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# zeta functions

Yoshifumi Tsuchimoto

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

is a prime ideal of $A$

For any subset of we define

Then we may topologize in such a way that the closed sets are sets of the form for some . Namely,

closed

We refer to the topology as the Zariski topology.

EXERCISE 10.1   Prove that Zariski topology is indeed a topolgy. That means, the collection satisfies the axiom of closed sets.

EXERCISE 10.2   Let be a ring. Then:
1. Show that for any , is an open set of .
2. Show that given a point of and an open set which contains , we may always find an element such that . (In other words, forms an open base of the Zariski topology.

LEMMA 10.2   For any ring , the following facts holds.
1. For any subset of , we have

2. For any subset of , let us denote by the ideal of generated by . then we have

PROPOSITION 10.3   For any ring homomorphism , we have a map

It is continuous with respect to the Zariski topology.

PROPOSITION 10.4   For any ring , the following statements hold.
1. For any ideal of , let us denote by the canonical projection. Then gives a homeomorphism between and .
2. For any element of , let us denote by be the canonical map. Then gives a homeomorphism between and .

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

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

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

such that

PROPOSITION 10.7   Let be a ring. Then;
1. For any ideal of , we have .
2. For two ideals , of , holds if and only if .
3. For an ideal of , is irreducible if and only if is a prime ideal.

It is knwon that 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 be a -graded ring. We put .

We define

It is known that carries a ringed space strucure on it and that it is a scheme.

DEFINITION 10.9   Let be a ring. Let be an ideal of . The scheme associated to the graded ring is called the blowing up of with respect to .

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2015-06-29