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# Cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 01.1   A (unital associative) ring is a set equipped with two binary operations (addition (+'') and multiplication ( '')) such that the following axioms are satisfied.
Ring1.
is an additive group with respect to the addition.
Ring2.
distributive law holds. Namely, we have

Ring3.
The multiplcation is associative.
Ring4.
has a multiplicative unit.

For any ring , we denote by (respectively, ) the zero element of (respectively, the unit element of ). Namely, and are elements of characterized by the following rules.

• ,      .
• ,      .
When no confusion arises, we omit the subscript  ' and write instead of .

DEFINITION 01.2   Let be a unital associative ring. An -module is an additive group with -action

which satisfies
Mod1.
Mod2.
Mod3.
.
Mod4.
.

EXAMPLE 01.3   Let us give some examples of -modules.
1. If is a field, then the concepts  -vector space" and  -module'' are identical.
2. Every abelian group is a module over the ring of integers in a unique way.

DEFINITION 01.4   An subset of an -module is said to be an -submodule of if itself is an -module and the inclusion map is an -module homomorphism.

DEFINITION 01.5   Let be modules over a ring . Then a map is called an -module homomorphism if it is additive and preserves the -action.

The set of all module homomorphisms from to is denoted by . It has an structure of an module in an obvious manner.

DEFINITION 01.6   An subset of an -module is said to be an -submodule of if itself is an -module and the inclusion map is an -module homomorphism.

DEFINITION 01.7   Let be a ring. Let be an -submodule of an -module . Then we may define the quotient by

where the equivalence relation is defined as follows:

It may be shown that the quotient so defined is actually an -module and that the natural projection

is an -module homomorphism.

DEFINITION 01.8   Let be an -module homomorphism between -modules. Then we define its kernel as follows.

The kernel and the image of an -module homomorphism are -modules.

THEOREM 01.9   Let be an -module homomorphism between -modules. Then

DEFINITION 01.10   Let be a ring. An `sequence''

is said to be an exact sequence of -modules if the following conditions are satisfied
Exact1.
are -modules.
Exact2.
are -module homomorphisms.
Exact3.
.

For any -submodule of an -module , we have the following exact sequence.

EXERCISE 01.1   Compute the following modules.
1. .
2. .
3. .

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2010-04-15