# Commutative algebra

Yoshifumi Tsuchimoto

Recall that for any commutative ring , we define its (Krull) dimension as the Krull dimension of .

DEFINITION 04.1   Let be a commutative ring. For any -module , we define its dimension as

where .

DEFINITION 04.2   For any -module of a ring , we define its length as the supremum of the lenths of descending chains of submodules of .

EXAMPLE 04.3   Let be a commutative field. A -module is a vector space over . The lengh of is then equal to the dimension of as a -vector space. In what follows, we denote it as .

EXERCISE 04.1   Compute the lenth of a -module .

DEFINITION 04.4   Let be a local ring. Let be an -module. we define to be the smallest value of such that there exist for which .

Let us recall the definition of Noetherian ring.

DEFINITION 04.5   A ring is called Noetherian if any asscending chain

stops after a finite number of steps. (That means, There exists a number such that .)

PROPOSITION 04.6   A commutative ring is Noetherina if and only if its ideals are always finitely generated.

DEFINITION 04.7   Let be a be a Noetherian local ring. Let be an ideal of . We say that is an ideal of definition if there exists an integer such that . Then for any finite -module , we define

It is known that there exists a polynomal such that for . We define as the degree of the polynomial . does not depend on the choice of the ideal of definition.

PROPOSITION 04.8   For any Noetherian local ring and for any finite -module , we have

DEFINITION 04.9   For any local ring , we define its embedding dimension as .

DEFINITION 04.10   A Noetherian local ring is said to be regular if its embedding dimension is equal to the dimension of .