Abelian Groups - László Fuchs [Springer]

258 9 Groups of Extensions and Cotorsion Groups
commutative. There is no difficulty in showing that in this way we have indeed
defined a category which will be denoted E.
In accordance with our definition of equivalent extensions above, we will say that
the exact sequences e and e 0 (qua extensions) are equivalent (notation: e e 0 )if
A D A 0 ; C D C 0 and there is a morphism .1 A ;ˇ;1 C / from e to e 0 .Thatˇ is then an
isomorphism follows at once from Lemma 2.6 in Chapter 1.
The beauty of treating extensions as short exact sequences lies in the fact that we
can work with commutative diagrams, making most proofs more transparent. But
first we must learn the basic facts about the category E.
To start with, we concentrate on extensions of a fixed group A.If W C 0 ! C is a
homomorphism, then there is a pull-back square
0 −−−−→ A
β
B ′ ν ′
⏐
↓
μ
−−−−→ B
−−−−→ C ′
⏐
γ↓
ν
−−−−→ C −−−−→ 0
From Sect. 3(a) in Chapter 2 we know that 0 is an epimorphism (since so is ), and
the pull-back property shows that Ker 0 Š Ker Š A. Hence there exists a monic
map 0 W A ! B 0 acting as 0 W a 7! .a;0/2 B 0 . B ˚ C 0 /, so that the diagram
μ ′
−−−−→ A −−−−→ B ′ ν
−−−−→ ′
C ′ −−−−→ 0
⏐ ⏐
∥
β↓
γ↓
−−−−→ A
μ
−−−−→ B
ν
−−−−→ C −−−−→ 0
has exact rows and commutative squares. The top row is an extension of A by C 0
which we have denoted by e to indicate its origin from e via . Note that D
.1 A ;ˇ;/is a morphism e ! e in the category E.
Now suppose we have a similar commutative diagram
μ
: 0 −−−−→ A −−−−→ ′′
B ′′ ν
−−−−→ ′′
C ′ −−−−→ 0
⏐ ⏐↓ ⏐
∥ β ′ γ↓
: 0 −−−−→ A
μ
−−−−→
B
ν
−−−−→ C −−−−→ 0
with exact rows, for another group B 00 . By the pull-back property there is a unique
W B 00 ! B 0 such that 0 D 00 and ˇ D ˇ0. Since the maps 00 ; 0 W A ! B 0
are such that ˇ. 00 / D ˇ0 00 D D ˇ 0 and 0 . 00 / D 00 00 D 0 D 0 0 ,
the uniqueness assertion on pull-backs implies 00 D 0 . Hence .1 A ;;1 C 0/ is an