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# Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 10.1   Let be a commutative ring. Let be its subset. We say that is multiplicative if
holds.

DEFINITION 10.2   Let be a multiplicative subset of a commutative ring . Then we define as

where in the above notation is a indeterminate prepared for each element .) We denote by a canonical map .

LEMMA 10.3   Let be a multiplicative subset of a commutative ring . Then the ring is characterized by the following property:

Let be a ring, be a ring homomorphism such that is invertible in for any . Then there exists a unique ring homomorphism such that

holds.

COROLLARY 10.4   Let be a multiplicative subset of a commutative ring . Let be an ideal of given by

such that

Then (1) is an ideal of . Let us put , the canonical projection. Then:

(2) is multiplicatively closed.

(3) We have

(4) is injective.

DEFINITION 10.5   Let be a multiplicative subset of a commutative ring . Let be an -module we may define as

where the equivalence relation is defined by

We may introduce a -module structure on in an obvious manner.

thus constructed satisfies an universality condition which the reader may easily guess.

LEMMA 10.6   Let be a commutative ring. Let be an -module. Then we have a canonical isomorphism of module

We may also localize categories, but we need to deal with non commutativity of composition. To simplify the situation we only deal with a localization with some nice properties as follows:

1. .
1. Let . Let , . Then there exist and morphisms , and such that the diagram

commutes.

In a simpler (but not rigorous) words, for each composable '', there exists such . Similarly, for each composable , there exists such that holds.

2. Let , . Then the following conditions are equivalent:
1. There exists and such that .
2. There exists and such that .
3. If and if then .

LEMMA 10.7   Let be a family of morphisms in which satisfies the properties above. Then one may construct a localization of with respect to . Furthermore, if is additive, then is also additive.

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2009-07-09