Localization (commutative algebra): Difference between revisions

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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 ring/module ''R'', 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 field <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 that 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.fcf. the example given at [[local ring]]).

== Localization of a ring ==

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* 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]].

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*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>

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::<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''^&minus;1''R'' and the set of prime ideals of ''R'' that doare not[[Disjoint intersectsets|disjoint]] from ''S''. This bijection is induced by the given homomorphism ''R'' → ''S''^{&nbsp;&minus;1}''R''.

=== Saturation of a multiplicative set ===

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* {{cite book|last=Cohn|first=P. M.|title=Algebra |volume=3 |edition=2nd |year=1991|publisher=John Wiley & Sons Ltd|___location=Chichester|pages=xii+474|chapter=§ 9.1|isbn=0-471-92840-2 |mr=1098018 }}

* {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1|mr=1322960 | year=1995 | volume=150}}

*[[Hideyuki Matsumura]]. Commutative Algebra. Benjamin-Cummings

* {{cite book|last=Stenström|first=Bo|title=Rings and modules of quotients|year=1971|publisher=Springer-Verlag|series=Lecture Notes in Mathematics, Vol. 237|___location=Berlin|pages=vii+136|isbn=978-3-540-05690-4|mr=0325663 }}

* [[Serge Lang]], "Algebraic Number Theory," Springer, 2000. pages 3&ndash;4.