Cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 01.1   A (unital associative) ring is a set $R$ equipped with two binary operations (addition (``+'') and multiplication (``$\cdot$ '')) such that the following axioms are satisfied.

Ring1.

$R$ is an additive group with respect to the addition.

Ring2.

distributive law holds. Namely, we have

$$a(b+c)=ab + bc,\quad (a+b)c=ac+bc \qquad (\forall a,\forall b,\forall c\in R).$$

Ring3.

The multiplcation is associative.

Ring4.

$R$ has a multiplicative unit.

For any ring $R$ , we denote by $0_R$ (respectively, $1_R$ ) the zero element of $R$ (respectively, the unit element of $R$ ). Namely, $0_R$ and $1_R$ are elements of $R$ characterized by the following rules.

• $a + 0_R =a$ ,      $0_R + a =a$ $\forall a \in R$ .
• $a \cdot 1_R =a$ ,      $1_R \cdot a =a$ $\forall a \in R$ .

When no confusion arises, we omit the subscript `${}_R$ ' and write $0,1$ instead of $0_R,1_R$ .

DEFINITION 01.2   Let $R$ be a unital associative ring. An $R$ -module $M$ is an additive group $M$ with $R$ -action

$$R\times M\to M$$

which satisfies

Mod1.

$(r_1 r_2). m= r_1.(r_2.m)$ $\quad (\forall r_1, \forall r_2\in R, \forall m\in M)$

Mod2.

$1.m=m$ $\quad (\forall m \in M)$

Mod3.

$(r_1+r_2).m=r_1.m+r_2.m$ $\quad (\forall r_1,\forall r_2\in R, \forall m \in M)$ .

Mod4.

$r.(m_1+m_2)=r.m_1+r.m_2$ $\quad (\forall r\in R, \forall m_1,\forall m_2 \in M)$ .

EXAMPLE 01.3   Let us give some examples of $R$ -modules.

1. If $k$ is a field, then the concepts ``$k$ -vector space" and ``$k$ -module'' are identical.
2. Every abelian group is a module over the ring of integers $\mathbb{Z}$ in a unique way.

DEFINITION 01.4   An subset $M$ of an $R$ -module $N$ is said to be an $R$ -submodule of $N$ if $M$ itself is an $R$ -module and the inclusion map $j:M\to N$ is an $R$ -module homomorphism.

DEFINITION 01.5   Let $M,N$ be modules over a ring $R$ . Then a map $f:M\to N$ is called an $R$ -module homomorphism if it is additive and preserves the $R$ -action.

The set of all module homomorphisms from $M$ to $N$ is denoted by $\operatorname{Hom}_R (M,N)$ . It has an structure of an module in an obvious manner.

DEFINITION 01.6   An subset $N$ of an $R$ -module $M$ is said to be an $R$ -submodule of $M$ if $N$ itself is an $R$ -module and the inclusion map $j:N\to M$ is an $R$ -module homomorphism.

DEFINITION 01.7   Let $R$ be a ring. Let $N$ be an $R$ -submodule of an $R$ -module $M$ . Then we may define the quotient $M/N$ by

$$M/N=M/\sim_N$$

where the equivalence relation $\sim_N$ is defined as follows:

$$m_1 \sim_N m_2 \quad \iff \quad m_1-m_2\in N.$$

It may be shown that the quotient $M/N$ so defined is actually an $R$ -module and that the natural projection

$$\pi: M\to M/N$$

is an $R$ -module homomorphism.

DEFINITION 01.8   Let $f:M\to N$ be an $R$ -module homomorphism between $R$ -modules. Then we define its kernel as follows.

$$\operatorname{Ker}(f)=f^{-1}(0)=\{ m\in M; f(m)=0\}.$$

The kernel and the image of an $R$ -module homomorphism $f$ are $R$ -modules.

THEOREM 01.9   Let $f:M\to N$ be an $R$ -module homomorphism between $R$ -modules. Then

$$M/\operatorname{Ker}(f) \cong f(N).$$

DEFINITION 01.10   Let $R$ be a ring. An ``sequence''

$$M_1\overset {f}{\to} M_2 \overset{g}{\to} M_3$$

is said to be an exact sequence of $R$ -modules if the following conditions are satisfied

Exact1.

$M_1,M_2$ are $R$ -modules.

Exact2.

$f,g$ are $R$ -module homomorphisms.

Exact3.

$\operatorname{Ker}(g)=\operatorname{Image}(f)$ .

For any $R$ -submodule $N$ of an $R$ -module $M$ , we have the following exact sequence.

$$0\to N\to M\to M/N \to 0$$

EXERCISE 01.1   Compute the following modules.

1. $\operatorname{Hom}_\mathbb{Z}(\mathbb{Z}/3\mathbb{Z},\mathbb{Z})$ .
2. $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Z})$ .
3. $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Z}/5\mathbb{Z})$ .