Localization (commutative algebra): Difference between revisions

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==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>

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.

Properties of a ring that can be characterized on its local rings are called ''local properties'', and are often the algebraic counterpart of geometric [[local property|local properties]] of [[algebraic varieties]], which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property thethat refers to localization to Zariski open sets; see {{slink||Localization to Zariski open sets}}, below.)

Many local properties are a consequence of the fact that the ring

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=== Examples of local properties ===

A property {{mvar|P}} of an {{mvar|R}}-module {{mvar|M}} is a ''local property'' if the following conditions are equivalent:

* {{mvar|P}} holds for {{mvar|M}}.

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* {{mvar|M}} is a [[flat module]].

* {{mvar|M}} is an [[invertible module]] (in the case where {{mvar|R}} is a commutative ___domain, and {{mvar|M}} is a submodule of the [[field of fractions]] of {{mvar|R}}).

* <math>f:\colon M \to N</math> is injective (resp. surjective), where {{mvar|N}} is another {{mvar|R}}-module.

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.