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Cohomologies.

Yoshifumi Tsuchimoto

\fbox{03. cohomology of a complex.} 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

$\displaystyle 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

$\displaystyle 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

    $\displaystyle \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)'')

    $\displaystyle \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

    % latex2html id marker 832
$\displaystyle \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

    $\displaystyle (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

$\displaystyle 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:

% latex2html id marker 871
$\displaystyle \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$ .

    $\displaystyle \begin{CD}
M @>\hat f >>I \\
@A \iota AA @\vert \\
N @>f >> I
\end{CD}$

  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$ .

    $\displaystyle \begin{CD}
P @>\hat f >>M \\
@\vert @V\pi VV \\
P @>f >> N
\end{CD}$

EXERCISE 03.1   Let $ R$ be a ring. Let

$\displaystyle 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

$\displaystyle 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.


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Next: Bibliography
2010-04-20