COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

04. Dimension Recall that for any commutative ring A, we define its (Krull) dimen-

sion dim(A) as the Krull dimension of Spec(A).

Definition 4.1. Let A be a commutative ring. For any A-module M, we define its dimension as

dim(M) = dim(A/ Ann(M)).

where Ann(M) = {x ∈ A; x.M = 0}.

Definition 4.2. For any A-module M of a ring A, we define its length l(M) as the supremum of the lenths of descending chains of submodules of M.

Example 4.3. Let k be a commutative field. A k-module V is a vector space over k. The lengh of V is then equal to the dimension of V as a k-vector space. In what follows, we denote it as rank_{k}(V).

Exercise 4.1. Compute the lenth of a ℤ-module ℤ/9ℤ ⊕ ℤ/6ℤ.

Definition 4.4. Let A = (A, 𝔪) be a local ring. Let M be an Amodule. we define δ(A) to be the smallest value of n such that there exist x_{1}, x_{2}, …, x_{n} ∈ 𝔪 for which l(M/x_{1}M + ⋯ + x_{n}M) < ∞.

Let us recall the definition of Noetherian ring.

Definition 4.5. A ring is called Noetherian if any asscending chain

I_{1} ⊂ I_{2} ⊂ I_{3} ⊂ …

stops after a finite number of steps. (That means, There exists a number N such that I_{N} = I_{N+1} = I_{N+2} = . . ..)

Proposition 4.6. A commutative ring A is Noetherina if and only if its ideals are always finitely generated.

Definition 4.7. Let (A, 𝔪) be a be a Noetherian local ring. Let I be an ideal of A. We say that I is an ideal of definition if there exists an integer j such that I ⊃ 𝔪^{j}. Then for any finite A-module M, we define

χ_{M}^{I}(n) = l(M/I^{n+1}M).

It is known that there exists a polynomal p_{M}^{I} such that χ_{M}^{I}(n) = p_{M}^{I}(n) for n >> 0. We define d(M) as the degree of the polynomial p. d does not depend on the choice of the ideal I of definition.

Proposition 4.8. For any Noetherian local ring A and for any finite A-module M, we have

d(M) = dim(M) = δ(M).

Definition 4.9. For any local ring A, we define its embedding dimension as rank_{A/𝔪} 𝔪/𝔪^{2}.

Definition 4.10. A Noetherian local ring is said to be regular if its embedding dimension is equal to the dimension of A.