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zeta functions

Yoshifumi Tsuchimoto

\fbox{ Spec and Proj.}

DEFINITION 10.1   For any commutative ring $ A$ , we define its spectrum as

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

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

$\displaystyle 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,

$\displaystyle F:$closed$\displaystyle \ \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

    $\displaystyle 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

    $\displaystyle V(S)=V(\langle S \rangle)
$

PROPOSITION 10.3   For any ring homomorphism $ \varphi: A \to B$ , we have a map

$\displaystyle \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

% latex2html id marker 2260
$\displaystyle 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

% latex2html id marker 2273
$\displaystyle \sqrt{I}=\{ x \in A; \exists N\in \mathbb{Z}_{>0}$    such that $\displaystyle x^N \in I\}.
$

PROPOSITION 10.7   Let $ A$ be a ring. Then;
  1. For any ideal $ I$ of $ A$ , we have % latex2html id marker 2287
$ V(I)=V(\sqrt{I})$ .
  2. For two ideals $ I$ , $ J$ of $ A$ , $ V(I)=V(J)$ holds if and only if % latex2html id marker 2297
$ \sqrt{I}=\sqrt{J}$ .
  3. For an ideal $ I$ of $ A$ , $ V(I)$ is irreducible if and only if % latex2html id marker 2305
$ \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

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

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$ .

\fbox{sheaves}




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2015-06-29