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.
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. .