Affine scheme: Difference between revisions
imported>Giovanni Antonio DiMatteo |
imported>Giovanni Antonio DiMatteo (→The Structural Sheaf: adding def of structural sheaf) |
||
| Line 16: | Line 16: | ||
# 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>. | ||
# For all <math>f\in A</math>, <math>\Gamma(D(f),O_X)\simeq A_f</math>, where <math>A_f</math> is the localization of <math>A</math> by the multiplicative set <math>S=\{1,f,f^2,\ldots\}</math>. In particular, <math>\Gamma(X,O_X)\simeq A</math>. | # For all <math>f\in A</math>, <math>\Gamma(D(f),O_X)\simeq A_f</math>, where <math>A_f</math> is the localization of <math>A</math> by the multiplicative set <math>S=\{1,f,f^2,\ldots\}</math>. In particular, <math>\Gamma(X,O_X)\simeq A</math>. | ||
Explicitly, the structural sheaf <math>O_X=</math> may be constructed as follows. To each open set <math>U</math>, associate the set of functions <math>O_X(U):=\{s:U\to \coprod_{p\in U\} A_p|s(p)\in A_p, \text{ and }s\text{ is locally constant}\}</math>; that is, <math>s</math> is locally constant if for every <math>p\in U</math>, there is an open neighborhood <math>V</math> contained in <math>U</math> and elements <math>a,f\in A</math> such that for all <math>q\in V</math>, <math>s(q)=a/f\in A_q</math> (in particular, <math>f</math> is required to not be an element of any <math>q\in V</math>). 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== | ==The Category of Affine Schemes== | ||
Revision as of 17:03, 12 December 2007
Definition
For a commutative 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} , the 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 Spec(A)} (called the prime spectrum 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} ) 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
- 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(E)=\{p\in Spec(A)| p\supseteq E\}}
for any subset 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\subseteq A} . This topology of closed sets is called 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)} . It is easy to check that 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(E)=V\left((E)\right)=V(\sqrt{(E)})} , 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 (E)} 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} .
Some Topological Properties
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 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.
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 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=} , called the structural sheaf of X. 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 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(U):=\{s:U\to \coprod_{p\in U\} A_p|s(p)\in A_p, \text{ and }s\text{ is locally constant}\}} ; 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.