Rational mapping: Difference between revisions

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In [[algebraic geometry]], a '''noetherian scheme''' is a [[scheme (mathematics)|scheme]] that admits a finite covering by open affine subsets <math>\operatorname{Spec} A_i</math>, <math>A_i</math> [[noetherian rings]]. More generally, a scheme is '''locally noetherian''' if it is covered by spectra of noetherian rings. Thus, a scheme is noetherian if and only if it is locally noetherian and quasi-compact. As with noetherian rings, the concept is named after [[Emmy Noether]].
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It can be shown that, in a locally noetherian scheme, if&nbsp; <math>\operatorname{Spec} A</math> is an open affine subset, then ''A'' is a noetherian ring. In particular, <math>\operatorname{Spec} A</math> is a noetherian scheme if and only if ''A'' is a noetherian ring. Let ''X'' be a locally noetherian scheme. Then the local rings <math>\mathcal{O}_{X, x}</math> are noetherian rings.
 
A noetherian scheme is a [[noetherian topological space]]. But the converse is false in general; consider, for example, the spectrum of a non-noetherian valuation ring.
 
The definitions extend to [[formal scheme]]s.
 
== References ==
* [[Robin Hartshorne]], ''Algebraic geometry''.
 
 
 
 
 
 
{{geometry-stub}}
 
[[Category:Algebraic geometry]]

Latest revision as of 23:11, 25 September 2014

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