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In [[commutative algebra]] and [[algebraic geometry]], '''localization''' is a formal way to introduce the "denominators" to a given [[ring (mathematics)|ring]] or [[module (mathematics)|module]]. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of [[algebraic fraction|fractions]] <math>\frac{m}{s},</math> 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
The technique has become fundamental, particularly in [[algebraic geometry]], as it provides a natural link to [[sheaf (mathematics)|sheaf]] theory. In fact, the term ''localization'' originated in [[algebraic geometry]]: if ''R'' is a ring of [[function (mathematics)|function]]s 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 <math>S^{-1}R</math> contains information about the behavior of ''V'' near ''p'', and excludes information that is not "local", such as the [[zero of a function|zeros of functions]] that are outside ''V'' (
== Localization of a ring ==
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Localization is commonly done with respect to a [[multiplicatively closed set]] {{mvar|S}} (also called a ''multiplicative set'' or a ''multiplicative system'') of elements of a ring {{mvar|R}}, that is a subset of {{mvar|R}} that is [[closure (mathematics)|closed]] under multiplication, and contains {{math|1}}.
The requirement that {{mvar|S}} must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to {{mvar|S}}. The localization by a set {{mvar|U}} that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of {{mvar|U}}. However, the same localization is obtained by using the multiplicatively closed set {{mvar|S}} of all products of elements of {{mvar|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 {{mvar|s}} introduces fractions of the form <math>\tfrac a s,</math> but also products of such fractions, such as <math>\tfrac {ab} {s^2}.</math> So, the denominators will belong to the multiplicative set <math>\{1, s, s^2, s^3,\ldots\}</math> of the powers of {{mvar|s}}. Therefore, one generally talks of "the localization by the
The localization of a ring {{mvar|R}} by a multiplicative set {{mvar|S}} is generally denoted <math>S^{-1}R,</math> but other notations are commonly used in some special cases: if <math>S= \{1, t, t^2,\ldots \}</math> consists of the powers of a single element, <math>S^{-1}R</math> is often denoted <math>R_t;</math> if <math>S=R\setminus \mathfrak p</math> is the [[complement (set theory)|complement]] of a [[prime ideal]] <math>\mathfrak p</math>, then <math>S^{-1}R</math> is denoted <math>R_\mathfrak p.</math>
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=== Integral domains ===
When the ring {{mvar|R}} is an [[integral ___domain]] and {{mvar|S}} does not contain {{math|0}}, the ring <math>S^{-1}R</math> is a subring of the [[field of fractions]] of {{mvar|R}}. As such, the localization of a ___domain is a ___domain.
More precisely, it is the [[subring]] of the field of fractions of {{mvar|R}}, that consists of the fractions <math>\tfrac a s</math> such that <math>s\in S.</math> This is a subring since the sum <math>\tfrac as + \tfrac bt = \tfrac {at+bs}{st},</math> and the product <math>\tfrac as \, \tfrac bt = \tfrac {ab}{st}</math> of two elements of <math>S^{-1}R</math> are in <math>S^{-1}R.</math> This results from the defining property of a multiplicative set, which implies also that <math>1=\tfrac 11\in S^{-1}R.</math> In this case, {{mvar|R}} is a subring of <math>S^{-1}R.</math> It is shown below that this is no longer true in general, typically when {{mvar|S}} contains [[zero divisor]]s.
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=== General construction ===
In the general case, a problem arises with [[zero divisor]]s. Let {{mvar|S}} be a multiplicative set in a commutative ring {{mvar|R}}.
Given {{mvar|R}} and {{mvar|S}} as above, one considers the [[equivalence relation]] on <math>R\times S</math> that is defined by <math>(r_1, s_1) \sim (r_2, s_2)</math> if there exists a <math>t\in S</math> such that <math>t(s_1r_2-s_2r_1)=0.</math>
The localization <math>S^{-1}R</math> is defined as the set of the [[equivalence class]]es for this relation. The class of {{math|(''r'', ''s'')}} is denoted as <math>\frac rs,</math> <math>r/s,</math> or <math>s^{-1}r.</math> So, one has <math>\tfrac{r_1}{s_1}=\tfrac{r_2}{s_2}</math> if and only if there is a <math>t\in S</math> such that <math>t(s_1r_2-s_2r_1)=0.</math> The reason for the <math>t</math> is to handle cases such as the above <math>\tfrac a1 = \tfrac 01,</math> where <math>s_1r_2-s_2r_1</math> is nonzero even though the fractions should be regarded as equal.
The localization <math>S^{-1}R</math> is a commutative ring with addition
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defines a [[ring homomorphism]] from <math>R</math> into <math>S^{-1}R,</math> which is [[injective function|injective]] if and only if {{mvar|S}} does not contain any zero divisors.
If <math>0\in S,</math> then <math>S^{-1}R</math> is the [[zero ring]] that has only one unique element {{math|0}}
If {{mvar|S}} is the set of all [[zero divisor|regular elements]] of {{mvar|R}} (that is the elements that are not zero divisors), <math>S^{-1}R</math> is called the [[total ring of fractions]] of {{mvar|R}}.
=== Universal property ===
The (above defined) ring homomorphism <math>j\colon R\to S^{-1}R</math> satisfies a [[universal property]] that is described below. This characterizes <math>S^{-1}R</math> up to an [[ring isomorphism|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
The universal property satisfied by <math>j\colon R\to S^{-1}R</math> is the following:
:If <math>f\colon R\to T</math> is a ring homomorphism that maps every element of {{mvar|S}} to a [[unit (ring theory)|unit]] (invertible element) in {{mvar|T}}, there exists a unique ring homomorphism <math>g\colon S^{-1}R\to T</math> such that <math>f=g\circ j.</math> 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 <math>\mathcal C</math> and <math>\mathcal D</math> be the categories whose objects are [[ordered pair|pairs]] of a commutative ring and a [[submonoid]] of, respectively, the multiplicative [[
Then the factorization <math>f=g\circ j</math> of the universal property defines a bijection
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*If {{mvar|R}} is an [[integral ___domain]], and <math>S=R\setminus \{0\},</math> then <math>S^{-1}R</math> is the [[field of fractions]] of {{mvar|R}}. The preceding example is a special case of this one.
*If {{mvar|R}} is a [[commutative ring]], and if {{mvar|S}} is the subset of its elements that are not [[zero divisor]]s, then <math>S^{-1}R</math> is the [[total ring of fractions]] of {{mvar|R}}. In this case, {{mvar|S}} is the largest multiplicative set such that the homomorphism <math>R\to S^{-1}R</math> is injective. The preceding example is a special case of this one.
*If
*If <math>\mathfrak p</math> is a [[prime ideal]] of a commutative ring {{mvar|R}}, the [[set complement]] <math>S=R\setminus \mathfrak p</math> of <math>\mathfrak p</math> in {{mvar|R}} is a multiplicative set (by the definition of a prime ideal). The ring <math>S^{-1}R</math> is a [[local ring]] that is generally denoted <math>R_\mathfrak p,</math> and called ''the local ring of {{mvar|R}} at'' <math>\mathfrak p.</math> This sort of localization is fundamental in [[commutative algebra]], because many properties of a commutative ring can be read on its local rings. Such a property is often called a [[local property]]. For example, a ring is [[regular ring|regular]] if and only if all its local rings are regular.
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* The ring <math>S^{-1}R</math> is a [[flat module|flat {{mvar|R}}-module]] (see {{slink||Localization of a module}} for details).
* If <math>S=R\setminus \mathfrak p</math> is the [[complement (set theory)|complement]] of a prime ideal <math>\mathfrak p</math>, then <math>S^{-1} R,</math> denoted <math>R_\mathfrak p,</math> is a [[local ring]]; that is, it has only one [[maximal ideal]].
*Localization commutes with formations of finite sums, products, intersections and radicals;<ref>{{harvnb|Atiyah|
▲''Properties to be moved in another section''
▲*Localization commutes with formations of finite sums, products, intersections and radicals;<ref>{{harvnb|Atiyah|MacDonald|1969|loc=Proposition 3.11. (v).}}</ref> e.g., if <math>\sqrt{I}</math> denote the [[radical of an ideal]] ''I'' in ''R'', then
::<math>\sqrt{I} \cdot S^{-1}R = \sqrt{I \cdot S^{-1}R}\,.</math>
:In particular, ''R'' is [[reduced ring|reduced]] if and only if its total ring of fractions is reduced.<ref>Borel, AG. 3.3</ref>
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::<math>R = \bigcap_\mathfrak{p} R_\mathfrak{p} = \bigcap_\mathfrak{m} R_\mathfrak{m}</math>
:where the first intersection is over all prime ideals and the second over the maximal ideals.<ref>Matsumura, Theorem 4.7</ref>
* There is a [[bijection]] between the set of prime ideals of ''S''<sup>−1</sup>''R'' and the set of prime ideals of ''R'' that
=== Saturation of a multiplicative set ===
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If {{mvar|S}} is not saturated, and <math>rs \in S,</math> then <math>\frac s{rs}</math> is a [[multiplicative inverse]] of the image of {{mvar|r}} in <math>S^{-1}R.</math> So, the images of the elements of <math>\hat S</math> are all invertible in <math>S^{-1}R,</math> and the universal property implies that <math>S^{-1}R</math> and <math>\hat {S}{}^{-1}R</math> are [[canonical isomorphism|canonically isomorphic]], that is, there is a unique isomorphism between them that fixes the images of the elements of {{mvar|R}}.
If {{mvar|S}} and {{mvar|T}} are two multiplicative sets, then <math>S^{-1}R</math> and <math>T^{-1}R</math> are isomorphic if and only if they have the same saturation, or, equivalently, if {{mvar|s}} belongs to one of the multiplicative
Saturated multiplicative sets are not widely used explicitly, since, for verifying that a set is saturated, one must know ''all'' [[unit (ring theory)|units]] of the ring.
== Terminology explained by the context ==
The term ''localization'' originates in the general trend of modern mathematics to study [[geometry|geometrical]] and [[topology|topological]] objects ''locally'', that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of [[manifold]]s, [[germ (mathematics)|germs]] and [[sheaf (mathematics)|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 ideal]]s of the ring (this is [[Hilbert's Nullstellensatz]]). This correspondence has been generalized for making the set of the [[prime ideal]]s of a [[commutative ring]] a [[topological space]] equipped with the [[Zariski topology]]; this topological space is called the [[spectrum of a ring|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.
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* The multiplicative set consists of all powers of an element {{mvar|t}} of a ring {{mvar|R}}. The resulting ring is commonly denoted <math>R_t,</math> and its spectrum is the Zariski open set of the prime ideals that do not contain {{mvar|t}}. 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).
{{anchor|away from}}In [[number theory]] and [[algebraic topology]], when working over the ring <math>\Z</math> of
== Localization and saturation of ideals ==
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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, {{mvar|S}} is a multiplicative set in a ring {{mvar|
* <math>1 \in S^{-1}I \quad\iff\quad 1\in \operatorname{sat}(I) \quad\iff\quad S\cap I \neq \emptyset</math>
* <math>I \subseteq J \quad\ \implies \quad\ S^{-1}I \subseteq
* <math>S^{-1}(I \cap J) = S^{-1}I \cap S^{-1}J,\qquad\, \operatorname{sat}(I \cap J) = \operatorname{sat}(I) \cap \operatorname{sat}(J)</math>
* <math>S^{-1}(I + J) = S^{-1}I + S^{-1}J,\qquad \operatorname{sat}(I + J) = \operatorname{sat}(I) + \operatorname{sat}(J)</math>
* <math>S^{-1}(I \cdot J) = S^{-1}I \cdot S^{-1}J,\qquad\quad \operatorname{sat}(I \cdot J) = \operatorname{sat}(I) \cdot \operatorname{sat}(J)</math>
* If <math>\mathfrak p</math> is a [[prime ideal]] such that <math>\mathfrak p \cap S = \emptyset,</math> then <math>S^{-1}\mathfrak p</math> is a prime ideal and <math>\mathfrak p = \operatorname{sat}(\mathfrak p)</math>; if the intersection is nonempty, then <math>S^{-1}\mathfrak p = S^{-1}R</math> and <math>\operatorname{sat}(\mathfrak p)=R.</math>
==Localization of a module ==
Let
:<math>u(sn-tm)=0.</math>
Addition and scalar multiplication are defined as for usual fractions (in the following formula, <math>r\in R,</math> <math>s,t\in S,</math> and <math>m,n\in M</math>):
:<math>\frac{m}{s} + \frac{n}{t} = \frac{tm+sn}{st},</math>
:<math>\frac rs \frac{m}{t} = \frac{r m}{st}.</math>
Moreover,
:<math> r\, \frac{m}{s} = \frac r1 \frac ms = \frac{rm}s.</math>
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 product of modules|tensor products]]:
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is an isomorphism. If <math>M</math> is a [[finitely presented module]], the natural map
:<math>S^{-1} \operatorname{Hom}_R (M, N) \to \operatorname{Hom}_{S^{-1}R} (S^{-1}M, S^{-1}N)</math>
is also an isomorphism.<ref>{{
If a module ''M'' is a [[finitely generated module|finitely generated]] over ''R'', one has
:<math>S^{-1}(\operatorname{Ann}_R(M)) = \operatorname{Ann}_{S^{-1}R}(S^{-1}M),</math>
where <math>\operatorname{Ann}</math> denotes [[annihilator (ring theory)|annihilator]], that is the ideal of the elements of the ring that map to zero all elements of the module.<ref>{{harvnb|Atiyah|
:<math>S^{-1} M = 0\quad \iff \quad S\cap \operatorname{Ann}_R(M) \ne \emptyset,</math>
that is, if <math>t M = 0</math> for some <math>t \in S.</math><ref>Borel, AG. 3.1</ref>
==Localization at primes==
The definition of a [[prime ideal]] implies immediately that the [[set complement|complement]] <math>S=R\setminus \mathfrak p</math> of a prime ideal <math>\mathfrak p</math> in a commutative ring {{mvar|R}} is a multiplicative set. In this case, the localization <math>S^{-1}R</math> is commonly denoted <math>R_\mathfrak p.</math> The ring <math>R_\mathfrak p</math> is a [[local ring]], that is called ''the local ring of {{mvar|R}}'' at <math>\mathfrak p.</math> This means that <math>\mathfrak p\,R_\mathfrak p=\mathfrak p\otimes_R R_\mathfrak p</math> is the unique [[maximal ideal]] of the ring <math>R_\mathfrak p.</math> Analogously one can define the localization of a module {{mvar|M}} at a prime ideal <math>\mathfrak p</math> of {{mvar|R}}. Again, the localization <math>S^{-1}M</math> is commonly denoted <math>M_{\mathfrak p}</math>.
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 ring|regular]] if and only if all its local rings are [[regular local ring]]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 property|local properties]] of [[algebraic varieties]], 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 {{slink||Localization to Zariski open sets}}, below.)
Many local properties are a consequence of the fact that the module
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On the other hand, some properties are not local properties. For example, an infinite [[direct product]] of [[field (mathematics)|fields]] is not an [[integral ___domain]] nor a [[Noetherian ring]], while all its local rings are fields, and therefore Noetherian integral domains.
== Non-commutative case ==
Localizing [[non-commutative ring]]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]].
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''<sup>−1</sup> for a differentiation operator ''D''. This is done in many contexts in methods for [[differential equation]]s. There is now a large mathematical theory about it, named [[microlocal analysis|microlocalization]], connecting with numerous other branches. The ''micro-'' tag is to do with connections with [[Fourier theory]], in particular.
== See also ==
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{{refbegin}}
* {{Cite book|last1=Atiyah|first1=Michael
*[[Armand Borel|Borel, Armand]]. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. {{ISBN|0-387-97370-2}}.
* {{cite book|last=Cohn|first=P. M.|title=Algebra |volume=
* {{cite book|last=Cohn|first=P. M.|title=Algebra |volume=
* {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1|mr=1322960 | year=1995 | volume=150}}
* {{Matsumura CA}}
* {{cite book|last=Stenström|first=Bo|title=Rings and modules of quotients|year=1971|publisher=Springer-Verlag|series=Lecture Notes in Mathematics, Vol. 237|___location=Berlin|pages=vii+136|isbn=978-3-540-05690-4|mr=0325663 }}
* [[Serge Lang]], "Algebraic Number Theory," Springer, 2000. pages 3–4.
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