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# Cohomologies.

Yoshifumi Tsuchimoto

We mainly follow the treatment in [1].

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

such that . The -th cohomology of the cochain complex is defined to be the -module

Elements of (respectively, ) are often referred to as cocycles (respectively, coboundaries).

A bit of category theory:

DEFINITION 03.2   A category is a collection of the following data
1. A collection of objects of .
2. For each pair of objects , a set

of morphisms.
3. For each triple of objects , a map(composition (rule)'')

satisfying the following axioms
1. unless .
2. (Existence of an identity) For any , there exists an element such that

holds for any ( ).
3. (Associativity) For any objects , and for any morphisms , we have

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 is said to be abelian if it satisfies the following axioms.
A4-1.
Every morphism in has a kernel .
A4-2.
Every morphism in has a cokernel .
A4-3.
For any given morphism , we have a suitably defined isomorphism

in . More precisely, is a morphism which is defined by the following relations:

DEFINITION 03.4   Let be an abelian category.
1. An object in is said to be injective if it satisfies the following condition: For any morphism and for any monic morphism , extends'' to a morphism .

2. An object in is said to be projective if it satisfies the following condition: For any morphism and for any epic morphism , lifts'' to a morphism .

EXERCISE 03.1   Let be a ring. Let

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

is exact.

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2010-04-20