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== Localization to Zariski open sets ==
Let <math>R</math> be a commutative ring. ItsWe can define 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 [[topologicalringed 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 <math>\operatorname{Spec}(R)</math>.whose:
* ''Points'' are the prime ideals of <math>R</math>,
Recall that 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>.
* c''losed sets'' correspond in some way (which we shall describe below) to the [[ideals]] of <math>R</math>,
* and which has a [[sheaf of rings]] <math>\mathcal O</math> attached to it.
TheA ring[[Zariski topology|Zariski closed sets]] <math>\Gamma(U,^C</math>of <math>\operatorname{Spec}(R))</math> iscorresponds definedin tosome beway theto localisationan ofideal <math>I \subset R</math>. byMore theprecisely, multiplicative<math>U^C</math> consists of the set of prime ideals <math>\{fP \insubset R:</math> \forallwhich \text{maximalare idealssupersets }of M<math>I</math>. \text{Therefore not<math>U</math>, containingwhich }Iis Zariski ''open'', fconsists of the prime ideals <math>P</math> which are \not \insupersets M\}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>.
Under the above definition, an affine scheme becomes a sheaf. When we glue these sheaves together, we obtain [[Scheme (mathematics)|schemes]] in the general sense.
== Non-commutative case ==
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