Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

We first review definition and some basic properties of modules.

DEFINITION 04.1   Let $R$ be a (unital associative) ring. A set $M$ is said to be an $R$ -module if there is given a binary map (``action'')

$$R\times M \ni (r,m)\mapsto r.m \in M$$

such that the following properties hold.

1. $\forall r \in R \forall m_1,\forall m_2 \in M \quad r. (m_1+m_2) =r. m_1 + r. m_2$ .
2. $\forall r_1,\forall r_2 \in R \forall m \in M \quad (r_1+ r_2).m= r_1.m + r_2 .m$ .
3. $\forall m\in M \quad 1_R.m= m$ .
4. $\forall r_1,\forall r_2 \in R \forall m \in M \quad (r_1 r_2). m=r_1.(r_2.m)$

DEFINITION 04.2   Let $R$ be a (unital associative) ring. A map $\varphi$ from an $R$ -module $M$ to another $R$ -module $N$ is an $R$ -module homomorphism if the following conditions are satisfied.

1. $\varphi$ is additive. That means, we have
   $$\forall m_1\forall m_2\in M \quad \varphi(m_1+m_2)=\varphi(m_1)+\varphi(m_2).$$

2. $\varphi$ preserves the $R$ -action. That means,
   $$\forall r \in R\forall m\in M \quad \varphi(r.m)=r.\varphi(m).$$

PROPOSITION 04.3   For any given ring $R$ , The category (R-mod) of $R$ -modules is an abelian category.

EXERCISE 04.1   Let $A$ be a $2\times 2$ matrix

$$A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$

over $\mathbb{C}$ . We define a structure of $\mathbb{C}[X]$ -module on $V=\mathbb{C}^2$ by putting

$$f(X). v= f(A) v \qquad (v\in \mathbb{C}^2)$$

Then:

1. Show that $V$ has a proper $\mathbb{C}[X]$ -submodule $W$ . (That means, $\mathbb{C}[X]$ submodule such that $W\neq V, 0$ .
2. Show that there is no other proper submodule of $V$ .

DEFINITION 04.4  

A cochain complex in an abelian category $\mathcal{C}$ is a sequence of objects and morphisms in $\mathcal{C}$

$$C^\bullet: \dots \overset{d^{n-1}}{\to} C^n \overset{d^{n}}{\to} C^{n+1} \overset{d^{n+1}}{\to} \dots$$

such that $d^{n}\circ d^{n-1}=0$ .

Cohomology objects of the cochain complex are

$$H^{n}(C^\bullet)=\operatorname{Ker}(d^{n})/\operatorname{Image}(d^{n-1}).$$