Abelian Groups - László Fuchs [Springer]

560 14 Butler Groups
0 −−−−→ B ′ −−−−→ C ′ −−−−→ G −−−−→ 0
⏐ ⏐
↓ ↓φ ∥
0 −−−−→ B −−−−→ C −−−−→ G −−−−→ 0.
Ker is balanced in C 0 , and hence also in B 0 . By Theorem 7.2 it is a B 2 -group, and
thus B 0 is B 2 as a balanced extension of the B 2 -group Ker by B 0 . By Lemma 2.2(iv),
the top row is prebalanced-exact. Hence Theorem 7.2 implies that G admits a solid
chain.
ut
We have a noteworthy corollary to this theorem:
Corollary 7.4. If a torsion-free group admits a solid chain, then its summands also
admit solid chains.
Proof. Suppose G D G 1 ˚ G 2 has a solid chain. Choose balanced-projective
resolutions 0 ! A i ! C i ! G i ! 0.i D 1; 2/, wheretheC i are completely
decomposable. Then 0 ! A 1 ˚ A 2 ! C 1 ˚ C 2 ! G ! 0 is a balanced-projective
resolution of G. By Theorem 7.3, A 1 ˚ A 2 is a B 2 -group, so by Corollary 5.5 the
groups A i are also B 2 -groups. Apply Theorem 7.3 again to conclude that the groups
G i admit solid chains.
ut
Next we verify an analogue of Lemma 6.5.
Lemma 7.5. Suppose B < A are pure subgroups of the torsion-free G. Then the
following holds:
(a) If there are solid chains from B to A and from A to G, then there is also one from
BtoG.
(b) Assume B is balanced in G. There is a solid chain from A to G if and only if
there is one from A=BtoG=B.
Proof. The proofs are similar for both claims, making use of Theorem 7.2.Wegive
details for (b) only.
(b) Starting from the relative balanced-exact resolution in the top row, we factor
out B to obtain the bottom row in the commutative diagram
0 −−−−→ K −−−−→ A ⊕ C −−−−→ G −−−−→ 0
⏐
⏐
∥
↓
↓
0 −−−−→ K −−−−→ A/B ⊕ C −−−−→ G/B −−−−→ 0.
As B is balanced in G, the bottom row is also balanced-exact, so it is a relative
balanced-exact resolution of A=B in G=B. Both solid chains in question exist exactly
if K is a B 2 -group (Theorem 7.2).
ut
The following lemma is a crucial ingredient in the proof of Corollary 7.7.