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# Cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 05.1   A (covariant) functor from a category to a category consists of the following data:
1. An function which assigns to each object of an object of .
2. An function which assigns to each morphism of an morphism of .
The data must satisfy the following axioms:
functor-1.
for any object of .
functor-2.
for any composable morphisms of .

By employing the following axiom instead of the axiom (functor-2) above, we obtain a definition of a contravariant functor:

(functor- ) for any composable morphisms

DEFINITION 05.2   Let be a functor between additive categories. We call additive if for any objects in ,

DEFINITION 05.3   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.

LEMMA 05.4   Let be a (unital associative but not necessarily commutative) ring. Then for any -module , the following conditions are equivalent.
1. is a direct summand of free modules.
2. is projective

COROLLARY 05.5   For any ring , the category of -modules have enough projectives. That means, for any object , there exists a projective object and a surjective morphism .

DEFINITION 05.6   Let be a commutative ring. We assume is a domain (that means, has no zero-divisors except for 0 .)

An -module is said to be divisible if for any , the multplication map

is surjective.

DEFINITION 05.7   Let be a commutative ring. We assume is a domain (that means, has no zero-divisors except for 0 .)

An -module is said to be divisible if for any , the multplication map

is epic.

DEFINITION 05.8   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 05.9   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 05.10   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 05.11   The derived functor is indeed a functor.

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2010-05-27