We recommend the book of Lang  as a good reference.
The treatment here follows the book for the most part.
be an abelian category with enough injectives, and let
be a covariant additive left functor to another abelian
- For each short exact sequence
and for each
there is a natural homomorphism
such that we obtain a long exact sequence
is natural. That means, for a morphism of short exact sequences
's give a commutative diagram:
- For each injective objective object
and for each
is a ``universal delta functor''. See .
Under the assumption of the previous theorem,
for any exact sequence
of objects in
there exists injective resolutions
respectively and a commutative diagram
such that the diagram of resolutions is exact.
Thus we obtain a diagram
such that each row in the last line is exact.
-th cohomology of the complex
Using the resolution given in the lemma above, we may prove
Let us describe the map
in more detail when
category of modules by ``diagram chasing''.
, let us show how to compute
may be represented as a class
of a cocycle
- We take a ``lift''
. Note that
is no longer a cocycle.
It is a coboundary and we have
- There thus exists an element
is no longer a coboundary but it is a cocycle.
- The cohomology class
is the required
Such computation appears frequently when we deal with cohomologies.
be a ring. Let
-modules. Then an extension
is a module
with a exact sequence
be another extension. Then the two extensions are said to be isomorphic
if there exists a commutative diagram
See [1, XX,Exercise 27]