Next: Bibliography

# Cohomologies.

Yoshifumi Tsuchimoto

LEMMA 04.1   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 04.2   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 04.3   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 04.4   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 04.5   Let be a (commutative) principal ideal domain (PID). Then an -module is injective if and only if it is divisible.

PROPOSITION 04.6   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 .

For the proof of the proposition above, we need the followin lemmas.

LEMMA 04.7   For any -module , let us denote by the module where . Then:
1. For any free -module , is divisible (hence is -injective).
2. For any -module , there is a canonical injective -homomorphism .
3. Any -module may be embeded in a divisible module .

LEMMA 04.8   Let be a divisible module. Then for any ring ,

is -injective.

Next: Bibliography
2010-05-12