Editing Ring (mathematics) (section)

=== Localization ===
The [[localization of a ring|localization]] generalizes the construction of the [[field of fractions]] of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring {{mvar|R}} and a subset {{mvar|S}} of {{mvar|R}}, there exists a ring <math>R[S^{-1}]</math> together with the ring homomorphism <math>R \to R\left[S^{-1}\right]</math> that "inverts" {{mvar|S}}; that is, the homomorphism maps elements in {{mvar|S}} to unit elements in <math>R\left[S^{-1}\right],</math> and, moreover, any ring homomorphism from {{mvar|R}} that "inverts" {{mvar|S}} uniquely factors through <math>R\left[S^{-1}\right].</math>{{sfnp|Cohn|1995|loc=Proposition 1.3.1|ps=}} The ring <math>R\left[S^{-1}\right]</math> is called the '''localization''' of {{mvar|R}} with respect to {{mvar|S}}. For example, if {{mvar|R}} is a commutative ring and {{mvar|f}} an element in {{mvar|R}}, then the localization <math>R\left[f^{-1}\right]</math> consists of elements of the form <math>r/f^n, \, r \in R , \, n \ge 0</math> (to be precise, <math>R\left[f^{-1}\right] = R[t]/(tf - 1).</math>){{sfnp|Eisenbud|1995|loc=Exercise 2.2|ps=}}

The localization is frequently applied to a commutative ring {{mvar|R}} with respect to the complement of a prime ideal (or a union of prime ideals) in&nbsp;{{mvar|R}}. In that case <math>S = R - \mathfrak{p},</math> one often writes <math>R_\mathfrak{p}</math> for <math>R\left[S^{-1}\right].</math> <math>R_\mathfrak{p}</math> is then a [[local ring]] with the [[maximal ideal]] <math>\mathfrak{p} R_\mathfrak{p}.</math> This is the reason for the terminology "localization". The field of fractions of an integral domain {{mvar|R}} is the localization of {{mvar|R}} at the prime ideal zero. If <math>\mathfrak{p}</math> is a prime ideal of a commutative ring&nbsp;{{mvar|R}}, then the field of fractions of <math>R/\mathfrak{p}</math> is the same as the residue field of the local ring <math>R_\mathfrak{p}</math> and is denoted by <math>k(\mathfrak{p}).</math><!-- In algebraic geometry, the field of fractions is the localization at the "[[generic point]]" -->

If {{mvar|M}} is a left {{mvar|R}}-module, then the localization of {{mvar|M}} with respect to {{mvar|S}} is given by a [[change of rings]] <math>M\left[S^{-1}\right] = R\left[S^{-1}\right] \otimes_R M.</math><!-- intuitively, {{mvar|M{{sub|p}}}} corresponds to the fiber over {{mvar|p}} in&nbsp;{{mvar|M}}. In particular, there is the canonical map <math>M \to M\left[S^{-1}\right], \, m \mapsto m / 1.</math> Its kernel consists of elements {{mvar|m}} such that {{mvar|1=''sm'' = 0}} for some {{mvar|s}} in {{mvar|S}}. -->

The most important properties of localization are the following: when {{mvar|R}} is a commutative ring and {{mvar|S}} a multiplicatively closed subset
* <math>\mathfrak{p} \mapsto \mathfrak{p}\left[S^{-1}\right]</math> is a bijection between the set of all prime ideals in {{mvar|R}} disjoint from {{mvar|S}} and the set of all prime ideals in <math>R\left[S^{-1}\right].</math>{{sfnp|Milne|2012|loc=Proposition 6.4|ps=}}
* <math>R\left[S^{-1}\right] = \varinjlim R\left[f^{-1}\right],</math> {{mvar|f}} running over elements in {{mvar|S}} with partial ordering given by divisibility.{{sfnp|Milne|2012|loc=end of Chapter 7|ps=}}
* The localization is exact: <math display="block">0 \to M'\left[S^{-1}\right] \to M\left[S^{-1}\right] \to M''\left[S^{-1}\right] \to 0</math> is exact over <math>R\left[S^{-1}\right]</math> whenever <math>0 \to M' \to M \to M'' \to 0</math> is exact over&nbsp;{{mvar|R}}.
* Conversely, if <math>0 \to M'_\mathfrak{m} \to M_\mathfrak{m} \to M''_\mathfrak{m} \to 0</math> is exact for any maximal ideal <math>\mathfrak{m},</math> then <math>0 \to M' \to M \to M'' \to 0</math> is exact.
* A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.)

In [[category theory]], a [[localization of a category]] amounts to making some morphisms isomorphisms. An element in a commutative ring {{mvar|R}} may be thought of as an endomorphism of any {{mvar|R}}-module. Thus, categorically, a localization of {{mvar|R}} with respect to a subset {{mvar|S}} of {{mvar|R}} is a [[functor]] from the category of {{mvar|R}}-modules to itself that sends elements of {{mvar|S}} viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, {{mvar|R}} then maps to <math>R\left[S^{-1}\right]</math> and {{mvar|R}}-modules map to <math>R\left[S^{-1}\right]</math>-modules.)