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# Commutative algebra

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.
Ring-1.
Ring-2.
distributive law holds. Namely, we have

Ring-3.
The multiplcation is associative.
Ring-4.
has a multiplicative unit.

In this lectuer we are mainly interested in commutative rings, that means, rings on which the multiplication satisfies the commutativity law.

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   A map from a unital associative ring to another unital associative ring is said to be ring homomorphism if it satisfies the following conditions.
Ringhom-1.
Ringhom-2.
Ringhom-3.

DEFINITION 01.3   Let be a unital associative ring. An -module is an additive group with -action

which satisfies
Mod-1.
Mod-2.
Mod-3.
.
Mod-4.
.

EXAMPLE 01.4   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 01.5   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. Furthermore, when is a commutative ring, then it has a structure of an -module.

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   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 01.8   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 01.9   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 01.10   Let be an -module homomorphism between -modules. Then

DEFINITION 01.11   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. .

DEFINITION 01.12   Let be an associative unital (but not necessarily commutative) ring. Let be a right -module. Let be a left -module. For any ( -)module , an map

is called an -balanced biadditive map if
1.      .
2.      .
3.      .

PROPOSITION 01.13   Let be an associative unital (but not necessarily commutative) ring. Then for any right -module and for any left -module , there exists a ( -)module together with a -balanced map

which is universal amoung -balanced maps.

DEFINITION 01.14   We employ the assumption of the proposition above. By a standard argument on universal objects, we see that such object is unique up to a unique isomorphism. We call it the tensor product of and and denote it by

LEMMA 01.15   Let be an associative unital ring. Then:
1. .
2. For any left -module , the functor is a right exact functor. Namely, for any exact sequence

the sequence

is also exact.
3. For any right ideal of and for any -module , we have

In particular, if the ring is commutative, then for any ideals of , we have

EXERCISE 01.2   Compute and .

DEFINITION 01.16   A left -module is said to be flat if is an exact functor. Namely, for any exact sequence

of left -modules, the sequence

is also exact.

**The following two facts may give some intuitive idea of what flatness means.

THEOREM 01.17   If is a Noetherian ring and is a finitely-generated -module, then is flat over if and only if the associated sheaf on is locally free.

THEOREM 01.18   [1, Theorem 23.1+Theorem 15.1] Let be a regular local ring. Let be a Cohen-Macaulay local ring. Let be a local ring homomorphism.We set

for the fiber ring of over . Then an equality

holds if and only if is flat over .

**

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2011-04-14