Localization (commutative algebra): Difference between revisions

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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 field <math>\Q</math> of [[rational number]]s from the ring <math>\Z</math> of [[integer]]s.

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'' (c.fcf. the example given at [[local ring]]).

== Localization of a ring ==

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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}} as unique element.

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 togethermore technical, straightforward and boring.

The universal property satisfied by <math>j\colon R\to S^{-1}R</math> is the following:

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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 {{mvar|<math>x}}</math> is an element of a commutative ring {{mvar|R}} and <math>S=\{1, x, x^2, \ldots\},</math> then <math>S^{-1}R</math> can be identified (is [[canonical isomorphism|canonically isomorphic]] to) <math>R[x^{-1}]=R[s]/(xs-1).</math> (The proof consists of showing that this ring satisfies the above universal property.) The ring <math>S^{-1}R</math> is generally denoted <math>R_x</math>.<ref>This definition makes sense even if ''x'' is [[nilpotent]], which would make ''S'' a finite set that contains 0, but in that case, <math>R_x=S^{-1}R =0</math>.</ref> This sort of localization plays a fundamental role in the definition of an [[affine scheme]].

*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]].

''<!--Properties to be moved in another section''-->

*Localization commutes with formations of finite sums, products, intersections and radicals;<ref>{{harvnb|Atiyah|MacDonaldMacdonald|1969|loc=Proposition 3.11. (v).}}</ref> e.g., if <math>\sqrt{I}</math> denote the [[radical of an ideal]] ''I'' in ''R'', then

''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''^&minus;1''R'' and the set of prime ideals of ''R'' that doare not[[Disjoint intersectsets|disjoint]] from ''S''. This bijection is induced by the given homomorphism ''R'' → ''S''^{&nbsp;&minus;1}''R''.

=== 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 setsets, then there exists <math>t\in R</math> such that {{mvar|st}} belongs to the other.

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.

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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 the [[integer]]s, one refers to a property relative to an integer {{mvar|n}} as a property true ''at'' {{mvar|n}} or ''away'' from {{mvar|n}}, depending on the localization that is considered. "'''Away from''' {{mvar|n}}" means that the property is considered after localization by the powers of {{mvar|n}}, and, if {{mvar|p}} is a [[prime number]], "at {{mvar|p}}" means that the property is considered after localization at the prime ideal <math>p\Z</math>. This terminology can be explained by the fact that, if {{mvar|p}} is prime, the nonzero prime ideals of the localization of <math>\Z</math> are either the [[singleton set]] {{math|{{mset|p}}}} or its complement in the set of prime numbers.

== Localization and saturation of ideals ==

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==Localization of a module ==

Let {{mvar|<math>R}}</math> be a [[commutative ring]], {{mvar|<math>S}}</math> be a [[multiplicative set]] in {{mvar|<math>R}}</math>, and {{mvar|<math>M}}</math> be an {{mvar|<math>R}}</math>-[[module (mathematics)|module]]. The '''localization of the module''' {{mvar|<math>M}}</math> by {{mvar|<math>S}}</math>, denoted {{<math|''>S''<sup>−1^{-1}M</supmath>''M''}}, is an {{<math|''>S''<sup>−1^{-1}R</supmath>''R''}}-module that is constructed exactly as the localization of {{mvar|<math>R}}</math>, except that the numerators of the fractions belong to {{mvar|<math>M}}</math>. That is, as a set, it consists of [[equivalence class]]es, denoted <math>\frac ms</math>, of pairs {{<math|>(''m'', ''s'')}}</math>, where <math>m\in M</math> and <math>s\in S,</math> and two pairs {{<math|>(''m'', ''s'')}}</math> and {{<math|>(''n'', ''t'')}}</math> are equivalent if there is an element {{mvar|<math>u}}</math> in {{mvar|<math>S}} </math> such that

:<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|''>S''<sup>−1^{-1}M</supmath>''M''}} is also an {{mvar|<math>R}}</math>-module with scalar multiplication

:<math> r\, \frac{m}{s} = \frac r1 \frac ms = \frac{rm}s.</math>

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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>{{harvnb|Eisenbud|1995|loc=Proposition 2.10}}</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|MacDonaldMacdonald|1969|loc=Proposition 3.14.}}</ref> In particular,

:<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.

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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.

== Localization to Zariski open sets ==

{{empty section|date=April 2021}}

== 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''^&minus;1 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 andFrancis|author1-link=Michael MacDonaldAtiyah|last2=Macdonald|first2=I.G.|author2-link=Ian G. Macdonald|title=Introduction to Commutative Algebra.|publisher=Westview AddisonPress|isbn=978-Wesley.0-201-40751-8|year=1969}}

*[[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=2|edition=2nd |year=1989|publisher=John Wiley & Sons Ltd|___location=Chichester|pages=xvi+428|chapter=§ 9.3|isbn=0-471-92234-X |mr=1006872 }}

* {{cite book|last=Cohn|first=P. M.|title=Algebra |volume=3 |edition=2nd |year=1991|publisher=John Wiley & Sons Ltd|___location=Chichester|pages=xii+474|chapter=§ 9.1|isbn=0-471-92840-2 |mr=1098018 }}

* {{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}}

*Matsumura. Commutative Algebra. Benjamin-Cummings

* {{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&ndash;4.