zeta functions

Yoshifumi Tsuchimoto

DEFINITION 10.1   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

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

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.

EXERCISE 10.1   Prove that Zariski topology is indeed a topolgy. That means, the collection $\{V(S)\}$ satisfies the axiom of closed sets.

EXERCISE 10.2   Let $A$ be a ring. Then:

1. Show that for any $f\in A$ , $D(f)=\{\mathfrak{p}\in \operatorname{Spec}(A); f \notin \mathfrak{p}\}$ is an open set of $\operatorname{Spec}(A)$ .
2. Show that given a point $\mathfrak{p}$ of $\operatorname{Spec}(A)$ and an open set $U$ which contains $\mathfrak{p}$ , we may always find an element $f\in A$ such that $\mathfrak{p}\in D(f)\subset U$ . (In other words, $\{D(f)\}$ forms an open base of the Zariski topology.

LEMMA 10.2   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 10.3   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 10.4   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 $\complement V_{\operatorname{Spec}_A}(\{s\})$ .

DEFINITION 10.5   Let $X$ be a topological space. A closed set $F$ of $X$ is said to be reducible if there exist closed sets $F_1$ and $F_2$ such that

$$F=F_1 \cup F_2,\quad F_1\neq F, F_2 \neq F$$

holds. $F$ is said to be irreducible if it is not reducible.

DEFINITION 10.6   Let $I$ be an ideal of a ring $A$ . Then we define its radical to be

$$\sqrt{I}=\{ x \in A; \exists N\in \mathbb{Z}_{>0}$$    such that $$x^N \in I\}.$$

PROPOSITION 10.7   Let $A$ be a ring. Then;

1. For any ideal $I$ of $A$ , we have $V(I)=V(\sqrt{I})$ .
2. For two ideals $I$ , $J$ of $A$ , $V(I)=V(J)$ holds if and only if $\sqrt{I}=\sqrt{J}$ .
3. For an ideal $I$ of $A$ , $V(I)$ is irreducible if and only if $\sqrt{I}$ is a prime ideal.

It is knwon that $\operatorname{Spec}A$ has a structure of ``locally ringed space''. A locally ringed space which locally lookes like an affine spectrum of a ring is called a scheme.

DEFINITION 10.8   Let $S=\bigoplus_{n \in \mathbb{N}} S_n$ be a $\mathbb{N}$ -graded ring. We put $S_{+}=\bigoplus_ {n>0} S_n$ .

We define

$$\operatorname{Proj}(S)=\{\mathfrak{p}\subset S; \text{$\math... ...geneous prime ideal of S, $\mathfrak{p}\not \supset S_{+}$}\}.$$

It is known that $\operatorname{Proj}(S)$ carries a ringed space strucure on it and that it is a scheme.

DEFINITION 10.9   Let $R$ be a ring. Let $I$ be an ideal of $R$ . The scheme $\tilde X=\operatorname{Proj}(S)$ associated to the graded ring $S=\bigoplus_{n\in \mathbb{N}} I^n$ is called the blowing up of $X$ with respect to $I$ .