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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:
1. .
2. For each short exact sequence

and for each there is a natural homomorphism

such that we obtain a long exact sequence

3. is natural. That means, for a morphism of short exact sequences

the 's give a commutative diagram:

4. 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 .

1. may be represented as a class of a cocycle .
2. We take a lift'' such that . Note that is no longer a cocycle.
3. Consider . It is a coboundary and we have .
4. There thus exists an element such that . is no longer a coboundary but it is a cocycle.
5. 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

PROPOSITION 07.4   There exists a bijection between the isomorphism classs of the extensions and elements of the . The bijection is given by corresponding the extension ( ) to the class of the identity map by associated to the exact sequence .

See [1, XX,Exercise 27]

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2010-06-15