Abelian Groups - László Fuchs [Springer]

260 9 Groups of Extensions and Cotorsion Groups
Proposition 1.1 (Baer). The sum of the extensions e 1 and e 2 of A by C is the
extension
e 1 C e 2 Dr A .e 1 ˚ e 2 / C :
Proof. What we have to establish is that if f i W C C ! A is a factor set
belonging to e i .i D 1; 2/, thenf 1 C f 2 belongs to r A .e 1 ˚ e 2 / C : It is clear that
.f 1 .c 1 ; c 2 /; f 2 .c 0 1 ; c0 2 // with c i; c 0 i 2 C is a factor set belonging to the direct sum e 1˚e 2 ,
and .f 1 .c 1 ; c 2 /; f 2 .c 1 ; c 2 // is one corresponding to .e 1 ˚e 2 / C : An application of r A
yields the factor set f 1 .c 1 ; c 2 / C f 2 .c 1 ; c 2 /.
ut
It is now easy to conclude that for the homomorphisms ˛i W A ! A 0 and
i W C 0 ! C the following equivalences hold for the extensions e; e 1 ; e 2 of A by C:
˛.e 1 C e 2 / ˛e 1 C ˛e 2 ; .e 1 C e 2 / e 1 C e 2 ; (9.3)
.˛1 C ˛2/e ˛1e C ˛2e; e. 1 C 2 / e 1 C e 2 : (9.4)
The equivalences (9.3) express the fact that ˛ W e 7! ˛e and W e 7! e are group
homomorphisms
˛ W Ext.C; A/ ! Ext.C; A 0 /;
W Ext.C; A/ ! Ext.C 0 ; A/;
while (9.4) asserts that .˛1 C˛2/ D .˛1/ C.˛2/ and . 1 C 2 / D . 1 / C. 2 / ,
i.e. the correspondence ExtW C A ! Ext.C; A/ with ˛ 7! ˛ D ˛ is an
additive bifunctor Ab Ab ! Ab. This fact is important enough to be recorded as
a theorem:
Theorem 1.2 (Eilenberg–MacLane [1]). Ext is an additive bifunctor Ab Ab !
Ab, contravariant in the first, and covariant in the second variable.
ut
In order to be consistent with the functorial notation for homomorphisms, we
shall also use the notation Ext.; ˛/ for ˛ D ˛ , thus Ext.; ˛/W Ext.C; A/ !
Ext.C 0 ; A 0 / acting as e 7! ˛e. In the sequel, we will often deal with diagrams of
extensions, emphasizing that in all of these diagrams extensions can be replaced by
equivalent extensions without causing any harm to commutativity or exactness.
A useful observation: if C is a p-group, then there are natural isomorphisms
Ext.C; A/ Š Ext.C; Z .p/ ˝ A/ and Ext.A; C/ Š Ext.Z .p/ ˝ A; C/:
In both cases the right-hand sides can also be interpreted as Ext Z.p/ .
F Notes. If we wish to develop extensions solely qua short exact sequences, then the Baer sum
would serve as the definition of the sum of two extensions, and then we have to verify: (1) e 1 C e 2
is indeed an extension of A by C which stays in the same equivalence class if e 1 ; e 2 are replaced
by equivalent extensions, and (2) the equivalence classes form a group under this operation. For
details of this approach, we refer to MacLane [M]. A third method of introducing Ext is as the