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# 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

.

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2012-05-11