Affine scheme: Difference between revisions
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==Definition== | ==Definition== | ||
For a commutative ring <math>A</math>, the set <math>Spec(A)</math> (called the ''prime spectrum of ''<math>A</math>) denotes the set of prime ideals of | For a commutative ring <math>A</math>, the set <math>Spec(A)</math> (called the ''prime spectrum of ''<math>A</math>) denotes the set of prime ideals of ''A''. This set is endowed with a [[Topological pace|topology]] of closed sets, where closed subsets are defined to be of the form | ||
:<math>V(E)=\{p\in Spec(A)| p\supseteq E\}</math> | :<math>V(E)=\{p\in Spec(A)| p\supseteq E\}</math> | ||
for any subset <math>E\subseteq A</math>. This topology of closed sets is called the ''Zariski topology'' on <math>Spec(A)</math>. It is easy to check that <math>V(E)=V\left((E)\right)=V(\sqrt{(E)})</math>, where | for any subset <math>E\subseteq A</math>. This topology of closed sets is called the ''Zariski topology'' on <math>Spec(A)</math>. It is easy to check that <math>V(E)=V\left((E)\right)=V(\sqrt{(E)})</math>, where | ||
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==The functor V and the Zariski topology== | ==The functor V and the Zariski topology== | ||
The Zariski topology on <math>Spec(A)</math> satisfies some properties: it is quasi-compact and <math>T_0</math>, but is rarely Hausdorff. <math>Spec(A)</math> is not, in general, a Noetherian topological space (in fact, it is a Noetherian topological space if and only if <math>A</math> is a | The Zariski topology on <math>Spec(A)</math> satisfies some properties: it is quasi-compact and <math>T_0</math>, but is rarely [[Hausdorff space|Hausdorff]]. <math>Spec(A)</math> is not, in general, a [[Noetherian space|Noetherian topological space]] (in fact, it is a Noetherian topological space if and only if <math>A</math> is a [[Noetherian ring]]. | ||
==The Structural Sheaf== | ==The Structural Sheaf== | ||
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==The Category of Affine Schemes== | ==The Category of Affine Schemes== | ||
Regarding <math>Spec(\cdot)</math> as a contravariant functor between the [[commutative ring|category of commutative rings]] and the category of affine schemes, one can show that it is in fact an [[anti-equivalence]] of categories. | Regarding <math>Spec(\cdot)</math> as a contravariant functor between the [[commutative ring|category of commutative rings]] and the category of affine schemes, one can show that it is in fact an [[Category of functors|anti-equivalence]] of categories. | ||
==Curves==[[Category:Suggestion Bot Tag]] | |||
==Curves== | |||
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Latest revision as of 06:00, 7 July 2024
Definition
For a commutative ring , the set (called the prime spectrum of ) denotes the set of prime ideals of A. This set is endowed with a topology of closed sets, where closed subsets are defined to be of the form
for any subset . This topology of closed sets is called the Zariski topology on . It is easy to check that , where is the ideal of generated by .
The functor V and the Zariski topology
The Zariski topology on satisfies some properties: it is quasi-compact and , but is rarely Hausdorff. is not, in general, a Noetherian topological space (in fact, it is a Noetherian topological space if and only if is a Noetherian ring.
The Structural Sheaf
has a natural sheaf of rings, denoted by and called the structural sheaf of X. The pair is called an affine scheme. The important properties of this sheaf are that
- The stalk is isomorphic to the local ring , where is the prime ideal corresponding to .
- For all , , where is the localization of by the multiplicative set . In particular, .
Explicitly, the structural sheaf may be constructed as follows. To each open set , associate the set of functions
; that is, is locally constant if for every , there is an open neighborhood contained in and elements such that for all , (in particular, is required to not be an element of any ). This description is phrased in a common way of thinking of sheaves, and in fact captures their local nature. One construction of the sheafification functor makes use of such a perspective.
The Category of Affine Schemes
Regarding as a contravariant functor between the category of commutative rings and the category of affine schemes, one can show that it is in fact an anti-equivalence of categories.
==Curves==