Localization (commutative algebra)

In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field of rational numbers from the ring of integers.
The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term localization originated in algebraic geometry: if R is a ring of functions defined on some geometric object V, and one wants to study this variety "locally" near a point p, then one considers the set S of all functions that are not zero at p and localizes R with respect to S. The resulting ring contains information about the behavior of V near p, and excludes information that is not "local", such as the zeros of functions that are outside V.

Localization of a ring

The localization of a commutative ring by a multiplicatively closed set is a new ring whose elements are fractions with numerators in and denominators in.
If the ring is an integral domain the construction generalizes and follows closely that of the field of fractions, and, in particular, that of the rational numbers as the field of fractions of the integers. For rings that have zero divisors, the construction is similar but requires more care.

Multiplicative set

Localization is commonly done with respect to a multiplicatively closed set of elements of a ring, that is a subset of that is closed under multiplication, and contains.
The requirement that must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to. The localization by a set that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of. However, the same localization is obtained by using the multiplicatively closed set of all products of elements of. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets.
For example, the localization by a single element introduces fractions of the form but also products of such fractions, such as So, the denominators will belong to the multiplicative set of the powers of. Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element".
The localization of a ring by a multiplicative set is generally denoted but other notations are commonly used in some special cases: if consists of the powers of a single element, is often denoted if is the complement of a prime ideal, then is denoted
''In the remainder of this article, only localizations by a multiplicative set are considered.''

Integral domains

When the ring is an integral domain and does not contain, the ring is a subring of the field of fractions of. As such, the localization of a domain is a domain.
More precisely, it is the subring of the field of fractions of, that consists of the fractions such that This is a subring since the sum and the product of two elements of are in This results from the defining property of a multiplicative set, which implies also that In this case, is a subring of It is shown below that this is no longer true in general, typically when contains zero divisors.
For example, the decimal fractions are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case, consists of the rational numbers that can be written as where is an integer, and is a nonnegative integer.

General construction

In the general case, a problem arises with zero divisors. Let be a multiplicative set in a commutative ring. Suppose that and is a zero divisor with Then is the image in of and one has Thus some nonzero elements of must be zero in The construction that follows is designed for taking this into account.
Given and as above, one considers the equivalence relation on that is defined by if there exists a such that
The localization is defined as the set of the equivalence classes for this relation. The class of is denoted as or So, one has if and only if there is a such that The reason for the is to handle cases such as the above where is nonzero even though the fractions should be regarded as equal.
The localization is a commutative ring with addition
multiplication
additive identity and multiplicative identity
The function
defines a ring homomorphism from into which is injective if and only if does not contain any zero divisors.
If then is the zero ring that has only one unique element.
If is the set of all regular elements of , is called the total ring of fractions of.

Universal property

The ring homomorphism satisfies a universal property that is described below. This characterizes up to an isomorphism. So all properties of localizations can be deduced from the universal property, independently from the way they have been constructed. Moreover, many important properties of localization are easily deduced from the general properties of universal properties, while their direct proof may be more technical.
The universal property satisfied by is the following:
Using category theory, this can be expressed by saying that localization is a functor that is left adjoint to a forgetful functor. More precisely, let and be the categories whose objects are pairs of a commutative ring and a submonoid of, respectively, the multiplicative monoid or the group of units of the ring. The morphisms of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let be the forgetful functor that forgets that the elements of the second element of the pair are invertible.
Then the factorization of the universal property defines a bijection
This may seem a rather tricky way of expressing the universal property, but it is useful for showing easily many properties, by using the fact that the composition of two left adjoint functors is a left adjoint functor.

Examples

If is the ring of integers, and then is the field of the rational numbers.
If is an integral domain, and then is the field of fractions of. The preceding example is a special case of this one.
If is a commutative ring, and if is the subset of its elements that are not zero divisors, then is the total ring of fractions of. In this case, is the largest multiplicative set such that the homomorphism is injective. The preceding example is a special case of this one.
If is an element of a commutative ring and then can be identified The ring is generally denoted. This sort of localization plays a fundamental role in the definition of an affine scheme.
If is a prime ideal of a commutative ring, the set complement of in is a multiplicative set. The ring is a local ring that is generally denoted and called the local ring of at This sort of localization is fundamental in commutative algebra, because many properties of a commutative ring can be read on its local rings. Such a property is often called a local property. For example, a ring is regular if and only if all its local rings are regular.
Ring properties

Localization is a rich construction that has many useful properties. In this section, only the properties relative to rings and to a single localization are considered. Properties concerning ideals, modules, or several multiplicative sets are considered in other sections.

if and only if contains.
The ring homomorphism is injective if and only if does not contain any zero divisors.
The ring homomorphism is an epimorphism in the category of rings, that is not surjective in general.
The ring is a flat -module.
If is the complement of a prime ideal, then denoted is a local ring; that is, it has only one maximal ideal, , and is the residue field of R at.
Localization commutes with formations of finite sums, products, intersections and radicals; e.g., if denote the radical of an ideal in, then
The ideals of are the extension of ideals in by ; that is, they take the form, where is an ideal of ; the ideal is a proper ideal if and only if.
Localization commutes with taking quotients; that is, if I is an ideal of R, then, where is the image of S in.
Let be an integral domain with the field of fractions. Then its localization at a prime ideal can be viewed as a subring of. Moreover,
There is a bijection between the set of prime ideals of and the set of prime ideals of that are disjoint from. This bijection is induced by the given homomorphism.
Saturation of a multiplicative set

Let be a multiplicative set. The saturation of is the set
The multiplicative set is saturated if it equals its saturation, that is, if, or equivalently, if implies that and are in.
If is not saturated, and then is a multiplicative inverse of the image of in So, the images of the elements of are all invertible in and the universal property implies that and are canonically isomorphic, that is, there is a unique isomorphism between them that fixes the images of the elements of.
If and are two multiplicative sets, then and are isomorphic if and only if they have the same saturation, or, equivalently, if belongs to one of the multiplicative sets, then there exists such that belongs to the other.
Saturated multiplicative sets are not widely used explicitly, since, for verifying that a set is saturated, one must know all units of the ring.