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Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

\fbox{Categories of modules.} 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'')

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

such that the following properties hold.
  1. % latex2html id marker 636
$ \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. % latex2html id marker 638
$ \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. % latex2html id marker 640
$ \forall m\in M \quad 1_R.m= m $ .
  4. % latex2html id marker 642
$ \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

    % latex2html id marker 665
$\displaystyle \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,

    % latex2html id marker 671
$\displaystyle \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

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

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

% latex2html id marker 699
$\displaystyle 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 % latex2html id marker 709
$ 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}$

$\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$ .

Cohomology objects of the cochain complex are

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


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2009-05-28