0, ⋯}

for i = 1, …, k(l). □

Step I of our construction. The following left R-module Q ⊆ P will serve as the principal building block for the construction of a non-trivial ℵ_{1}-separable module inside the product M^{Ω}.

Definition. Q = S + ∑_{j≥1} Rm^{(j)}, where S = ⨁_{n∈ℕ} M_{n}.

Lemma 2. S is not a direct summand of Q.

Proof. (Compare [7, Lemma 2.1], [22, Lemma 3.2], [10, Théorème].)

Suppose that, to the contrary, Q = S ⨁ T, and split up m^{(1)} ∈ Q with respect to this decomposition, say

m^{(1)} = s + t with s ∈ S, t ∈ T.

Next we compare this decomposition of m^{(1)} with the following ones:

m^{(1)} = (m^{(1,1)}, …, m^{(l,1)}, 0, …) + (0, …, 0, m^{(l+1,1)}, m^{(l+2,1)}, …), for l ≥ 1.

Call the first summand on the right-hand side u^{(l)}, the second v^{(l)}, and observe that u^{(l)} ∈ S and v^{(l)} ∈ Q. Decompose v^{(l)} further, namely

v^{(l)} = s^{(l)} + t^{(l)} with s^{(l)} ∈ S, t^{(l)} ∈ T.

In view of m^{(1)} = (u^{(l)} + s^{(l)}) + t^{(l)}, we obtain

t = t^{(l)} for all l ≥ 1.

This means, in particular, that the elements m^{(1)} and t^{(l)} of P coincide in all positions outside the support of s. Let q be any index not belonging to the support of s. If we can show that the q-th entry t^{(q,q)} of t^{(q)} lies in [A_{q+1}, 1]M we are done, because we have that m^{(q,1)} = t^{(q,q)} by the preceding discussion, which will contradict our choice of m^{(q,1)} outside [A_{q+1}, 1]M.

To prove that t^{(q,q)} ∈ [A_{q+1}, 1]M, we show first that v^{(q)} ∈ [A_{q+1}, 1]Q. Indeed, truncating the m^{(j)} to

\underline{m}^{(j)} = m^{(j)} − (m^{(1,j)}, …, m^{(q,j)}, 0, …),

we clearly obtain

\underline{m}^{(1)} = v^{(q)} and \underline{m}^{(j)} ∈ Q for all j ≥ 1.

Moreover, Lemma 1 and the truncated form of the \underline{m}^{(j)} guarantee that

∑_{j=1}^{k(q+1)} a_{ij}\underline{m}^{(j)} = 0 for i = 1, …, k(q + 1),

and thus indeed, v^{(q)} belongs to [A_{q+1}, 1]Q.

But t^{(q)} being the projection of v^{(q)} on T along S, we conclude that

t^{(q)} ∈ [A_{q+1}, 1]T ⊆ [A_{q+1}, 1]P,

whence all entries of t^{(q)} belong to the subgroup [A_{q+1}, 1]M. In particular, t^{(q,q)} ∈ [A_{q+1}, 1]M. □

Step II, Gluing layers. To obtain a non-trivial ℵ_{1}-separable submodule C of the uncountable direct product M^{Ω} = ∏_{α<Ω} M_{α}, we proceed as follows. For each limit ordinal β < Ω, we pick a sequence σ_{β}(1) < σ_{β}(2) < σ_{β}(3) < ⋯ of ordinal numbers below β such that β = sup_{l∈ℕ} σ_{β}(l), and we set J_{β} = {σ_{β}(k) | k ∈ ℕ}. Next we define elements

x_{β}(l) = (x_{β}(l)_{α})_{α<Ω} ∈ M^{Ω}

for l ∈ ℕ via

(x_{β}(l))_{α} = \{ \begin{matrix}0 & if α ∉ J_{β} \\ m^{(k,l)} & if α = σ_{β}(k). \end{matrix}

In more intuitive terms: For each limit ordinal β < Ω, the module

⨁_{α∈J_{β}} M_{α} + ∑_{l∈ℕ} Rx_{β}(l)

is a ‘carbon copy’ of the module Q which we considered in Step I. The idea of layering uncountably many copies of an ‘interesting’ object inside a large direct product by tying them to limit ordinals goes back to Hill [13].

Definition. Let C be the following submodule of M^{Ω}:

C = ⨁_{α<Ω} M_{α} + ∑_{β<Ω β a limit ordinal} ∑_{l∈ℕ} Rx_{β}(l).

Lemma 3. C is not a direct sum of countably generated modules.

Proof. Recall that we may assume without loss of generality that M is countably generated (see the remark preceding Lemma 1).

Start by observing that C = ⋃_{γ<Ω} C_{γ}, where

C_{γ} = ⨁_{α<γ} M_{α} + ∑_{β<γ β a limit ordinal} ∑_{l∈ℕ} Rx_{β}(l) for γ < Ω.

In particular, we note that each C_{γ} is countably generated, and that C_{γ} = ⋃_{α<γ} C_{α}, whenever γ < Ω is a limit ordinal.

Now suppose that our claim fails, i.e., that C = ⨁_{i∈I} D_{i}, where each D_{i} is countably generated. Going back and forth between the D_{i}’s and the C_{γ}’s, where γ runs through the limit ordinals < Ω (in the tradition of Kaplansky’s [16, proof of Theorem 1]), we obtain a limit ordinal δ < Ω and a countable subset \overline{I} of I such that C_{δ} = ⨁_{i∈\overline{I}} D_{i}. This implies, in particular, that C_{δ} is a direct summand of C, and, a fortiori, of C_{δ+1}.

To prove this splitting incompatible with our construction, we write

K_{δ} = {α | α an ordinal, α < δ} \ J_{δ},

and verify that

C_{δ} = (C_{δ} ∩ ∏_{α∈K_{δ}} M_{α}) ⊕ ( ⨁_{α∈J_{δ}} M_{α}).

Indeed, one inclusion is obvious, because ⨁_{α∈J_{δ}} M_{α} ⊆ ⨁_{α<δ} M_{α} ⊆ C_{δ}; to see that, conversely, C_{δ} is contained in the right-hand side, we note that for every limit ordinal β < δ, the set J_{β} has finite intersection with J_{δ}, whence each x_{β}(l), l ∈ ℕ, belongs to the right-hand side.

Analogously,

C_{δ+1} = (C_{δ} ∩ ∏_{α∈K_{δ}} M_{α}) ⊕ ( ⨁_{α∈J_{δ}} M_{α} + ∑_{l∈ℕ} Rx_{δ}(l)).

Thus splitting of C_{δ} inside C_{δ+1} entails splitting of the inclusion

⨁_{α∈J_{δ}} M_{α} ⊆ ⨁_{α∈J_{δ}} M_{α} + ∑_{l∈ℕ} Rx_{δ}(l).

But the latter inclusion is merely a replica of the inclusion S ⊆ Q with which we dealt in Lemma 2. Since we ruled out the possibility of splitting there, the proof of Lemma 3 is complete. □

Lemma 4. C is ℵ_{1}-separable.

Proof. As in the proof of Lemma 3, we assume that M is countably generated.

Let (y_{n})_{n∈ℕ} be a generating set for M, and for each pair (α, n), where α is an ordinal below Ω and n ∈ ℕ, denote by y_{α}(n) the element of M^{Ω} which carries y_{n} in the α-th slot and zeros in the other positions. Then clearly, the elements

x_{β}(l), y_{α}(n), l, n ∈ ℕ, α, β < Ω, β a limit ordinal,

belong to the submodule C of M^{Ω} and generate it. It thus suffices to show that each countable subset of this generating set is contained in a countably generated direct summand of C.

Let Y be such a countable subset, say

Y = {x_{β_{s}}(l_{s}), y_{α_{s}}(n_{s}) | s ∈ ℕ},

where we permit the equalities α_{s} = α_{t}, resp. β_{s} = β_{t}, for s ≠ t. Moreover, let γ be the supremum of all the ordinal numbers α_{s}, β_{s}, s ∈ ℕ. Then, clearly, γ < Ω, and the submodule C ∩ ∏_{α≤γ} M_{α} of C contains Y. This submodule is countably generated, e.g., by the elements x_{β}(l), y_{α}(n) for l, n ∈ ℕ, α, β ≤ γ, β a limit ordinal. Finally, we have

C = (C ∩ ∏_{α≤γ} M_{α}) ⊕ (C ∩ ∏_{γ<α<Ω} M_{α}),

since for every limit ordinal β with γ < β < Ω, the critical set J_{β} contains at most finitely many ordinals α ≤ γ. This completes the proof of Lemma 4. □

We summarize our insights in

Theorem 5. If M is a left R-module which is not Σ-pure injective, then M^{Ω} contains a non-trivial ℵ_{1}-separable submodule. □

Remark 6. We can do slightly better than we did in Theorem 5. Namely, whenever M is a left R-module such that M^{Ω} contains a non-Σ-pure injective submodule, the product M^{Ω} contains a non-trivial ℵ_{1}-separable submodule. For the proof of this upgraded statement, it suffices to note that (M^{Ω})^{Ω} ≅ M^{Ω}.

This last remark provides some extra mileage in the classical case where R is a Dedekind domain.

Corollary 7. Given a module M over a Dedekind domain R, the following statements are equivalent:

(1) M is unbounded (i.e., the annihilator of M in R is zero).

(2) M^{Ω} contains a non-trivial ℵ_{1}-separable submodule.

Proof. The implication ‘(2)⇒(1)’ is obvious, because, if the annihilator of M in R is nonzero, then so is the annihilator of any direct product of copies of M, and consequently every submodule of M^{Ω} is a direct sum of cyclic modules.

To check ‘(1)⇒(2)’, we use the well-known fact that M is Σ-pure injective if and only if M is a direct sum of a divisible and a bounded module (see e.g. [22]). Since we have already dealt with the case where M is not Σ-pure injective, it suffices to prove (2) in case M is divisible. But if M contains the quotient field of R as a submodule, then M contains the non-Σ-pure injective R-module R, and if M contains a Prüfer module \underrightarrow{lim}_{n∈ℕ} R/P^{n} for some prime ideal P of R, then M^{ℕ} contains the sum ⨁_{n∈ℕ} R/P^{n}, for instance, which, in turn, fails to be Σ-pure injective. □

Remark. For R = ℤ, there are two (distinct) other usages of the attribute ‘ℵ_{1}-separable’ − see [4, p. 121] and [8, p. 102].

Another immediate consequence of Theorem 5 is

Theorem 8. For a ring R the following statements are equivalent:

(1) The left pure global dimension of R is zero (i.e., every pure inclusion of left R-modules splits).

(2) Every left R-module is a direct sum of finitely generated submodules.

(3) Every ℵ_{1}-separable left R-module is trivial.

Proof. While ‘(1)⇔(2)’ is well-known ([21, Satz], see also [11, Proposition 10.7]) and ‘(2)⇒(3)’ is trivial, the implication ‘(3)⇒(1)’ follows from Theorem 5 above. □

We conclude this section by placing some well-known results into the framework of our construction.

Theorem 9. ([10, Théorème] and [22, Folgerung 3.4]) If M is a left R-module which is not Σ-pure injective, then the canonical embedding M^{(ℕ)} ↪ M^{ℕ} does not split.

Proof. In our previous notation, we have M^{(ℕ)} ⊆ Q ⊆ M^{ℕ}. By Lemma 2, M^{(ℕ)} is not a direct summand of Q, if M fails to be Σ-pure injective, and, a fortiori, M^{(ℕ)} is not a direct summand of M^{ℕ}. □

The following statement is a very slightly strengthened version of a theorem of Griffith’s [7, Theorem 2.2]: There exists a flat, torsionless non-trivial ℵ_{1}- separable left R-module precisely if R fails to be left perfect. (One implication is, of course, an immediate consequence of Bass’s result saying that all flat left modules over a left perfect ring are projective.)

In [14] a left R-module M was called locally projective if for each finite set of elements x_{1}, …, x_{k} ∈ M there exist m_{1}, …, m_{n} ∈ M and f_{1}, …, f_{n} ∈ Hom_{R}(M, R) with x_{j} = ∑_{i=1}^{n} f_{i}(x_{j})m_{i} for j = 1, …, k. These same modules were labelled flat strict Mittag-Leffler modules by Gruson and Raynaud [12], trace modules by Ohm and Rush [17], and universally torsionless modules by Garfinkel [6].

Theorem 10. The following statements are equivalent for a ring R:

(1) R is not left perfect.

(2) There exists a locally projective left R-module which is not projective.

(3) There exists a non-trivial ℵ_{1}-separable left R-module which is locally projective.

Proof. (Compare [7, Theorem 2.2].) The implication ‘(3)⇒(2)’ is trivial, and in view of the fact that locally projective modules are flat [14], so is ‘(2)⇒(1)’.

To prove ‘(1)⇒(3)’, we start by observing that the left R-module M = R is not Σ-pure injective, since clearly each finitely generated right ideal is a finite matrix subgroup of M. We will construct a non-trivial ℵ_{1}-separable submodule C ⊆ M^{Ω} = R^{Ω} as before, after ‘normalizing’ the corresponding matrices A_{n} to make sure that C will be locally projective.

Let b_{1}R ⊋ b_{1}b_{2}R ⊋ ⋯ ⊋ b_{1}b_{2} ⋯ b_{n}R ⊋ ⋯ be a strictly descending chain of principal right ideals, and let A_{n} be the following (n + 1) × (n + 1) matrix over R:

A_{n} = ( \begin{matrix}−1 & b_{1} & 0 & ⋯ \\ −1 & 0 & b_{1}b_{2} & 0 & ⋯ \\ −1 & 0 & 0 & b_{1}b_{2}b_{3} & 0 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ −1 & 0 & 0 & ⋯ & ⋯ & 0 & b_{1}b_{2} ⋯ b_{n} \\ 0 & 0 & 0 & ⋯ & ⋯ & 0 & 0 \end{matrix} ) .

(The last row of zeros has the sole purpose of making the matrix square).

Next observe that the following rows m^{(j)} ∈ R^{ℕ} satisfy the requirements of the construction presented above:

m^{(1)} = ( b_{1}, b_{1}b_{2}, ⋯, b_{1}b_{2} ⋯ b_{n}, b_{1}b_{2} ⋯ b_{n+1}, ⋯)

and

m^{(j)} = \binom{-j-}{0, ⋯, 0, 1, b_{j}, b_{j}b_{j+1}, ⋯, b_{j}b_{j+1} ⋯ b_{j+n}, ⋯}

for j ≥ 2. Indeed,

∑_{j=1}^{l+1} a_{ij}m^{(j)} = \binom{-l-}{⋆, ⋯, ⋆, 0, ⋯}

for l ≥ 1 and i = 1, …, l + 1, where the a_{ij} are the entries of the matrices A_{n}. Let C ⊆ R^{Ω} denote the resulting non-trivial ℵ_{1}-separable (left) submodule of R^{Ω} = ∏_{α<Ω} R_{α} with R_{α} = R for all α. To prove local projectivity of C, it clearly suffices to check that each finite subset Y = {x_{β_{i}}(l_{i}), y_{α_{i}}(n_{i}), 1 ≤ i ≤ m} of C − notation as in the proof of Lemma 4 − is contained in a free direct summand of C.

Let β_{1}, …, β_{r} be the distinct limit ordinals appearing among the β_{1}, …, β_{m}. Now use the fact that the intersection J_{β_{i}} ⋂ J_{β_{j}} is finite for i ≠ j, 1 ≤ i, j ≤ r, to find pairwise disjoint cofinite subsets I_{β_{i}} ⊆ J_{β_{i}} and a finite set F of ordinals < Ω, disjoint from the I_{β_{i}}, such that

{α_{1}, …, α_{m}} ∪ ⋃_{1≤i≤r} J_{β_{i}} = F ∪ ⋃_{1≤i≤r} I_{β_{i}}.

Denote the last union of ordinal numbers by G, and observe that

C = [⨁_{α∈F} R_{α} ⊕ ⨁_{i=1}^{r}(C ∩ ∏_{α∈I_{β_{i}}} M_{α})] ⊕ [C ∩ ∏_{{α<Ω \atop α∉G}} M_{α}].

Clearly, our finite set Y is contained in the left-hand term of this direct sum. It is free, since by Griffith’s [7, Lemma 2.1], each of the intersections C ∩ ∏_{α∈I_{β_{i}}} M_{α} is free. □

3. Connection between locally split exact sequences and ℵ_{1}-separa-ble modules.

We will call an R-module monomorphism f : A ⟶ B (resp., an R-module epimorphism g : B ⟶ C) locally split if, for each finite subset A′ ⊆ A (resp., for each finite C′ ⊆ C), there exists a homomorphism f′ : B ⟶ A (resp., g′ : C ⟶ B) such that f′f|_{A′} = id_{A′} (resp., gg′|_{C′} = id_{C′}). Correspondingly, an exact sequence

0 ⟶ A ⟶^{f} B ⟶^{g} C ⟶ 0

is labelled locally split in case f is a locally split monomorphism and g a locally split epimorphism. In the special case R = ℤ, these sequences are called strongly balanced; in [18] they were shown to be of crucial importance in the study of separable abelian groups.

The relevance of locally split homomorphisms in more general contexts lies in the fact that, in investigating properties of modules which are defined via finite subsets, these maps may be as helpful as split maps, while being far more readily available. We point towards two specific well-known illustrations. In [3, Proposition 2.2], Chase used the fact that, given a projective resolution

0 ⟶ K ⟶ P ⟶ M ⟶ 0

of a flat module M, the inclusion K ⊆ P is locally split, in characterizing the rings over which flatness is passed on to direct products. In [12, Part 2] Gruson and Raynaud studied the descent of projectivity in ring extensions via the descent of the property ‘flat strict Mittag-Leffler’. Their flat strict Mittag-Leffler modules (= locally projective modules in the terminology of Section 2) are intimitely linked to local splitness of maps: In fact, M is locally projective if and only if every epimorphism onto M is locally split. For a rich supply of examples of non-projective locally projective modules we refer to [12] and [14]; for further illustrations of natural occurrence and applicability of locally projective modules and locally split maps we refer to [17] (here ‘locally projective module’ = ‘trace module’), [6] (here ‘locally projective’ = ‘universally torsionless’), and [1].

The purpose of this section is twofold. First we will assign to each ℵ_{1}-separable module M an exact sequence

0 ⟶ A ⟶ B ⟶ M ⟶ 0

which is locally split in a particularly strong sense. Then we will combine this observation with Theorem 8 of Section 2 to characterize the rings over which all locally split exact sequences of left modules split. Again, we will find that this occurs extremely rarely, namely only when the left pure global dimension of R vanishes.

Our first proposition along this line is well-known.

Proposition 11. (See [12, Corollaire 2.2.2] and [17, Proposition 4.1].) If g : B ⟶ C is a locally split epimorphism and C is countably generated, then g splits. □

The following fact was independently observed by Rangaswamy and the author.

Proposition 12. Suppose that 0 ⟶ A ⟶^{f} B ⟶^{g} C ⟶ 0 is an exact sequence, where g is locally split and B a direct sum of countably generated modules. Then, given any countable subset A′ of A, there exists a homomorphism f′ : B ⟶ A such that f′f|_{A′} = id_{A′}.

In particular, the above exact sequence is locally split.

Proof. Write B = ⨁_{i∈I} B_{i}, where each B_{i} is countably generated. Letting A′ be a countable subset of A, we pick a countable subset I_{1} ⊆ I such that f(A′) ⊆ ⨁_{i∈I_{1}} B_{i}, as well as a countable generating set (c_{n}^{(1)})_{n∈ℕ} for the image g(⨁_{i∈I_{1}} B_{i}). Now we use local splitness of g to choose, for each finite subset F ⊆ ℕ, a homomorphism g_{F}^{(1)} : C ⟶ B such that gg_{F}^{(1)}(c_{n}^{(1)}) = c_{n}^{(1)} for all n ∈ F. Since the set

{g_{F}^{(1)}(c_{n}^{(1)}) | F a finite subset of ℕ, n ∈ F }

is in turn countable, it is contained in another countable subsum ⨁_{i∈I_{2}} B_{i} of B. We may clearly assume that I_{1} ⊆ I_{2}.

Now repeat this procedure, namely, pick a countable generating set (c_{n}^{(2)})_{n∈ℕ} for the image g(⨁_{i∈I_{2}} B_{i}) and maps g_{F}^{(2)} : C ⟶ B, where F runs through the finite subsets of ℕ, such that gg_{F}^{(2)}(c_{n}^{(2)}) = c_{n}^{(2)} whenever n ∈ F.

Iteration of this process yields a sequence I_{1} ⊆ I_{2} ⊆ I_{3} ⊆ ⋯ of countable subsets of I with the property that, for each k ∈ ℕ and each finite subset C′ of g(⨁_{i∈I_{k}} B_{i}), there exists a map g′ : C ⟶ ⨁_{i∈I_{k+1}} B_{i} such that gg′|_{C′} = id_{C′}. Define J = ⋃_{k∈ℕ} I_{k}, and consider the exact sequence

0 ⟶ f^{−1}(⨁_{j∈J} B_{j}) ⟶^{\overline{f}} ⨁_{j∈J} B_{j} ⟶^{\overline{g}} g(⨁_{j∈J} B_{j}) ⟶ 0,

where \overline{f}, resp. \overline{g}, is the appropriate restriction of f, resp. g. By construction, the epimorphism \overline{g} retains local splitness, and therefore splits in view of Proposition 11.

Noting that A′ is contained in f^{−1}(⨁_{j∈J} B_{j}) by our choice of I_{1}, we thus obtain a homomorphism f′′ : ⨁_{j∈J} B_{j} ⟶ f^{−1}(⨁_{j∈J} B_{j}) with f′′\overline{f}|_{A′} = id_{A′}. Its natural extension f′ : ⨁_{i∈I} B_{i} ⟶ A then clearly satisfies the requirement f′f|_{A′} = id_{A′}. □

We briefly interrupt our main line with an example showing that neither of the two preceding propositions has a direct analog for locally split monomorphisms.

Example 13. (K. M. Rangaswamy) Let R = ℤ, and choose a free resolution of ℚ, say

0 ⟶ K ⟶ ℤ^{(ℕ)} ⟶ ℚ ⟶ 0.

Due to the flatness of ℚ over ℤ, the inclusion K ↪ ℤ^{(ℕ)} is then locally split. On the other hand, Hom_{ℤ}(ℚ, ℤ^{(ℕ)}) = 0; in particular, the above exact sequence fails to be locally split.

Assembling our previous insights is all that is left to be done to prove the main result of this section.

Theorem 14. The following statements are equivalent for a ring R:

(1) Each left R-module is a direct sum of finitely generated modules (i.e., the left pure global dimension of R is zero).

(2) Each ℵ_{1}-separable left R-module is trivial.

(3) Each locally split exact sequence of left R-modules splits.

Proof. The equivalence ‘(1)⇔(2)’ we have already established in Theorem 8. Moreover, the implication ‘(1)⇒(3)’ is obvious, since, clearly, every locally split exact sequence is pure exact.

To show ‘(3)⇒(2)’, let M be an ℵ_{1}-separable left R-module, and (M_{i})_{i∈I} the family of all countably generated direct summands of M, and observe that then the canonical map

g : ⨁_{i∈I} M_{i} ⟶ M

is a locally split epimorphism. Using Proposition 12, we conclude that the exact sequence

0 ⟶ ker g ⟶ ⨁_{i∈I} M_{i} ⟶^{g} M ⟶ 0

is locally split. By (3), this sequence splits, which shows that M is a direct summand of a direct sum of countably generated modules. But now Kaplansky’s theorem [16, Theorem 1] guarantees that M, in turn, is a direct sum of countably generated components as claimed. □

We conclude by noting that the status of the rings with vanishing left pure global dimension has not yet been satisfactorily resolved. While it is well-known that the rings whose left and right pure global dimension is zero are precisely those of finite representation type ([5] and [19]), it is still open in general whether one-sided vanishing of the pure global dimension forces this condition to hold on the other side as well. The following equivalence may therefore just be a clumsy rephrasing of Theorem 14: R has finite representation type if and only if each ℵ_{1}-separable R-module (left or right) is trivial, if and only if each locally split exact sequence of R-modules splits.

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Department of Mathematics, University of Utah, Salt Lake City, Utah 84112