Commutative algebra

Yoshifumi Tsuchimoto

LEMMA 06.1 (Artin-Rees)   Let $I$ be an ideal of a Noetherian ring $A$ . Let $M$ be an $A$ module with a submodule $N$ . Then there exists an integer $c>0$ such that

$$I^n M \cap N = I^{n-c}(I^c M \cap N)$$

holds for all $n>c$ .

THEOREM 06.2 (Krull)   Let $A$ be a ring with an ideal $I$ . Let $M$ be a finite $A$ -module. We set $N=\cap_{k=1}^\infty I^k M$ . Then there exists $a\in A$ such that $a \equiv 1 \mod I$ and $a N=0$ .

THEOREM 06.3 (the Krull intersection theorem)   Let $A$ be a Noetherian ring.

1. If $I$ is in the Jacobson radical $\operatorname{rad} A$ of $A$ , then for any finite $A$ -module $M$ , we have $\cap_n I^n M=0$ . Furthermore, for any submodule $N$ of $M$ , we have $\cap_n I^n M \cap N=0$ .
2. If $A$ is an integral domain and $I$ is a proper ideal of $A$ , then we have $\cap_n I^n=0$ .

PROPOSITION 06.4   Let $A$ be a local ring. The following conditions are equivalent:

1. $l(A)< \infty$ (which is also equivalent to saying that $d(A)=0$ or that $\delta(A)=0$ ).
2. $\dim(A)=0$ .
3. Any descending chain
   $$I_0\supset I_1 \supset I_2 \supset \dots$$
   of ideals of $A$ stops.

LEMMA 06.5   Let $A$ be a ring with an ideal $I$ . Let $M$ be an $A/I$ -module. then we may (of course) consider $M$ as an $A$ -module. The dimensions $\dim(M),d(M),\delta(M)$ are irrelevant of whether we consider $M$ as an $A/I$ -module or as an $A$ -module.

LEMMA 06.6   Let $0\to M' \to M\to M''\to 0$ be an exact sequence of finite $A$ -modules over a Noetherian local ring $A$ . Then:

1. $d(M)=\max(d(M'),d(M''))$ .
2. For any ideal $I$ of definition of $A$ , The leading coefficient of $\chi_M^I-\chi_{M''}^I$ coincides with that of $\chi_{M'}^I$ .

THEOREM 06.7   Let $(A,\mathfrak{m})$ be a $d$ -dimensional regular local ring with the residue field $k=A/\mathfrak{m}$ . Then

$$\operatorname{gr}_{\mathfrak{m}}(A)\cong k[X_1,\dots, X_d].$$