COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

06. Integral elements, normal closure

Definition 6.1. Let S be a ring which contains a subring R. An element s of S is said to be integral over R if it is a root of a monic polynomial in R[X].

Lemma 6.2. Let S be a ring which contains a subring R. For any element s of S, the following conditions are equivalent:

(1) s is integral over R.

(2) R[s] is a finite R-module.

(3) There exists a subring S_{1} of S which contains R[s] as a subset such that S_{1} is a finite module over R.

Proposition 6.3. Let S be a ring which contains a subring R. Then the set of all elements of S which are integral over R is a subring of S. (We call it the integral closure of R in S.)

Example 6.4. Each element of ℂ which is integral over ℤ is said to be an algebraic integer. The set of algebraic integers forms a subring of ℂ.

Definition 6.5. Let R be an integral domain. Let us denote its field of quotients by Q(R). The integral closure of R in Q(R) is called the normalization of R. R is called normal if it is equal to its normalizaiton.

By using the Gauss’s lemma, we see that every PID is normal.

Normalizations are useful to reduce singularities.

Example 6.6. Let us put

R = ℂ[X, Y]/(Y^{2} − X^{2}(X + 1))

and denote the class of X, Y in R by x, y respectively. R is not normal. Indeed, z = y/x satisfies a monic equation

z^{2} − (x + 1) = 0.

Thus the normalization R̄ of R contains the element z. Now, let us note that equation

x = z^{2} − 1, y = zx = z(z^{2} − 1)

holds so that R[z] = ℂ[z] holds. Since ℂ[z] is normal, we see that R̄ = R[z] = ℂ[z]. Note that Ω_{R/ℂ}^{1} is not locally free whereas Ω_{R̄/ℂ}^{1} is free.

Example 6.7. Let us consider a ring R = ℤ[X]/(u(X)) where u is a monic element in ℤ[X]. Let us denote by α the residue class of X in R.

Ω_{R/ℤ}^{1} = RdX/u′(X)RdX ≅ R/(u′(α)).

YOSHIFUMI TSUCHIMOTO

Exercise 6.1. The normalization of R = ℤ[\sqrt{−3}] is equal to R̄ = ℤ[\sqrt{−3}]. Compute Ω_{R/ℤ}^{1} and Ω_{R̄/ℤ}^{1}.

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Theorem 6.8 (Matsumura, Corollary of Theorem 23.9). Let A, B are Noetherian local ring Let φ : A → B is be a local homormophism. If φ is flat morphism, and if B is normal, then A is also normal.

In other words, a normalization of a ring A can never be flat (unless the trivial case where A itself is normal).

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Definition 6.9. For any commutative ring A, we define its spectrum as

Spec(A) = {𝔭 ⊂ A; 𝔭 is a prime ideal of A.

For any subset S of A we define

V (S) = V_{Spec A}(S) = {𝔭 ∈ Spec A; 𝔭 ⊂ S}

Then we may topologize Spec(A) in such a way that the closed sets are sets of the form V (S) for some S. Namely,

F : closed ⟺ ∃S ⊂ A(F = V (S))

We refer to the topology as the Zariski topology.

Lemma 6.10. For any ring A, the following facts holds.

(1) For any subset S of A, we have

V (S) = ⋂_{s∈S} V ({s}).

(2) For any subset S of A, let us denote by ⟨S⟩ the ideal of A generated by S. then we have

V (S) = V (⟨S⟩)

Proposition 6.11. For any ring homomorphism φ : A → B, we have a map

Spec(φ) : Spec(B) ∋ 𝔭 ↦ φ^{−1}(𝔭) ∈ Spec(A).

It is continuous with respect to the Zariski topology.

Proposition 6.12. For any ring A, the following statements hold.

(1) For any ideal I of A, let us denote by π_{I} : A → A/I the canonical projection. Then Spec(π_{I}) gives a homeomorphism between Spec(A/I) and V_{Spec A}(I).

(2) For any element s of A, let us denote by ι_{s} : A → A[s^{−1}] be the canonical map. Then Spec(ι_{s}) gives a homeomorphism between Spec(A[s^{−1}]) and ∁V_{Spec_{A}}({s}).

Proposition 6.13. Let A, B be a ring. Let φ : A → B be a ring homomoprhism. We regard B as an A module via φ. If B is a finite A-module, then

Spec(φ) : Spec(B) → Spec(A)

is a closed map with respect to the Zariski topology.