Editing Localization (commutative algebra) (section)

== Localization of a ring ==
The localization of a [[commutative ring]] {{mvar|R}} by a [[multiplicatively closed set]]  {{mvar|S}} is a new ring <math>S^{-1}R</math> whose elements are fractions with numerators in {{mvar|R}} and denominators in {{mvar|S}}.

If the ring is an [[integral domain]] the construction generalizes and follows closely that of the [[field of fractions]], and, in particular, that of the [[rational numbers]] as the field of fractions of the integers. For rings that have [[zero divisor]]s, the construction is similar but requires more care.

===Multiplicative set===
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 powers of an element" rather than of "the localization by an element".

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>

''In the remainder of this article, only localizations by a multiplicative set are considered.''

=== 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.

For example, the [[decimal fraction]]s are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case, <math>S^{-1}R</math> consists of the rational numbers that can be written as <math>\tfrac n{10^k},</math> where {{mvar|n}} is an integer, and {{mvar|k}} is a nonnegative integer.

=== 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}}. Suppose that <math>s\in S,</math> and <math>0\ne a\in R</math> is a zero divisor with <math>as=0.</math> Then <math>\tfrac a1</math> is the image in <math>S^{-1}R</math> of <math>a\in R,</math> and one has <math>\tfrac a1 = \tfrac {as}s = \tfrac 0s = \tfrac 01.</math> Thus some nonzero elements of {{mvar|R}} must be zero in <math>S^{-1}R.</math> The construction that follows is designed for taking this into account.

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 
:<math>\frac {r_1}{s_1}+\frac {r_2}{s_2} = \frac{r_1s_2+r_2s_1}{s_1s_2},</math>
multiplication
:<math>\frac {r_1}{s_1}\,\frac {r_2}{s_2} = \frac{r_1r_2}{s_1s_2},</math>
[[additive identity]] <math>\tfrac 01,</math> and [[multiplicative identity]] <math>\tfrac 11.</math>

The [[function (mathematics)|function]] 
:<math>r\mapsto \frac r1</math>
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 more technical.

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 [[monoid]] or the [[group of units]] of the ring. The [[morphism]]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 <math>\mathcal F\colon \mathcal D \to \mathcal C</math> be the forgetful functor that forgets that the elements of the second element of the pair are invertible.

Then the factorization <math>f=g\circ j</math> of the universal property defines a bijection
:<math>\hom_\mathcal C((R,S), \mathcal F(T,U))\to \hom_\mathcal D ((S^{-1}R, j(S)), (T,U)).</math>

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 <math>R=\Z</math> is the ring of [[integer]]s, and <math>S=\Z\setminus \{0\},</math> then <math>S^{-1}R</math> is the field <math>\Q</math> of the [[rational number]]s.
*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.) 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.

=== 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 [[ideal (ring theory)|ideals]], [[module (mathematics)|modules]], or several multiplicative sets are considered in other sections.

* <math>S^{-1}R = 0</math> [[if and only if]] {{math|''S''}} contains {{math|0}}.
* The [[ring homomorphism]] <math>R\to S^{-1}R</math> is injective if and only if {{math|''S''}} does not contain any [[zero divisor]]s.
* The ring homomorphism <math>R\to S^{-1}R</math> is an [[epimorphism]] in the [[category of rings]], that is not [[surjective]] in general.
* 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|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>
*Let ''R'' be an integral domain with the field of fractions ''K''. Then its localization <math>R_\mathfrak{p}</math> at a prime ideal <math>\mathfrak{p}</math> can be viewed as a subring of ''K''. Moreover,
::<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>&minus;1</sup>''R'' and the set of prime ideals of ''R'' that are [[Disjoint sets|disjoint]] from ''S''. This bijection is induced by the given homomorphism ''R'' → ''S''<sup>&nbsp;&minus;1</sup>''R''.

=== Saturation of a multiplicative set ===
Let <math>S \subseteq R</math> be a multiplicative set. The ''saturation'' <math>\hat{S}</math> of <math>S</math> is the set 
:<math>\hat{S} = \{ r \in R \colon \exists s \in R, rs \in S \}.</math>
The multiplicative set {{mvar|S}} is ''saturated'' if it equals its saturation, that is, if <math>\hat{S}=S</math>, or equivalently, if  <math>rs \in S</math> implies that {{mvar|r}} and {{mvar|s}} are in {{mvar|S}}.

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 sets, 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.