Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 06.1   Let be a ring which contains a subring . An element of is said to be integral over if it is a root of a monic polynomial in .

LEMMA 06.2   Let be a ring which contains a subring . For any element of , the following conditions are equivalent:
1. is integral over .
2. is a finite -module.
3. There exists a subring of which contains as a subset such that is a finite module over .

PROPOSITION 06.3   Let be a ring which contains a subring . Then the set of all elements of which are integral over is a subring of . (We call it the integral closure of in .)

EXAMPLE 06.4   Each element of which is integral over is said to be an algebraic integer. The set of algebraic integers forms a subring of .

DEFINITION 06.5   Let be an integral domain. Let us denote its field of quotients by . The integral closure of in is called the normalization of . 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

and denote the class of in by respectively. is not normal. Indeed, satisfies a monic equation

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

holds so that holds. Since is normal, we see that . Note that is not locally free whereas is free.

EXAMPLE 06.7   Let us consider a ring where is a monic element in . Let us denote by the residue class of in .

EXERCISE 06.1   The normalization of is equal to . Compute and .

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THEOREM 06.8 (Matsumura, Corollary of Theorem 23.9)   Let are Noetherian local ring Let is be a local homormophism. If is flat morphism, and if is normal, then is also normal.

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

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DEFINITION 06.9   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.

LEMMA 06.10   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 06.11   For any ring homomorphism , we have a map

It is continuous with respect to the Zariski topology.

PROPOSITION 06.12   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 .

PROPOSITION 06.13   Let be a ring. Let be a ring homomoprhism. We regard as an module via If is a finite -module, then

is a closed map with respect to the Zariski topology.