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Cohomologies.
Yoshifumi Tsuchimoto
We recommend the book of Lang [1] as a good reference.
The treatment here follows the book for the most part.
THEOREM 07.1
Let
be an abelian category with enough injectives, and let
be a covariant additive left functor to another abelian
category
. Then:

.
 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
the
's give a commutative diagram:
 For each injective objective object
of
and for each
we have
.
The collection
of functors
is a ``universal delta functor''. See [1].
LEMMA 07.2
Under the assumption of the previous theorem,
for any exact sequence
of objects in
,
there exists injective resolutions
of
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.
Note that
th cohomology of the complex
(respectively,
)
gives the
.)
Using the resolution given in the lemma above, we may prove
Theorem 7.1.
Let us describe the map
in more detail when
is a
category of modules by ``diagram chasing''.
Namely, for
, let us show how to compute
.

may be represented as a class
of a cocycle
.
 We take a ``lift''
such that
. Note that
is no longer a cocycle.
 Consider
.
It is a coboundary and we have
.
 There thus exists an element
such that
.
is no longer a coboundary but it is a cocycle.
 The cohomology class
of
is the required
.
Such computation appears frequently when we deal with cohomologies.
DEFINITION 07.3
Let
be a ring. Let
be
modules. Then an
extension
of
by
is a module
with a exact sequence
(E) 

of
modules.
Let
be another extension. Then the two extensions are said to be isomorphic
if there exists a commutative diagram
See [1, XX,Exercise 27]
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