Cohomology and Extensions
The Baer sum is the natural addition operation on abelian group extensions as well on the extensions of -modules, for fixed ring .
For a group and an abelian group, the extensions of by are classified by the degree-2 group cohomology
On cocycles there is a canonical addition operation coming from the additive structure of , and the Baer sum is the corresponding operation on the extensions that these cocycles classify.
Below are discussed several different equivalent ways to define the Baer sum
On concrete cocycles
A cocycle in degree-2 group cohomology is a function
satisfying the cocycle property.
Given two coycles their sum is the composite
the diagonal on ;
the direct product ;
the group operation .
Hence for all this sum is the function that sends
On abstract cocycles
As discussed at group cohomology, a cocycle is equivalently a morphism of 2-groupoids from the delooping groupoid of to the double-delooping 2-groupoid of :
Since is an abelian group, is naturally an abelian 3-group, equipped with a group operation .
With respect to this the sum operation is
On short exact sequences
In any category with products, for any object there is a diagonal morphism ; in a category with coproducts there is a codiagonal morphism (addition in the case of modules). Every additive category is, in particular, a category with finite biproducts, so both morphisms are there. Short exact sequences in the category of -modules, or in arbitrary abelian category , form an additive category (morphisms are commutative ladders of arrows) in which the biproduct for is .
Now if is any extension, call it , and a morphism, then there is a morphism from an extension of the form to , where the pair s unique up to isomorphism of extensions, and it is called . In fact, the diagram
is a pullback diagram. Every morphism of abelian extensions in a unique way decomposes as
for some with as above. In short, the morphism of extensions factorizes through .
Dually, for any morphism , there is a morphism to an extension of the form ; the pair is unique up to isomorphism of extensions and it is called .
In fact, the diagram
is a pushout diagram. Every morphism of abelian extensions in a unique way decomposes as
for some , with as above. In short, the morphism of extensions factorizes through .
There are the following isomorphisms of extensions: , , , , .
The Baer’s sum of two extensions of the form (i.e. with the same and ) is given by ; this gives the structure of the abelian group on and is a biadditive (bi)functor. This is also related to the isomorphisms of extensions , , , .
In different notation, if for are two short exact sequences of abelian groups, their Baer sum is
The first step forms the pullback of the short exact sequence along rhe diagonal on :
The second forms the pushout along the addition map on :
S. MacLane, Homology, 1963
Patrick Morandi, Ext groups and Ext functors (pdf)