RESOLUTIONS OF SINGULARITIES.

YOSHIFUMI TSUCHIMOTO

05. supplement

Proposition 5.1. For any ring A, the map ℙ^{n}(A) → Spec A is proper.

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

Corollary 5.2. For any ring A and for any ℕ-graded ring S = ⊕_{n}S_{n} which is generated by a finite subset of S_{1} over the ring S_{0} ≅ A, the map Proj(S) → Spec A is proper.

Proposition 5.3. Let I be an ideal of a ring A generated by elements f_{0}, … f_{s}. Assume that each of the elements f_{0}, f_{1}, …, f_{s} is not a zero divisor in A. Then:

(1) ⊕_{n∈ℕ}I^{n} ⊂ ⊕_{n∈ℕ}A is isomorphic to A[I · T] ⊂ A[T] for an indeterminate T.

(2)

Proj(⊕A[IṪ]) = Proj(⊕A[f_{0}T, f_{1}T, …, f_{s}T]) = ∪_{i=0}^{s}D_{+}(f_{i}T)

(3) D_{+}(f_{i}T) ≅ Spec A[f_{0}/f_{i}, f_{1}/f_{i}, …, f_{s}/f_{i}].

Definition 5.4. Let A be a commutative ring. Let 𝔭 be its prime ideal. Then we define the localization of A with respect to 𝔭 by

A_{𝔭} = A[(A \ 𝔭)^{−1}]

Definition 5.5. A commutative ring A is said to be a local ring if it has only one maximal ideal.

Example 5.6. We give examples of local rings here.

• Any field is a local ring.

• For any commutative ring A and for any prime ideal 𝔭 ∈ Spec(A), the localization A_{𝔭} is a local ring with the maximal ideal 𝔭A_{𝔭}.

Lemma 5.7. (1) Let A be a local ring. Then the maximal ideal

of A coincides with A \ A^{×}.

(2) A commutative ring A is a local ring if and only if the set A\A^{×} of non-units of A forms an ideal of A.

Proof. (1) Assume A is a local ring with the maximal ideal 𝔪. Then for any element f ∈ A \ A^{×}, an ideal I = fA + 𝔪 is an ideal of A. By Zorn’s lemma, we know that I is contained in a maximal ideal of A. From the assumption, the maximal ideal should be 𝔪. Therefore, we have

fA ⊂ 𝔪

which shows that

A \ A^{×} ⊂ 𝔪.

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

A \ A^{×} = 𝔪.

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

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Corollary 5.8. Let A be a commutative ring. Let 𝔭 its prime ideal. Then A_{𝔭} is a local ring with the only maximal ideal 𝔭A_{𝔭}.

Definition 5.9. Let A, B be local rings with maximal ideals 𝔪_{A}, 𝔪_{B} respectively. A local homomorphism φ : A → B is a homomorphism which preserves maximal ideals. That means, a homomorphism φ is said to be loc al if

φ^{−1}(𝔪_{B}) = 𝔪_{A}

Example 5.10 (of NOT being a local homomorphism).

ℤ_{(p)} → ℚ

is not a local homomorphism.

Lemma 5.11 (Zorn’s lemma). Let 𝒮 be a partially ordered set. Assume that every totally ordered subset of 𝒮 has an upper bound in 𝒮. Then 𝒮 has at least one maximal element.

Proposition 5.12. Let A be a commutative ring. let I be an ideal of A such that A ≠ I. Then there exists a maximal ideal 𝔪 of A which contains I.

Theorem 5.13 (Nakayama’s lemma, or NAK). Let A be a commutative ring. Let M be an A-module. We assume that M is finitely generated (as a module) over A. That means, there exists a finite set of elements {m_{i}}_{i=1}^{t} such that

M = ∑_{i=1}^{t} Am_{i}

holds. If an ideal I of A satisfies

IM = M (that is, M/IM = 0),

then there exists an element c ∈ I such that

cm = m (∀m ∈ M)

holds. If furthermore I is contained in ∩_{𝔪∈Spm(A)}𝔪 (the Jacobson radical of A), then we have M = 0.

Proof. Since IM = M, there exists elements b_{il} ∈ I such that

a_{i} = ∑_{l=1}^{t} b_{il}a_{l} (1 ≤ i ≤ t)

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

v = Bv

with v =^{t} (m_{1}, …, m_{n}), B = (b_{ij}) ∈ M_{t}(I). Using the unit matrix 1_{t} ∈ M_{t}(A) one may also write :

(1_{t} − B)v = 0.

Now let R be the adjugate matrix of 1_{t} − B. In other words, it is a matrix which satisfies

R(1_{t} − B) = (1_{t} − B)R = (det(1_{t} − B))1_{t}.

Then we have

det(1_{t} − B) · v = R(1_{t} − B)v = 0.

On the other hand, since 1_{t} − B = 1_{t} modulo I, we have det(1_{t} − B) = 1 − c for some c ∈ I. This c clearly satisfies

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

Theorem 5.14. (Matsumura“commutative ring theory” Theorem14.2) Let (R, 𝔪) be a regular local ring. then the following two statements are equivalent:

(1) The images in 𝔪/𝔪^{2} of x_{1}, …, x_{i} are linearly independent over R/𝔪.

(2) R/(x_{1}, …, x_{i}) is an (n − i)-dimensional regular local ring.