Abelian Groups - László Fuchs [Springer]

160 5 Purity and Basic Subgroups
˛
ˇ
(c) 0 ! AŒn !BŒn !CŒn ! 0 is exact for every n;
(d) 0 ! A=AŒn
˛!B=BŒn !C=CŒn ˇ
! 0 is exact for every n.
Moreover, the sequences (b) and (c) are splitting exact.
Proof. We prove only (a) and (c), because then (b) and (d) will follow from the
3 3-lemma. That the compositions of ˛ and ˇ are 0 throughout is evident.
(a) It is clear that, for all n, ˛ is monic if and only if it is monic in (5.3). Ker ˇ is
˛A \ nB which is ˛.nA/ if and only if (5.3) is exact at B. Finally, for all n, ˇ is
epic exactly it is epic in (5.3).
(c) Again, for all n, ˛ is monic if and only if it is monic in (5.3), and ˇ is epic for
every n exactly if every element of order n is the image of an element of order
n in B.Kerˇ is equal to ˛.nA/ if and only if (5.3) is exact at B.
The last claim follows straightforwardly by showing that (b) and (c) are pureexact
and the groups are bounded.
ut
The next theorem characterizes pure-exact sequences in terms of their injective
and projective properties.
Theorem 3.2. An exact sequence (5.3) is pure-exact if and only if the finite cyclic
groups have the injective property relative to it if and only if the finite cyclic groups
have the projective property relative to it.
Proof. We prove the claim only for injectivity, a dual proof applies to projectivity.
Let W A ! H be a homomorphism into a cyclic group H; without loss of generality
we may assume is epic. The existence of a W B ! H with ˛ D is equivalent
to the extensibility of the isomorphism A= Ker Š H to a map B ! H, i.e.to
the fact that ˛.A= Ker / is a summand of B=˛ Ker . An appeal to Theorem 3.1
completes the proof.
ut
Direct Limits and Purity We turn our attention to the behavior of pure-exact
sequences towards direct limits. Interestingly, direct limits of pure-exact sequences
are again pure-exact.
Theorem 3.3. Let A D fA i .i 2 I/I j i g, B D fB i .i 2 I/I j i g and C D
fC i .i 2 I/I j i
g be direct systems of groups, and let ˆ W A ! B;‰ W B ! C
be homomorphisms between them such that, for every i 2 I, the sequence 0 !
i i
A i !B i !C i ! 0 is pure-exact. Then the sequence
of direct limits is likewise pure-exact.
ˆ ‰
0 ! A !B !C ! 0 (5.4)
Proof. In view of Theorem 4.6 in Chapter 2, we need only check the purity of
ˆ.A / in B .Letnb D ˆa for a 2 A ; b 2 B , and for some n 2 N. Then there
are a i 2 A i ; b i 2 B i for some i 2 I such that i a i D a; i b i D b for the canonical
maps i W A i ! A ; i W B i ! B . Because of the commutativity of the diagram