COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

02. Localization

Definition 2.1. Let A be a commutative ring. Let S be its subset. We say that S is multiplicative if

(1) 1 ∈ S

(2) x, y ∈ S ⟹ xy ∈ S holds.

Definition 2.2. Let S be a multiplicative subset of a commutative ring A. Then we define A[S^{−1}] as

A[{X_{s}; s ∈ S}]/({sX_{s} − 1; s ∈ S})

where in the above notation X_{s} is a indeterminate prepared for each element s ∈ S.) We denote by ι_{S} a canonical map A → A[S^{−1}].

Lemma 2.3. Let S be a multiplicative subset of a commutative ring A. Then the ring B = A[S^{−1}] is characterized by the following property:

Let C be a ring, φ : A → C be a ring homomorphism such that φ(s) is invertible in C for any s ∈ S. Then there exists a unique ring homomorphism ψ = ϕ[S^{−1}] : B → C such that

φ = ψ ∘ ι_{S}

holds.

Corollary 2.4. Let S be a multiplicative subset of a commutative ring A. Let I be an ideal of A given by

I = {x ∈ I; ∃s ∈ S such that sx = 0}

Then I is an ideal of A. Let us put Ā = A/I, π : A → Ā the canonical projection. Then:

(1) S̄ = π(S) is multiplicatively closed.

(2) We have

A[S^{−1}] ≅ Ā[S̄^{−1}]

(3) ιS̄ : Ā → Ā[S̄^{−1}] is injective.

There is another description of A[S^{−1}]. Namely, We consider an equivalence relateion ∼_{S} on a set S × A by

(s_{1}, a_{1}) ∼_{S} (s_{2}, a_{2}) ⟺ t(s_{1}a_{2} − s_{2}a_{1}) = 0(∃t ∈ S)

We call the quotient space space S × A/ ∼_{S} as S^{−1}A. The equivalence class of (s, a) ∈ S × A in S^{−1}A is denoted by s^{−1}a. Then it is easy to introduce a ring structure of S^{−1}A and see that S^{−1}A actually satisfies the universal property of A[S^{−1}]. We thus have a canonical isomorphism S^{−1}A ≅ A[S^{−1}].

Example 2.5. A_{f} = A[S^{−1}] for S = {1, f, f^{2}, f^{3}, f^{4}, … }. The total ring of quotients Q(A) is defined as A[S^{−1}] for

S = {x ∈ A; x is not a zero divisor of A}.

When A is an integral domain, then Q(A) is the field of quotients of A.

Definition 2.6. 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 2.7. Let S be a multiplicative subset of a commutative ring A. Let M be an A-module we may define S^{−1}M as

{(m/s); m ∈ M, s ∈ S}/ ∼

where the equivalence relation ∼ is defined by

(m_{1}/s_{1}) ∼ (m_{2}/s_{2}) ⟺ t(m_{1}s_{2} − m_{2}s_{1}) = 0 (∃t ∈ S).

We may introduce a S^{−1}A-module structure on S^{−1}M in an obvious manner.

S^{−1}M thus constructed satisfies an universality condition which the reader may easily guess.

By a universality argument, we may easily see the following proposition.

Proposition 2.8. Let A be a commutative ring. Let S be a multiplicative subet of A. Let M be an A-module. Then we have an isomorphism

S^{−1}A ⊗_{A} M ≅ S^{−1}M

of S^{−1}A-modules.

Proposition 2.9. Let A be a commutative ring. Let S be a multiplicative subet of A. Then the natural homomorphism A → S^{−1}A is flat.

2.1. local rings.

Definition 2.10. A commutative ring A is said to be a local ring if it has only one maximal ideal.

Example 2.11. 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 2.12. (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 2.13. 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 2.14. 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 2.15 (of NOT being a local homomorphism).

ℤ_{(p)} → ℚ

is not a local homomorphism.

In the argument above, we have used the following lemma.

Lemma 2.16 (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.

We prove here another consequence of the lemma.

Proposition 2.17. 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.