Commutative algebra

Yoshifumi Tsuchimoto

PROPOSITION 10.1   Let be a local ring with the residue field . Let be -modules. We assume is a finite module. Then an -module homomorphism is surjective if and only if is surjective.

DEFINITION 10.2   Let

be a resolution of . We say that it is minimal free resolution if the following conditions are satisfied.
1. each is a finite free -module.
2. .
3. is an isomorphism.

LEMMA 10.3   A finite module over a Noetherian local ring has a minimal free resolution.

DEFINITION 10.4   Let be an -module over a ring . We define the projective dimension of ( ) to be the minimal length of the projective resolution of .

PROPOSITION 10.5   Let be a local ring with the residue field . Then:
1. If we have a minimal resolution of an -module , then we have .
2. .

DEFINITION 10.6   We define the global dimension of a ring by

LEMMA 10.7   Let be a local ring. Let be its residue field. Then we have

THEOREM 10.8 (Serre)   Let be a Noetherian local ring. Then:

is regular

THEOREM 10.9 (Serre)   A localization of a regular local ring at a prime ideal is also a regular local ring.

DEFINITION 10.10   A Noetherian ring is said to be regular if its localilzation at every prime is a regular local ring.