Abelian Groups - László Fuchs [Springer]

184 6 Algebraically Compact Groups
(d) a system of equations over A is solvable in A provided that each of its finite
subsystems has a solution in A;
(e) A is pure-injective.
Proof. (a) ) (b) By Lemma 4.6 in Chapter 5, A can be embedded as a pure
subgroup in a direct product of cocyclic groups. Hence it is clear that (a)
implies (b).
(b) ) (c) Since the direct product of compact groups is compact in the product
topology, and since the property of being a summand is transitive, (c) will be
established as soon as we can show that cocyclic groups share property (c).
But this is evident, since the cyclic groups Z.p k /.k < 1/ are compact in the
discrete topology, while the quasi-cyclic groups Z.p 1 / are summands of the
circle group T D R=Z (the reals mod 1) which is a well-known compact group.
(c) ) (d) Consider the system
X
n ijx j D a i 2 A .i 2 I/ (6.1)
j2J
of linear equations where n ij 2 Z such that, for a fixed i, almost all the
coefficients n ij are 0, and suppose that every finite subsystem is solvable in A.By
hypothesis (c), for some group B, the group C D A ˚ B admits a compact group
topology. We may equally well view our system (6.1) to be over C, and finitely
solvable in C. A solution of the ith equation can be regarded as an element
.:::;c j ;:::/in the cartesian power C J such that x j D c j 2 C .j 2 J/ satisfies the
ith equation. The set of all solutions to the ith equation is thus a subset X i of the
compact space C J ; moreover, it is a closed subset, since it is defined in terms
of an equation. The hypothesis that every finite subsystem of (6.1) is solvable
in C amounts to the condition that every finite set of the X i .i 2 I/ has a nonempty
intersection. By the compactness of C J , the intersection \ i2I X i is not
vacuous. This means that the entire system (6.1) admits a solution in C. The
A-coordinates of a solution yield a solution of (6.1)inA.
(d) ) (e) Let B denote a pure subgroup of C, and W B ! A. Letfc j g j2J be a
generating system of C mod B,and
X
n ij c j D b i 2 B .n ij 2 Z; i 2 I/
j2J
the list of all the relations between these c j and elements of B. Wepasstothe
corresponding system
X
n ij x j D b i 2 A .i 2 I/ (6.2)
j2J
of equations. A finite subsystem of (6.2) contains but a finite number of
unknowns x j1 ;:::;x jk explicitly. By purity, B is a direct summand of B 0 D
hB; c j1 ;:::;c jk i, B 0 D B ˚ C 0 (cf. Theorem 2.9 in Chapter 5), and the images