Abelian Groups - László Fuchs [Springer]

558 14 Butler Groups
(i) there exists a solid chain from A to G;
(ii) there is a balanced-exact sequence
0 ! B ! A ˚ C !G ! 0 (14.7)
where C is completely decomposable, A is the inclusion, and B is a B 2 -
group;
(iii) the same as (ii) with prebalanced-exact (14.7).
Proof. (i) ) (ii) We place ourselves in the following setting: we have a solid chain
A D G 0 < < G < G C1 < < G with factors of rank 1. We build a relative
balanced-projective resolution of G with the aid of this chain as follows: from a
relative balanced-projective resolution 0 ! B ! A ˚ C ! G ! 0 of G
we form a relative balanced-projective resolution 0 ! B C1 ! A ˚ C C1 !
G C1 ! 0 of G C1 by choosing C C1 D C ˚ C for a suitable completely
decomposable group C and mapping C into G C1 such that in an arbitrarily
chosen direct decomposition of C , no rank 1 summand maps into G .
In this way, we obtain commutative diagrams with exact rows and columns:
0 0 0
⏐
⏐
⏐
↓
↓
↓
0 −−−−→ B σ −−−−→ A ⊕ C σ −−−−→ G σ −−−−→ 0
⏐
⏐
⏐
↓
↓
↓
0 −−−−→ B σ+1 −−−−→ A ⊕ C σ ⊕ C ∗ −−−−→ G σ+1 −−−−→ 0
⏐
⏐
⏐
↓
↓
↓
0 −−−−→ B σ+1 /B σ −−−−→ C ∗ −−−−→ G σ+1 /G σ −−−−→ 0
⏐
⏐
⏐
↓
↓
↓
0 0 0
Evidently, B is balanced in B C1 , since it is balanced in A˚C which is a summand
of A ˚ C ˚ C . It is routine to check that at limit ordinals the direct limits provide
relative balanced-projective resolutions.
Now, if G is solid in G C1 , then the ideal I generated by the types of summands
in C has a countable set of generators, thus jC j D @ 0 may be assumed. In
the diagram, B C1 =B is a corank one pure subgroup of the countable completely
decomposable group C , so Theorem 6.8 guarantees that B C1 =B is a B 2 -group.
Consequently, B is the union of a smooth chain 0 D B 0 < < B < B C1 < :::
of subgroups such that each is balanced in its successor, and each of the factors