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## localization of a commutative ring

.

DEFINITION 1.6   Let be an element of a commutative ring . Then we define the localization of with respect to as a ring defined by

where is a indeterminate.

In the ring , the residue class of plays the role of the inverse of . Therefore, we may write instead of if there is no confusion.

One may define localization in much more general situation. The reader is advised to read standard books on commutative algebras.

LEMMA 1.7   Let be an element of a commutative ring . Then there is a canonically defined homeomorphism between and . (It is usual to identify these two via this homeomorphism.)

PROOF.. Let be the natural homomorphism. We have already seen that we have a continuous map

We need to show that it is injective, and that it gives a homeomorphism between and .

Let us do this by considering representations.

1. corresponds to a representation .

2. corresponds to a representation .

3. corresponds to a restriction map .

Now, for any , extends to if and only if the image of is invertible, that means, . In such a case, the extension is unique. (We recall the fact that the inverse of an element of a field is unique.)

It is easy to prove that is a homeomorphism.

Let be a ring. Let . It is important to note that each element of is written as a fraction''

One may introduce as a set of such formal fractions which satisfy ordinary computation rules. In precise, we have the following Lemma.

LEMMA 1.8   Let be a ring, be its element. Let us consider the following set

We introduce on the following equivalence law.

Then we may obtain a ring structure on by introducing the following sum and product.

where we have denoted by the equivalence class of .

COROLLARY 1.9   Let be a ring, be its element. Then we have if and only if is nilpotent.

Likewise, for any -module , we may define as a set of formal fractions

which satisfy certain computation rules.

Subsections

Next: Existence of a point Up: (Usual) affine schemes Previous: ring homomorphism and spectrum
2007-12-11