Abelian Groups - László Fuchs [Springer]

118 3 Direct Sums of Cyclic Groups
Theorem 8.14 (Raynaud–Gruson). A group is Mittag-Leffler if and only if it is
@ 1 -cyclic.
Proof. We start the proof by showing that M is Mittag-Leffler if and only if each
of its countable pure subgroups is Mittag-Leffler. One direction the claim follows
from Lemma 8.13.
For the converse, assume maps P n
jD1 .x j ˝ b j /.x j 2 M; b j 2 Q A i / to 0. Then
by Lemma 1.12 in Chapter 8 and by the remark after it, the same sum vanishes in
.M 0 ˝ Q A i / for a countable pure subgroup M 0 M containing the x j ’s. Hence
M cannot be Mittag-Leffler if its countable pure subgroups are not, but it is if its
countable pure subgroups are Mittag-Leffler.
It remains to prove that countable Mittag-Leffler groups are †-cyclic. Suppose
N is a countable p-group which has elements ¤ 0 of infinite p-heights. Clearly, all
non-zero elements of Q n2N .N ˝Z.pn // have finite p-heights. However, Q n2N Z.pn /
has summands Š J p ,soN ˝ Qn2N Z.pn / has summands Š N ˝ J p Š N with
elements of infinite heights. Thus N is not Mittag-Leffler. A countable Mittag-
Leffler group must therefore have separable p-components, so its torsion subgroup
is †-cyclic.
Next, let M be of finite torsion-free rank n >0such that M=tM is not finitely
generated. Then M contains a subgroup N such that N=tM Š Z n and M=N is an
infinite torsion group. First assume M=N is reduced. Then it is †-cyclic of the form
˚i