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. | 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== | ||
<math>X=Spec(A)</math> has a natural sheaf of rings, denoted <math>O_X | <math>X=Spec(A)</math> has a natural sheaf of rings, denoted by <math>O_X</math> and called the ''structural sheaf'' of ''X''. The pair <math>(Spec(A),O_X)</math> is called an ''affine [[Scheme|scheme]]''. The important properties of this sheaf are that | ||
# The [[ringed space|stalk]] <math>O_{X,x}</math> is isomorphic to the local ring <math>A_{\mathfrak{p}}</math>, where <math>\mathfrak{p}</math> is the prime ideal corresponding to <math>x\in X</math>. | # The [[ringed space|stalk]] <math>O_{X,x}</math> is isomorphic to the local ring <math>A_{\mathfrak{p}}</math>, where <math>\mathfrak{p}</math> is the prime ideal corresponding to <math>x\in X</math>. | ||
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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== | |||
[[Category: | |||
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} generated by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} .
The functor V and the Zariski topology
The Zariski topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Spec(A)} satisfies some properties: it is quasi-compact and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_0} , but is rarely Hausdorff. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Spec(A)} is not, in general, a Noetherian topological space (in fact, it is a Noetherian topological space if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a Noetherian ring.
The Structural Sheaf
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=Spec(A)} has a natural sheaf of rings, denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O_X} and called the structural sheaf of X. The pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Spec(A),O_X)} is called an affine scheme. The important properties of this sheaf are that
- The stalk Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O_{X,x}} is isomorphic to the local ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{\mathfrak{p}}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{p}} is the prime ideal corresponding to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} .
- For all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\in A} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(D(f),O_X)\simeq A_f} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_f} is the localization of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} by the multiplicative set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\{1,f,f^2,\ldots\}} . In particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(X,O_X)\simeq A} .
Explicitly, the structural sheaf Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O_X=} may be constructed as follows. To each open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , associate the set of functions
; that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} is locally constant if for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in U} , there is an open neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} and elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,f\in A} such that for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\in V} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(q)=a/f\in A_q} (in particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is required to not be an element of any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\in V} ). 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Spec(\cdot)} 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==