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}}.

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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 <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''-->

''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>

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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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==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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* {{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.