Localization (commutative algebra): Difference between revisions

Revision as of 21:24, 6 September 2022

Revision as of 15:12, 24 November 2023

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''Properties to be moved in another section''

*Localization commutes with formations of finite sums, products, intersections and radicals;<ref>{{harvnb|Atiyah|MacDonaldMacdonald|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>

:In particular, ''R'' is [[reduced ring|reduced]] if and only if its total ring of fractions is reduced.<ref>Borel, AG. 3.3</ref>

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is an isomorphism. If <math>M</math> is a [[finitely presented module]], the natural map

:<math>S^{-1} \operatorname{Hom}_R (M, N) \to \operatorname{Hom}_{S^{-1}R} (S^{-1}M, S^{-1}N)</math>

is also an isomorphism.<ref>{{harvnb|Eisenbud|1995|loc=Proposition 2.10}}</ref>

If a module ''M'' is a [[finitely generated module|finitely generated]] over ''R'', one has

:<math>S^{-1}(\operatorname{Ann}_R(M)) = \operatorname{Ann}_{S^{-1}R}(S^{-1}M),</math>

where <math>\operatorname{Ann}</math> denotes [[annihilator (ring theory)|annihilator]], that is the ideal of the elements of the ring that map to zero all elements of the module.<ref>{{harvnb|Atiyah|MacDonaldMacdonald|1969|loc=Proposition 3.14.}}</ref> In particular,

:<math>S^{-1} M = 0\quad \iff \quad S\cap \operatorname{Ann}_R(M) \ne \emptyset,</math>

that is, if <math>t M = 0</math> for some <math>t \in S.</math><ref>Borel, AG. 3.1</ref>

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{{refbegin}}

* {{Cite book|last1=Atiyah|first1=Michael andFrancis|author1-link=Michael MacDonaldAtiyah|last2=Macdonald|first2=I.G.|author2-link=Ian G. Macdonald|title=Introduction to Commutative Algebra.|publisher=Westview AddisonPress|isbn=978-Wesley.0-201-40751-8|year=1969}}

*[[Armand Borel|Borel, Armand]]. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. {{ISBN|0-387-97370-2}}.

* {{cite book|last=Cohn|first=P. M.|title=Algebra |volume=2|edition=2nd |year=1989|publisher=John Wiley & Sons Ltd|___location=Chichester|pages=xvi+428|chapter=§ 9.3|isbn=0-471-92234-X |mr=1006872 }}