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Commutative algebra

Yoshifumi Tsuchimoto

\fbox{04. Dimension}

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

$\displaystyle \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

$\displaystyle 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

$\displaystyle \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

$\displaystyle 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$ .

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