Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 06.1   Let be a functor between additive categories. We call additive if for any objects in ,

DEFINITION 06.2   Let be an additive functor from an abelian category to .
1. is said to be left exact (respectively, right exact ) if for any exact sequence

the corresponding map

(respectively,

is exact
2. is said to be exact if it is both left exact and right exact.

DEFINITION 06.3   Let , be complexes of objects of an additive category .
1. A morphism of complex is a family

of morphisms in such that commutes with . That means,

holds.
2. A homotopy between two morphisms of complexes is a family of morphisms

such that holds.

LEMMA 06.4   Let be an abelian category that has enough injectives. Then:
1. For any object in , there exists an injective resolution of . That means, there exists an complex and a morphism such that

2. For any morphism of , and for any injective resolutions , of and (respectively), There exists a morphism of complexes which commutes with . Forthermore, if there are two such morphisms and , then the two are homotopic.

DEFINITION 06.5   Let be an abelian category which has enough injectives. Let be a left exact functor to an abelian category. Then for any object of we take an injective resolution of and define

and call it the derived functor of .

LEMMA 06.6   The derived functor is indeed a functor.