Localization (commutative algebra): Difference between revisions

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{{short description|Construction of a ring of fractions, in commutative algebra}}

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 one, 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 ring '''<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 which 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.f. the example given at [[local ring]]).

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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 power 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|1>S=''S'' - \{{mset|1, ''t'', ''t''{{sup|^2}},\ldots ...}}}\}</math> consists of the powers of a single element, <math>S^{-1}R</math> is often denoted <math>S_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}}.

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In the general case, a problem arises with [[zero divisor]]s. Let {{mvar|S}} be a multiplicative set in a commutative ring {{mvar|R}}. If <math>\tfrac a1</math> is the image in <math>S^{-1}R</math> of <math>a\in R,</math> and if {{math|1=''as'' = 0}} with <math>s\in S,</math> then one must have <math>\tfrac 0s = \tfrac {as}{s} = \tfrac a1,</math> and 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>

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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^{{math|-1}R =''S''<sup> −10</supmath>''R'' = {0}}} [[if and only if]] {{math|''S''}} contains {{math|0}}.

* The [[ring homomorphism]] {{<math|''>R''\to → ''S''<sup> −1^{-1}R</supmath>''R''}} is injective if and only if {{math|''S''}} does not contain any [[zero divisor]]s.

* The ring homomorphism {{<math|''>R''\to → ''S''<sup> −1^{-1}R</supmath>''R''}} is an [[epimorphism]] in the [[category of rings]], that is not [[surjective]] in general.

* The ring {{<math|''>S''<sup> −1^{-1}R</supmath>''R''}} 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]].