Affine scheme: Difference between revisions
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imported>Aleksander Stos m (Affine Scheme moved to Affine scheme: convention) |
imported>Giovanni Antonio DiMatteo |
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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 Space|topology]] of closed sets, where closed subsets are defined to be of the form | 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 Space|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>. | 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 | ||
<math>(E)</math> is the ideal of <math>A</math> generated by <math>E</math>. | |||
==Some Topological Properties== | ==Some Topological Properties== | ||
Revision as of 13:42, 9 December 2007
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 .
Some Topological Properties
is Hausdorff
The Structural Sheaf
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.
