Abelian Groups - László Fuchs [Springer]

422 12 Torsion-Free Groups
and
0 ! Hom.C; G/ ! Hom.A; G/ ˛! Hom.B; G/ !
! Bext 1 .C; G/ ! Bext 1 .A; G/ ˛! Bext 1 .B; G/ ! :::
are exact.
Proof. This follows from a general theorem on relative homological algebra; see,
e.g., Mac Lane [M].
ut
There are more functors Bext i which continue the long exact sequences in
the same fashion. As far as their definition is concerned, let us observe that if
0 ! B ! C ! G ! 0 is a balanced-projective resolution of G (with completely
decomposable C, see next section), then
Bext iC1 .G; A/ D Bext i .B; A/
for every torsion-free group A, and for every i 1. This follows from the
continuation of the exact sequences in the preceding theorem by making use
of Bext i .C; A/ D 0 for completely decomposable groups C. Thus with the aid
of repeated balanced-projective resolutions the higher Bexts can be computed
recursively.
F Notes. The systematic study of balancedness started with Hunter [1], though several
sporadic results on balancedness had been proved earlier under different or no names, see, e.g.,
Lyapin [1]. The first balanced-projective resolutions appear in Albrecht–Hill [1].
Specific subgroups of various kind were subjects of publications, but none of them turned out to
be able to compete in importance with balancedness. For example, Bican [1] studied subgroups G
of a torsion-free A, called regular to mean t G .g/ D t A .g/ for all g 2 G. Regular subgroups display
a few interesting properties, and are useful in certain proofs. We also mention the dual concept: an
exact sequence 0 ! B ! A ! C ! 0 is said to be cobalanced if all rank 1 torsion-free groups
R have the injective property relative to it, i.e. if the induced map Hom.A; R/ ! Hom.B; R/ is
always surjective. Cobalancedness has been investigated by several authors, see, e.g., Goeters [2].
So far, cobalancedness has not proved to be very useful.
Exercises
(1) Localization preserves balancedness.
(2) The group J p of the p-adic integers (p any prime) is a torsion-free group of
cardinality 2 @ 0
that has only two balanced subgroups, viz. 0andJ p .
(3) Let F be a free group, and G a pure subgroup such that F=G is homogeneous of
type .0;:::;0;:::/, but not free (e.g., an infinite direct product of copies of Z).
Show that G is balanced, but not a summand in F.
(4) (Kravchenko) If B is a balanced subgroup of A, then for all types s and t,
.B C A.s// \ .B C A.t// D B C A.s ^ t/.
(5) Let G D A C B be a torsion-free group, where A; B are pure subgroups of G.
If A \ B is balanced in A, thenB is balanced in G. [Hint:G=B Š A=.A \ B/