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LOCALLY ^-PROJECTIVE ABELIAN GROUPS AND GENERALIZATIONS 217
where PQ is ^4-projective, and U c P o
satisfies S A
(U) = U because of
(c). Since U is ^4-solvable by Lemma 3.1, and we have already verified
the implication (a) => (c), we obtain an ^4-balanced exact sequence
with Pi ^4-projective. Because of the validity of the implication (a) =>*
(b), S A
(V) = V. Consequently, an inductive argument completes the
proof.
(d) =» (a): The sequence in (d) induces an exact sequence
0 - T A
H A
(U)
Θ
^U^ T A
H A
{G) Θ Λ G -> 0
by Lemma 3.1 where U = imφ\. Because of S A
(U) = U 9
the map θjj
is onto; and ΘQ is an isomorphism.
EXAMPLE 3.3. Let A be a countable abelian group of infinite rank
with E(A) = Z. Every free subgroup F of A which has infinite rank
contains a subgroup F x
such that F/F x
= T A
(Q) = 0 ω
Q. Thus,
F/F\ is a direct summand of A/F\ 9
and there exists a non-zero proper
subgroup U of A with A/U = 0 ω
Q.
We now show that S A
(U) = 0. If this is not the case, then H A
(U) Φ
0; and H A
(A)/H A
(U) is a bounded abelian group. However, since the
latter is isomorphic to a subgroup of the torsion-free group H A
(® ω
Q),
this is only possible if H A
{A) = H A
(U). Because this contradicts the
condition A Φ U 9
we obtain S A
(U) = 0. On the other hand, every free
resolution 0 —• 0 ω
Z -• 0 ω
Z -• Q -• 0 yields an exact sequence
Finally, we construct an 4-balanced exact sequence 0 -> V —•
φ 7
4 -• 0 ω
Q -^ 0 such that S A
{V) φ V: For this, we observe
HA(Q) — Θ2 N o Q Hence, there exists an 4-balanced exact sequence
0 -> F ^ 0 2
κ o
4 -^ 0 ω
Q -• 0. By Proposition 3.2, ^(F) ^ F.
In the next part of this section, we introduce the concept of an A-
projective dimension for 4-solvable groups. In view of Proposition
3.2, we assume, that A is self-small and flat as an 2s(4)-module, and
consider two 4-balanced 4-projective resolutions 0 —• I// —• PiG —> 0
(/ = 1,2) of an ^4-solvable group G. They induce exact sequences
0 -• H A
(Ui) -+ H Λ
(Pi) -> H A
{G) -+ 0 of right E{A)-moάxx\es for
/ = 1,2. By ShanueΓs Lemma [R, Theorem 3.62], H A
(P {
) ®H A
(U 2
) =
HA(PI)®H A
(U\). Since both, the C//'s and the P/'s, are ^-solvable, we
ω