Cohomologies.

Yoshifumi Tsuchimoto

We mainly follow the treatment in [1].

DEFINITION 03.1   Let $R$ be a ring. A cochain complex of $R$ -modules is a sequence of $R$ -modules

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

such that $d^{n}\circ d^{n-1}=0$ . The $n$ -th cohomology of the cochain complex is defined to be the $R$ -module

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

Elements of $\operatorname{Ker}(d^{n})$ (respectively, $\operatorname{Image}(d^{n-1})$ ) are often referred to as cocycles (respectively, coboundaries).

A bit of category theory:

DEFINITION 03.2   A category $\mathcal{C}$ is a collection of the following data

1. A collection $\operatorname{Ob}(\mathcal{C})$ of objects of $\mathcal{C}$ .
2. For each pair of objects $X,Y \in \operatorname{Ob}(\mathcal{C})$ , a set
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)$$
   of morphisms.
3. For each triple of objects $X,Y,Z \in \operatorname{Ob}(\mathcal{C})$ , a map(``composition (rule)'')
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)\times \operatorname{Hom}_{\mathcal{C}}(Y,Z)\to \operatorname{Hom}_{\mathcal{C}}(X,Z)$$

satisfying the following axioms

1. $\operatorname{Hom}(X,Y)\cap \operatorname{Hom}(Z,W) =\emptyset$ unless $(X,Y)=(Z,W)$ .
2. (Existence of an identity) For any $X\in \operatorname{Ob}(\mathcal{C})$ , there exists an element $\operatorname{id}_X\in \operatorname{Hom}(X,X)$ such that
   $$\operatorname{id}_X \circ f=f,\quad g \circ \operatorname{id}_X=g$$
   holds for any $f \in \operatorname{Hom}(S,X), g \in \operatorname{Hom}(X,T)$ ( $\forall S,T \in \operatorname{Ob}(\mathcal{C})$ ).
3. (Associativity) For any objects $X,Y,Z,W \in \operatorname{Ob}(\mathcal{C})$ , and for any morphisms $f\in \operatorname{Hom}(X,Y), g\in \operatorname{Hom}(Y,Z), h \in \operatorname{Hom}(Z,W)$ , we have
   $$(f\circ g)\circ h =f\circ (g\circ h).$$

Morphisms are the basic actor/actoress in category theory.

An additive category is a category in which one may ``add'' some morphisms.

DEFINITION 03.3   An additive category $\mathcal{C}$ is said to be abelian if it satisfies the following axioms.

A4-1.

Every morphism $f:X\to Y$ in $\mathcal{C}$ has a kernel $\ker(f):\operatorname{Ker}(f)\to X$ .

A4-2.

Every morphism $f:X\to Y$ in $\mathcal{C}$ has a cokernel $\operatorname{coker}(f):Y\to \operatorname{Coker}(f)$ .

A4-3.

For any given morphism $f:X\to Y$ , we have a suitably defined isomorphism

$$l: \operatorname{Coker}(\ker(f))\cong \operatorname{Ker}(\operatorname{coker}(f))$$

in $\mathcal{C}$ . More precisely, $l$ is a morphism which is defined by the following relations:

$$\ker(\operatorname{coker}(f))\circ \o... ...exists \overline{f}),\quad \overline{f}=l \circ \operatorname{coker}(\ker(f)).$$

DEFINITION 03.4   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 $\hat f: M\to I$ .
   $$\begin{CD} P @>\hat f >>M \\ @\vert @V\pi VV \\ P @>f >> N \end{CD}$$

EXERCISE 03.1   Let $R$ be a ring. Let

$$0\to M_1 \to M_2 \to M_3 \to 0$$

be an exact sequence of $R$ -modules. Assume furthermore that $M_3$ is projective. Then show that the sequence

$$0 \to \operatorname{Hom}_R (N,M_1) \overset{\operatorname{Hom}_R(... ..._2) \overset{\operatorname{Hom}_R(N,g)}{\to} \operatorname{Hom}_R(N,M_3) \to 0$$

is exact.