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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. Prominent ...
and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, 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 A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
\frac, such that the
denominator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
''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 se ...
, then the localization is the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
: this case generalizes the construction of the field \Q 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 ratio ...
s from the ring \Z 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 languag ...
s. The technique has become fundamental, particularly in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, as it provides a natural link to
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * '' The Sheaf'', a student-run newspaper s ...
theory. In fact, the term ''localization'' originated in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
: if ''R'' is a ring of functions defined on some geometric object (
algebraic variety 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 numbers. M ...
) ''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 S^R 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 In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
).


Localization of a ring

The localization of a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
by a
multiplicatively closed 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 ...
is a new ring S^R 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 se ...
the construction generalizes and follows closely that of the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
, and, in particular, that of the
rational numbers 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 rat ...
as the field of fractions of the integers. For rings that have
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s, the construction is similar but requires more care.


Multiplicative set

Localization is commonly done with respect to a
multiplicatively closed 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 ...
(also called a ''multiplicative set'' or a ''multiplicative system'') of elements of a ring , that is a subset of that is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
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 \tfrac a s, but also products of such fractions, such as \tfrac . 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 S^R, but other notations are commonly used in some special cases: if S= \ consists of the powers of a single element, S^R is often denoted R_t; if S=R\setminus \mathfrak p is the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
of a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
\mathfrak p, then S^R is denoted R_\mathfrak p. ''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 se ...
and does not contain , the ring S^R is a subring of the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of . As such, the localization of a domain is a domain. More precisely, it is the
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of the field of fractions of , that consists of the fractions \tfrac a s such that s\in S. This is a subring since the sum \tfrac as + \tfrac bt = \tfrac , and the product \tfrac as \, \tfrac bt = \tfrac of two elements of S^R are in S^R. This results from the defining property of a multiplicative set, which implies also that 1=\tfrac 11\in S^R. In this case, is a subring of S^R. It is shown below that this is no longer true in general, typically when contains
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s. For example, the
decimal fraction The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
s are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case, S^R consists of the rational numbers that can be written as \tfrac n, where is an integer, and is a nonnegative integer.


General construction

In the general case, a problem arises with
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s. Let be a multiplicative set in a commutative ring . Suppose that s\in S, and 0\ne a\in R is a zero divisor with as=0. Then \tfrac a1 is the image in S^R of a\in R, and one has \tfrac a1 = \tfrac s = \tfrac 0s = \tfrac 01. Thus some nonzero elements of must be zero in S^R. 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 R\times S that is defined by (r_1, s_1) \sim (r_2, s_2) if there exists a t\in S such that t(s_1r_2-s_2r_1)=0. The localization S^R 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 a ...
es for this relation. The class of is denoted as \frac rs, r/s, or s^r. So, one has \tfrac=\tfrac if and only if there is a t\in S such that t(s_1r_2-s_2r_1)=0. The reason for the t is to handle cases such as the above \tfrac a1 = \tfrac 01, where s_1r_2-s_2r_1 is nonzero even though the fractions should be regarded as equal. The localization S^R is a commutative ring with addition :\frac +\frac = \frac, multiplication :\frac \,\frac = \frac, additive identity \tfrac 01, and
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
\tfrac 11. The function :r\mapsto \frac r1 defines a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserv ...
from R into S^R, which is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
if and only if does not contain any zero divisors. If 0\in S, then S^R 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 fo ...
that has as unique element. If is the set of all regular elements of (that is the elements that are not zero divisors), S^R is called the total ring of fractions of .


Universal property

The (above defined) ring homomorphism j\colon R\to S^R satisfies a universal property that is described below. This characterizes S^R 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 j\colon R\to S^R is the following: :If f\colon R\to T 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 g\colon S^R\to T such that f=g\circ j. Using
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, this can be expressed by saying that localization is a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
that is
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to a
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
. More precisely, let \mathcal C and \mathcal D 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 ar ...
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 a ...
or the
group of units In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for thi ...
of the ring. The
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let \mathcal F\colon \mathcal D \to \mathcal C be the forgetful functor that forgets that the elements of the second element of the pair are invertible. Then the factorization f=g\circ j of the universal property defines a bijection :\hom_\mathcal C((R,S), \mathcal F(T,U))\to \hom_\mathcal D ((S^R, j(S)), (T,U)). 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 R=\Z 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 languag ...
s, and S=\Z\setminus \, then S^R is the field \Q 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 ratio ...
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 se ...
, and S=R\setminus \, then S^R is the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of . The preceding example is a special case of this one. *If is a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, and if is the subset of its elements that are not
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s, then S^R is the total ring of fractions of . In this case, is the largest multiplicative set such that the homomorphism R\to S^R is injective. The preceding example is a special case of this one. *If is an element of a commutative ring and S=\, then S^R can be identified (is canonically isomorphic to) R ^R (xs-1). (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 In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
. *If \mathfrak p is a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
of a commutative ring , the
set complement In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is ...
S=R\setminus \mathfrak p of \mathfrak p in is a multiplicative set (by the definition of a prime ideal). The ring S^R is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
that is generally denoted R_\mathfrak p, and called ''the local ring of at'' \mathfrak p. 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. Prominent ...
, because many properties of a commutative ring can be read on its local rings. Such a property is often called a
local property In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). Pr ...
. 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 Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
, or several multiplicative sets are considered in other sections. * S^R = 0
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 bic ...
contains . * The
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserv ...
R\to S^R is injective if and only if does not contain any
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s. * The ring homomorphism R\to S^R is an
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
in the
category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is ...
, that is not
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
in general. * The ring S^R is a flat -module (see for details). * If S=R\setminus \mathfrak p is the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
of a prime ideal \mathfrak p, then S^ R, denoted R_\mathfrak p, is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
; that is, it has only one
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals con ...
. ''Properties to be moved in another section'' *Localization commutes with formations of finite sums, products, intersections and radicals; e.g., if \sqrt denote the
radical of an ideal In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called ...
''I'' in ''R'', then ::\sqrt \cdot S^R = \sqrt\,. :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 R_\mathfrak at a prime ideal \mathfrak can be viewed as a subring of ''K''. Moreover, ::R = \bigcap_\mathfrak R_\mathfrak = \bigcap_\mathfrak R_\mathfrak :where the first intersection is over all prime ideals and the second over the maximal ideals. * There is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 S \subseteq R be a multiplicative set. The ''saturation'' \hat of S is the set :\hat = \. The multiplicative set is ''saturated'' if it equals its saturation, that is, if \hat=S, or equivalently, if rs \in S implies that and are in . If is not saturated, and rs \in S, then \frac s 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''/'' ...
of the image of in S^R. So, the images of the elements of \hat S are all invertible in S^R, and the universal property implies that S^R and \hat ^R 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 S^R and T^R are isomorphic if and only if they have the same saturation, or, equivalently, if belongs to one of the multiplicative set, then there exists t\in R 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 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'' (al ...
of the ring.


Terminology explained by the context

The term ''localization'' originates in the general trend of modern mathematics to study
geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
and
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
objects ''locally'', that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s, germs and sheafs. In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, an
affine algebraic set Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine com ...
can be identified with a
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
of a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variab ...
in such a way that the points of the algebraic set correspond to the
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals con ...
s of the ring (this is
Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
). This correspondence has been generalized for making the set of the
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s of a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
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 In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
; 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 A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
of a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
\mathfrak p of a ring . In this case, one speaks of the "localization at \mathfrak p", or "localization at a point". The resulting ring, denoted R_\mathfrak p is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
, 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 R_t, 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 In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbou ...
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, "Mathe ...
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 classify u ...
, when working over the ring \Z 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 languag ...
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 (mathematics), 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 ...
, "at " means that the property is considered after localization at the prime ideal p\Z. This terminology can be explained by the fact that, if is prime, the nonzero prime ideals of the localization of \Z are either the
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
or its complement in the set of prime numbers.


Localization and saturation of ideals

Let be a multiplicative set in a commutative ring , and j\colon R\to S^R be the canonical ring homomorphism. Given an ideal in , let S^I the set of the fractions in S^R whose numerator is in . This is an ideal of S^R, which is generated by , and called the ''localization'' of by . The ''saturation'' of by is j^(S^I); it is an ideal of , which can also defined as the set of the elements r\in R such that there exists s\in S with sr\in I. 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 \operatorname_S (I), or, when the multiplicative set is clear from the context, \operatorname(I). * 1 \in S^I \quad\iff\quad 1\in \operatorname(I) \quad\iff\quad S\cap I \neq \emptyset * I \subseteq J \quad\ \implies \quad\ S^I \subseteq S^J \quad\ \text \quad\ \operatorname(I)\subseteq \operatorname(J)
(this is not always true for strict inclusions) * S^(I \cap J) = S^I \cap S^J,\qquad\, \operatorname(I \cap J) = \operatorname(I) \cap \operatorname(J) * S^(I + J) = S^I + S^J,\qquad \operatorname(I + J) = \operatorname(I) + \operatorname(J) * S^(I \cdot J) = S^I \cdot S^J,\qquad\quad \operatorname(I \cdot J) = \operatorname(I) \cdot \operatorname(J) * If \mathfrak p is a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
such that \mathfrak p \cap S = \emptyset, then S^\mathfrak p is a prime ideal and \mathfrak p = \operatorname(\mathfrak p); if the intersection is nonempty, then S^\mathfrak p = S^R and \operatorname(\mathfrak p)=R.


Localization of a module

Let be a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, be a multiplicative set 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 a ...
es, denoted \frac ms, of pairs , where m\in M and s\in S, and two pairs and are equivalent if there is an element in such that :u(sn-tm)=0. Addition and scalar multiplication are defined as for usual fractions (in the following formula, r\in R, s,t\in S, and m,n\in M): :\frac + \frac = \frac, :\frac rs \frac = \frac. Moreover, is also an -module with scalar multiplication : r\, \frac = \frac r1 \frac ms = \fracs. 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: :S^M=S^R \otimes_R M. The proof of equivalence (up to a
canonical isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
) can be done by showing that the two definitions satisfy the same universal property.


Module properties

If is a
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the m ...
of an -module , and is a multiplicative set in , one has S^M\subseteq S^N. This implies that, if f\colon M\to N is an
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
module homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an '' ...
, then :S^R\otimes_R f : \quad S^R\otimes_R M\to S^R\otimes_R N is also an injective homomorphism. Since the tensor product is a right exact functor, this implies that localization by maps
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...
s of -modules to exact sequences of S^R-modules. In other words, localization is an
exact functor In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
, and S^R 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 :S^(M \otimes_R N) \to S^M \otimes_ S^N is an isomorphism. If M is a
finitely presented module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts inc ...
, the natural map :S^ \operatorname_R (M, N) \to \operatorname_ (S^M, S^N) is also an isomorphism. If a module ''M'' is a finitely generated over ''R'', one has :S^(\operatorname_R(M)) = \operatorname_(S^M), where \operatorname denotes annihilator, that is the ideal of the elements of the ring that map to zero all elements of the module. In particular, :S^ M = 0\quad \iff \quad S\cap \operatorname_R(M) \ne \emptyset, that is, if t M = 0 for some t \in S.Borel, AG. 3.1


Localization at primes

The definition of a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
implies immediately that the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
S=R\setminus \mathfrak p of a prime ideal \mathfrak p in a commutative ring is a multiplicative set. In this case, the localization S^R is commonly denoted R_\mathfrak p. The ring R_\mathfrak p is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
, that is called ''the local ring of '' at \mathfrak p. This means that \mathfrak p\,R_\mathfrak p=\mathfrak p\otimes_R R_\mathfrak p is the unique
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals con ...
of the ring R_\mathfrak p. 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. 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 if and only if all its local rings are
regular local ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...
s. 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 numbers. M ...
, 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 :\bigoplus_\mathfrak p R_\mathfrak p is a faithfully flat module when the direct sum is taken over all prime ideals (or over all
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals con ...
s of ). See also
Faithfully flat descent Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, that are flat and surjective, are common, one example coming from an open c ...
.


Examples of local properties

A property of an -module is a ''local property'' if the following conditions are equivalent: * holds for . * holds for all M_\mathfrak, where \mathfrak is a prime ideal of . * holds for all M_\mathfrak, where \mathfrak 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 In algebra, a flat module over a ring ''R'' is an ''R''- module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact ...
. * is an invertible module (in the case where is a commutative domain, and is a submodule of the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of ). * f\colon M \to N is injective (resp. surjective), where is another -module. On the other hand, some properties are not local properties. For example, an infinite
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
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 se ...
nor a
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
, while all its local rings are fields, and therefore Noetherian integral domains.


Localization to Zariski open sets


Non-commutative case

Localizing
non-commutative ring In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s 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 In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or ...
. 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 equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. 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 Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ...
, in particular.


See also

*
Local analysis In mathematics, the term local analysis has at least two meanings, both derived from the idea of looking at a problem relative to each prime number ''p'' first, and then later trying to integrate the information gained at each prime into a 'global ...
*
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 genera ...
*
Localization of a topological space In mathematics, well-behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in . The rea ...


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


Localization
from
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...
. Ring theory Module theory Localization (mathematics)