NYJM Monographs

Volume 1 2003

A Pext primer: Pure extensions and lim^{1} for infinite abelian groups

Claude L. Schochet

Abstract. The abelian group Pext_{ℤ}^{1}(G, H) of pure extensions has recently attracted the interest of workers in non-commutative topology, especially those using KK-theory, since under minimal hypotheses the closure of zero in the Kasparov group KK_{∗}(A, B) (for separable C^{∗}- algebras A and B) is isomorphic to the group

Pext_{ℤ}^{1}(K_{∗}(A), K_{∗}(B)).

As K_{∗}(A) and K_{∗}(B) can take values in all countable abelian groups, assuming that G and H are countable is natural.

In this mostly expository work we survey the known (and not so wellknown) properties of Pext and its relationship to lim^{1} and develop some new results on their computation.

Mathematics Subject Classification. Primary: 20K35, 19K35, 46L80. Secondary: 18E25, 18G15, 20K40, 20K45, 47L80, 55U99.

Key words and phrases. pure extensions, Pext_{ℤ}^{1}(G, H), lim^{1}, Jensen’s Theorem, infinite abelian groups, quasidiagonality, phantom maps.

Contents

1. Introduction 1 2. First facts on Hom and Ext 3 3. First facts on lim^{1} 8 4. Exact sequences, divisibility results 17 5. First facts on Pext 23 6. Jensen’s theorem and Roos’s theorem 29 7. Ranges and inverse limits 33 8. Decoupling 38 9. Pext_{ℤ}^{1}(G, H) for H torsionfree 45 10. Pext_{ℤ}^{1}(G, H) for G torsionfree and H torsion 49 11. Pext_{ℤ}^{1}(G, H) for both G and H torsion 51 12. Pext_{ℤ}^{1}(G, H) and phantom maps 58 13. Pext_{ℤ}^{1}(G, H) and quasidiagonality 61 References 64

1. Introduction

The abelian group Pext_{ℤ}^{1}(G, H) has recently appeared in non-commutative topology, specifically in the Kasparov KK-theory, since under minimal hypotheses the closure of zero in the Kasparov group

KK_{∗}(A, B)

(for separable C^{∗}-algebras A and B) is isomorphic to the group

Pext_{ℤ}^{1}(K_{∗}(A), K_{∗}(B))

and this subgroup of KK_{∗}(A, B) is the subgroup of quasidiagonal elements (see §13 for details.) The groups K_{∗}(A) and K_{∗}(B) range over all countable abelian groups, so assuming that G and H are countable is natural.

Coming from outside the world of infinite abelian groups, the functional analyst tends to ask elementary questions:

1. What are the typical examples of Pext_{ℤ}^{1}(G, H) ?

2. When does the group vanish?

3. Which nonzero values does it take?

4. How does one compute the group?

5. Does this group appear elsewhere in mathematics as an obstruction group?

The present work is designed to answer these questions.

The most complete source for the theory of infinite abelian groups is the two-volume work by L. Fuchs [24], [25] which unfortunately is out of print. We have thus included more elementary material in this paper to compensate for the relative unavailability of these books. We also strongly recommend the little red book of Kaplansky [40] which has been so influential in this area.

C. U. Jensen’s Theorem 6.1 is central to this paper. It asserts that if G is an abelian group written as the union of an increasing sequence of finitely generated subgroups G_{i} then there is a natural isomorphism

\underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) ≅ Pext_{ℤ}^{1}(G, H).

This connects up Pext and the theory of infinite abelian groups with \underleftarrow{lim}^{1}, a classic (and difficult) functor from homological algebra and algebraic topology.

Typically, \underleftarrow{lim}^{1} in topology detects phantom behavior. It notices maps X → Y (where X and Y are locally finite CW -complexes or, better, spectra) which are not null-homotopic but whose restriction to finite CW -complexes (or spectra) are null-homotopic. See §12 for details.

As mentioned before, Pext is also related to quasidiagonality for C^{∗}- algebras (see §13 for details). There are in fact certain similarities between the two phenomena which we hope to examine in the future.

When writing a paper that is mostly expository, one must make a pedagogical decision: what level of generality is appropriate to the audience. We have written this paper with the specialists in operator algebras in mind as potential consumers. So we have stayed resolutely within the context of the category of abelian groups, even though we are well aware that much of this paper generalizes to categories of R-modules and further.

This decision may be somewhat short-sighted. For instance, one may well ask for equivariant versions of this work. If G is a locally compact group then the groups KK_{∗}^{G}(A, B) are modules over the ring KK_{∗}^{G}(ℂ, ℂ) which (for G compact) is the complex representation ring R(G). One must draw the line somewhere, though, so equivariant Pext will have to wait.

We have attempted to attribute each result in this work to the appropriate source. However, we are not expert in the area of pure homological algebra. It may be that all of the results in this work are known- possibly as trivial corollaries of major results in much more rarified contexts. We apologize in advance for incorrect attributions and we ask each reader’s help in correcting these (and all other) errors. Errata will be posted at http://nyjm.albany.edu:8000/m/2003/1-info.htm.

We have been fortunate in having R. G. Bruner, John Irwin, and C. McGibbon as colleagues and we acknowledge with pleasure their continuing assistance. Likewise we thank Baruch Solel and the faculty of the Technion for their hospitality during our sabbatical visit. Finally, we gratefully acknowledge the assistance of H. P. Goeters.

All groups which appear in this paper are abelian unless specified otherwise.

2. First facts on Hom and Ext

We shall have occasion to use a number of terms from the theory of infinite abelian groups that were not familiar to the author and perhaps are not familiar to the reader. We begin with enough definitions to get started. We will place the rest as footnotes in the paper so as to facilitate easy retrieval.

Let ℤ_{p} denote the integers localized at the prime p (that is, all primes except p have been inverted) and let ℤ(p^{∞}) denote the p-group P/ℤ, where P is the group of rational numbers with denominator some power of the prime p. Let ℤ̂_{p} = \underleftarrow{lim} ℤ/p^{n} denote the p-adic integers. We note that

ℤ̂_{p} ≅ Hom_{ℤ}(ℤ(p^{∞}), ℤ(p^{∞})).

A group is reduced if it has no divisible subgroups other than 0. Let

nG = {ng : g ∈ G} ⊆ G.

A subgroup S of a group G is pure if

nS = S ∩ nG

for all natural numbers n.

A group is algebraically compact ([24], §38) if it is a direct summand in every group that contains it as a pure subgroup. Equivalently ([24], §38.1) it is algebraically compact if and only if it is algebraically a direct summand in a group which admits a compact topology. (If the group is abelian then the compact group may be taken to be abelian). Examples include compact groups, divisible groups, and bounded groups. A group is algebraically compact if and only if it is of the form

D ⊕ Π_{p}D^{p}

where D is divisible and for each prime p D^{p} is the completion in the p-adic topology of the direct sum of cyclic p-groups and groups of p-adic integers. If {G_{i}} is any sequence of abelian groups, then the group Π_{i}G_{i}/ ⊕ G_{i} is algebraically compact, by [37], and its structure is determined in [28].

We now give some basic and not so basic observations on the abelian groups Hom_{ℤ}(G, H) and Ext_{ℤ}^{1}(G, H).

Proposition 2.1. Suppose that G and H are abelian groups.

1. Hom_{ℤ}(G, H) is a functor to abelian groups, contravariant in G and covariant in H.

2. Hom is additive in each variable: there are natural isomorphisms

Hom_{ℤ}(⊕_{i}G_{i}, H) ≅ Π_{i}Hom_{ℤ}(G_{i}, H)

and

Hom_{ℤ}(G, Π_{i}H_{i}) ≅ Π_{i}Hom_{ℤ}(G, H_{i}).

3. If G is divisible or H is torsionfree, then Hom_{ℤ}(G, H) is torsionfree.

4. If G or H is both divisible and torsionfree, then Hom_{ℤ}(G, H) is divisible and torsionfree.

5. If G and H are finitely generated, then Hom_{ℤ}(G, H) is finitely generated.

6. If G is a torsion group, then Hom_{ℤ}(G, H) is reduced and algebraically compact.

7. If H is algebraically compact then so is Hom_{ℤ}(G, H).

8. If H is compact then G ↦ Hom_{ℤ}(G, H) is a contravariant functor from groups and homomorphisms to compact groups and continuous homomorphisms.

Proof. Most of these results are elementary and are found in ([24], §§43 - 46). Part 6) is a theorem of Fuchs and Harrison (cf. ([24], §46.1). For Parts 7) and 8) we note that if H is compact then so is H^{G} with the product topology, and Hom_{ℤ}(G, H) is a closed subset of H^{G}, hence compact. If H is algebraically compact, with H ⊕ H′ compact, then Hom_{ℤ}(G, H) is algebraically a direct summand of the compact group Hom_{ℤ}(G, H ⊕ H′), hence algebraically compact. The rest is immediate. □

If G is a direct sum of cyclic groups C_{i} (for instance, if it is finitely generated), then of course the computation of Hom_{ℤ}(G, H) is elementary:

Hom_{ℤ}(G, H) ≅ Hom_{ℤ}(⊕_{i}C_{i}, H) ≅ Π_{i}Hom_{ℤ}(C_{i}, H)

and one then observes that Hom_{ℤ}(ℤ, H) ≅ H and that Hom_{ℤ}(ℤ/n, H) ≅ H[n].^{1}

Definition 2.2. The support of a torsionfree group G, denoted Supp(G), is defined by

Supp(G) = {primes p: pG ≠ G}.

Thus p ∈ Supp(G) if and only if p : G → G is not an automorphism. We define ℤ_{G} to be the localization of ℤ obtained by inverting all primes p ∉

A[n] = {a ∈ A: na = 0}.

1The group A[n] is defined by

Supp(G). (This is denoted R(G) in the papers of Warfield and elsewhere.) Thus ℤ_{G} is the greatest subring of ℚ such that G is a ℤ_{G}-module.

Warfield analyzes Hom as follows.

Proposition 2.3 (Warfield [62]). Suppose that G is a torsionfree abelian group of finite rank^{2} and H is a divisible, countable torsion group. Then:

1. The group Hom_{ℤ}(G, H) is divisible, hence the sum of n_{o} copies of ℚ and for each p, n_{p} copies of ℤ(p^{∞}).

2. Hom_{ℤ}(G, H) is a torsion group if and only if G ⊗ ℤ_{H} is a free ℤ_{H}- module. Otherwise, n_{o} = 𝔠.

3. The p-torsion subgroup of the group Hom_{ℤ}(G, H) is isomorphic to the direct sum of r_{p}(G) copies^{3} of the group H_{p}.^{4}

We turn now to elementary properties of Ext.

Proposition 2.4. Let G and H be abelian groups.

1. If

0 → H′ → H → H′′ → 0

is a short exact sequence of abelian groups then there is an associated natural six-term exact sequence

0 → Hom_{ℤ}(G, H′) → Hom_{ℤ}(G, H) → Hom_{ℤ}(G, H′′) →
→ Ext_{ℤ}^{1}(G, H′) → Ext_{ℤ}^{1}(G, H) → Ext_{ℤ}^{1}(G, H′′) → 0

and similarly in the other variable.

2. There are natural isomorphisms

Ext_{ℤ}^{1}(⊕G_{i}, H) ≅ Π_{i}Ext_{ℤ}^{1}(G_{i}, H)

and

Ext_{ℤ}^{1}(G, Π_{i}H_{i}) ≅ Π_{i}Ext_{ℤ}^{1}(G, H_{i}).

rank(G) = dim_{ℚ}(G ⊗ ℚ).

r_{p}(G) = dim_{ℤ/p} (G/pG).

G_{p} = G ⊗ ℤ_{p}.

2If G is torsionfree then the rank of G is defined by Note that G ≅ G ⊗ 1 ⊂ G ⊗ ℚ and hence G is isomorphic to a subgroup of a ℚ-vector space of dimension rank(G). Thus every torsionfree group may be realized as a subgroup of a ℚ-vector space. There is no general classification of torsionfree groups.

3For any torsion group G, let For example, if G is divisible and torsion, then pG = G, and hence r_{p}(G) = 0.

4Here, for any group G and prime p, its localization G_{p} is defined to be

3. If G and H are finitely generated then Ext_{ℤ}^{1}(G, H) is finite.

4. If G is torsionfree then Ext_{ℤ}^{1}(G, H) is divisible.

5. If H is algebraically compact then Ext_{ℤ}^{1}(G, H) is reduced and algebraically compact.

6. If H is compact then the group Ext_{ℤ}^{1}(G, H) is reduced and compact, and Ext_{ℤ}^{1}(−, H) is a functor from groups and homomorphisms to reduced compact groups and continuous homomorphisms.

7. If G and H are torsionfree and

Supp(G) ∩ Supp(H) = ϕ

then Ext_{ℤ}^{1}(G, H) is torsionfree.^{5} If Hom_{ℤ}(G, H) is divisible then the converse is true.

8. If G is a torsion group then Ext_{ℤ}^{1}(G, H) is reduced.

9. The group Ext_{ℤ}^{1}(G, H) = 0 for all G if and only if H is divisible.

10. The group Ext_{ℤ}^{1}(G, H) = 0 for all H if and only if G is free abelian.

Proof. Parts 1) and 2) are due to Cartan-Eilenberg [8]. Part 3) comes down to the two cases

Ext_{ℤ}^{1}(ℤ/p, ℤ/q) = ℤ/r

where r is the gcd of p and q, and

Ext_{ℤ}^{1}(ℤ/p, ℤ) = ℤ/p.

Part 4) is established in ([24], p. 223) and Part 5) in ([24], p. 225). To prove the compactness conclusions of Part 6), argue as follows. Let

0 → F ⟶^{ι} F′ → G → 0

be a free resolution of the group G. Then

Hom_{ℤ}(F′, H) → Hom_{ℤ}(F, H) → Ext_{ℤ}^{1}(G, H) → 0

is exact by the Hom-Ext exact sequence. The groups Hom_{ℤ}(F′, H) and Hom_{ℤ}(F, H) are compact since H is compact, and the map

ι^{∗}: Hom_{ℤ}(F′, H) → Hom_{ℤ}(F, H)

Ext_{ℤ}^{1}(ℤ_{p}, ℤ) ≅ ℚ^{ℵ_{o}} ⊕ ℤ(p^{∞})

5The group Ext_{ℤ}^{1}(G, H) may well have torsion in general even if G and H are torsionfree.

Countable examples are not hard to find. For instance, which has p^{r}-torsion for all r. Complicated necessary and sufficient conditions have been found to ensure that Ext_{ℤ}^{1}(G, H) is torsionfree (cf. [26]). The conditions stated in 7) barely scratch the surface of the literature. We return to this point in Section 9.

is a continuous homomorphism, by Proposition 2.1. Thus the quotient group Ext_{ℤ}^{1}(G, H) is compact.^{6} Further, we see that a homomorphism G → G′ induces a map of resolutions and hence a continuous homomorphism

Ext_{ℤ}^{1}(G′, H) → Ext_{ℤ}^{1}(G, H)

which completes the proof of Part 6). Part 8) is found in ([24], §55.3).

To prove Part 7) we follow [26]. Fix a prime p. Applying Hom_{ℤ}(G, −) to the short exact sequence

0 → H ⟶^{p} H → H/pH → 0

and identifying

Hom_{ℤ}(G, H/pH) ≅ Hom_{ℤ}(G/pG, H/pH)

leads to the surjection

Hom_{ℤ}(G/pG, H/pH) → Ext_{ℤ}^{1}(G, H)[p] → 0

The support condition implies that Hom_{ℤ}(G/pG, H/pH) = 0 and then exactness implies that Ext_{ℤ}^{1}(G, H)[p] = 0. Thus Ext_{ℤ}^{1}(G, H)[p] = 0 for each prime p which implies that Ext_{ℤ}^{1}(G, H) is torsionfree.

For the converse, note that if Hom_{ℤ}(G, H) is divisible then multiplication by p is an isomorphism, which implies in turn that the natural map

Hom_{ℤ}(G/pG, H/pH) → Ext_{ℤ}^{1}(G, H)[p]

is an isomorphism. So if Ext_{ℤ}^{1}(G, H) is torsionfree then

Hom_{ℤ}(G/pG, H/pH) = 0

for each prime p, which is equivalent to the support condition

Supp(G) ∩ Supp(H) = ϕ.

The final two statements 9) and 10) may be proved directly, but they also are part of the axiomatic description of Ext thought of as the derived functor of Hom, since the divisible groups are the injective groups and the free abelian groups are the projective groups. See MacLane [41] for details. □

6Note that Im(ι^{∗}) is not necessarily a closed subgroup, and hence Ext_{ℤ}^{1}(G, H) is not necessarily Hausdorff.

3. First facts on lim^{1}

Next we record some well-known observations about \underleftarrow{lim}^{1} for abelian groups.^{7} We review an important example, we define and explain the Mittag-Leffler condition, and we examine the behavior of \underleftarrow{lim}^{1} with respect to tensor product.

An inverse sequence {G_{i}} of abelian groups is a collection of abelian groups indexed by a countable^{8} partially ordered set (which we may take to be the positive integers without loss of generality) together with a coherent family of maps f_{ji}: G_{j} → G_{i} for j ≥ i. We let f_{i} = f_{i,i−1}. The functor \underleftarrow{lim}^{1}G_{i} may be defined categorically as the first derived functor of \underleftarrow{lim} , but for countable index sets the following description, due to Eilenberg [18], is available. Let

Ψ : Π_{i}G_{i} → Π_{i}G_{i}

be defined by

Ψ(g_{i}) = (g_{i} − f_{i+1}(g_{i+1})) (3.1)

so that Ker(Ψ) ≅ \underleftarrow{lim} G_{i}. Then \underleftarrow{lim}^{1}G_{i} is given by

(3.2) \underleftarrow{lim}^{1}G_{i} = Coker(Ψ).

Here are some of the resulting elementary properties. For proofs the reader may consult, e.g., [38] (which also deals with the much harder case of general index sets.) Some of these results were first established in the context of derived functors and abelian categories by Cartan-Eilenberg [8] and the rest by Yeh [64], Eilenberg-Moore [18], and Roos [52].

Proposition 3.3. 1. The functors \underleftarrow{lim} and \underleftarrow{lim}^{1} are covariant functors from the category of inverse sequences of abelian groups to the category of abelian groups.

2. The functors \underleftarrow{lim} and \underleftarrow{lim}^{1} are left unchanged by passage to cofinal subsequences.

7This is all the generality we need, but we note in passing that one could work in the context of modules over a commutative ring or even more generally in an abelian category.

8If one allows index sets of higher cardinality then the entire theory becomes very much more complex. The functor \underleftarrow{lim}^{1} is only the first of a sequence of derived functors \underleftarrow{lim}^{n} which are nonzero in general. The six term \underleftarrow{lim} -\underleftarrow{lim}^{1} sequence of Part 5) of Proposition 3.3 becomes a long exact sequence. There is no explicit description of the functors \underleftarrow{lim}^{n} for n > 1 analogous to Eilenberg’s description for \underleftarrow{lim}^{1}. We shall have no need for those functors, as in our applications the index sets are in fact countable (corresponding to the fact that we concentrate upon separable C^{∗}-algebras.)

3. The functors \underleftarrow{lim} and \underleftarrow{lim}^{1} respect finite direct sums.

4. If each f_{i} : G_{i} → G_{i−1} is an isomorphism then \underleftarrow{lim} G_{i} ≅ G_{1} and \underleftarrow{lim}^{1}G_{i} = 0.

5. If

0 → {G′_{i}} → {G_{i}} → {G′′_{i}} → 0

is a short exact sequence of inverse sequences then there is an associated \underleftarrow{lim} -\underleftarrow{lim}^{1} exact sequence

0 → \underleftarrow{lim} G′_{i} → \underleftarrow{lim} G_{i} → \underleftarrow{lim} G′′_{i} → \underleftarrow{lim}^{1}G′_{i} → \underleftarrow{lim}^{1}G_{i} → \underleftarrow{lim}^{1}G′′_{i} → 0

which is natural with respect to morphisms of short exact sequences of inverse sequences.^{9}

The following proposition is due to Warfield [62], p. 434.

Proposition 3.4. For any inverse sequence {G_{i}}, the group

\underleftarrow{lim}^{1}G_{i}

is a cotorsion^{10} group.

Proof. Let

Ψ : Π_{i}G_{i} → Π_{i}G_{i}

be the Eilenberg map. It is easy to see that

⊕_{i}G_{i} ⊆ Im(Ψ)

and hence there is an exact sequence

\frac{Π_{i}G_{i}}{⊕G_{i}} ⟶ \underleftarrow{lim}^{1}G_{i} → 0.

Now the group \frac{Π_{i}G_{i}}{⊕G_{i}} is algebraically compact for any choice of \{G_{i}\}, by [37], (and see the material at the beginning of Section 2) hence cotorsion, and any quotient of a cotorsion group is again cotorsion. □

We mention in passing one general result on the size of \underleftarrow{lim}^{1}, a theme to which we will return in connection with Pext.

9Formally we say that \underleftarrow{lim}^{1} is the first derived functor of \underleftarrow{lim} .

10A group G is cotorsion if and only if Ext_{ℤ}^{1}(ℚ, G) = 0. For instance, algebraically compact groups are cotorsion, and any group of the form G = Ext_{ℤ}^{1}(H, K) is cotorsion. A group is cotorsion if and only if it is the quotient of an algebraically compact group ([24], §54.1). On the other hand, ℤ is not cotorsion. See ([24] §§54-58.) Fuchs remarks that the concept of a cotorsion group is due to Harrison [31] and independently by Nunke [49] and Fuchs [23].

Proposition 3.5 (B. Gray [29]). Suppose given an inverse sequence {G_{i}} with each G_{i} finite or countable. Then the group

\underleftarrow{lim}^{1}G_{i}

either is zero or uncountable.

Example 3.6. The simplest nontrivial example of \underleftarrow{lim}^{1} arises from the inverse sequence

ℤ ⟵^{2} ℤ ⟵^{2} ℤ ⟵^{2} …

denoted {ℤ, 2} where each map is just multiplication by 2. (This example arose in the first use of \underleftarrow{lim}^{1} by Steenrod; see our discussion after 3.12.)

We compute as follows. The collection of short exact sequences

0 → ℤ ⟶^{2^{n}} ℤ → ℤ/2^{n} → 0

combine to yield a natural short exact sequence of inverse sequences

0 → {ℤ, 2} → {ℤ, 1} → {ℤ/2^{n}, π} → 0.

Take the associated \underleftarrow{lim} -\underleftarrow{lim}^{1} sequence. Using the fact that \underleftarrow{lim} \{ℤ, 1\} ≅ ℤ, \underleftarrow{lim} \{ℤ, 2\} = 0, and \underleftarrow{lim}^{1}\{ℤ, 1\} = 0, we obtain the sequence

0 → ℤ → \underleftarrow{lim} ℤ/2^{n} → \underleftarrow{lim}^{1}\{ℤ, 2\} → 0

and hence

\underleftarrow{lim}^{1}\{ℤ, 2\} ≅ ℤ̂_{2}/ℤ.

The example may be expanded as follows. Let ℤ[\frac{1}{2}] denote the subring of the rational numbers generated by ℤ and by \frac{1}{2}. Write ℤ[\frac{1}{2}] = \underrightarrow{lim} G_{i}, where G_{i} ≅ ℤ and the maps are multiplication by 2. Then there is an obvious isomorphism of inverse sequences

{ℤ, 2} ≅ {Hom_{ℤ}(G_{i}, ℤ)}.

We may replace ℤ[\frac{1}{2}] by any subring R of the rational numbers which is not divisible^{11} and obtain a similar isomorphism.

See the remarks at the end of Section 6 for further development of this example.

11To obtain nontrivial examples we must be in a situation where the map R → R is not an isomorphism, by 3.3. For instance, we cannot take R = ℚ.

The following three examples are taken from McGibbon [44], p. 1238.

Example 3.7. Suppose that {G_{i}} is an inverse sequence of abelian groups, p is a prime, and G_{ip} ≅ G_{i} ⊗ ℤ_{p} denotes the localization. Then there is a natural map

δ_{∗} : \underleftarrow{lim}^{1}G_{i} ⟶ Π_{p} \underleftarrow{lim}^{1} G_{ip}.

McGibbon points out that this map is seldom an isomorphism. For instance, if

G_{i} = Hom_{ℤ}(A, ℤ)

with A = \underrightarrow{lim} A_{i} and each A_{i} is finitely generated then this map corresponds via Jensen’s isomorphism (6.1) to the natural map

δ_{∗} : Ext_{ℤ}^{1}(A, ℤ) ⟶ Π_{p} Ext_{ℤ}^{1}(A, ℤ_{p}).

This map is always surjective. If Ext_{ℤ}^{1}(A, ℤ) ≠ 0 then δ_{∗} always has nontrivial kernel. For instance, in Example 3.6, Ker(δ_{∗}) is the countable group ℤ_{2}/ℤ.

Example 3.8. Suppose that G_{n} = m_{n}ℤ where m_{n} is the product of the first n primes. Then

\underleftarrow{lim}^{1}G_{n} ≅ ℝ ⊕ ℚ/ℤ

and δ_{∗} = 0.

Example 3.9. Suppose that G_{n} = n!ℤ. Then \underleftarrow{lim}^{1}G_{n} ≅ ℝ and Ker(δ_{∗}) is isomorphic to the sum of uncountably many copies of ℚ.

We use the following notation. If {G_{i}} is an inverse sequence of abelian groups with structural maps

f_{ji}: G_{j} → G_{i}

for each j > i, then define

G_{j,i} = Im(f_{ji}: G_{j} → G_{i})

so that for each fixed i there is a sequence of subgroups

G_{i} ⊇ G_{i+1,i} ⊇ G_{i+2,i} ⊇ · · · .

Definition 3.10. The inverse sequence^{12} {G_{i}} satisfies the Mittag-Leffler condition [30, 2, 3] if for each i there exists some integer ϕ(i) ≥ i such that

G_{j,i} = G_{ϕ(i),i}

12For inverse systems indexed by a directed set the Mittag-Leffler condition is not helpful: cf. [60], page 369.

for all j ≥ ϕ(i).

For instance, if we are given an inverse system of finitely generated modules over a ring satisfying the descending chain condition, then the Mittag-Leffler condition is satisfied if and only if the original inverse sequence is isomorphic as a pro-object to an inverse sequence whose structural maps are surjective. It is then fairly easy to prove the first part of the following theorem.

Theorem 3.11. 1. If an inverse sequence {G_{i}} of abelian groups satisfies

the Mittag-Leffler condition then \underleftarrow{lim}^{1}G_{i} = 0.

2. If an inverse sequence {G_{i}} of countable abelian groups satisfies

\underleftarrow{lim}^{1}G_{i} = 0

then it satisfies the Mittag-Leffler condition.

Proof. The first statement is a classic result established in [30], Ch. 0, §13. The second statement is due to B. Gray ([29] p. 242). □

Example 3.12 (McGibbon [44]). It is easy to see that the inverse sequence

ℤ ←^{3} ℤ ←^{5} ℤ ←^{7} ℤ ←^{11} …

is not Mittag-Leffler.

The earliest study of \underleftarrow{lim}^{1} was by Steenrod [59] who was interested in the difference between Vietoris (more or less the same as Čech) homology (which did not satisfy the exactness axiom, and hence was not really a homology theory) and what is now called Steenrod homology (which did indeed satisfy the axioms.) Steenrod homology maps onto Čech homology, and the kernel is a suitable \underleftarrow{lim}^{1} group which Steenrod computed in the case of a solenoid, demonstrating that the two theories were really different. He also introduced a topology for Steenrod homology and showed that the closure of zero was the \underleftarrow{lim}^{1} subgroup. At that time it was not clear that \underleftarrow{lim}^{1} was a derived functor (indeed, the word functor was not even in use mathematically), which makes Steenrod’s contribution all the more impressive.

Eilenberg and Steenrod pursued this theme systematically in their seminal work [19]. They showed that the axioms were satisfied by Ȟ_{∗}(X; G) when G is compact or when G is a finite-dimensional vector space over a field. We understand now that the obstruction to exactness (the axiom that failed) is exactly the group \underleftarrow{lim}^{1}H_{∗}(X_{α}; G).

The work of Eilenberg and Steenrod was followed by results of various people, particularly Yeh [64], Eilenberg and Moore [18], and Roos [52]. J.

Milnor [47], studied the behavior of limits on representable cohomology theories. If h^{∗} is such a theory and if X is an infinite CW-complex with h^{∗}(X) of finite type then for each n there is a Milnor sequence of the form

0 → \underleftarrow{lim}^{1}h^{n−1}(X_{j}) → h^{n}(X) → \underleftarrow{lim} h^{n}(X_{j}) → 0. (3.13)

This is in a sense dual to Steenrod’s initial use of \underleftarrow{lim}^{1}. Steenrod dealt with a compact space written as an inverse limit of finite complexes. Milnor deals with an infinite CW-complex written as the direct limit of finite CWcomplexes. Milnor’s setting is typical in modern algebraic topology, whereas Steenrod’s setting generalizes to C^{∗}-algebras cf. [55].

Milnor [48] pursued Steenrod’s original example in his paper “On the Steenrod homology theory” which was first distributed in 1961 and was published more than 30 years later.

Remark 3.14. C. McGibbon has observed that it is possible in general to have an inverse sequence which is not Mittag-Leffler and yet for which \underleftarrow{lim}^{1} vanishes. Here is his example. Let K be the direct product of countably many copies of ℤ/2 where each factor has the discrete topology and K has the product topology. Let K_{n} be the kernel of the projection of K onto the first n factors. These projections are continuous, and the K_{n} are both open and closed (hence compact) subgroups of K. There are evident inclusion maps K_{n} → K_{n−1} and the associated sequence is not Mittag-Leffler. Nevertheless, \underleftarrow{lim}^{1}K_{n} = 0 by the following result.

Proposition 3.15. An inverse sequence \{G_{i}\} satisfies \underleftarrow{lim}^{1}G_{i} = 0 if either of the following conditions holds:

1. {G_{i}} consists of compact Hausdorff (not necessarily abelian) groups and continuous homomorphisms.

2. {G_{i}} consists of finite-dimensional vector spaces over a field and linear maps.

Proof. This is established by Eilenberg-Steenrod [19] in the abelian setting and by McGibbon [44] in general. □

McGibbon notes that the Hausdorff assumption is essential. For instance, take any example with \underleftarrow{lim}^{1}G_{i} ≠ 0 and place the indiscrete topology on each group. Then each G_{i} is compact (but not Hausdorff), maps are continuous, and obviously \underleftarrow{lim}^{1}G_{i} ≠ 0!

Next we consider the relationship between the groups

\underleftarrow{lim}^{1}G_{i}  and  \underleftarrow{lim}^{1}(G_{i} ⊗ M).

Proposition 3.16. Let {G_{i}} be an inverse sequence of abelian groups, and let M be an abelian group. Then:

1. If M is finitely generated, then

(\underleftarrow{lim}^{1}G_{i}) ⊗ M ≅ \underleftarrow{lim}^{1}(G_{i} ⊗ M).

2. If {G_{i}} is an inverse sequence of countable abelian groups and if

\underleftarrow{lim}^{1}G_{i} = 0

then

\underleftarrow{lim}^{1}(G_{i} ⊗ M) = 0.

3. If {G_{i}} is an inverse sequence of countable abelian groups, M is a countable, faithfully flat^{13} abelian group, and if

\underleftarrow{lim}^{1}(G_{i} ⊗ M) = 0

then

\underleftarrow{lim}^{1}G_{i} = 0.

Proof. To establish Part 1), we note that since \underleftarrow{lim}^{1} respects finite sums this comes down to checking the case M = ℤ, which is trivial, and the case M = ℤ/n, which is very simple. For Part 2), suppose that \underleftarrow{lim}^{1}G_{i} = 0. Then the sequence {G_{i}} satisfies the Mittag-Leffler condition, by Theorem 3.11, so that there is some function ϕ(i) such that

G_{j,i} = G_{ϕ(i),i}

for all j ≥ ϕ(i). Fix some index i. Then

Im(G_{j,i} ⊗ M → G_{i} ⊗ M) ≅ \frac{G_{j,i} ⊗ M}{T or^{ℤ}(G_{i}/G_{j,i}, M)}
≅ \frac{G_{ϕ(i),i} ⊗ M}{T or^{ℤ}(G_{i}/G_{ϕ(i),i}, M)}
≅ Im(G_{ϕ(i),i} ⊗ M → G_{i} ⊗ M)

and hence the inverse sequence {G_{i} ⊗ M} satisfies the Mittag-Leffler condition, so that \underleftarrow{lim}^{1}(G_{i} ⊗ M) = 0. This proves Part 2).

For Part 3) we need the following fact:

Fact. If M is faithfully flat and α: G → G′ is a homomorphism such that

α ⊗ 1 : G ⊗ M → G′ ⊗ M

13An abelian group M is faithfully flat if for any group G, if G ⊗ M = 0 then G = 0.

For instance ℚ is not faithfully flat, while ℚ ⊕ ℤ is faithfully flat. This coincides with the usual definition when M is torsionfree.

is an isomorphism, then α itself is an isomorphism.

This fact is immediate from the definition of faithful flatness and the isomorphisms

0 ≅ Ker(α ⊗ 1) ≅ Ker(α) ⊗ M

and

0 ≅ Coker(α ⊗ 1) ≅ Coker(α) ⊗ M.

Let H_{i} = G_{i} ⊗ M and define

H_{j,i} = Im(H_{j} → H_{i}) ≅ G_{j,i} ⊗ M.

Suppose that \underleftarrow{lim}^{1}H_{i} = 0. Then Theorem 3.11(2) implies that \{H_{i}\} satisfies the Mittag-Leffler condition. Let α : G_{j,i} → G_{ϕ(i),i} be the canonical map. We have a commuting diagram

G_{j,i} ⊗ M ⟶^{α⊗1} G_{ϕ(i),i} ⊗ M
↓≅  ↓≅
H_{j,i} ⟶ H_{ϕ(i),i}

and the map H_{j,i} → H_{ϕ(i),i} is an isomorphism by the Mittag-Leffler assumption. Thus α ⊗ 1 is an isomorphism and hence α is itself an isomorphism. Thus {G_{i}} satisfies the Mittag-Leffler condition. □

I. Emmanouil [21] gives the definitive result in this direction. His result is in the context of modules over a fixed ring; we state it for abelian groups.

Proposition 3.17. For an inverse sequence {G_{i}} the following are equivalent:

1. {G_{i}} satisfies the Mittag-Leffler condition.

2. For all abelian groups M,

\underleftarrow{lim}^{1}(G_{i} ⊗ M) = 0.

3. There is some free abelian group F of infinite rank such that

\underleftarrow{lim}^{1}(G_{i} ⊗ F) = 0.

Finally, anticipating our study of Pext and Jensen’s theorem, we record the following.

Proposition 3.18. Suppose that G = \underrightarrow{lim} G_{i} is a direct limit of finitely generated abelian groups G_{i} and suppose that H is also a finitely generated abelian group. Then:

1.

\underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) ≅ [\underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, ℤ)] ⊗ H.

2. If

\underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, ℤ) = 0

then

\underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) = 0.

3. If fH ≠ 0^{14} and

\underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) = 0

then \underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, ℤ) = 0.

Proof. Since each G_{i} is free there are natural isomorphisms

Hom_{ℤ}(G_{i}, H) ≅ Hom_{ℤ}(G_{i}, ℤ) ⊗ H

for each i and hence Proposition 3.16 implies Part 1). Part 2) is a consequence of Part 1). For Part 3) we note that a finitely generated abelian group H is faithfully flat if and only if fH ≠ 0. □

14We let tG denote the torsion subgroup of G and fG = G/tG denote the maximal torsionfree quotient.

4. Exact sequences, divisibility results

In this section we search for general results which imply that \underleftarrow{lim}^{1} = 0 or that \underleftarrow{lim}^{1} is divisible. We pay special attention to algebraically compact groups, and we show that if {G_{i}} is an inverse sequence of abelian groups such that {tG_{i}} is a continuous algebraically compact inverse sequence and each fG_{i} is torsion-free of finite rank, then \underleftarrow{lim}^{1}G_{i} is divisible. This implies that in many cases of interest the Milnor \underleftarrow{lim}^{1} sequence 3.13 splits unnaturally.

Definition 4.1. An inverse sequence {G_{i}} of algebraically compact abelian groups is continuous if there exists an inverse sequence {K_{i}} of compact abelian groups and continuous maps such that the inverse sequence {G_{i}} is a direct summand^{15} of {K_{i}}.

Proposition 4.2. If {G_{i}} is a continuous inverse sequence of algebraically compact abelian groups then \underleftarrow{lim}^{1}G_{i} = 0.

Proof. Since {G_{i}} is a direct summand of an inverse sequence {K_{i}} of compact groups and continuous homomorphisms, there is an isomorphism

\underleftarrow{lim}^{1}K_{i} ≅ \underleftarrow{lim}^{1}G_{i} ⊕ \underleftarrow{lim}^{1}K_{i}/G_{i}

and \underleftarrow{lim}^{1}K_{i} = 0 by Proposition 3.15. This implies that \underleftarrow{lim}^{1}G_{i} = 0. □

The following example shows that the assumption that the structural maps be continuous is essential.

Example 4.3 (C. McGibbon). Let B_{o} be a vector space over ℤ/2 of countably infinite dimension, with basis {e_{1}, e_{2}, …, e_{n}, … }. Let C_{n} be the span of {e_{1}, …, e_{n}} and let B_{o} → C_{n} be the canonical projection with kernel A_{n}. Then there is a natural commuting diagram

0 ⟶ A_{n} ⟶ B_{n} ⟶ C_{n} ⟶ 0
↓  ↓  ↓
0 ⟶ A_{n−1} ⟶ B_{n−1} ⟶ C_{n−1} ⟶ 0

(where in the middle sequence B_{n} = B_{o} and the structural map is the identity) and hence a short exact sequence of inverse sequences. This gives an exact sequence

15as inverse sequences. That is, not only is {G_{i}} an inverse subsequence of {K_{i}} and each G_{i} a direct summand of K_{i}, but the retraction maps K_{i} → G_{i} must respect the structural maps of the inverse sequences. See Example 4.3 for an illustration of what can go wrong otherwise.

0 → \underleftarrow{lim} B_{n} → \underleftarrow{lim} C_{n} → \underleftarrow{lim}^{1}A_{n} → 0

since \underleftarrow{lim} A_{n} = 0 and \underleftarrow{lim}^{1}B_{n} = 0. This implies that

\underleftarrow{lim}^{1}A_{n} ≅ \frac{\underleftarrow{lim} C_{n}}{\underleftarrow{lim} B_{n}} ≅ \frac{Π_{1}^{∞}(ℤ/2)}{⊕_{1}^{∞}(ℤ/2)} ≅ (ℤ/2)^{ℵ_{o}}

which is the product of countably many copies of ℤ/2 and is isomorphic as a vector space to the sum of uncountably many copies of ℤ/2. Thus \underleftarrow{lim}^{1}A_{n} is an uncountable reduced group with every element of order 2. This shows that although each group A_{n} is algebraically compact (it embeds as a pure subgroup of the compact group Π_{1}^{∞}(ℤ/2)), the structural maps are not continuous (in the sense of the definition above). The inverse sequence \{A_{n}\} is not a direct summand of a compact inverse sequence and \underleftarrow{lim}^{1} is highly nontrivial.^{16}

Note that in the example above that the group ℤ/2 may be replaced by any commutative ring R, in which case

\underleftarrow{lim}^{1}A_{n} ≅ \frac{Π_{1}^{∞}R}{⊕_{1}^{∞}R}.

For instance, if we insert ℚ then we obtain an example of an inverse sequence of divisible groups with nontrivial \underleftarrow{lim}^{1}. In fact, \underleftarrow{lim}^{1} = ℚ^{ℵ_{o}}.

Proposition 4.4. Let {G_{i}} be an inverse sequence of abelian groups satisfying the following two conditions:

1. The inverse sequence {tG_{i}} is a continuous inverse sequence of algebraically compact^{17} groups (e.g., tG_{i} = 0 for all i).

2. For each prime p,

\underleftarrow{lim}^{1} \frac{G_{i}/pG_{i}}{tG_{i}/ptG_{i}} = 0.

Then the group \underleftarrow{lim}^{1}G_{i} is divisible.

16It is instructive to experiment with this directly. Take A_{n} ⊆ B_{n} ⊆ K_{n} where B_{n} = B_{o} ⊆ K = K_{n} is the canonical inclusion. Then it is not hard to see that there is no “chain map”s : K_{∗} → A_{∗} which is a retraction for the canonical inclusion.

17An algebraically compact group which is also torsion is the direct sum of cyclic groups and of groups of the type ℤ(p^{∞}), by results of Fuchs [24], §40.3, Prüfer [51] , and Baer [4].

Condition 2) of Proposition 4.4 is satisfied whenever fG_{i} is torsionfree of finite rank since any torsionfree group G of finite rank has the property (cf. [1] 0.3) that for each prime p,

dim_{ℤ/p}(G/pG) ≤ rank(G)

which implies that for each i, G_{i}/nG_{i} is a finite group.

Both Conditions 1) and 2) are satisfied in each of the following special cases:

1. Each G_{i} is finitely generated.^{18}

2. Each G_{i} is torsionfree, and for each prime p, the group G_{i}/pG_{i} is finite.

3. Each G_{i} is torsionfree and divisible.

4. Each G_{i} is torsionfree of finite rank.

Proof. The first hypothesis implies that \underleftarrow{lim}^{1}tG_{i} = 0 by Proposition 4.2 and hence

\underleftarrow{lim}^{1}G_{i} ≅ \underleftarrow{lim}^{1}G_{i}/tG_{i}

by the long exact \underleftarrow{lim} -\underleftarrow{lim}^{1} sequence. Further,

\frac{fG_{i}}{pfG_{i}} ≅ \frac{G_{i}/tG_{i}}{p(G_{i}/tG_{i})} ≅ \frac{G_{i}/pG_{i}}{tG_{i}/ptG_{i}}

for each i by the Snake Lemma, and hence

\underleftarrow{lim}^{1} \frac{G_{i}/tG_{i}}{p(G_{i}/tG_{i})} ≅ \underleftarrow{lim}^{1} \frac{G_{i}/pG_{i}}{tG_{i}/ptG_{i}} .

So without loss of generality we may assume that each G_{i} is torsionfree.

Fix some prime p and let ζ : G_{i} → G_{i} denote multiplication by p. This induces a short exact sequence of inverse sequences of the form

0 → {G_{i}} ⟶^{ζ} {G_{i}} → {G_{i}/pG_{i}} → 0

and hence a six term \underleftarrow{lim} -\underleftarrow{lim}^{1} sequence, the last three terms of which are

\underleftarrow{lim}^{1}G_{i} ⟶^{ζ_{∗}} \underleftarrow{lim}^{1}G_{i} ⟶ \underleftarrow{lim}^{1}G_{i}/pG_{i} ⟶ 0.

It is easy to show that ζ_{∗} is still multiplication by p. We have assumed that \underleftarrow{lim}^{1}(G_{i}/pG_{i}) = 0, and hence the map

ζ_{∗}: \underleftarrow{lim}^{1}G_{i} ⟶ \underleftarrow{lim}^{1}G_{i}

is surjective. Thus

p(\underleftarrow{lim}^{1}G_{i}) = \underleftarrow{lim}^{1}G_{i}.

18This case of the Proposition is well-known and is proved, e.g., in [38].

This is true for each prime p and hence \underleftarrow{lim}^{1}G_{i} is divisible. □

Note that if each G_{i} is a divisible group then \underleftarrow{lim}^{1}G_{i} is divisible, since it is a quotient of the divisible group Π_{i}G_{i}. We may extend this result slightly as follows. Let R_{i} denote the maximal reduced quotient of G_{i}.

Corollary 4.5 (C. McGibbon). Suppose that {G_{i}} is an inverse sequence of abelian groups. Then the following conditions are equivalent:

1. \underleftarrow{lim}^{1}G_{i} is divisible.

2. For each prime p, \underleftarrow{lim}^{1}(G_{i}/pG_{i}) = 0.

3. For each prime p, \underleftarrow{lim}^{1}(R_{i}/pR_{i}) = 0.

Proof. The sequence

0 → pG_{i} → G_{i} → G_{i}/pG_{i} → 0 (4.6)

is exact and there is an associated short exact sequence of inverse sequences obtained by varying i. The \underleftarrow{lim} -\underleftarrow{lim}^{1} sequence concludes with

(4.7) \underleftarrow{lim}^{1}pG_{i} ⟶ \underleftarrow{lim}^{1}G_{i} ⟶ \underleftarrow{lim}^{1}G_{i}/pG_{i} → 0.

Suppose that \underleftarrow{lim}^{1}G_{i} is divisible. Then \underleftarrow{lim}^{1}(G_{i}/pG_{i}) is divisible as well, by the exact sequence 4.7. In particular, division by p is possible. On the other hand, \underleftarrow{lim}^{1}(G_{i}/pG_{i}) is a quotient of the ℤ/p-module Π_{i}(G_{i}/pG_{i}) and these together imply that \underleftarrow{lim}^{1}(G_{i}/pG_{i}) = 0. Thus Part 1) implies Part 2).

To prove the converse, suppose that for each prime p we have

\underleftarrow{lim}^{1}(G_{i}/pG_{i}) = 0.

Sequence 4.7 becomes

\underleftarrow{lim}^{1}pG_{i} ⟶ \underleftarrow{lim}^{1}G_{i} ⟶ 0. (4.8)

The short exact sequences

0 → T or_{1}^{ℤ}(G_{i}, ℤ/p) → G_{i} ⟶^{π} pG_{i} → 0

give rise to a short exact sequence of inverse sequences and hence a \underleftarrow{lim} -\underleftarrow{lim}^{1} sequence which concludes as

\underleftarrow{lim}^{1}G_{i} ⟶^{π_{∗}} \underleftarrow{lim}^{1}pG_{i} → 0

and hence the composite map

\underleftarrow{lim}^{1}G_{i} ⟶^{π_{∗}} \underleftarrow{lim}^{1}pG_{i} ⟶ \underleftarrow{lim}^{1}G_{i}

is surjective. This composite is induced by the natural maps

G_{i} ⟶^{π} G_{i}[p] → G_{i}

and this is the canonical factorization of the multiplication by p map. Hence

ζ_{∗} : \underleftarrow{lim}^{1}G_{i} → \underleftarrow{lim}^{1}G_{i}

is surjective. This simply says that

p(\underleftarrow{lim}^{1}G_{i}) = \underleftarrow{lim}^{1}G_{i}

for each p, which implies that \underleftarrow{lim}^{1}G_{i} is divisible. Thus Part 2) implies Part 1).

Finally, the equivalence of Part 2) and Part 3) is immediate from the natural isomorphism

G_{i}/pG_{i} ≅ R_{i}/pR_{i}.
□

Let us examine Condition 2) more closely. Note that the group G_{i}/pG_{i} is a ℤ/p-module, that is, a vector space over the field ℤ/p, and hence it is determined up to isomorphism by its ℤ/p-dimension.

Theorem 4.9. Suppose that {G_{i}} is an inverse sequence of abelian groups such that the following conditions hold:

1. For each i, tG_{i} is algebraically compact and the sequence {tG_{i}} is continuous.

2. Each G_{i}/tG_{i} is torsionfree of finite rank.

Then

\underleftarrow{lim}^{1}G_{i} ≅ \underleftarrow{lim}^{1}(G_{i}/tG_{i})

and this group is divisible.

Proof. The first assumption implies that

\underleftarrow{lim}^{1}tG_{i} = 0.

The \underleftarrow{lim} -\underleftarrow{lim}^{1}sequence implies that

\underleftarrow{lim}^{1}G_{i} ≅ \underleftarrow{lim}^{1}G_{i}/tG_{i}

and the group \underleftarrow{lim}^{1}G_{i}/tG_{i} is divisible by Proposition 4.4. □

Remark 4.10. It is possible that these assumptions may fail even if the G_{i} are countable. A countable abelian group is algebraically compact if and only if it is a direct sum of a divisible group and a bounded group [24] §40; this follows from the general classification of algebraic compact groups. For instance, the group ⊕(ℤ/p) (where the sum runs over all primes) is countable, torsion, and yet not algebraically compact. Similarly, it is easy to construct examples of countable torsionfree abelian groups of infinite rank — a countable-dimensional ℚ-vector space provides the simplest example.

Corollary 4.11. Suppose that {G_{i}} is an inverse sequence of abelian groups and that one of the following conditions holds:

1. Each G_{i} is divisible.

2. Each G_{i} is torsionfree of finite rank.

3. Each G_{i} is finitely generated. Then \underleftarrow{lim}^{1}G_{i} is divisible. In particular, any short exact sequence of the form

0 → \underleftarrow{lim}^{1}G_{i} → K → L → 0

splits.

Short exact sequences of this sort are very common in classical topology as well as in non-commutative topology. So the corollary gives us very general circumstances under which they split.

Proof. Part 1) follows as indicated before, since \underleftarrow{lim}^{1}G_{i} is a quotient of Π_{i}G_{i}, and quotients and products of divisible groups are again divisible. Parts 2) and 3) follow from Theorem 4.9. □

Remark 4.12. We note in passing that if the groups {G_{i}} are allowed to be nonabelian then the terrain is quite different. (This is the situation of greatest interest when studying homotopy classes of phantom maps in algebraic topology.) The interested reader should consult [46, 44].

5. First facts on Pext

In this section we recall the basic definitions and elementary properties of the Ulm subgroups and Pext, and we show that Ext = Pext in certain special cases.

Definition 5.1. Suppose that G is an abelian group. The first Ulm subgroup of G is defined by

U^{1}(G) = ∩_{n}nG

and the higher Ulm subgroups are defined inductively by

U^{n+1}(G) = U^{1}(U^{n}(G))

and by taking limits for limit ordinals. These form a decreasing family of subgroups of G. The Ulm length of G is the first ordinal τ such that

U^{τ+1}(G) = U^{τ} (G).

Recall that a subgroup G′ of a group G is said to be pure if G′ ∩nG = nG′ for all positive integers n.

Definition 5.2. An extension

0 → G′ → G → G′′ → 0

is said to be pure if G′ is a pure subgroup of G. If an extension is pure then any other extension in its equivalence class in Ext is also pure. Equivalence classes of pure extensions form a subgroup of Ext_{ℤ}^{1}(G, H) denoted Pext_{ℤ}^{1}(G, H).

If G′ is a direct summand of G then G′ is a pure subgroup, but not conversely. For example, the torsion subgroup tG is always a pure subgroup of G (this is a routine check) but it is not necessarily a direct summand of G. (See §8).

Proposition 5.3. Suppose that G is an abelian group. Then:

1. If G is divisible then U^{σ}(G) = G for all σ.

2. If τ is the Ulm length of G, then U^{τ} (G) is the maximal divisible subgroup of G.

3. If G is torsionfree then

U^{τ} (G) = U^{1}(G)

so that U^{1}(G) is the maximal divisible subgroup of G.

4. If G is algebraically compact then U^{τ} (G) = U^{1}(G) so that U^{1}(G) is the maximal divisible subgroup of G.

5. For any abelian group H,

Pext_{ℤ}^{1}(G, H) ≅ U^{1}(Ext_{ℤ}^{1}(G, H)).

Proof. An abelian group A is divisible if and only if A = nA for each positive integer n. This condition is clearly satisfied by U^{τ} (G). Any divisible group H ⊆ G lies in U^{1}(G), since

H = nH ⊆ nG

for each n, and using induction we see that H ⊆ U^{τ} (G). Thus U^{τ} (G) contains all divisible subgroups of G. This proves Parts 1) and 2). Part 3) is stated in ([24], Page 101, Exer. 2.) Part 4) is stated in ([24], Page 162, Exer. 7). Part 5) is a theorem of Nunke and Fuchs (cf. [24], §53.3) and it will follow from Jensen’s Theorem 6.1. □

Proposition 5.4. 1. For each pair of abelian groups G and H, the group Pext_{ℤ}^{1}(G, H) is a subgroup of the group Ext_{ℤ}^{1}(G, H). The functor is contravariant in G and covariant in H.

2. A pure short exact sequence induces long exact Hom-Pext sequences in either variable.

3. Pext_{ℤ}^{1}(G, H) = 0 for all G if and only if H is algebraically compact.

4. Pext_{ℤ}^{1}(G, H) = 0 for all H if and only if G is a direct sum of cyclic groups.

5. There is a natural isomorphism

Pext_{ℤ}^{1}(⊕_{i}G_{i}, H) ≅ Π_{i}Pext_{ℤ}^{1}(G_{i}, H).

6. There is a natural isomorphism

Pext_{ℤ}^{1}(G, Π_{i}H_{i}) ≅ Π_{i}Pext_{ℤ}^{1}(G, H_{i}).

Proof. These are foundational facts found in [24], §53. Part 1) is due to Fuchs [22] and Baer [5]. Part 2) is a result of Harrison [31]. Part 3) follows immediately by the definition of an algebraically compact group. Part 4) is due to Maranda [24]. Parts 5) and 6) are exercises in Fuchs [24]. If one has already shown that Pext is really defined in terms of relative homological algebra then all of these properties follow at once for elementary reasons. □

Here is an easy corollary of our previous work.

Corollary 5.5. If G is a torsionfree abelian group then

Pext_{ℤ}^{1}(G, H) ≅ Ext_{ℤ}^{1}(G, H).

In particular, Pext_{ℤ}^{1}(G, H) is divisible.

Proof. If G is torsionfree then Ext_{ℤ}^{1}(G, H) is divisible, by Proposition 2.4, and then Proposition 5.3 implies the result. □

Proposition 5.6. Let G and H be abelian groups. Then:

1. The group Pext_{ℤ}^{1}(G, H) satisfies

Pext_{ℤ}^{1}(G, H) ≅ ⋂_{G′} Ker[Ext_{ℤ}^{1}(G, H) ⟶ Ext_{ℤ}^{1}(G′, H)]

where G′ runs over all finitely generated subgroups of G.

2. The group Pext_{ℤ}^{1}(G, H) satisfies

Pext_{ℤ}^{1}(G, H) ≅ ⋂_{G′} Ker[Ext_{ℤ}^{1}(G, H) ⟶ Ext_{ℤ}^{1}(G′, H)]

where G′ runs over all finite subgroups of G.

3. If G is written as an increasing union of finitely generated subgroups G_{i}, then there is a natural short exact sequence

(5.7) 0 → Pext_{ℤ}^{1}(G, H) ⟶ Ext_{ℤ}^{1}(G, H) ⟶ \underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H) → 0.

Proof. Fuchs [24], §20.3 proves that an extension

0 → H → J → G → 0

is pure exact if and only if for each cyclic group C and for each homomorphism ζ: C → G the induced extension

0 → H → J′ → C → 0

splits. Translated into terms of Ext, this says that

Pext_{ℤ}^{1}(G, H) ≅ ⋂_{G′} Ker[Ext_{ℤ}^{1}(G, H) ⟶ Ext_{ℤ}^{1}(G′, H)]

where G′ runs over all cyclic subgroups of G. Equivalently, we may let G′ run over all finitely generated subgroups of G, or over all finite subgroups of G. This establishes Parts 1) and 2). Part 3) is immediate from Part 1) and elementary definitions. □

Let θ: tG → G denote the natural inclusion of the torsion subgroup, and let

θ_{E}^{∗} : Ext_{ℤ}^{1}(G, H) → Ext_{ℤ}^{1}(tG, H)

denote the induced map. Similarly, let

θ_{P}^{∗} : Pext_{ℤ}^{1}(G, H) → Pext_{ℤ}^{1}(tG, H)

denote the induced map.

Theorem 5.8. Let G and H be abelian groups. Then

Ker(θ_{E}^{∗}) ≅ Ker(θ_{P}^{∗}).

Further,

(5.9) Ker(θ_{E}^{∗}) ≅ Pext_{ℤ}^{1}(G, H) ⟺ Pext_{ℤ}^{1}(tG, H) = 0.

Thus if H is algebraically compact then θ_{E}^{∗} is a monomorphism. Suppose further that one of the following hypotheses hold:

1. tG is the direct sum of cyclic groups.

2. tG is countable and has no nonzero elements of infinite height^{19} (for

instance, if tG is bounded). Then

Ker(θ_{E}^{∗}) ≅ Pext_{ℤ}^{1}(G, H). (5.10)

Proof. The short exact sequence

0 → tG → G → G/tG → 0

is pure and hence there is a commutative diagram with exact columns of the form

Hom_{ℤ}(tG, H) ⟶^{1} Hom_{ℤ}(tG, H)
↓  ↓
Pext_{ℤ}^{1}(G/tG, H) ⟶^{≅} Ext_{ℤ}^{1}(G/tG, H)
↓^{λ_{P}}  ↓^{λ_{E}}
Pext_{ℤ}^{1}(G, H) ⟶ Ext_{ℤ}^{1}(G, H)
↓^{θ_{P}^{∗}}  ↓^{θ_{E}^{∗}}
Pext_{ℤ}^{1}(tG, H) ⟶ Ext_{ℤ}^{1}(tG, H)
↓  ↓
0  0.

(5.11)

The inclusion

Pext_{ℤ}^{1}(G/tG, H) → Ext_{ℤ}^{1}(G/tG, H)

19An element g ∈ G has infinite height if for all primes p and natural numbers r the equation p^{r}x = g has a solution x ∈ G.

is an isomorphism by Corollary 5.5, since G/tG is torsionfree. A standard diagram chase shows that

Ker(θ_{P}^{∗}) ≅ Im(λ_{P}) ≅ Im(λ_{E}) ≅ Ker(θ_{E}^{∗})

which proves the first assertion.

If H is algebraically compact then Pext_{ℤ}^{1}(−, H) = 0 and hence θ_{E}^{∗} is mono. Moving to the last part of the Theorem, suppose that tG ≅ ⊕_{i}C_{i} where

each C_{i} is a finite cyclic group. Then

Ext_{ℤ}^{1}(tG, H) ≅ Ext_{ℤ}^{1}(⊕_{i}C_{i}, H) ≅ Π_{i} Ext_{ℤ}^{1}(C_{i}, H)

so that

θ_{E}^{∗} ≅ Π_{i} θ_{i}^{∗} : Ext_{ℤ}^{1}(G, H) ⟶ Π_{i}Ext_{ℤ}^{1}(C_{i}, H).

Thus

Pext_{ℤ}^{1}(G, H) ≅ ⋂ Ker[Ext_{ℤ}^{1}(G, H) ⟶ Ext_{ℤ}^{1}(G′, H)]

with intersection over all finite subgroups G′ ⊆ G

⊆ ⋂ Ker[Ext_{ℤ}^{1}(G, H) ⟶ Ext_{ℤ}^{1}(C_{i}, H)]

with intersection over all cyclic summands C_{i},

⊆ ⋂ Ker(θ_{E}^{∗}) ≅ Ker(θ_{E}^{∗})

and hence

Pext_{ℤ}^{1}(G, H) ≅ Ker(θ_{E}^{∗})

so Equation 5.10 holds.

Any countable abelian group with no nonzero elements of infinite height is a sum of cyclic groups, by a theorem of Prüfer (cf. [24], §17.2). Hence Part 2) is implied by Part 1). □

Remark 5.12. Pext may be brought into the context of relative homological algebra. Let S be the class of pure short exact sequences. This is a proper class in the sense of Buchsbaum-MacLane [41] where S-projectives are direct sums of cyclic groups and S-injectives are algebraically compact groups. It is not hard to prove (cf. [61]) that

Ext_{S}^{1}(G, H) ≅ Pext_{ℤ}^{1}(G, H).

This is the beginning of a long and deep story which goes under the rubric of “pure homological algebra”. There are extensive results on pure extensions over commutative rings. As indicated in the Introduction, we do not intend to discuss these results, as they take us too far afield. Pure global dimension and the Ziegler spectrum of indecomposable pure-injective modules are 28 Claude L. Schochet active areas of research. See the survey article by Ivo Herzog [32] for more information.

6. Jensen’s theorem and Roos’s theorem

This section is devoted to Jensen’s theorem and its application to a proof of a well-known result of Roos.

C. U. Jensen [38] has discovered a remarkable connection between homological algebra and infinite abelian groups. Here is his result, specialized to our situation.

Theorem 6.1 ([38]). Let G be a countable abelian group written as an increasing union of finitely generated subgroups G_{i} and let H be any abelian group. Then there is a natural isomorphism

\underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) ≅ Pext_{ℤ}^{1}(G, H).

We believe that this theorem deserves more attention, since anything which sheds light upon \underleftarrow{lim}^{1} is welcome, and especially because of the crosscultural nature of the result.

Sklaryenko observes [60] that the theorem has as a consequence the Stein-Serre theorem: If G is a countable abelian group with Ext_{ℤ}^{1}(G, ℤ) = 0, then G is free abelian.

Jensen proves Theorem 6.1 in a much more general context. He shows that if R is a ring, G_{α} and H are R-modules with G = \underrightarrow{lim} G_{α} where α runs over an arbitrary directed set, then there is a spectral sequence with

E_{2}^{p,q} ≅ \underleftarrow{lim}^{p}Pext_{R}^{q}(G_{α}, H)

and which converges to

Pext_{R}^{n}(G, H).

If the R-modules G_{α} are of finite type then

Pext_{R}^{q}(G_{α}, H) = 0

for all q > 0 and all α and hence the spectral sequence collapses, with the edge homomorphism producing an isomorphism

Pext_{R}^{n}(G, H) ≅ \underleftarrow{lim}^{n}Hom_{R}(G_{α}, H).

Taking R = ℤ and n = 1 yields the theorem.

We present here a very simple alternate proof of Jensen’s Theorem, using only the Eilenberg definition of \underleftarrow{lim}^{1} and the most elementary facts about Pext. (Our longer argument was rendered obsolete by an excellent suggestion of an anonymous referee.)

Proof. Suppose that G is a countable abelian group written as the union of an increasing sequence of finitely generated subgroups

G_{1} ⟶^{ξ_{1}} G_{2} ⟶^{ξ_{2}} G_{3} ⟶ ⋯ ⟶ G.

Define ϕ_{i} : G_{i} → ⊕G_{n} by

ϕ_{i}(g_{i}) = (0, …, 0, g_{i}, −ξ_{i}(g_{i}), 0, …).

Then {ϕ_{i}} coalesce to form ϕ : ⊕_{n}G_{n} → ⊕_{n}G_{n} and there is a natural short exact sequence

0 → ⊕_{n}G_{n} ⟶^{ϕ} ⊕_{n}G_{n} ⟶ G → 0

expressing G as the direct limit of the groups {G_{n}}. This is a pure short exact sequence. Apply the functor Hom_{ℤ}(−, H) to the short exact sequence. The resulting Hom-Pext long exact sequence degenerates to give an exact sequence whose final terms are

Hom_{ℤ}(⊕_{n}G_{n}, H) ⟶^{ϕ^{∗}} Hom_{ℤ}(⊕_{n}G_{n}, H) ⟶ Pext_{ℤ}^{1}(G, H) → 0.

The diagram

Hom_{ℤ}(⊕_{n}G_{n}, H) ⟶^{ϕ^{∗}} Hom_{ℤ}(⊕_{n}G_{n}, H)
↓≅  ↓≅
Π_{n}Hom_{ℤ}(G_{n}, H) ⟶^{ψ} Π_{n}Hom_{ℤ}(G_{n}, H)

commutes by a direct check, and so

Pext_{ℤ}^{1}(G, H) ≅ Coker(ϕ^{∗}) ≅ Coker(ψ) ≅ \underleftarrow{lim}^{1}Hom_{ℤ}(G_{n}, H)

which proves Theorem 6.1. □

Theorem 6.1 has as a consequence an important theorem^{20} of Roos [52]. We supply two proofs. The first (of the first part of the theorem) uses Jensen’s Theorem. The second proof is a sketch of Roos’s high-tech proof.

Theorem 6.2 (Roos). Let {G_{i}} be a direct sequence of abelian groups with G = \underrightarrow{lim} G_{i} and let H be an abelian group. Then there is a natural short exact sequence

(6.3) 0 → \underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) → Ext_{ℤ}^{1}(G, H) ⟶^{Γ} \underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H) → 0

and in addition

\underleftarrow{lim}^{1}Ext_{ℤ}^{1}(G_{i}, H) ≅ 0. (6.4)

20Roos actually establishes this result in the context of inverse systems in an abelian category; the result below is a simple and very special case of his result.

Proof (low tech). Equation 5.7 yields a natural short exact sequence

0 → Pext_{ℤ}^{1}(G, H) → Ext_{ℤ}^{1}(G, H) ⟶^{Γ} \underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H) → 0

and Jensen’s Theorem 6.1 yields a natural isomorphism

\underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) ≅ Pext_{ℤ}^{1}(G, H).

Combining these two results yields the first part (6.3) of Roos’s theorem. □

Proof (high tech). Roos shows that for any ring R, left R-module H, and directed system (countable or not) of left R-modules G_{i}, that there is a spectral sequence which converges to Ext_{R}^{n}(G, H) with

E_{2}^{p,q} ≅ \underleftarrow{lim}^{p}Ext_{R}^{q}(G_{∗}, H).

Take R = ℤ. This is a principal ideal domain, and hence Ext_{ℤ}^{q} = 0 for q > 1. Further, insist that the index set be countable. Then \underleftarrow{lim}^{p} = 0 for p > 1. Thus there are at most four nontrivial terms at the E_{2}-level. There are no further differentials since E_{2}^{p,∗} = 0 for p > 1. Hence E_{2} = E_{∞}. Here is E_{∞}:

In total degree p + q = 0 there is the term

E_{∞}^{0,0} ≅ \underleftarrow{lim} Hom_{ℤ}(G_{i}, H) ≅ Hom_{ℤ}(G, H)

as required.

In total degree one there are two terms, namely

E_{∞}^{1,0} ≅ \underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H)

and

E_{∞}^{0,1} ≅ \underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H)

and these give a composition series for Ext_{ℤ}^{1}(G, H), namely sequence 6.3, where Γ is the edge homomorphism in the spectral sequence.

In total degree two there is only one^{21} possible nonzero term, namely

E_{∞}^{1,1} ≅ \underleftarrow{lim}^{1}Ext_{ℤ}^{1}(G_{i}, H).

E_{2}^{2,0} ≅ \underleftarrow{lim}^{2}Hom_{ℤ}(G_{i}, H) d^{2} : E_{2}^{2,0} → E_{2}^{0,1}.

21Here is a good example where sticking to countable index sets really pays off. If the index set were allowed to be uncountable then the term would also be present. There would be a possible differential The kernel of this differential would survive to E_{∞}^{2,0} to contribute a possibly nontrivial component to \underleftarrow{lim}^{1}Ext_{ℤ}^{1}(G_{i}, H) to cause trouble at this point, and the cokernel would replace \underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H) in the sequence 6.3.

However, E_{∞}^{1,1} must be the associated graded group associated to some filtration of the group Ext_{ℤ}^{2}(G, H), since the spectral sequence converges. This group vanishes, since

Ext_{ℤ}^{2}(−, −) ≅ 0

and this implies that

\underleftarrow{lim}^{1}Ext_{ℤ}^{1}(G_{i}, H) = 0

as required. This verifies 6.4 and completes the proof. □

We conclude this section by reexamining Example 3.6, in which we demonstrated that

\underleftarrow{lim}^{1}\{ℤ, 2\} ≅ ℤ̂_{2}/ℤ.

Let us revisit this calculation using Jensen’s Theorem 6.1. Let {G_{i}} be an increasing sequence of copies of ℤ in ℤ[\frac{1}{2}] whose union is equal to ℤ[\frac{1}{2}]. Apply Hom_{ℤ}(−, ℤ) and we obtain an inverse sequence {Hom_{ℤ}(G_{i}, ℤ)} which is obviously isomorphic to a cofinal subsequence of {ℤ, 2} and hence has the same \underleftarrow{lim}^{1}. Thus

\underleftarrow{lim}^{1}\{ℤ, 2\} ≅ \underleftarrow{lim}^{1}\{Hom_{ℤ}(G_{i}, ℤ)\} ≅ Pext_{ℤ}^{1}(ℤ[\frac{1}{2}], ℤ)

by Jensen’s Theorem 6.1

≅ Ext_{ℤ}^{1}(ℤ[\frac{1}{2}], ℤ)

since ℤ[\frac{1}{2}] is torsionfree, by Corollary 5.5. This is a divisible group, and by Jensen’s calculations it is the direct sum of 2^{ℵ_{o}} copies of ℚ and for each odd prime p one copy of ℤ(p^{∞}) ([38], and see 9.7). This is a very round-about way of determining the underlying structure of ℤ̂_{2}/ℤ as an abelian group!

7. Ranges and inverse limits

In this section we consider certain classes of abelian groups determined by \underleftarrow{lim}^{1} and by Pext. Then we turn our attention briefly to an examination of \underleftarrow{lim} G_{i}.

Which abelian groups can be realized as \underleftarrow{lim}^{1} groups? Here is a good answer to that question.

Proposition 7.1 ([63], Thm. 1). The following are equivalent for an abelian group L:

1. L is a cotorsion group.

2. There exists abelian groups \{G_{i}\}_{i∈ℕ} such that \underleftarrow{lim}^{1}G_{i} = L.

3. There exists a sequence {G_{i}}_{i∈ℕ} such that L is a homomorphic image of the group ΠG_{n}/ ⊕ G_{n}.

4. There exists an abelian group H and a countable abelian group G with Pext_{ℤ}^{1}(G, H) = L.

We note that cotorsion groups have some interesting properties that help in various applications. For example, a countable cotorsion group is a direct sum of a divisible group and a bounded group. For more, see [24], §§54, 55.

If one relaxes the assumption on the index set and allows inverse systems which are indexed by the set of all ordinals less than the smallest uncountable ordinal, then Warfield and Huber also show ([63], Theorem 3) that every abelian group is of the form \underleftarrow{lim}^{1}G_{i}.

Proposition 7.1 shows that the collection of abelian groups of the form \underleftarrow{lim}^{1}G_{i} is exactly the cotorsion groups. It is an open problem (cited, for instance by McGibbon [44]) to determine which cotorsion groups are of the form \underleftarrow{lim}^{1}G_{i} when the G_{i} must be countable. Denote this collection by G. Of course any G ∈ G must be cotorsion, and it also must be uncountable, by Proposition 3.5.

Similarly, let P denote those groups which are of the form Pext_{ℤ}^{1}(G, H) with both G and H countable. Then P ⊆ G by Jensen’s Theorem 6.1. It is an open problem to determine this class as well. Determination of P would also determine under weak hypotheses which groups can appear as the closure of zero in the Kasparov group KK_{∗}(A, B).

Is studying Pext more restrictive than studying arbitrary \underleftarrow{lim}^{1}? Any inverse sequence {L_{i}} can be written as {Hom_{ℤ}(G_{i}, H)} simply by taking G_{i} to be the character group of L_{i} and H = ℝ/ℤ, and hence

\underleftarrow{lim}^{1}L_{i} ≅ \underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) ≅ Pext_{ℤ}^{1}(G, H)

so, for instance, the class of abelian groups of the form \underleftarrow{lim}^{1}L_{i} is the same as the class of abelian group of the form Pext_{ℤ}^{1}(G, H), namely the cotorsion groups, by Proposition 7.1.

Suppose that we require L_{i}, G and H to be countable abelian groups. Then there are two possible questions.

One possibility would be to ask that given any inverse system {L_{i}} of countable abelian groups, does there exist some countable abelian groups G and H such that

\underleftarrow{lim}^{1}L_{i} ≅ Pext_{ℤ}^{1}(G, H).

We don’t know the answer to this question.

Suppose that we restrict further: we can insist that the entire inverse system {L_{i}} coincide with the inverse system {Hom_{ℤ}(G_{i}, H)}. Here we have an answer: in general this cannot be done. In fact, Example 4.3 does not arise in this way. Recall that it was of the form \underleftarrow{lim}^{1}A_{n}, with each A_{n} a countable sum of copies of ℤ/2 and each map A_{n} → A_{n−1} was mono and had cokernel one copy of ℤ/2. Here is a proof that this cannot be realized.

Proposition 7.2. Example 4.3 cannot be realized as an inverse sequence Hom_{ℤ}(G_{i}, H) with each G_{i} finitely generated, each G_{i−1} → G_{i} an inclusion, and H a countable group.

Proof. Let {A_{i}} denote the sequence defined in 4.3. Suppose that the increasing sequence {G_{i}} and countable group H satisfied

A_{i} ≅ Hom_{ℤ}(G_{i}, H)

and the connecting maps respected the isomorphisms so that

\underleftarrow{lim}^{1}A_{i} ≅ \underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H).

We have

⊕_{1}^{∞}(ℤ/2) ≅ A_{i} ≅ Hom_{ℤ}(G_{i}, H)

for each i. Write G_{i} = tG_{i} ⊕ fG_{i}, the sum of its torsion subgroup and torsionfree quotient. Then

⊕_{1}^{∞}(ℤ/2) ≅ Hom_{ℤ}(tG_{i}, H) ⊕ Hom_{ℤ}(fG_{i}, H).

If fG_{i} = ℤ^{t(i)} with t(i) > 0 for at least one value of i then

Hom_{ℤ}(fG_{i}, H) ≅ H^{t(i)} ⊆ ⊕_{1}^{∞}(ℤ/2)

which implies that H is also a direct sum of a countable number of copies of ℤ/2. On the other hand, if tG_{i} is nontrivial then without loss of generality we may assume that it is a sum of u(i) < ∞ copies of ℤ/2, and hence

Hom_{ℤ}(tG_{i}, H) ≅ ⊕_{1}^{u(i)}H[2] ⊆ ⊕_{1}^{∞}(ℤ/2).

In either situation it is clear that we may assume without loss of generality that H is a torsion group with all torsion of order 2. Thus H is a vector space over ℤ/2. It must be of infinite dimension for us to have a chance of success, and so without loss of generality we may assume H = ⊕_{1}^{∞}(ℤ/2).

So we are reduced down to consideration of the inverse sequence

A_{i} ≅ Hom_{ℤ}(G_{i}, ⊕_{1}^{∞}(ℤ/2))

as the only remaining possibility.

Consider the commuting diagram 0

↓

0 ⟶ Hom_{ℤ}(G_{i}/G_{i−1}, ⊕_{1}^{∞}(ℤ/2))
↓  ↓
A_{i} ⟶^{≅} Hom_{ℤ}(G_{i}, ⊕_{1}^{∞}(ℤ/2))
↓  ↓
A_{i−1} ⟶^{≅} Hom_{ℤ}(G_{i−1}, ⊕_{1}^{∞}(ℤ/2)).

The right column is exact as it is part of the 6-term Hom-Ext sequence. The left column is exact by the definition of the A_{i} and the two lower horizontal arrows are isomorphisms making the diagram commute by assumption. Thus

Hom_{ℤ}(G_{i}/G_{i−1}, ⊕_{1}^{∞}(ℤ/2)) = 0

for each i, and since the G_{i} are finitely generated abelian groups we may assume that each G_{i}/G_{i−1} is a finite group of odd order. By induction we see that G_{i}/G_{0} must also be a finite group of odd order. So G_{i} = G_{0} ⊕ K_{i} with each K_{i} of odd order. Then

Hom_{ℤ}(K_{i}, H) ≅ 0

since H has only 2-torsion. So there is a natural isomorphism

Hom_{ℤ}(G_{i}, H) ≅ Hom_{ℤ}(G_{0}, H)

which respects the given maps G_{i−1} → G_{i}. Thus

\underleftarrow{lim}^{1}A_{i} ≅ \underleftarrow{lim}^{1}Hom_{ℤ}(G_{i}, H) ≅ \underleftarrow{lim}^{1}Hom_{ℤ}(G_{0}, H) = 0

because the inverse system Hom_{ℤ}(G_{0}, H) is constant. □

In the remainder of this section we focus upon the structure of the groups

\underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H).

These groups arise in applications together with \underleftarrow{lim}^{1} as in Jensen’s Theorem.

We note first that the group depends only upon tG and not upon G itself, since Ext_{ℤ}^{1}(G, H) ≅ Ext_{ℤ}^{1}(tG, H) whenever G is finitely generated:

Proposition 7.3. Suppose that G = \underrightarrow{lim} G_{i} with each G_{i} finitely generated. Then

\underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H) ≅ \underleftarrow{lim} Ext_{ℤ}^{1}(tG_{i}, H).

Proposition 7.4. Suppose that

G = \underrightarrow{lim} G_{i}

with each G_{i} finitely generated and that H is finitely generated. Then

\underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H)

is profinite and reduced.

Proof. The group

\underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H)

is a subgroup of the group

Π_{i}Ext_{ℤ}^{1}(G_{i}, H).

Thus to prove the result it is enough to show that if G and H are cyclic groups then Ext_{ℤ}^{1}(G, H) is finite and reduced, and this is clear. □

Proposition 7.5. Suppose that {G_{i}} is a direct sequence of abelian groups and H is an (algebraically) compact abelian group. Then the group

\underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H)

is reduced and (algebraically) compact.

Proof. Suppose first that H is compact. Then each Ext_{ℤ}^{1}(G_{i}, H) is compact and the maps

Ext_{ℤ}^{1}(G_{i+1}, H) ⟶ Ext_{ℤ}^{1}(G_{i}, H)

are continuous, by Proposition 2.4(4). A continuous inverse limit of compact groups is again compact. If H is only algebraically compact then we may write H ⊕ H′ ≅ K for some compact group K. Then

Ext_{ℤ}^{1}(G_{i}, H) ⊕ Ext_{ℤ}^{1}(G_{i}, H′) ≅ Ext_{ℤ}^{1}(G_{i}, K).

As before, the groups Ext_{ℤ}^{1}(G_{i}, K) are compact and the structural maps are continuous, so that \underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, K) is compact. Further, the projection K → H induces a continuous retraction of the inclusion

{Ext_{ℤ}^{1}(G_{i}, H)} ↪ {Ext_{ℤ}^{1}(G_{i}, K)}

A Pext primer 37

and hence {Ext_{ℤ}^{1}(G_{i}, H)} is a continuous inverse sequence of algebraically compact groups. Arguing as in Proposition 4.2 we see that \underleftarrow{lim} Ext_{ℤ}^{1}(G_{i}, H) is reduced and algebraically compact. (See also [24], §39.4). □

8. Decoupling

In this section we consider the computation of Pext_{ℤ}^{1}(G, H) when G and H are general abelian groups. The goal is to reduce down to calculations that only involve torsion subgroups and torsionfree quotient groups, and, up to group extension, this is the result. Theorem 8.11 is the complete statement.

If G and H are split^{22} groups, (and this is, of course, a common situation), then these results are unnecessary; one proceeds at once to the subsequent sections.

We believe that the non-attributed results in Sections 8-11 are new, but please keep in mind our general caution regarding attribution.

Definition 8.1. Given abelian groups G and H, let

D(G, H) ⊂ Pext_{ℤ}^{1}(G, H)

denote the maximal divisible subgroup of Pext_{ℤ}^{1}(G, H) and let

R(G, H) = Pext_{ℤ}^{1}(G, H)/D(G, H)

denote the maximal reduced quotient group of Pext_{ℤ}^{1}(G, H).

The canonical short exact sequence

0 → D(G, H) → Pext_{ℤ}^{1}(G, H) → R(G, H) → 0

splits (unnaturally), of course, and so Pext_{ℤ}^{1}(G, H) is determined up to (unnatural) isomorphism by the groups D(G, H) and R(G, H).

The short exact sequence

0 → tG → G → fG → 0

is pure and hence it induces both long exact Hom-Ext and Hom-Pext sequences. Let ∂_{E} and ∂_{P} denote the connecting homomorphisms in the two sequences respectively.

G = Π_{p∈P} ℤ/p

tG = ⊕_{p∈P} ℤ/p

22A group is said to be split if its torsion subgroup is a direct summand. For instance, finitely generated groups, divisible groups, and (obviously) torsionfree groups have this property. Kaplansky has shown that if tG is bounded then G is split. (Actually he proved this for modules over Dedekind domains.) It is a major open problem to characterize and classify all such groups (cf. [25], §100.) Here is an example of a non-split group from Fuchs [25]. Let P be an infinite collection of distinct primes and define Then and G is not split.

Proposition 8.2. For all groups G and H,

(8.3) D(G, H) ≅ Coker(∂_{P}) ≅ Coker(∂_{E})

(8.4) R(G, H) ≅ Pext_{ℤ}^{1}(tG, H) ≅ Pext_{ℤ}^{1}(tG, tH) ≅ R(tG, tH).

Proof. Consider the following commutative diagram, where the horizontal arrows indicate the inclusion of Pext into Ext:

Hom_{ℤ}(tG, H) ⟶^{≅} Hom_{ℤ}(tG, H)
↓^{∂_{P}}  ↓^{∂_{E}}
Pext_{ℤ}^{1}(fG, H) ⟶^{≅} Ext_{ℤ}^{1}(fG, H)
↓  ↓
Pext_{ℤ}^{1}(G, H) ⟶ Ext_{ℤ}^{1}(G, H)
↓^{θ_{P}^{∗}}  ↓^{θ_{E}^{∗}}
Pext_{ℤ}^{1}(tG, H) ⟶ Ext_{ℤ}^{1}(tG, H)
↓  ↓
0  0.

(8.5)

Each column is exact, and so it is immediate that

Coker(∂_{P}) ≅ Coker(∂_{E}).

Furthermore, we may unsplice the left column to obtain the short exact sequence

(8.6) 0 → Coker(∂_{P}) → Pext_{ℤ}^{1}(G, H) ⟶^{θ_{P}^{∗}} Pext_{ℤ}^{1}(tG, H) → 0.

The group Ext_{ℤ}^{1}(tG, H) is reduced by Proposition 2.4, since tG is a torsion group. The group Pext_{ℤ}^{1}(tG, H) is a subgroup of a reduced group, hence reduced. The group

Pext_{ℤ}^{1}(fG, H) ≅ Ext_{ℤ}^{1}(fG, H)

is divisible, since fG is torsionfree. The group Coker(∂_{P}) is thus divisible, since it is a quotient of the divisible group Pext_{ℤ}^{1}(fG, H). Furthermore, it must be the maximal divisible subgroup of the group Pext_{ℤ}^{1}(G, H), since the quotient group

\frac{Pext_{ℤ}^{1}(G, H)}{Coker(∂_{P})} ≅ Pext_{ℤ}^{1}(tG, H)

is reduced. This implies that

D(G, H) ≅ Coker(∂_{P})

and that

R(G, H) ≅ Pext_{ℤ}^{1}(tG, H)

as required. The final isomorphism is immediate from the long exact sequence for Pext in the second variable and the fact that

Pext_{ℤ}^{1}(tG, fH) = 0

for all groups G and H. □

We note as a consequence that Pext_{ℤ}^{1}(G, H) is divisible if and only if

Pext_{ℤ}^{1}(tG, H) = 0,

which is a theorem of Huber and Meier [34].

Proposition 8.7. For all groups G and H, suppose that one of the following conditions holds:

1. G or H is torsionfree.

2. tG is the sum of cyclic groups.

3. tG is countable with no elements of infinite height. Then

Pext_{ℤ}^{1}(G, H) ≅ D(G, H).

In general,

D(G, H) ≅ Ker(θ_{E}^{∗}) ≅ Ker(θ_{P}^{∗}).

Proof. Part 1) is immediate from (8.4). For Part 2) we note that

R(G, H) = R(tG, tH) ⊆ Pext_{ℤ}^{1}(tG, tH)

which is trivial by Proposition 5.4. Part 3) follows from 2) and the fact (due to Prüfer [51] and Baer [4] and demonstrated in [24], §17.2) that a countable group with no elements of infinite height must be a sum of cyclic groups.

In the general situation we refer to Diagram 8.5. Then

D(G, H) ≅ Coker(∂_{E}) by Proposition 8.2
≅ Ker(θ_{P}^{∗})  by Diagram 8.5
≅ Ker(θ_{E}^{∗})  by Theorem 5.8

as required. □

Proposition 8.8. For all abelian groups G and H, there is a natural long exact sequence

0 → \frac{Hom_{ℤ}(G, fH)}{Im Hom_{ℤ}(G, H)} → D(G, tH) ⟶^{σ_{D}} D(G, H) → D(G, fH) → 0.

If H is split then σ_{D} is a monomorphism which splits unnaturally.

Proof. The commutative diagram Hom_{ℤ}(G, H)

↓

Hom_{ℤ}(G, fH)

↓

0 ⟶ D(G, tH) ⟶ Pext_{ℤ}^{1}(G, tH) ⟶ R(G, tH) ⟶ 0
↓^{σ_{D}}  ↓^{σ_{P}}  ↓^{σ_{R}}
0 ⟶ D(G, H) ⟶ Pext_{ℤ}^{1}(G, H) ⟶ R(G, H) ⟶ 0

↓

Pext_{ℤ}^{1}(G, fH)

↓

0

has exact rows and columns, and the map σ_{R} is an isomorphism. The six term Snake sequence degenerates to the isomorphisms

Ker(σ_{D}) ⟶^{≅} Ker(σ_{P})

and

Coker(σ_{D}) ⟶^{≅} Coker(σ_{P}).

Now

Ker(σ_{P}) ≅ \frac{Hom_{ℤ}(G, fH)}{Im Hom_{ℤ}(G, H)}

and

Coker(σ_{P}) ≅ Pext_{ℤ}^{1}(G, fH) ≅ D(G, fH)

by Proposition 8.2 and hence after making identifications the natural long exact sequence

0 → Ker(σ_{D}) → D(G, tH) ⟶^{σ_{D}} D(G, H) → Coker(σ_{D}) → 0

becomes the sequence 8.8 as desired. □

Proposition 8.9. For all abelian groups G and H there is a short exact sequence

0 ⟶ \frac{Hom_{ℤ}(tG, tH)}{Im Hom_{ℤ}(G, tH)} ⟶ D(fG, tH) ⟶^{τ_{D}} D(G, tH) ⟶ 0,

and in addition there is a natural isomorphism

D(G, fH) ≅ D(fG, fH).

Proof. Consider the commuting diagram

0 ⟶ D(fG, tH) ⟶ Pext_{ℤ}^{1}(fG, tH) ⟶ R(fG, tH) ⟶ 0
↓^{τ_{D}}  ↓^{τ_{P}}  ↓^{τ_{R}}
0 ⟶ D(G, tH) ⟶ Pext_{ℤ}^{1}(G, tH) ⟶ R(G, tH) ⟶ 0.

The group R(fG, tH) = 0 since fG is torsionfree and hence the associated six term Snake sequence degenerates to the isomorphism

Ker(τ_{D}) ≅ Ker(τ_{P})

and the short exact sequence

(8.10) 0 → Coker(τ_{D}) → Coker(τ_{P}) → R(G, tH) → 0.

As τ_{P} lies in the long exact Hom-Pext sequence, both of the groups Ker(τ_{P}) and Coker(τ_{P}) are easily identified:

Ker(τ_{P}) ≅ \frac{Hom_{ℤ}(tG, tH)}{Im Hom_{ℤ}(G, tH)}

and

Coker(τ_{P}) ≅ Pext_{ℤ}^{1}(tG, tH).

Thus the sequence 8.10 is isomorphic to the sequence

0 → Coker(τ_{D}) → Pext_{ℤ}^{1}(tG, tH) →^{τ′} R(G, tH) → 0.

The map τ′ is an isomorphism by Proposition 8.2 and this implies that

Coker(τ_{D}) ≅ Coker(τ_{P}) ≅ 0

so the natural sequence

0 → Ker(τ_{D}) → D(fG, tH) ⟶^{τ_{D}} D(G, tH) → 0

becomes the exact sequence

0 → \frac{Hom_{ℤ}(tG, tH)}{Im Hom_{ℤ}(G, tH)} → D(fG, tH) ⟶^{τ_{D}} D(G, tH) → 0,

as required. □

The results above combine as follows:

Theorem 8.11. Let G and H be abelian groups. Define obstruction groups Ψ_{1}(G, H) and Ψ_{2}(G, H) by

Ψ_{1}(G, H) = \frac{Hom_{ℤ}(tG, tH)}{Im Hom_{ℤ}(G, tH)}

and

Ψ_{2}(G, H) = \frac{Hom_{ℤ}(G, fH)}{Im Hom_{ℤ}(G, H)} .

Then there is a diagram with exact rows and columns of the form

0

↓

Ψ_{1}(G, H)

↓

D(fG, tH)  0  D(fG, fH)
↓  ↓  ↓≅
0 → Ψ_{2}(G, H) ⟶ D(G, tH) ⟶^{σ_{D}} D(G, H) ⟶ D(G, fH) → 0
↓  ↓
0  Pext_{ℤ}^{1}(G, H)

↓

R(tG, tH)

↓

0

and thus Pext_{ℤ}^{1}(G, H) is determined up to group extension by the three groups

D(fG, fH),  D(fG, tH),  and  R(tG, tH)

and by the two obstruction groups Ψ_{1}(G, H) and Ψ_{2}(G, H).

Remark 8.12. We note that the following are equivalent:

1. The group G is split.

2. The group Ψ_{1}(G, G) = 0.

3. For all groups H, Ψ_{1}(G, H) = 0.

Similarly that the following are equivalent:

1. The group H is split.

2. The group Ψ_{2}(H, H) = 0.

3. For all groups G, Ψ_{2}(G, H) = 0.

Thus the two obstruction groups vanish if and only if both G and H are split.

Proof of Theorem 8.11. The horizontal long exact sequence is the conclusion of Proposition 8.8 and the left vertical sequence is an application of Proposition 8.9. The middle vertical sequence is exact by the definition of D and R. Finally, the right vertical isomorphism is immediate from the fact that R(G, K) = 0 whenever K is torsionfree and similarly D(fG, fH) = 0. □

Finally, we note that Goeters and Keef have found complicated necessary and sufficient conditions for Pext_{ℤ}^{1}(G, H) = 0. We note a nice corollary of their result.

Theorem 8.13 ([27]). Suppose that G and H are countable abelian groups. Then the following are equivalent:

1. Pext_{ℤ}^{1}(G, H) = 0.

2. Pext_{ℤ}^{1}(G, fH) = 0 and Pext_{ℤ}^{1}(G, tH) = 0.

The implication 2) ⇒ 1) is immediate from the long exact sequence for Pext. If 1) holds then Pext_{ℤ}^{1}(G, fH) = 0 also by the long exact sequence, so it suffices to show that Pext_{ℤ}^{1}(G, tH) = 0. Our Theorem 8.11 implies that

D(tG, H) = D(fG, tH)  and  R(G, tH) = R(tG, tH) = 0

so it suffices to show that D(fG, tH) = 0. Theorem 10.1 implies that

D(fG, tH) ≅ ℚ^{n}

with n either zero or countably infinite. To show that n = 0 is harder, and for this we refer the reader to [27].

9. Pext_{ℤ}^{1}(G, H) for H torsionfree

This section is devoted to the calculation of the group Pext_{ℤ}^{1}(G, H) when H is assumed to be torsionfree. In that case,

Pext_{ℤ}^{1}(G, H) ≅ D(G, H) ≅ D(fG, H)

by Theorem 8.11. Since fG is torsionfree, we have

Pext_{ℤ}^{1}(fG, H) ≅ Ext_{ℤ}^{1}(fG, H)

by Corollary 5.5. Finally, any divisible group must be a direct sum of copies of the groups ℚ and ℤ(p^{∞}). Thus we have established the following result:

Theorem 9.1. Suppose that G and H are abelian groups with H torsionfree. Then

Pext_{ℤ}^{1}(G, H) ≅ Ext_{ℤ}^{1}(fG, H) ≅ (⊕_{n_{o}}ℚ) ⊕_{p} (⊕_{n_{p}}ℤ(p^{∞}))

where p runs over all primes and the cardinals n_{o} and n_{p} count the multiplicities of the summands ℚ and the various ℤ(p^{∞}).

Thus the computation of Pext_{ℤ}^{1}(G, H) for H torsionfree reduces down to the determination of Ext_{ℤ}^{1}(G, H) where both G and H are torsionfree. Unfortunately this is an unsolved problem in general.^{23}

Suppose that G and H are torsionfree, G is countable, and

Ext_{ℤ}^{1}(G, H) = Pext_{ℤ}^{1}(G, H) ≠ 0.

Then this group must be uncountable. This follows by combining Jensen’s Theorem 6.1 with Brayton Gray’s result 3.5. There is a parallel, very overlapping result for Ext_{ℤ}^{1}(G, H) under somewhat different hypotheses due to Warfield [62], Theorem 4.

Proposition 9.2 (Warfield). Suppose that G and H are torsionfree groups of finite rank and Ext_{ℤ}^{1}(G, H) ≠ 0. Then this group has cardinality ≥ 𝔠.

We want more specific results, and we start with some examples (all found in [24] (or assigned there as homework)).

Proposition 9.3. 1. Pext_{ℤ}^{1}(ℚ, ℤ) ≅ ℚ^{ℵ_{o}}.

2. Pext_{ℤ}^{1}(ℤ_{p}, ℤ) ≅ ℚ^{ℵ_{o}} ⊕ ℤ(p^{∞}).

3. Pext_{ℤ}^{1}(ℤ̂_{p}, ℤ) ≅ ℚ^{2^{ℵo}} ⊕ ℤ(p^{∞}).

4. Pext_{ℤ}^{1}(ℤ̂_{p}, ℤ̂_{p}) ≅ ℚ^{2^{ℵo}} ⊕ ℤ(p^{∞}).

23One should keep in mind that torsionfree groups themselves are not classified!

5. Pext_{ℤ}^{1}(ℤ_{p}, ℤ_{p}) = 0.

6. Pext_{ℤ}^{1}(ℚ, ℤ_{p}) ≅ ℚ^{ℵ_{o}}.

Proof. We shall prove 1), leaving the others as exercises (cf. [24], p. 226). As ℚ is torsionfree,

Pext_{ℤ}^{1}(ℚ, ℤ) ≅ Ext_{ℤ}^{1}(ℚ, ℤ)

and this group is divisible. The injective resolution of ℤ

0 → ℤ → ℚ → ℚ/ℤ → 0 (9.4)

yields a long exact sequence

0 → Hom_{ℤ}(ℚ, ℚ) → Hom_{ℤ}(ℚ, ℚ/ℤ) → Ext_{ℤ}^{1}(ℚ, ℤ) → 0

and since Hom_{ℤ}(ℚ, ℚ) ≅ ℚ,

Ext_{ℤ}^{1}(ℚ, ℤ) ≅ \frac{Hom_{ℤ}(ℚ, ℚ/ℤ)}{ℚ}.

The group Hom_{ℤ}(ℚ, ℚ/ℤ) is divisible by Proposition 2.1, hence the group Ext_{ℤ}^{1}(ℚ, ℤ) is torsionfree and divisible, and so

Ext_{ℤ}^{1}(ℚ, ℤ) ≅ ⊕_{m}ℚ

and it remains to determine m.

Using the sequence 9.4 in the other variable yields a short exact sequence

0 → Hom_{ℤ}(ℚ/ℤ, ℚ/ℤ) → Hom_{ℤ}(ℚ, ℚ/ℤ) → Hom_{ℤ}(ℤ, ℚ/ℤ) → 0

which simplifies to

0 → Π_{p}ℤ̂_{p} → Hom_{ℤ}(ℚ, ℚ/ℤ) → ℚ/ℤ → 0.

and since Π_{p}ℤ̂_{p} has cardinality 2^{ℵ_{o}} it follows that m = 2^{ℵ_{o}}. □

There are two general results which improve the early result of B. Gray 3.11.

Theorem 9.5 (Warfield [62]). Suppose that G and H are countable abelian groups, H is torsionfree, and

Pext_{ℤ}^{1}(G, H) ≠ 0.

Then Pext_{ℤ}^{1}(G, H) contains n_{o} = 2^{ℵ_{o}} summands of the group ℚ.

Theorem 9.6 (Warfield, [62], Theorem 2). Suppose that G and H are torsionfree abelian groups of finite rank. Then

1. The group Ext_{ℤ}^{1}(G, H) = 0 if and only if G ⊗ ℤ_{H} is a free ℤ_{H}-module. Otherwise, the group has n_{o} = c.

2. The p-torsion subgroup of Ext_{ℤ}^{1}(G, H) has p-rank

r_{p}(G)r_{p}(H) − r_{p}(Hom_{ℤ}(G, H)).

The case of Pext_{ℤ}^{1}(G, ℤ) for G countable has been analyzed completely by C. U. Jensen, as follows.

Theorem 9.7 ([38]). Suppose that {Γ_{i}} is an inverse sequence of finitely generated abelian groups. Let

G = \underrightarrow{lim} Hom_{ℤ}(Γ_{i}/tΓ_{i}, ℤ).

Then G is countable, torsionfree, and

\underleftarrow{lim}^{1}Γ_{i} ≅ Pext_{ℤ}^{1}(G, ℤ) ≅ Ext_{ℤ}^{1}(G, ℤ).

Further:

1. For any countable torsionfree group G, if Pext_{ℤ}^{1}(G, ℤ) ≠ 0, then the number n_{o} of direct summands of type ℚ that it contains is 2^{ℵ_{o}}.

2. For each prime p, the number n_{p} of summands of type ℤ(p^{∞}) that Pext_{ℤ}^{1}(G, H) contains is either finite or 2^{ℵ_{o}}.

Conversely, given cardinals n_{o}, n_{p} which satisfy Conditions 1) and 2), there is a countable torsionfree abelian group G such that Pext_{ℤ}^{1}(G, H) has this decomposition.

The following result generalizes Jensen’s result.

Theorem 9.8 (Huber and Warfield [35]). Suppose that G and H are torsionfree abelian groups with G of countable rank and H of rank 1. If

Ext_{ℤ}^{1}(G, H) ≠ 0

then for each prime p, the number r_{p}(Ext_{ℤ}^{1}(G, H)) is either finite or 2^{ℵ_{o}}.

The case Ext_{ℤ}^{1}(G, ℤ) for G uncountable has been analyzed by Hiller, Huber, and Shelah [33]. Their work uses similar tools to those used by Shelah in his solution to Whitehead’s problem using Gödel’s Axiom of Constructibility. For instance, they show:

Proposition 9.9. Suppose that G is an abelian group with Ext_{ℤ}^{1}(G, ℤ) nontrivial and divisible. Then its torsionfree rank is 2^{μ} for some infinite cardinal μ.

Shelah also constructs an example of an uncountable group G such that Ext_{ℤ}^{1}(G, ℤ) does not satisfy the conditions of Theorem 9.7.

Finally, we note their compactness results. The assumption V = L is a logic assumption.

Theorem 9.10 ([33]). 1. Suppose that V = L. If Hom_{ℤ}(G, ℤ) = 0 (or, equivalently, if G has no free summand), then the group Ext_{ℤ}^{1}(G, ℤ) admits a compact topology.

2. If A is a compact abelian group then there exists an abelian group G with Hom_{ℤ}(G, ℤ) = 0 and Ext_{ℤ}^{1}(G, ℤ) = A.

Harrison showed [31] that any compact abelian group is a product of groups of the form ℤ/n, ℤ̂_{p}, ℝ, ℤ(p^{∞}), with the condition that for each prime p the number of copies of ℤ(p^{∞}) must not exceed the number of copies of ℝ.

10. Pext_{ℤ}^{1}(G, H) for G torsionfree and H torsion

If the group G is torsionfree then we know (Theorem 8.11) that

Pext_{ℤ}^{1}(G, H) ≅ Ext_{ℤ}^{1}(G, H) ≅ D(G, H)

and hence once again is determined by the multiplicity numbers. If G is of finite rank^{24} then the following result gives a good general picture of the situation.

Theorem 10.1. Suppose that G is a torsionfree abelian group and that H is a torsion group. In addition, suppose either:

a) G is of finite rank, or

b) H is countable. Then

P ext_{ℤ}^{1}(G, H) ≅ ℚ^{n}

with either n = 0 or n ≥ ℵ_{o}.

Proof. For Part a), See [24], p. 232, Ex. 14bc. For Part b), see [20], p. 448, who credit Baer. □

Eklof and Huber also give the following sharp information:

Theorem 10.2. Suppose that G is a countable torsionfree abelian group and H is an abelian group. Then the following are equivalent:

1. Ext_{ℤ}^{1}(G, H) = 0.

2. There exists an ascending sequence of pure subgroups of G

G_{0} ⊆ G_{i} ⊆ ⋯ ⊆ G_{n} ⋯,  ∪_{i}G_{i} = G,

with G_{0} = 0, rank(G_{n+1}/G_{n}) = 1, and

Ext_{ℤ}^{1}(G_{n+1}/G_{n}, H) = 0.

Eklof and Huber note the following corollary.

Corollary 10.3. Suppose that G is a countable torsionfree group and H is arbitrary. If

Pext_{ℤ}^{1}(G′, H) = 0

for all finite rank subgroups G′ ⊂ G, then

Pext_{ℤ}^{1}(G, H) = 0.

24Recall that if G is torsionfree then its (torsionfree) rank is the ℚ-dimension of the vector space G ⊗ ℚ. The map ℚ → G ⊗ ℚ is an inclusion, and the rank may also be described as the dimension of the ℚ-vector space of minimal dimension which contains a copy of G as a subgroup.

It is striking that the structure of H seems to play no role at this level. Of course we have no way in Theorem 10.1 to determine which of the extreme alternatives occurs — perhaps this depends on some deep properties of H.

There is an obvious conjecture that emerges from these results which would clarify the general situation.

Conjecture 10.4. Suppose that G is a torsionfree abelian group and H is a torsion group. Then

Pext_{ℤ}^{1}(G, H) ≅ ℚ^{n}

with either n = 0 or n ≥ ℵ_{o}.

We are not expert in these matters, and so perhaps there is an elementary counterexample. What we would need to discover is some information concerning the map

Θ : Pext_{ℤ}^{1}(G, H) ⟶ \underleftarrow{lim} Pext_{ℤ}^{1}(G′, H)

where G′ runs over all finite rank subgroups of G. This might be of some interest for all abelian groups G and H. Note that Corollary 10.3 tells us that if G is torsionfree then Θ is mono.

11. Pext_{ℤ}^{1}(G, H) for both G and H torsion

The remaining case is when G and H are both torsion groups. Our principal result is Theorem 11.7 which determines Pext_{ℤ}^{1}(G, H) when G and H are p-groups and G has finite Ulm length.

If G is a torsion group then Ext_{ℤ}^{1}(G, H) is reduced by Proposition 2.4 which implies that Pext_{ℤ}^{1}(G, H) is reduced. Thus we have the following elementary fact.

Proposition 11.1. If G is a torsion group then

Pext_{ℤ}^{1}(G, H) = R(G, H).

Assume that G and H are torsion groups, and let G(p) and H(p) denote their p-primary subgroups. Then

Pext_{ℤ}^{1}(G, H) ≅ Pext_{ℤ}^{1}(⊕_{p}G(p), H)
≅ Π_{p}Pext_{ℤ}^{1}(G(p), H)
≅ Π_{p}Pext_{ℤ}^{1}(G(p), H(p))

so that the determination of the group Pext_{ℤ}^{1}(G, H) comes down to a determination of the various groups Pext_{ℤ}^{1}(G(p), H(p)). Thus without loss of generality we may focus attention upon the case where both G and H are torsion groups with all torsion having orders various powers of some fixed prime p. For brevity we refer to such a group as a p-group.

Let

Ĥ ≅ \underleftarrow{lim} H/p^{n}H

denote the p-adic completion of the abelian group H. Note that the p-adic and ℤ-adic topologies coincide on p-groups, so that Ĥ is isomorphic to the ℤ-adic completion of H (and the notation is not ambiguous). Let

μ : H → Ĥ

denote the natural map. It is elementary that

Ker(μ) ≅ ∩_{n}p^{n}H = H^{1}

where H^{1} denotes the first Ulm subgroup of H. Thus μ : H → Ĥ is an inclusion if and only if H is Hausdorff. If so then the resulting short exact sequence

0 → H → Ĥ → Ĥ/H → 0

is pure ([24], §39.5) and Ĥ/H is divisible. For any abelian group G, let

Δ = Δ_{G}: Hom_{ℤ}(G, Ĥ/H) ⟶ Pext_{ℤ}^{1}(G, H)

denote the resulting connecting homomorphism.

Proposition 11.2. Let H be a p-group which is Hausdorff in the p-adic topology. Then

Δ_{ℚ/ℤ}: Hom_{ℤ}(ℚ/ℤ, Ĥ/H) ⟶ Pext_{ℤ}^{1}(ℚ/ℤ, H)

is an isomorphism, and for each prime q the map

Δ_{ℤ(q^{∞})}: Hom_{ℤ}(ℤ(q^{∞}), Ĥ/H) ⟶ Pext_{ℤ}^{1}(ℤ(q^{∞}), H)

is an isomorphism. Thus the maximal divisible subgroup of Ĥ/H has no q-torsion except possibly for p = q.

Proof. The map Δ_{ℚ/ℤ} is an isomorphism by [24], §53.8. The natural isomorphism

ℚ/ℤ ≅ ⊕_{q}ℤ(q^{∞})

induces the commutative diagram

Hom_{ℤ}(ℚ/ℤ, Ĥ/H) ⟶^{≅} Π_{q}Hom_{ℤ}(ℤ(q^{∞}, Ĥ/H)
Δ_{ℚ/ℤ}↓  ^{Π_{q}Δ_{ℤ(q^{∞})}}↓
Pext_{ℤ}^{1}(ℚ/ℤ, H) ⟶^{≅} Π_{q}Pext_{ℤ}^{1}(ℤ(q^{∞}), H)

and the naturality of Δ implies that the map

Π_{q}Δ_{ℤ(q^{∞})}

is an isomorphism. It follows that Δ_{ℤ(q^{∞})} is an isomorphism for each q. □

Next we consider the analogous question when H is not necessarily Hausdorff. We may assume without loss of generality that H is a reduced p-group. Then H has a p-basic subgroup B (cf. [24], §33). Such a subgroup is characterized up to isomorphism as a subgroup which satisfies the following three conditions:

1. B is a direct sum of cyclic groups.

2. B is a pure subgroup of H.

3. H/B is divisible. Since H is a p-group, B has the form

B = ⊕_{n}B_{n}

where n = 1, 2, 3, …, and B_{n} is the sum of k(n) copies of the group ℤ/p^{n}. (For convenience of notation we allow the case k(n) = 0.) Note that the group B is bounded if and only if k(n) = 0 for all but finitely many values of n.

The group H/B must be a direct sum of j copies of the group ℤ(p^{∞}). However, even though B is uniquely determined up to isomorphism, the copy of B that we choose in H is not: in general there are many choices for B. The effect of this is that the group H/B depends upon our choice of B (see [24], Page 150). Thus the number j depends upon our choice of B.

Proposition 11.3. Let B be a sum of cyclic p-groups as above. Then: 1. If B is bounded, then

Pext_{ℤ}^{1}(ℤ(p^{∞}), B) ≅ 0.

2. If B is unbounded then

Pext_{ℤ}^{1}(ℤ(p^{∞}), B) ≅ (⊕_{2^{ℵo}} ℤ̂_{p}) ≅ (ℤ̂_{p})^{ℵ_{o}}.

Proof. Any direct sum of cyclic groups is Hausdorff, so by Proposition 11.2 it suffices to determine the group

Hom_{ℤ}(ℤ(p^{∞}), B̂/B).

Fuchs ([24], p. 166, ex. 7) shows that B̂/B is the maximal divisible subgroup of the group

\frac{Π_{n}B_{n}}{⊕_{n}B_{n}} .

Fortunately this situation has been analyzed in great generality by Golema and Hulanicki [28] and we use their analysis below.

If B is bounded then B̂ = B and B̂/B = 0, establishing Part 1).

Now suppose that B is unbounded. Let

b_{n} = (1, 0, 0, …) ∈ B_{n}.

(If B_{n} = 0 then set b_{n} = 0.) The element b_{n} has order exactly p^{n} whenever B_{n} ≠ 0. Thus for infinitely many n the element p^{n−1}b_{n} has order exactly p and it is divisible by p^{n−1}. Define

b = Πp^{n−1}b_{n} ∈ Π_{n∈N}B_{n}.

Then b has order p and for any fixed t all but a finite number of the entries of b are divisible by p^{t}. Let b denote the image of b in the group

\frac{Π_{n}B_{n}}{⊕_{n}B_{n}} .

We see that b has order exactly p and that for any t, b is divisible by p^{t}. It follows easily that in fact b is divisible by any positive integer. This forces the group \frac{Π_{n}B_{n}}{⊕_{n}B_{n}} to contain at least one copy of the group ℤ(p^{∞}). In fact we have the following proposition.

Proposition 11.4 ([28]). With the above notation, suppose that the group B is unbounded. Then:

B̂/B ≅ ( ⊕_{2^{ℵo}}ℚ) ⊕ ( ⊕_{2^{ℵo}}ℤ(p^{∞})) .

Thus

Hom_{ℤ}(ℤ(p^{∞}), B̂/B) ≅ Hom_{ℤ}(ℤ(p^{∞}), ⊕_{2^{ℵo}} ℤ(p^{∞}))

and hence (by Fuchs [24] 44.3)

Hom_{ℤ}(ℤ(p^{∞}), B̂/B) ≅ (⨁_{2^{ℵo}} ℤ̂_{p}) .

Finally,

(⨁_{2^{ℵo}} ℤ̂_{p}) ≅ (ℤ̂_{p})^{ℵ_{o}}

We note that the final conclusion follows since both groups are torsionfree, p-adic algebraically compact, and with the same basic p-adic module.

This completes the proof of Part 2). □

With this preparation we are able to establish the following proposition.

Proposition 11.5. Suppose that H is a reduced p-group with p-basic subgroup B and H/B ≅ ⊕_{j}ℤ(p^{∞}). Then:

1. If B is bounded then

Pext_{ℤ}^{1}(ℤ(p^{∞}), H) = 0.

2. If B is unbounded, then

Pext_{ℤ}^{1}(ℤ(p^{∞}), H) ≅ \frac{(⨁_{2^{ℵo}} ℤ̂_{p})}{Φ_{B} ((⨁_{j} ℤ̂_{p}) )}

where Φ_{B} is a homomorphism defined below.

Proof. Applying the functor Pext_{ℤ}^{1}(ℤ(p^{∞}), −) to the pure short exact sequence

0 → B → H → H/B → 0

yields a long exact Hom-Pext sequence. However, the group

Hom_{ℤ}(ℤ(p^{∞}), H) = 0

since H is reduced and

Pext_{ℤ}^{1}(ℤ(p^{∞}), H/B) = 0

since H/B is divisible. Thus the sequence degenerates to the exact sequence

Hom_{ℤ}(ℤ(p^{∞}), H/B) → Pext_{ℤ}^{1}(ℤ(p^{∞}), B) → Pext_{ℤ}^{1}(ℤ(p^{∞}), H) → 0.

Using Proposition 11.3 and the natural isomorphism

Hom_{ℤ}(ℤ(p^{∞}), H/B) ≅ Hom_{ℤ}(ℤ(p^{∞}), ⊕_{j}ℤ(p^{∞})) ≅ (⊕_{j}ℤ̂_{p})

gives us the exact sequence

(⨁_{j} ℤ̂_{p}) ⟶^{Φ_{B}} (⨁_{2^{ℵo}} ℤ̂_{p}) ⟶ Pext_{ℤ}^{1}(ℤ(p^{∞}), H) → 0

as required. □

We emphasize that j and the map Φ_{B} depend upon the choice of the subgroup B in H. On the other hand, the group

Coker(Φ_{B}) ≅ Pext_{ℤ}^{1}(ℤ(p^{∞}), H)

is obviously independent of the choice of B.

We recall from Definition 5.1 that each p-torsion group G has an Ulm sequence

⋯ G^{3} ⊆ G^{2} ⊆ G^{1} ⊆ G

and G is said to have finite Ulm length if the sequence stabilizes after a finite number τ steps, in which case G^{τ} is the maximal divisible subgroup of G. For example, if G is algebraically compact then G has Ulm length 1, by Proposition 5.3.

There exist countable abelian groups of every finite Ulm length as well as countable abelian groups with infinite Ulm length (cf. [25], §76). Note that each G^{s}/G^{s+1} is a direct sum of cyclic groups. Thus

Pext_{ℤ}^{1}(G^{s}/G^{s+1}, H) = 0

for all s. It follows easily from the long exact sequence

Pext_{ℤ}^{1}(G^{s}/G^{s+1}, H) → Pext_{ℤ}^{1}(G^{s}, H) → Pext_{ℤ}^{1}(G^{s+1}, H) → 0

that

Pext_{ℤ}^{1}(G^{s}, H) ≅ Pext_{ℤ}^{1}(G^{s+1}, H)

via the natural map. Thus we have established the following theorem.

Theorem 11.6. Let G be a p-group with finite Ulm length and with maximal divisible subgroup ⊕_{i}ℤ(p^{∞}). Let H be a reduced p-group. Then

Pext_{ℤ}^{1}(G, H) ≅ Π_{i}Pext_{ℤ}^{1}(ℤ(p^{∞}), H).

In particular, if G is a reduced p-group with finite Ulm length and H is a reduced p-group, then

Pext_{ℤ}^{1}(G, H) ≅ 0.

These results on the torsion-torsion case may be gathered up into the following omnibus theorem. The theorem is a generalization of a result of Harrison (cf. [24], §57.3) which deals with the case where H is Hausdorff.

Theorem 11.7. Let G be a p-group with finite Ulm length. Let H be a reduced p-group with p-basic subgroup B and with

H/B ≅ ⊕_{j}ℤ(p^{∞}).

1. If B is bounded or if G is reduced then

Pext_{ℤ}^{1}(G, H) = 0.

2. On the other hand, if B is unbounded and if G has maximal divisible subgroup ⊕_{i}ℤ(p^{∞}) ≠ 0 then

Pext_{ℤ}^{1}(G, H) ≅ ∏_{i} \frac{(⨁_{2^{ℵo}} ℤ̂_{p})}{Φ_{B} ((⨁_{j} ℤ̂_{p}) )}.

We note that if we regard ℤ̂_{p} as a commutative ring then its indecomposable modules are ℤ̂_{p}, ℤ̂_{p}/p^{n}, the p-adic numbers ℚ̂_{p}, and ℚ̂_{p}/ℤ̂_{p} (cf. [40], p. 53). We suppose that the group

∏_{i} \frac{(⨁_{2^{ℵo}} ℤ̂_{p})}{Φ_{B} ((⨁_{j} ℤ̂_{p}) )}

is decomposable as a ring (with its obvious ring structure) and (based on naive optimism) we expect that it will be the sum of (many!) copies of ℤ̂_{p}.

Proof. As G has finite Ulm length, Proposition 11.6 reduces the computation down to the computation of the group Pext_{ℤ}^{1}(ℤ(p^{∞}), H), and this group is dealt with in Proposition 11.5. □

The assumption that H is reduced is not really restrictive, since

Pext_{ℤ}^{1}(G, H) ≅ Pext_{ℤ}^{1}(G, H/D)

where D is the maximal divisible subgroup of H.

The assumption that G has finite Ulm length is a real restriction. We know of no examples of explicit computations of Pext for groups G of infinite Ulm length.

We conclude this section with our strongest divisibility result on Pext.

Theorem 11.8. Let G and H be countable abelian groups. Assume that (tG)_{p} has finite Ulm length for each prime p and that tH is a reduced group. Let B(p) denote the p-basic subgroup of (tH)_{p}. Then the following are equivalent:

1. Pext_{ℤ}^{1}(G, H) is divisible.

2. For each prime p, either B(p) is bounded or (tG)_{p} is reduced.

Proof. Fix some prime p. Then the second condition is equivalent to

Pext_{ℤ}^{1}((tG)_{p}, (tH)_{p}) = 0

by Theorem 11.7. From Section 8 and elementary considerations there are isomorphisms

R(G, H) ≅ R(tG, tH) ≅ Pext_{ℤ}^{1}(G, H) ≅ Π_{p}Pext_{ℤ}^{1}((tG)_{p}, (tH)_{p})

and the corollary follows at once. □

Finally, we note that perhaps the earliest study of these problems is represented by the work of Kulikoff and Papp (cf. [22], §34.6.)

12. Pext_{ℤ}^{1}(G, H) and phantom maps

This section is a very brief introduction to phantom maps. It is based upon McGibbon’s fine survey article [44] and we encourage the reader to consult that article for a much better and more extensive discussion. We aim here simply to give a taste of the subject. Below all maps and homotopies are understood to be base-point preserving.

Let X be a connected locally finite^{25} CW -complex with n-skeleton X_{n}. A continuous map f : X → Y is said to be a phantom map if its restriction to each X_{n} is null-homotopic. If X is infinite-dimensional and Y has infinitely many nontrivial homotopy groups then there sometimes exist phantom maps which are nontrivial in homotopy. For example, there are uncountably many homotopy classes of phantom maps from ℂP^{∞} to S^{3}.

Phantom maps are difficult to detect. For instance, if f : X → Y is a phantom map and α : S^{k} → X is any map, then α is homotopic to a map that factors through some X_{n} by the cellular approximation theorem, and hence fα ≃ ∗. This implies that

f_{∗} = 0 : π_{∗}(X) ⟶ π_{∗}(Y).

Similarly, f_{∗} = 0 when a generalized homology theory is used.

Let [X, Y] denote homotopy classes of maps from X to Y and let P(X, Y) denote homotopy classes of phantom maps from X to Y. Then it is clear that P(X, Y) is by definition the “kernel” of the natural map

[X, Y] ⟶ \underleftarrow{lim} [X_{n}, Y].

The quotations are appropriate since in general these are not groups, but it still makes sense to take inverse limits of sets and to talk about the kernel of the map as the subset of [X, Y] sent to the trivial class. Generalizing the work of Milnor [47], Bousfield and Kan [7] identified the kernel of this map in terms of \underleftarrow{lim}^{1}. For this purpose they extended the definition of \underleftarrow{lim}^{1} to nonabelian groups. Note that [ΣX, Y] is a group, where Σ denotes pointed suspension.

Theorem 12.1. (Bousfield-Kan) For any pointed complexes X and Y,

P(X, Y) ≅ \underleftarrow{lim}^{1}[ΣX_{n}, Y].

25If X is not locally finite then this definition is incorrect. In the general situation one says that f : X → Y is a phantom map if for every finite CW -complex X′ and for every map X′ → X the composite X′ → X → Y is null-homotopic. If X is locally finite then this is equivalent to the definition above.

For example, take Y = BU × ℤ so that

[X, Y] = K^{0}(X)

for any compact space. Then we have

P(X, BU × ℤ) ≅ \underleftarrow{lim}^{1}K^{0}(ΣX_{n}) ≅ \underleftarrow{lim}^{1}K^{1}(X_{n}).

If we take X = ℂP^{∞} (or, more generally, if X has cells only in even dimensions), then

P(ℂP^{∞}, BU × ℤ) ≅ \underleftarrow{lim}^{1}K^{1}(ℂP^{n}) ≅ \underleftarrow{lim}^{1}0 ≅ 0

so there are no nontrivial phantom maps from ℂP^{∞} to BU × ℤ.

Huber and Meier [34] relate the group Pext to phantom maps in a way which is very similar to our study of quasidiagonality in [55]; see below. Suppose that X is a spectrum, E_{∗} is a homology theory of finite type, F^{∗} is a cohomology theory, and H is some abelian group. Suppose that there is a natural exact UCT sequence for E_{∗} and F^{∗} of the form

0 → Ext_{ℤ}^{1}(E_{n−1}(X), H) → F^{n}(X) → Hom_{ℤ}(E_{n}(X), H) → 0. (12.2)

Then the collection of phantom classes in F^{n}(X) is isomorphic to the subgroup Pext_{ℤ}^{1}(E_{n−1}(X), H).

There are analogous definitions and theorems at the level of spectra. For instance, if A and B are abelian groups with associated Eilenberg-MacLane spectra \underline{H}(A), \underline{H}(B), then Christensen-Strickland [10] show that phantom maps

Σ^{−1}\underline{H}(A) ⟶ \underline{H}(B)

correspond bijectively to elements of Pext_{ℤ}^{1}(A, B).

There is a universal phantom map

Θ : X ⟶ ⋁_{n=1}^{∞} ΣX_{n}

defined as follows (cf. [44], Pages 1214-5). First construct the mapping telescope Tel(X) of X as

Tel(X) = ⋃_{n} X_{n} × [n − 1, n]/ ∼

where each X_{n} ×n is identified with its image in X_{n+1} ×n. We may identify X with this space up to homotopy type. There is a natural collapse map

Tel(X) ⟶ ⋁_{n=1}^{∞} ΣX_{n}

obtained by collapsing each subset X_{n} × n to a point. It is easy to see that the resulting map Θ is a phantom map. In fact, under modest hypotheses every phantom map from X to some other space Y must factor through Θ up to homotopy.

We note in the following section that there is an analogous construction for C^{∗}-algebras.

13. Pext_{ℤ}^{1}(G, H) and quasidiagonality

As we mentioned in the introduction, our interest in Pext arose from our study of the topological structure of the Kasparov groups in C^{∗}-algebras. We discuss this matter in depth in [58]. Here we give a semi-heuristic introduction, showing by example how some knowledge of Pext helps in functional analysis. (For undefined terms in this section please see [58].)

A bounded operator T on a (complex separable) Hilbert space is said to be quasidiagonal if for some choice of basis the operator has the form T = D + K where D is a block-diagonal operator (with each block of finite dimension) and K is a compact operator.

For example, let the Hilbert space have basis {e_{o}, e_{1}, e_{2}, … } and define the unilateral shift by T e_{n} = e_{n+1}. This operator is not quasidiagonal. Here is a proof. Suppose that T were quasidiagonal, so that T = D + K for some choice of block diagonal matrix D and compact operator K. The operator T is Fredholm, and

index(T) = dim ker (T) − dim ker (T^{∗}) = 0 − 1 = −1.

On the other hand,

index(D + K) = index(D) = 0.

Thus T cannot be quasidiagonal. This argument is related to the γ invariant below.

A C^{∗}-algebra A is said to be quasidiagonal if it has a faithful representation on a Hilbert space such that for some choice of basis each of its elements is quasidiagonal. For instance, any commutative C^{∗}-algebra is quasidiagonal, and obviously the compact operators K themselves are quasidiagonal. However, no C^{∗}-algebra containing the unilateral shift can be quasidiagonal.

Let S be the C^{∗}-algebra generated by the identity, the compact operators, and the unilateral shift. It is not quasidiagonal, by the remarks above. Then

S/K ≅ C(S^{1})

and hence there is a natural short exact sequence

0 → K → S → C(S^{1}) → 0.

We see from this example that a extension of quasidiagonal C^{∗}-algebras is not necessarily quasidiagonal.

Davidson, Herrero, and Salinas established the following theorem. Let L denote the bounded operators on a separable Hilbert space and let K denote the compacts in L as is customary.

Theorem 13.1 (Davidson, Herrero, Salinas [17]). Suppose given the essential extension

0 → K ⟶ E_{τ} ⟶ A → 0

with associated faithful representation

e_{τ} : E_{τ} ⟶ L

with E_{τ} separable and nuclear. Suppose that e_{τ} is a quasidiagonal representation. Then A ≅ E/K is quasidiagonal.

The example of the extension generated by the unilateral shift demonstrates that the converse of this theorem is false. Here is the complete story within the category Ñ of separable nuclear C^{∗}-algebras which satisfy the UCT.

Theorem 13.2 ([58]). Suppose given the essential extension

0 → K ⟶ E_{τ} ⟶ A → 0

with associated faithful representation

e_{τ} : E_{τ} ⟶ L

with A ∈ Ñ and quasidiagonal. Then e_{τ} is a quasidiagonal representation (that is, the set τ(E_{τ}) is a quasidiagonal set) if and only if the following two conditions hold:

1. The K-theory boundary homomorphism

γ : K_{1}(A) → K_{0}(K) = ℤ

is the zero homomorphism.

2. The resulting K-theory short exact sequence

0 → ℤ ⟶ K_{0}(E_{τ}) ⟶ K_{0}(A) → 0

is a pure exact sequence. If in addition K_{0}(A) is torsionfree then Condition 2) is automatically satisfied, so that E_{τ} is quasidiagonal if and only if Condition 1) holds.

The analysis of this problem goes as follows: one considers all^{26} extensions of the form

0 → K ⟶ D ⟶ A → 0.

These form a topological group called the Kasparov group. The group is denoted KK_{1}(A, K) or just K^{1}(A). Salinas shows [54] that the quasidiagonal extensions form a subgroup, denoted QD(A, K) and this subgroup is just the closure of zero. The C^{∗}-algebra A satisfies the UCT, which says that there is a natural short exact UCT (Universal Coefficient Theorem) sequence [53]

26Well, not all — there are technical conditions.

0 → Ext_{ℤ}^{1}(K_{0}(A), ℤ) → K^{1}(A) ⟶^{γ} Hom_{ℤ}(K_{1}(A), ℤ) → 0. (13.3)

Under this identification we demonstrate in [58] that

QD(A, K) ≅ Pext_{ℤ}^{1}(K_{0}(A), ℤ)

from which the theorem follows easily.

In fact we consider the more general situation where K is replaced by K ⊗ B for some other C^{∗}-algebra B. Extensions of C^{∗}-algebras of the form

0 → K ⊗ B ⟶ E ⟶ A → 0

form the Kasparov group KK_{1}(A, B). This is a topological group and its closure of zero is again isomorphic to the “relatively quasidiagonal” extensions QD(A, B). For A ∈ N and quasidiagonal relative to B, we demonstrate [58] that

QD(A, B) ≅ Pext_{ℤ}^{1}(K_{∗}(A), K_{∗}(B))_{0}

as abelian groups.

If A ∈ N is not relatively quasidiagonal then Manuilov and Thomsen [42] have found it possible to identify the group Pext_{ℤ}^{1}(K_{∗}(A), K_{∗}(B)) as “weakly quasidiagonal” extensions.

We should like to emphasize the formal similarity between sequences 12.2 and 13.3. It is our hope that there is some deeper connection here between phantom behavior and quasidiagonality which remains to be uncovered.

Here is one more example of this connection. We have explained that in the context of phantom maps there is a universal phantom map

Θ : X ⟶ ⋁_{n=1}^{∞} ΣX_{n}.

It turns out that there is a natural analogue of this map in the context of C^{∗}-algebras. We learned the construction from L.G. Brown and used it in our general study of homology theories for C^{∗}-algebras [55], pages 431-432. One is given a direct sequence of C^{∗}-algebras A_{i} with A = \underrightarrow{lim} A_{i}. One constructs a contractible mapping telescope T = T ({A_{i}}) with a natural map e : T ({A_{i}}) → A. Inside of T one has the subalgebra ⊕_{i}SA_{i}, where SA denotes the suspension of A, and the induced map

Θ : ⨁_{n=1}^{∞} SA_{i} ⟶ A (13.4)

is very analogous to the universal phantom map.

64 Claude L. Schochet

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