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Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 08.1   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).

DEFINITION 08.2   Let be a ring.
1. An -module is said to be injective if it satisfies the following condition: For any -module homomorphism and for any monic -module homomorphism , extends'' to an -module homomorphism .

2. A -module is said to be projective if it satisfies the following condition: For any -module homomorphism and for any epic -module homomorphism , lifts'' to a morphism .

EXERCISE 08.1   Let be a ring. Let

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

is exact.

LEMMA 08.3   Let be a (unital associative but not necessarily commutative) ring. Then for any -module , the following conditions are equivalent.
1. is a direct summand of free modules.
2. is projective

COROLLARY 08.4   For any ring , the category of -modules have enough projectives. That means, for any object , there exists a projective object and a surjective morphism .

DEFINITION 08.5   Let be a commutative ring. We assume is a domain (that means, has no zero-divisors except for 0 .)

An -module is said to be divisible if for any , the multplication map

is surjective.

DEFINITION 08.6   Let be a commutative ring. We assume is a domain (that means, has no zero-divisors except for 0 .)

An -module is said to be divisible if for any , the multplication map

is epic.

LEMMA 08.7   Let be a (commutative) principal ideal domain (PID). Then an -module is injective if and only if it is divisible.

PROPOSITION 08.8   For any (not necessarily commutative) ring , the category of -modules has enough injectives. That means, for any object , there exists an injective object and an monic morphism .

A bit of category theory:

DEFINITION 08.9   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 08.10   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 08.11   A (covariant) functor from a category to a category consists of the following data:
1. An function which assigns to each object of an object of .
2. An function which assigns to each morphism of an morphism of .
The data must satisfy the following axioms:
functor-1.
for any object of .
functor-2.
for any composable morphisms of .

By employing the following axiom instead of the axiom (functor-2) above, we obtain a definition of a contravariant functor:

(functor- ) for any composable morphisms

DEFINITION 08.12   Let be a functor between additive categories. We call additive if for any objects in ,

DEFINITION 08.13   Let be an additive functor from an abelian category to .
1. is said to be left exact (respectively, right exact ) if for any exact sequence

the corresponding map

(respectively,

is exact
2. is said to be exact if it is both left exact and right exact.

LEMMA 08.14   Let be a (unital associative but not necessarily commutative) ring. Then for any -module , the following conditions are equivalent.
1. is a direct summand of free modules.
2. is projective

COROLLARY 08.15   For any ring , the category of -modules have enough projectives. That means, for any object , there exists a projective object and a surjective morphism .

DEFINITION 08.16   Let be a commutative ring. We assume is a domain (that means, has no zero-divisors except for 0 .)

An -module is said to be divisible if for any , the multplication map

is surjective.

DEFINITION 08.17   Let be a commutative ring. We assume is a domain (that means, has no zero-divisors except for 0 .)

An -module is said to be divisible if for any , the multplication map

is epic.

DEFINITION 08.18   Let , be complexes of objects of an additive category .
1. A morphism of complex is a family

of morphisms in such that commutes with . That means,

holds.
2. A homotopy between two morphisms of complexes is a family of morphisms

such that holds.

LEMMA 08.19   Let be an abelian category that has enough injectives. Then:
1. For any object in , there exists an injective resolution of . That means, there exists an complex and a morphism such that

2. For any morphism of , and for any injective resolutions , of and (respectively), There exists a morphism of complexes which commutes with . Forthermore, if there are two such morphisms and , then the two are homotopic.

DEFINITION 08.20   Let be an abelian category which has enough injectives. Let be a left exact functor to an abelian category. Then for any object of we take an injective resolution of and define

and call it the derived functor of .

LEMMA 08.21   The derived functor is indeed a functor.

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2012-06-28