Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 05.1  

1. A morphism $f: X\to Y$ in a category is said to be monic if for any object $Z$ of $\mathcal{C}$ and for any morphism $g_1,g_2: Z\to X$ , we have
   $$f\circ g_1 =f \circ g_2 \implies g_1=g_2$$

2. A morphism $f: X\to Y$ in a category is said to be epic if for any object $Z$ of $\mathcal{C}$ and for any morphism $g_1,g_2: Y\to Z$ , we have
   $$g_1\circ f = g_2 \circ f \implies g_1=g_2$$

PROPOSITION 05.2   Let $\mathcal{C}$ be an abelian category. Then for any morphism $f$ in $\mathcal{C}$ , we have:

1. $f$ :monic $\iff$ $\operatorname{Ker}(f)=0$ .
2. $f$ :epic $\iff$ $\operatorname{Coker}(f)=0$ .

DEFINITION 05.3   Let $\mathcal{C}$ be an abelian category.

1. An object $I$ in $\mathcal{C}$ is said to be injective if it satisfies the following condition: For any morphism $f:M \to I$ and for any monic morphism $\iota:N \to M$ , $f$ ``extends'' to a morphism $\hat f: M\to I$ .
   $$\begin{CD} M @>\hat f »I \\ @A \iota AA @\vert \\ N @>f » I \end{CD}$$

2. An object $P$ in $\mathcal{C}$ is said to be projective if it satisfies the following condition: For any morphism $f: P \to N$ and for any epic morphism $\pi:M \to N$ , $f$ ``lifts'' to a morphism $\hat f: M\to I$ .
   $$\begin{CD} P @>\hat f »M \\ @\vert @V\pi VV \\ P @>f » N \end{CD}$$

LEMMA 05.4   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 05.5   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 an epic morphism $f: P \to M$ .

DEFINITION 05.6   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 05.7   Let $R$ be a (commutative) principal ideal domain (PID). Then an $R$ -module $I$ is injective if and only if it is divisible.

PROPOSITION 05.8   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 $f:M \to I$ .