Abelian Groups - László Fuchs [Springer]

1 Group Extensions 257
μ
−−−−→ A −−−−→ B
⏐
∥
β↓
−−−−→ A
ν
−−−−→ C −−−−→ 0
∥
μ ′
−−−−→ B ′ ν ′
−−−−→ C −−−−→ 0
is commutative. In this case, the extensions e and e 0 themselves are called equivalent.
Thus, there is a bijective correspondence between the equivalence classes of
extensions of A by C and the equivalence classes of factor sets f W C C ! A.
In this correspondence, the splitting extensions form an equivalence class, and the
corresponding equivalence class of factor sets consists of factor sets of the form
h.u/ C h.v/ h.u C v/ for arbitrary functions hW C ! A (with h.0/ D 0 as agreed).
The Group Ext We have come to a leading idea: instead of being satisfied with
a survey of the collection of all extensions, one tries to furnish this set with a proper
algebraic structure which would provide a more powerful tool in the exploration.
This was done by R. Baer who introduced a group structure, creating a fascinating
theory.
If f ; f 0 W C C ! A are factor sets, then their sum f C f 0 defined as
.f C f 0 /.u;v/D f .u;v/C f 0 .u;v/
is again a factor set. Under this composition, the factor sets form a group, denoted
Z.C; A/. The coboundaries form a subgroup B.C; A/, and what has been concluded
above can be rephrased by saying that there is a bijective correspondence between
the equivalence classes of extensions of A by C and the elements of the factor group
Z.C; A/=B.C; A/. This factor group is generally called the group of extensions of
A by C:
Ext.C; A/ D Z.C; A/=B.C; A/:
Having defined Ext in terms of factor sets, we now proceed to another interpretation:
via short exact sequences (9.1). If we think of extensions of A by C as such
sequences, then it seems reasonable to create and to study first a category whose
objects are short exact sequences. In doing so, the first order of business is to define
the morphisms between two exact sequences, e and e 0 . The right definition is pretty
clear: it is a triple .˛;ˇ;/of group homomorphisms rendering the diagram
−−−−→ A −−−−→ B
⏐ ⏐
α↓
β↓
μ
ν
−−−−→ C −−−−→ 0
⏐
γ↓
−−−−→ A ′ μ ′
−−−−→ B ′ ν ′
−−−−→ C ′ −−−−→ 0