Localization in a commutative algebra

A transition from a commutative ring $ A $ to the ring of fractions (cf. Fractions, ring of) $ A [ S ^ {-1} ] $, where $ S $ is a subset of $ A $. The ring $ A [ S ^ {-1} ] $ can be defined as the solution of the problem of a universal mapping from $ A $ into a ring under which all elements of $ S $ become invertible. However, there are explicit constructions for $ A [ S ^ {-1} ] $:

1) as the set of fractions of the form $ a / s $, where $ a \in A $ and $ s $ is a product of elements of $ S $( two fractions $ a / s $ and $ a ^ \prime / s ^ \prime $ are regarded as equivalent if and only if there is an $ s ^ {\prime\prime} $ that is a product of elements of $ S $ and is such that $ s ^ {\prime\prime} ( s a ^ \prime - s ^ \prime a ) = 0 $; fractions are added and multiplied by the usual rules);

2) as the quotient ring of the ring of polynomials $ A [ X _ {s} ] $, $ s \in S $, with respect to the ideal generated by the polynomials $ s X _ {s} - 1 $, $ s \in S $;

3) as the inductive limit of an inductive system of $ A $- modules $ ( A _ {i} , \phi _ {ij} ) $, where $ i $ runs through a naturally-ordered free commutative monoid $ N ^ {(S)} $. All the $ A _ {i} $ are isomorphic to $ A $, and the homomorphisms $ \phi _ {ij} : A _ {i} \rightarrow A _ {j} $ with $ j = i + n _ {1} s _ {1} + \dots + n _ {k} s _ {k} $ coincide with multiplication by $ s _ {1} ^ {n _ {1} } {} \dots s _ {k} ^ {n _ {k} } \in A $.

The ring $ A $ is canonically mapped into $ A [ S ^ {-1} ] $ and converts the latter into an $ A $-algebra. This mapping $ A \rightarrow A [ S ^ {-1} ] $ is injective if and only if $ S $ does not contain any divisor of zero in $ A $. On the other hand, if $ S $ contains a nilpotent element, then $ A [ S ^ {-1} ] = 0 $.

Without loss of generality the set $ S $ can be assumed to be closed with respect to products (such a set is known as multiplicative, or as a multiplicative system). In this case the ring $ A [ S ^ {-1} ] $ is also denoted by $ S ^ {-1} A $ or $ A _ {S} $. The most important examples of multiplicative systems are the following:

a) the set $ \{ s ^ {n} \} $ of all powers of an element of $ A $;

b) the set $ A \setminus \mathfrak P $, that is, the complement of a prime ideal $ \mathfrak P $. The corresponding ring of fractions is local and is denoted by $ A _ {\mathfrak P } $;

c) the set $ R $ of all non-divisors of zero in $ A $.

The ring $ R ^ {-1} A $ is called the complete ring of fractions of $ A $. If $ A $ is integral, then $ R ^ {-1} A = A _ {(0)} $ is a field of fractions.

The operation of localization carries over with no difficulty to arbitrary $ A $- modules $ M $ if one sets

$$ M [ S ^ {-1} ] = M \otimes _ {A} A [ S ^ {-1} ] . $$

The transition from $ M $ to $ M [ S ^ {-1} ] $ is an exact functor. In other words, the $ A $- module $ A [ S ^ {-1} ] $ is flat. Localization commutes with direct sums and inductive limits.

From the geometrical point of view localization means transition to an open subset. More precisely, for $ s \in A $ the spectrum $ \mathop{\rm Spec} A [ s ^ {-1} ] $ is canonically identified with the open (in the Zariski topology) subset $ D ( s) \subset \mathop{\rm Spec} A $ consisting of the prime ideals $ \mathfrak P $ not containing $ s $. Moreover, this operation makes it possible to associate with each $ A $- module $ M $ a quasi-coherent sheaf $ \widetilde{M} $ on the affine scheme $ \mathop{\rm Spec} A $ for which

$$ \Gamma ( D ( s) , \widetilde{M} ) = M [ S ^ {-1} ] . $$

Localization can be regarded as an operation that makes it possible to invert morphisms of multiplication by an $ s \in S $ in the category of $ A $- modules. In this approach the operation of localization admits a wide generalization to arbitrary categories (see Localization in categories).

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)