Localization (commutative algebra): Difference between revisions

Revision as of 14:24, 17 April 2024

Revision as of 14:26, 17 April 2024

Line 183:

== Localization to Zariski open sets ==

Let <math>R</math> be a commutative ring. Its [[Spectrum of a ring|spectrum]] <math>\operatorname{Spec}(R)</math> is by definition an [[affine scheme]]. Initially, this is defined merely to be a [[topological space]] carrying a [[Zariski topology]]. However, this view loses too much information about <math>R</math>, which we can recover by attaching an appropriate [[sheaf]] <math>\Gamma(U, \operatorname{Spec}(R))</math> to it<math>\operatorname{Spec}(R)</math>.

Recall Athat a [[Zariski topology|Zariski closed set]] <math>U^C</math>of <math>\operatorname{Spec}(R)</math> corresponds in some 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>\Gamma(U, \operatorname{Spec}(R))</math> is 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>.