Abelian Groups - László Fuchs [Springer]

192 6 Algebraically Compact Groups
Proof. Apply the preceding corollary to B D Ker , and note that A= Ker is
complete by virtue of Lemma 7.2 in Chapter 2.
ut
Lemma 2.8 (Kaplansky [K]). Let G be a pure subgroup of a complete group C.
Then the Z-adic closure of G in C is a summand of C. In particular, a pure closed
subgroup is a summand.
Proof. A basic subgroup B 0 of G extends to a basic subgroup B D B 0 ˚ B 00 of C.
Then C D QB 0 ˚ QB 00 by Theorem 2.4,whereG QB 0 . Thus QB 0 is the closure of G in C.
ut
Completions We turn our attention to the completion process. In the next result
we expand Theorem 7.7(i) in Chapter 2.
Theorem 2.9. For any group A, the inverse limit
QA D lim.A=nA/
with the connecting maps a C knA 7! a C nA .a 2 AI n; k 2 N/ is a complete group.
The canonical map
A W a 7! .:::;a C nA;:::/2 QA
has A 1 for its kernel, and an isomorphic copy of A=A 1 for its image. A .A/ is pure
in QA, and the factor group QA= A .A/ is divisible.
Proof. Since the factor groups A=nA are bounded, and hence complete, Q .A=nA/
is complete, and its pure closed subgroup (the inverse limit) is complete by
Lemma 2.8. Clearly, a D 0 amounts to a 2 nA for every n, whence Ker D A 1
and Im Š A=A 1 .IfQb D .:::;b n C nA;:::/.b n 2 A/ satisfies mQb D a for some
a 2 A and m 2 N, thenmb n a 2 nA for every n, in particular, for n D m, whence
a 2 mA, and the purity of A in QA follows. To prove that QA=A is divisible, we show
that every Qb D .:::;b n C nA;:::/ .b n 2 A/ is divisible by every m 2 N mod A.
Since mjb m b km for each k 2 N,wehavemjQb b m , in fact. ut
An immediate consequence of the preceding theorem is the inequality
j QAj jAj @ 0
; (6.6)
which is evident in view of QA Q n A=nA.
The group QA is called the Z-adic completion of A. Themap A W A ! QA is
natural in the categorical sense, so the correspondence A 7! QA is functorial. Indeed,
we have
Proposition 2.10. Every homomorphism ˛ W A ! B induces a unique QZhomomorphism
Q˛ W QA ! QB making the diagram