# Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

We first review definition and some basic properties of modules.

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

such that the following properties hold.
1. .
2. .
3. .

DEFINITION 04.2   Let be a (unital associative) ring. A map from an -module to another -module is an -module homomorphism if the following conditions are satisfied.
1. is additive. That means, we have

2. preserves the -action. That means,

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

EXERCISE 04.1   Let be a matrix

over . We define a structure of -module on by putting

Then:
1. Show that has a proper -submodule . (That means, submodule such that .
2. Show that there is no other proper submodule of .

DEFINITION 04.4

A cochain complex in an abelian category is a sequence of objects and morphisms in

such that .

Cohomology objects of the cochain complex are