Localization (commutative algebra): Difference between revisions

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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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==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>