Abelian Groups - László Fuchs [Springer]

292 9 Groups of Extensions and Cotorsion Groups
Proof. Sufficiency being obvious, assume A is torsion and cotorsion. By Proposition
8.3 its initial Ulm factor is algebraically compact. By Corollary 3.5 in Chapter 6,
a torsion group that is reduced and algebraically compact is bounded. Hence the
reduced part of A is bounded.
ut
Our next result relates the Ulm length of the torsion group T to the corresponding
adjusted cotorsion group T D G. Specifically, we prove:
Proposition 8.5. Let T be a reduced torsion group of Ulm length . Then the
cotorsion group G D Ext.Q=Z; T/ has Ulm length or C 1. The subgroup G is
torsion-free, and there is a natural isomorphism
G Š Hom.Q=Z; G=.G C T//:
Proof. Since T < G, it is clear that the Ulm length of G cannot be less than . We
also have G \ T D T D 0,soG is torsion-free cotorsion. By Lemma 8.1(ii), G
is algebraically compact, so its first Ulm subgroup vanishes, G C1 D 0.
The sequence 0 ! T ! G=G ! G=.G C T/ ! 0 is exact, and the last factor
group is divisible as an epic image of G=T. Hence Theorem 2.3 leads us to the exact
sequence
0 ! Hom.Q=Z; G=.G C T// ! Ext.Q=Z; T/ Š G
! Ext.Q=Z; G=G / Š G=G ! 0;
where the last isomorphism holds because G=G is reduced cotorsion. The map
between the two Exts is natural, so its kernel must be G .
ut
Groups That Can be Ext The groups that appear in the form Ext.C; A/ when
the arguments belong to certain subclasses have been under close scrutiny by
abelian group theorists. There are numerous publications about this subject, not only
structural results in ZFC, but also under additional set-theoretical hypotheses, not
to mention several consistency theorems. These are deeper results whose proofs
demand more substantial machinery, so we cannot discuss them here. We only
include a result with a relatively easy proof.
Theorem 8.6 (Jensen [Je]). If A is torsion-free of countable rank, then Ext.A; Z/
is either 0 or is a direct sum of the following summands:
(1) a direct sum of continuously many copies of Q, and,
(2) for each prime p, a direct sum of finitely or continuously many copies of Z.p 1 /.
Proof. By Corollary 8.3 in Chapter 3, A is a direct sum of a free group and a torsionfree
group which has no non-trivial homomorphism into Z. Evidently, it suffices to
consider the second summand only, i.e. we may assume Hom.A; Z/ D 0: In this
case, for a prime p, the exact sequence 0 ! A !A Pp ! A=pA ! 0 induces the
exact sequence 0 ! Ext.A=pA; Z/ ! Ext.A; Z/ Pp ! Ext.A; Z/ ! 0; where in view
of Sect. 3(F), Ext.A; Z/ is a divisible group. The image of the first Ext in the second