In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
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
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, then the localization is the
field of fractions: this case generalizes the construction of the field
of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s from the ring
of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s.
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
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'' (c.f. the example given at
local ring).
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
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
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 (also called a ''multiplicative set'' or a ''multiplicative system'') 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
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on
that is defined by
if there exists a
such that
The localization
is defined as the set of the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es 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
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which f ...
that has as unique element.
If is the set of all
regular elements of (that is the elements that are not zero divisors),
is called the
total ring of fractions
In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction emb ...
of .
Universal property
The (above defined) 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 together technical, straightforward and boring.
The universal property satisfied by
is the following:
:If
is a ring homomorphism that maps every element of to a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
(invertible element) in , there exists a unique ring homomorphism
such that
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
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids a ...
of, respectively, the multiplicative
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
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
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, and
then
is the field
of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s.
*If is an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, 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
In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction emb ...
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 (is
canonically isomorphic to)
(The proof consists of showing that this ring satisfies the above universal property.) 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 (by the definition of a prime ideal). 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
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
, 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
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
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
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
,
modules, or several multiplicative sets are considered in other sections.
*
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
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 (see for details).
* If
is the
complement of a prime ideal
, then
denoted
is a
local ring; that is, it has only one
maximal ideal.
''Properties to be moved in another section''
*Localization commutes with formations of finite sums, products, intersections and radicals; e.g., if
denote the
radical of an ideal ''I'' in ''R'', then
::
:In particular, ''R'' is
reduced if and only if its total ring of fractions is reduced.
*Let ''R'' be an integral domain with the field of fractions ''K''. Then its localization
at a prime ideal
can be viewed as a subring of ''K''. Moreover,
::
:where the first intersection is over all prime ideals and the second over the maximal ideals.
* There is a
bijection between the set of prime ideals of ''S''
−1''R'' and the set of prime ideals of ''R'' that do not intersect ''S''. This bijection is induced by the given homomorphism ''R'' → ''S''
−1''R''.
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
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
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 set, 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.
Terminology explained by the context
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
algebraic geometry, an
affine algebraic set can be identified with a
quotient ring of a
polynomial ring in such a way that the points of the algebraic set correspond to the
maximal ideals of the ring (this is
Hilbert's Nullstellensatz). This correspondence has been generalized for making the set of the
prime ideals of a
commutative ring a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
equipped with the
Zariski topology; this topological space is called the
spectrum of the ring.
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:
* The multiplicative set is the
complement of a
prime ideal of a ring . In this case, one speaks of the "localization at
", or "localization at a point". The resulting ring, denoted
is a
local ring, and is the algebraic analog of a
ring of germs.
* The multiplicative set consists of all powers of an element of a ring . The resulting ring is commonly denoted
and its spectrum is the Zariski open set of the prime ideals that do not contain . Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a
neighborhood basis consisting of Zariski open sets of this form).
In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
, when working over the ring
of the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, one refers to a property relative to an integer as a property true ''at'' or ''away'' from , depending on the localization that is considered. "Away from " means that the property is considered after localization by the powers of , and, if is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
, "at " means that the property is considered after localization at the prime ideal
. This terminology can be explained by the fact that, if is prime, the nonzero prime ideals of the localization of
are either the
singleton set or its complement in the set of prime numbers.
Localization and saturation of ideals
Let be a multiplicative set in a commutative ring , and
be the canonical ring homomorphism. Given an
ideal in , let
the set of the fractions in
whose numerator is in . This is an ideal of
which is generated by , and called the ''localization'' of by .
The ''saturation'' of by is
it is an ideal of , which can also defined as the set of the elements
such that there exists
with
Many properties of ideals are either preserved by saturation and localization, or can be characterized by simpler properties of localization and saturation.
In what follows, is a multiplicative set in a ring , and and are ideals of ; the saturation of an ideal by a multiplicative set is denoted
or, when the multiplicative set is clear from the context,
*
*
(this is not always true for
strict inclusions)
*
*
*
* If
is a
prime ideal such that
then
is a prime ideal and
; if the intersection is nonempty, then
and
Localization of a module
Let be a
commutative ring, be a
multiplicative set In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold:
* 1 \in S,
* xy \in S for all x, y \in S.
In other words, ''S'' is closed under taking finite ...
in , and be an -
module. The localization of the module by , denoted , is an -module that is constructed exactly as the localization of , except that the numerators of the fractions belong to . That is, as a set, it consists of
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es, denoted
, of pairs , where
and
and two pairs and are equivalent if there is an element in such that
:
Addition and scalar multiplication are defined as for usual fractions (in the following formula,
and
):
:
:
Moreover, is also an -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.
Module properties
If is a
submodule of an -module , and is a multiplicative set in , one has
This implies that, if
is an
injective module homomorphism, then
:
is also an injective homomorphism.
Since the tensor product is a
right exact functor, this implies that localization by maps
exact sequences of -modules to exact sequences of
-modules. In other words, localization is an
exact functor, and
is a
flat -module.
This flatness and the fact that localization solves a
universal property make that localization preserves many properties of modules and rings, and is compatible with solutions of other universal properties. For example, the
natural map
:
is an isomorphism. If
is a
finitely presented module, the natural map
:
is also an isomorphism.
If a module ''M'' is a
finitely generated over ''R'', one has
:
where
denotes
annihilator, that is the ideal of the elements of the ring that map to zero all elements of the module. In particular,
:
that is, if
for some
[Borel, AG. 3.1]
Localization at primes
The definition of a
prime ideal implies immediately that the
complement of a prime ideal
in a commutative ring is a multiplicative set. In this case, the localization
is commonly denoted
The ring
is a
local ring, that is called ''the local ring of '' at
This means that
is the unique
maximal ideal of the ring
Such localizations are fundamental for commutative algebra and algebraic geometry for several reasons. One is that local rings are often easier to study than general commutative rings, in particular because of
Nakayama lemma
Nakayama (中山) may refer to:
People
*Nakayama (surname)
Places
*Nakayama, Ehime, a town in Ehime Prefecture
*Nakayama, Tottori, a town in Tottori Prefecture
*Nakayama, Yamagata, a town in Yamagata Prefecture
* Nakayama-dera, a temple in Hyōgo ...
. However, the main reason is that many properties are true for a ring if and only if they are true for all its local rings. For example, a ring is
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
if and only if all its local rings are
regular local rings.
Properties of a ring that can be characterized on its local rings are called ''local properties'', and are often the algebraic counterpart of geometric
local properties of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex number
...
, which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property that refers to localization to Zariski open sets; see , below.)
Many local properties are a consequence of the fact that the module
:
is a
faithfully flat module when the direct sum is taken over all prime ideals (or over all
maximal ideals of ). See also
Faithfully flat descent.
Examples of local properties
A property of an -module is a ''local property'' if the following conditions are equivalent:
* holds for .
* holds for all
where
is a prime ideal of .
* holds for all
where
is a maximal ideal of .
The following are local properties:
* is zero.
* is torsion-free (in the case where is a
commutative domain).
* is a
flat module.
* is an
invertible module In mathematics, particularly commutative algebra, an invertible module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the definition of invertible sheaves in algebraic geom ...
(in the case where is a commutative domain, and is a submodule of the
field of fractions of ).
*
is injective (resp. surjective), where is another -module.
On the other hand, some properties are not local properties. For example, an infinite
direct product of
fields is not an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
nor a
Noetherian ring, while all its local rings are fields, and therefore Noetherian integral domains.
Localization to Zariski open sets
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.
See also
*
Local analysis
*
Localization of a category In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in gen ...
*
Localization of a topological space
References
*Atiyah and MacDonald. Introduction to Commutative Algebra. Addison-Wesley.
*
Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. .
*
*
*
*Matsumura. Commutative Algebra. Benjamin-Cummings
*
*
Serge Lang, "Algebraic Number Theory," Springer, 2000. pages 3–4.
{{refend
External links
Localizationfrom
MathWorld.
Ring theory
Module theory
Localization (mathematics)