# Commutative algebra

Yoshifumi Tsuchimoto

LEMMA 06.1 (Artin-Rees)   Let be an ideal of a Noetherian ring . Let be an module with a submodule . Then there exists an integer such that

holds for all .

THEOREM 06.2 (Krull)   Let be a ring with an ideal . Let be a finite -module. We set . Then there exists such that and .

THEOREM 06.3 (the Krull intersection theorem)   Let be a Noetherian ring.
1. If is in the Jacobson radical of , then for any finite -module , we have . Furthermore, for any submodule of , we have .
2. If is an integral domain and is a proper ideal of , then we have .

PROPOSITION 06.4   Let be a local ring. The following conditions are equivalent:
1. (which is also equivalent to saying that or that ).
2. .
3. Any descending chain

of ideals of stops.

LEMMA 06.5   Let be a ring with an ideal . Let be an -module. then we may (of course) consider as an -module. The dimensions are irrelevant of whether we consider as an -module or as an -module.

LEMMA 06.6   Let be an exact sequence of finite -modules over a Noetherian local ring . Then:
1. .
2. For any ideal of definition of , The leading coefficient of coincides with that of .

THEOREM 06.7   Let be a -dimensional regular local ring with the residue field . Then