Resolutions of singularities.

Yoshifumi Tsuchimoto

PROPOSITION 05.1   For any ring , the map is proper.

PROOF.. See http://amathew.wordpress.com/2010/10/23/a-projective-morphism-is-proper/.

COROLLARY 05.2   For any ring and for any -graded ring which is generated by a finite subset of over the ring , the map is proper.

PROPOSITION 05.3

Let be an ideal of a ring generated by elements . Assume that each of the elements is not a zero divisor in . Then:

1. is isomorphic to for an indeterminate .
2. .

DEFINITION 05.4   Let be a commutative ring. Let be its prime ideal. Then we define the localization of with respect to by

DEFINITION 05.5   A commutative ring is said to be a local ring if it has only one maximal ideal.

EXAMPLE 05.6   We give examples of local rings here.
• Any field is a local ring.
• For any commutative ring and for any prime ideal , the localization is a local ring with the maximal ideal .

LEMMA 05.7
1. Let be a local ring. Then the maximal ideal of coincides with .
2. A commutative ring is a local ring if and only if the set of non-units of forms an ideal of .

PROOF.. (1) Assume is a local ring with the maximal ideal . Then for any element , an ideal is an ideal of . By Zorn's lemma, we know that is contained in a maximal ideal of . From the assumption, the maximal ideal should be . Therefore, we have

which shows that

The converse inclusion being obvious (why?), we have

(2) The only if'' part is an easy corollary of (1). The if'' part is also easy.

COROLLARY 05.8   Let be a commutative ring. Let its prime ideal. Then is a local ring with the only maximal ideal .

DEFINITION 05.9   Let be local rings with maximal ideals respectively. A local homomorphism is a homomorphism which preserves maximal ideals. That means, a homomorphism is said to be loc al if

EXAMPLE 05.10 (of NOT being a local homomorphism)

is not a local homomorphism.

LEMMA 05.11 (Zorn's lemma)   Let be a partially ordered set. Assume that every totally ordered subset of has an upper bound in . Then has at least one maximal element.

PROPOSITION 05.12   Let be a commutative ring. let be an ideal of such that . Then there exists a maximal ideal of which contains .

THEOREM 05.13 (Nakayama's lemma, or NAK)   Let be a commutative ring. Let be an -module. We assume that is finitely generated (as a module) over . That means, there exists a finite set of elements such that

holds. If an ideal of satisfies

that is,

then there exists an element such that

holds. If furthermore is contained in (the Jacobson radical of ), then we have .

PROOF.. Since , there exists elements such that

holds. In a matrix notation, this may be rewritten as

with , . Using the unit matrix one may also write :

Now let be the adjugate matrix of . In other words, it is a matrix which satisfies

Then we have

On the other hand, since modulo , we have for some . This clearly satisfies

We need a criterion for regularity. Instead of developing the vast theory of regular rings, we site here the following theorem:

THEOREM 05.14 (Matsumuracommutative ring theory'' Theorem14.2)   Let be a regular local ring. then the following two statements are equivalent:
1. The images in of are linearly independent over .
2. si an -dimensional regular local ring.