Commutative algebra

Yoshifumi Tsuchimoto

Recall that for any commutative ring $A$ , we define its (Krull) dimension $\dim(A)$ as the Krull dimension of $\operatorname{Spec}(A)$ .

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

$$\dim(M)=\dim(A/\operatorname{Ann}(M)).$$

where $\operatorname{Ann}(M)=\{x \in A; x.M=0\}$ .

DEFINITION 04.2   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 04.3   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 $\operatorname{rank}_k(V)$ .

EXERCISE 04.1   Compute the lenth of a $\mathbb{Z}$ -module $\mathbb{Z}/9\mathbb{Z}\oplus \mathbb{Z}/6\mathbb{Z}$ .

DEFINITION 04.4   Let $A=(A,\mathfrak{m})$ be a local ring. Let $M$ be an $A$ -module. we define $\delta(A)$ to be the smallest value of $n$ such that there exist $x_1,x_2,\dots,x_n\in \mathfrak{m}$ for which $l(M/x_1 M + \dots +x_n M)<\infty$ .

Let us recall the definition of Noetherian ring.

DEFINITION 04.5   A ring is called Noetherian if any asscending chain

$$I_1\subset I_2 \subset I_3\subset \dots$$

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

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

DEFINITION 04.7   Let $(A,\mathfrak{m})$ 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 $I\supset \mathfrak{m}^j$ . Then for any finite $A$ -module $M$ , we define

$$\chi_M^I(n)=l(M/I^{n+1}M).$$

It is known that there exists a polynomal $p_M^I$ such that $\chi_M^I(n)=p_M^I(n)$ 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 04.8   For any Noetherian local ring $A$ and for any finite $A$ -module $M$ , we have

$$d(M)=\dim(M)=\delta(M).$$

DEFINITION 04.9   For any local ring $A$ , we define its embedding dimension as $\operatorname{rank}_{A/\mathfrak{m}} \mathfrak{m} /\mathfrak{m}^2$ .

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