Abelian Groups - László Fuchs [Springer]

2 Complete Groups 191
is a Cauchy sequence in A, hence it converges to a limit in A which we define as a.
qJ p D J p implies qA D A for all primes q ¤ p. As a consequence, the p-adic and
the Z-adic topologies are identical on A.
ut
In a similar fashion, we can prove that every group that is complete in the Z-adic
topology is a QZ-module.
Recall that groups that are J p -modules were called p-adic groups. p-adic groups
have basic subgroups, unique up to isomorphism.
Theorem 2.4. (i) A complete p-adic group is the completion of any of its basic
subgroups.
(ii) A †-cyclic p-group is basic in its completion.
(iii) Two complete p-adic groups are isomorphic if and only if their basic subgroups
are isomorphic.
Proof. (i) This follows from the fact that a basic subgroup is dense and the
induced topology on it is the same as its p-adic topology.
(ii) is clear from (i).
(iii) One way the claim is trivial. Conversely, a complete p-adic group is determined
by a basic subgroup.
ut
Completeness and Algebraic Compactness The principal result on complete
groups is the following theorem, which is essentially due to Kaplansky [K].
Theorem 2.5. A group is complete in its Z-adic topology if and only if it is a
reduced algebraically compact group.
Proof. Assume A is reduced and algebraically compact. Owing to Corollary 1.4, A is
a summand of a direct product of cyclic groups Z.p k /. Each component is complete
in its Z-adic topology, so the same holds for the summands of their direct product
(see Lemma 7.10 in Chapter 2).
Conversely, suppose that A is complete in its Z-adic topology and is pure in the
group G. IfG 1 D 0, then we expand G to its completion QG; this contains G, and
hence also A as a pure subgroup. A basic subgroup B 0 of A is a summand of a basic
subgroup B D B 0 ˚ B 00 of QG, whence QG D QB 0 ˚ QB 00 D A ˚ QB 00 . Thus A is a summand
of G, too. If G 1 ¤ 0, then factoring out G 1 ,theimageofA remains pure in G=G 1 ,
so .A C G 1 /=G 1 Š A is a summand of G=G 1 .AsA \ G 1 D 0, A is a summand of
G. Consequently, A is algebraically compact. ut
Since the Z-adic topology on A induces a topology on a subgroup B of A that
is coarser than the Z-adic topology of B, we combine the preceding theorem with
Lemma 7.2 in Chapter 2 to derive at once:
Corollary 2.6. Let A be a reduced algebraically compact group, and B a subgroup
such that .A=B/ 1 D 0. Then both B and A=B are algebraically compact. ut
Another noteworthy observation is the following.
Corollary 2.7. If is an endomorphism of a complete group A, then both Ker
and Im are complete groups.