Abelian Groups - László Fuchs [Springer]

424 12 Torsion-Free Groups
(ii)
If 0 ! B 0 ! C 0 ! G ! 0 is another balanced-projective resolution of the
same group, then
B ˚ C 0 Š B 0 ˚ C:
Proof. (i) Let R˛ run over all rank one pure subgroups of G. For each ˛, picka
rank one group C˛ isomorphic to R˛ along with an isomorphism ˛ W C˛ ! R˛.
Let C be the direct sum of all these C˛, one for every R˛, and define W C ! G
to act like ˛ on C˛. Then, evidently, the map is surjective. Its kernel must be
balanced in C, because the criterion of Lemma 2.3 applies by construction.
(ii) (à la Schanuel’s lemma) The pull-back diagram
0 −−−−→ B −−−−→ H −−−−→ C ′ −−−−→ 0
⏐ φ ⏐
∥ ↓ ↓θ
0 −−−−→ B −−−−→ C
γ
−−−−→ G −−−−→ 0
can be completed by a map W C 0 ! C such that D by Theorem 3.1.This
means that the top exact sequence is splitting, i.e. H Š B ˚ C 0 . For reason of
symmetry, H Š B 0 ˚ C as well.
ut
Imitating the definition of projective dimension of modules, we can introduce
a dimension concept for balancedness. Accordingly, a group G is defined to have
balanced-projective dimension n if n is the smallest integer for which
Bext nC1 .G; A/ D 0 for all groups A:
We will indicate this by writing bpd G D n. A more explicit way to describe this
dimension is to form a long balanced-exact sequence
0 ! C n ! C n 1 !!C 1 ! C 0 ! G ! 0
(with completely decomposable groups C i ) that is shortest of this kind. Thus
bpd G D 0 if and only if G is completely decomposable. We will see later on in
Lemma 1.9 in Chapter 14 that for every n 0, there are finite rank torsion-free
groups G with bpd G D n.
Relative Balanced-Projective Resolutions A useful device in the discussion
of infinite rank Butler groups (Chapter 14) is a straightforward generalization of
balanced-projective resolutions: the relative balanced-projective resolution introduced
by Bican–Fuchs [2].
Let A be a pure subgroup of the torsion-free group G. We say that
0 ! K ! A ˚ C !G ! 0 (12.4)