Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 06.1   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 06.2   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 06.3   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 06.4   Each element of $\mathbb{C}$ which is integral over $\mathbb{Z}$ is said to be an algebraic integer. The set of algebraic integers forms a subring of $\mathbb{C}$ .

DEFINITION 06.5   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 06.6   Let us put

$$R=\mathbb{C}[X,Y]/(Y^2 -X^2 (X+1))$$

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 $\bar R$ of $R$ contains the element $z$ . Now, let us note that equation

$$x= z^2 -1,\quad y=z x = z(z^2-1)$$

holds so that $R[z]= \mathbb{C}[z]$ holds. Since $\mathbb{C}[z]$ is normal, we see that $\bar R= R[z]= \mathbb{C}[z]$ . Note that $\Omega^1_{R/\mathbb{C}}$ is not locally free whereas $\Omega^1_{\bar R/\mathbb{C}}$ is free.

EXAMPLE 06.7   Let us consider a ring $R=\mathbb{Z}[X]/(u(X))$ where $u$ is a monic element in $\mathbb{Z}[X]$ . Let us denote by $\alpha$ the residue class of $X$ in $R$ .

$$\Omega^{1}_{R/\mathbb{Z}}= R d X / u'(X) R d X \cong R/(u'(\alpha)).$$

EXERCISE 06.1   The normalization of $R=\mathbb{Z}[\sqrt{-3}]$ is equal to $\bar R=\mathbb{Z}[\sqrt{-3}]$ . Compute $\Omega^1_{R/\mathbb{Z}}$ and $\Omega^1_{\bar R/\mathbb{Z}}$ .

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THEOREM 06.8 (Matsumura, Corollary of Theorem 23.9)   Let $A,B$ are Noetherian local ring Let $\varphi:A\to B$ is be a local homormophism. If $\varphi$ 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 06.9   For any commutative ring $A$ , we define its spectrum as

$$\operatorname{Spec}(A)=\{ \mathfrak{p}\subset A; \mathfrak{p}$$ is a prime ideal of $A$$$.$$

For any subset $S$ of $A$ we define

$$V(S)=V_{\operatorname{Spec}A} (S) =\{\mathfrak{p}\in \operatorname{Spec}A; \mathfrak{p}\subset S\}$$

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

$$F:$$closed$$\ \iff \ \exists S \subset A ( F=V(S) )$$

We refer to the topology as the Zariski topology.

LEMMA 06.10   For any ring $A$ , the following facts holds.

1. For any subset $S$ of $A$ , we have
   $$V(S)=\bigcap_{s \in S} V(\{s\}).$$

2. For any subset $S$ of $A$ , let us denote by $\langle S \rangle$ the ideal of $A$ generated by $S$ . then we have
   $$V(S)=V(\langle S \rangle)$$

PROPOSITION 06.11   For any ring homomorphism $\varphi:A\to B$ , we have a map

$$\operatorname{Spec}(\varphi): \operatorname{Spec}(B) \ni \mathfrak{p}\mapsto \varphi^{-1}(\mathfrak{p})\in \operatorname{Spec}(A).$$

It is continuous with respect to the Zariski topology.

PROPOSITION 06.12   For any ring $A$ , the following statements hold.

1. For any ideal $I$ of $A$ , let us denote by $\pi_I :A\to A/I$ the canonical projection. Then $\operatorname{Spec}(\pi_I)$ gives a homeomorphism between $\operatorname{Spec}(A/I)$ and $V_{\operatorname{Spec}A}(I)$ .
2. For any element $s$ of $A$ , let us denote by $\iota_s: A \to A[s^{-1}]$ be the canonical map. Then $\operatorname{Spec}(\iota_s)$ gives a homeomorphism between $\operatorname{Spec}(A[s^{-1}])$ and $\complement V_{\operatorname{Spec}_A}(\{s\})$ .

PROPOSITION 06.13   Let $A,B$ be a ring. Let $\varphi:A\to B$ be a ring homomoprhism. We regard $B$ as an $A$ module via $\varphi.$ If $B$ is a finite $A$ -module, then

$$\operatorname{Spec}(\varphi): \operatorname{Spec}(B)\to \operatorname{Spec}(A)$$

is a closed map with respect to the Zariski topology.