Abelian Groups - László Fuchs [Springer]

274 9 Groups of Extensions and Cotorsion Groups
C C1 ! B .If … E, then select D 0. Define a direct system of splitting short
exact sequences e . < / with commutative diagrams
φ ν
ν :0 −−−−→ A ν −−−−→ A ν ⊕ C ν −−−−→ Cν −−−−→ 0
⏐ ⏐
⏐
↓ γ ν
⏐↓ ⏐↓
φ ν+1
ν+1 :0−−−−→ A ν+1 −−−−→ A ν+1 ⊕ C ν+1 −−−−→ Cν+1 −−−−→ 0
where the extremal vertical arrows are the injection maps, ; C1 are the obvious
projections, while is defined as follows:
W .a ; c / ! .a C c ; c /;
where a 2 A ; c 2 C . At limit ordinals , we form the direct limit of the system
fe j