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
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 (algebraic variety) 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
For rings that have zero divisors, the construction is similar but requires more care.
The requirement that S must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to S. The localization by a set U that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of U.
However, the same localization is obtained by using the multiplicatively closed set S of all products of elements of U.
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 s introduces fractions of the form
The localization of a ring R by a multiplicative set S is generally denoted
In the remainder of this article, only localizations by a multiplicative set are considered.
This results from the defining property of a multiplicative set, which implies also that
It is shown below that this is no longer true in general, typically when S 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 the general case, a problem arises with zero divisors.
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.
are isomorphic if and only if they have the same saturation, or, equivalently, if s belongs to one of the multiplicative sets, then there exists
The term localization originates in the general trend of modern mathematics to study geometrical and topological objects locally, that is in terms of their behavior near each point.
Examples of this trend are the fundamental concepts of manifolds, germs and sheafs.
In this context, a localization by a multiplicative set may be viewed as the restriction of the spectrum of a ring to the subspace of the prime ideals (viewed as points) that do not intersect the multiplicative set.
Two classes of localizations are more commonly considered: In number theory and algebraic topology, when working over the ring
"Away from n" means that the property is considered after localization by the powers of n, and, if p is a prime number, "at p" means that the property is considered after localization at the prime ideal
and two pairs (m, s) and (n, t) are equivalent if there is an element u in S such that Addition and scalar multiplication are defined as for usual fractions (in the following formula,
): Moreover, S−1M is also an R-module with scalar multiplication It is straightforward to check that these operations are well-defined, that is, they give the same result for different choices of representatives of fractions.
The localization of a module can be equivalently defined by using tensor products: The proof of equivalence (up to a canonical isomorphism) can be done by showing that the two definitions satisfy the same universal property.
is a finitely presented module, the natural map is also an isomorphism.
[6] The definition of a prime ideal implies immediately that the complement
For example, an infinite direct product of fields is not an integral domain nor a Noetherian ring, while all its local rings are fields, and therefore Noetherian integral domains.
While the localization exists for every set S of prospective units, it might take a different form to the one described above.
There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches.