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# Categories, abelian categories and 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 .

LEMMA 07.2   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.

DEFINITION 07.3   Let be a left exact additive functor. An object is called -acyclic if for all .

THEOREM 07.4   Let

be a resolution of by -acyclics. Let

be an injective resolution. Then there exists a morphism of complexes extending the identity on , and this morphism induces an isomorphism

for all .

Note: Our notation of denoting complexes such as differs from that in [1].

The book of Grivel [2] is also a good reference for our future arguments.

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2009-06-18