Localization (commutative algebra): Difference between revisions

Revision as of 14:57, 17 April 2024

Revision as of 14:57, 17 April 2024

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* and which has a [[sheaf of rings]] <math>\mathcal O</math> attached to it.

A [[Zariski topology|Zariski closed setsset]] <math>U^C</math>of <math>\operatorname{Spec}(R)</math> corresponds in somea certain way to an ideal <math>I \subset R</math>. More precisely, <math>U^C</math> consists of the set of prime ideals <math>P \subset R</math> which are supersets of <math>I</math>. Therefore <math>U</math>, which is Zariski ''open'', consists of the prime ideals <math>P</math> which are not supersets of <math>I</math>.

The ring <math>\mathcal O(U)</math> is then defined to be the localisation of <math>R</math> by the multiplicative set <math>\{f \in R: \forall \text{maximal ideals } M \text{ not containing }I, f \not \in M\}</math>.