Abelian Groups - László Fuchs [Springer]

4 Pure-Projectivity and Pure-Injectivity 163
(5) Let E denote a class of exact sequences.
(a) The class of groups that enjoy the injective property with respect to every
member of E is closed under taking direct products and summands.
(b) The same for ‘projective’ in place of ‘injective,’ but change ‘direct product’
to ‘direct sum.’
4 Pure-Projectivity and Pure-Injectivity
In Theorem 3.2 the pure-exact sequences were characterized by properties that the
finite cyclic groups have the projective as well as the injective property relative to
them. Our next goal is to find all groups that have the projective, resp. the injective
property relative to all pure-exact sequences.
Pure-Projective Groups A group P is called pure-projective if it enjoys the
projective property relative to the class of pure-exact sequences; i.e., if every
diagram
ψ
P
⏐
↓φ
0 −−−−→ A
α
−−−−→ B
β
−−−−→ C −−−−→ 0
with pure-exact row can be completed by a map W P ! B such that ˇ D .
Example 4.1. All cyclic groups are pure-projective: Z is because it is projective, and all finite
cyclic groups because of Theorem 3.2. Hence all †-cyclic groups are pure-projective.
In order to find all pure-projective groups, we prove a lemma (which can be
interpreted as the existence theorem on pure-projective resolutions; we say: there
are enough pure-projectives).
Lemma 4.2. Every group A can be embedded in a pure-exact sequence
0 ! B ! P ˛!A ! 0 (5.5)
where P is †-cyclic .hence B as well/.
Proof. For every a 2 A,lethc a iŠhai be a cyclic group, and define P D˚a2A hc a i.
Let ˛ W P ! A act via ˛ W c a 7! a for all a 2 A. This is a well-defined epimorphism,
and if we set B D Ker ˛, then we get the exact sequence (5.5). B is pure in P, since
by construction, every coset mod B is represented by an element of the same order
(cf. Lemma 2.8). Ker ˛ D B is also †-cyclic, since it is a subgroup in a †-cyclic
group (Theorem 5.7 in Chapter 3).
ut