# Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 05.1
1. A morphism in a category is said to be monic if for any object of and for any morphism , we have

2. A morphism in a category is said to be epic if for any object of and for any morphism , we have

PROPOSITION 05.2   Let be an abelian category. Then for any morphism in , we have:
1. :monic .
2. :epic .

DEFINITION 05.3   Let be an abelian category.
1. An object in is said to be injective if it satisfies the following condition: For any morphism and for any monic morphism , extends'' to a morphism .

2. An object in is said to be projective if it satisfies the following condition: For any morphism and for any epic morphism , lifts'' to a morphism .

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 an epic 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 epic.

LEMMA 05.7   Let be a (commutative) principal ideal domain (PID). Then an -module is injective if and only if it is divisible.

PROPOSITION 05.8   For any (not necessarily commutative) ring , the category of -modules has enough injectives. That means, for any object , there exists an injective object and an monic morphism .