Cohomologies.

Yoshifumi Tsuchimoto

LEMMA 04.1   Let $R$ be a (unital associative but not necessarily commutative) ring. Then for any $R$ -module $M$ , the following conditions are equivalent.

1. $M$ is a direct summand of free modules.
2. $M$ is projective

COROLLARY 04.2   For any ring $R$ , the category $(R \operatorname{-modules})$ of $R$ -modules have enough projectives. That means, for any object $M \in (R\operatorname{-modules})$ , there exists a projective object $P$ and a surjective morphism $f: P \to M$ .

DEFINITION 04.3   Let $R$ be a commutative ring. We assume $R$ is a domain (that means, $R$ has no zero-divisors except for 0 .)

An $R$ -module $M$ is said to be divisible if for any $r \in R\setminus \{0\}$ , the multplication map

$$M \overset{r \times }{\to} M$$

is surjective.

DEFINITION 04.4   Let $R$ be a commutative ring. We assume $R$ is a domain (that means, $R$ has no zero-divisors except for 0 .)

An $R$ -module $M$ is said to be divisible if for any $r \in R\setminus \{0\}$ , the multplication map

$$M \overset{r \times }{\to} M$$

is epic.

LEMMA 04.5   Let $R$ be a (commutative) principal ideal domain (PID). Then an $R$ -module $I$ is injective if and only if it is divisible.

PROPOSITION 04.6   For any (not necessarily commutative) ring $R$ , the category $(R \operatorname{-modules})$ of $R$ -modules has enough injectives. That means, for any object $M \in (R\operatorname{-modules})$ , there exists an injective object $I$ and an monic morphism $f: M \to I$ .

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

LEMMA 04.7   For any $\mathbb{Z}$ -module $M$ , let us denote by $\hat M$ the module $\operatorname{Hom}_\mathbb{Z}(M,\mathbb{T}_1)$ where $T_1=\mathbb{R}/\mathbb{Z}$ . Then:

1. For any free $\mathbb{Z}$ -module $F$ , $\hat F$ is divisible (hence is $\mathbb{Z}$ -injective).
2. For any $\mathbb{Z}$ -module $M$ , there is a canonical injective $\mathbb{Z}$ -homomorphism $M\to \widehat{(\hat M)}$ .
3. Any $Z$ -module $M$ may be embeded in a divisible module $T$ .

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

$$\operatorname{Hom}_\mathbb{Z}(A,T)$$

is $A$ -injective.