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 one 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 ring Q of rational numbers from the ring Z of rational 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 which are not zero at p and localizes R with respect to S. The resulting ring R* contains only information about the behavior of V near p.
An important related process is completion: one often localizes a ring/module, then completes.

Construction and properties for commutative rings

The set S is assumed to be a submonoid of the multiplicative monoid of R, i.e. 1 is in S and for s and t in S we also have st in S. A subset of R with this property is called a multiplicatively closed set, multiplicative set or multiplicative system. This requirement on S is natural and necessary to have since its elements will be turned into units of the localization, and units must be closed under multiplication.
It is standard practice to assume that S is multiplicatively closed. If S is not multiplicatively closed, it suffices to replace it by its multiplicative closure, consisting of the set of the products of elements of S. This does not change the result of the localization. The fact that we talk of "a localization with respect to the powers of an element" instead of "a localization with respect to an element" is an example of this. Therefore, we shall suppose S to be multiplicatively closed in what follows.

Construction

For integral domains

In the case R is an integral domain there is an easy construction of the localization. Since the only ring in which 0 is a unit is the trivial ring, the localization R* is if 0 is in S. Otherwise, the field of fractions K of R can be used: we take R* to be the subset of K consisting of the elements of the form r/s with r in R and s in S; as we have supposed S multiplicatively closed, R* is a subring of K. The standard embedding of R into R* is injective in this case, although it may be non-injective in a more general setting. For example, the dyadic fractions are the localization of the ring of integers with respect to the powers of two. In this case, R* is the dyadic fractions, R is the integers, the denominators are powers of 2, and the natural map from R to R* is injective. The result would be exactly the same if we had taken S = .

For general commutative rings

For general commutative rings, we don't have a field of fractions. Nevertheless, a localization can be constructed consisting of "fractions" with denominators coming from S; in contrast with the integral domain case, one can safely 'cancel' from numerator and denominator only elements of S.
This construction proceeds as follows: on R × S define an equivalence relation ~ by setting ~ if there exists t in S such that
We think of the equivalence class of as the "fraction" r/s and, using this intuition, the set of equivalence classes R* can be turned into a ring with operations that look identical to those of elementary algebra: and. The map that maps r to the equivalence class of is then a ring homomorphism. In general, this is not injective; if a and b are two elements of R such that there exists s in S with, then their images under j are equal.

Universal property

The ring homomorphism j : R → R* maps every element of S to a unit in R* = S −1R. The universal property is that if f : R → T is some other ring homomorphism into another ring T which maps every element of S to a unit in T, then there exists a unique ring homomorphism g : R* → T such that f = g∘j.
This can also be phrased in the language of category theory. If R is a ring and S is a subset, consider all R- A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. These algebras are the objects of a category, with R-algebra homomorphisms as morphisms. Then, the localization of R at S is the initial object of this category.

Examples

Some properties of the localization R* = S −1R:
The term localization originates 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 R* contains only information about the behavior of V near p. For more detail, see Ring of germs.
Two classes of localizations occur commonly in commutative algebra and algebraic geometry and are used to construct the rings of functions on open subsets in Zariski topology of the spectrum of a ring, Spec.
In number theory and algebraic topology, one refers to the behavior of a ring at a number n or away from n. "Away from n" means "in the ring localized by the set of the powers of n". If n is a prime number, "at n" means "in the ring localized by the set of the integers that are not a multiple of n".

Localization of a module

Let R be a commutative ring and S be a multiplicatively closed subset of R. Then the localization of M with respect to S, denoted S−1M, is defined to be the following module: as a set, it consists of equivalence classes of pairs, where m ∈ M and s ∈ S. Two such pairs and are considered equivalent if there is a third element u of S such that
It is common to denote the equivalence class of by.
To make this set an R-module, define
and
It is straightforward to check that these operations are well-defined, i.e. they give the same result for different choices of representatives of fractions. One interesting characterization of the equivalence relation is that it is the smallest relation such that cancellation laws hold for elements in S. That is, it is the smallest relation such that sm/st = m/t for all s,t in S and m in M.
One case is particularly important: if S equals the complement of a prime ideal p ⊂ R then the localization is denoted Mp instead of −1M. The support of the module M is the set of prime ideals p such that Mp ≠ 0. Viewing M as a function from the spectrum of R to R-modules, mapping
this corresponds to the support of a function.
Localization of a module at primes also reflects the "local properties" of the module. In particular, there are many cases where the more general situation can be reduced to a statement about localized modules. The reduction is because an R-module M is trivial if and only if all its localizations at primes or maximal ideals are trivial.
Remark:
From the definition, one can see that localization of modules is an exact functor, or in other words that S−1R is a flat module over R. This fact is foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of the open set Spec into Spec is a flat morphism.
The localization functor preserves Hom and tensor products in the following sense: the natural map
is an isomorphism and if is finitely presented, the natural map
is an isomorphism.
If a module M is a finitely generated over R,
If is an -module, the statement that property P holds for "at a prime ideal " has two possible meanings. The first is that P holds for, and the second is that P holds for a neighborhood of. The first interpretation is more common, but for many properties the first and second interpretations coincide. Explicitly, the second means the following conditions are equivalent:
Then the following are local properties in the second sense:
On the other hand, some properties are not local properties. For example, "noetherian" is in general not a local property: that is, to say there is a non-noetherian ring whose localization at every maximal ideal is noetherian: this example is due to Nagata.

(Quasi-)coherent sheaves

In terms of localization of modules, one can define quasi-coherent sheaves and coherent sheaves on locally ringed spaces. In algebraic geometry, the quasi-coherent OX-modules for schemes X are those that are locally modelled on sheaves on Spec of localizations of any R-module M. A coherent OX-module is such a sheaf, locally modelled on a finitely-presented module over R.

Non-commutative case

Localizing non-commutative rings is more difficult. While the localization exists for every set S of prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved is the Ore condition.
One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.

Localization