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Difference between revisions of "Noetherian scheme"

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(Category:Algebraic geometry)
 
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A [[Scheme|scheme]] admitting a finite open covering by spectra of Noetherian rings (cf. [[Noetherian ring|Noetherian ring]]). An affine Noetherian scheme is precisely the spectrum of a Noetherian ring. The topological space of a Noetherian scheme $X$ is a Noetherian topological space, and the local rings $\mathcal O_{X,x}$ are Noetherian. If every point of a scheme has an open affine Noetherian neighbourhood, the scheme is called locally Noetherian. A quasi-compact locally Noetherian scheme is a Noetherian scheme. An example of a Noetherian scheme is a scheme of finite type over a field (an algebraic variety) or over any Noetherian ring.
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A [[scheme]] admitting a finite open covering by [[Spectrum of a ring|spectra]] of [[Noetherian ring]]s. An affine Noetherian scheme is precisely the spectrum of a Noetherian ring. The topological space of a Noetherian scheme $X$ is a [[Noetherian space]], and the local rings $\mathcal O_{X,x}$ are Noetherian. If every point of a scheme has an open affine Noetherian neighbourhood, the scheme is called locally Noetherian. A quasi-compact locally Noetherian scheme is a Noetherian scheme. An example of a Noetherian scheme is a scheme of finite type over a field (an [[algebraic variety]]) or over any Noetherian ring.
  
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
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[[Category:Algebraic geometry]]

Latest revision as of 21:42, 31 October 2014

A scheme admitting a finite open covering by spectra of Noetherian rings. An affine Noetherian scheme is precisely the spectrum of a Noetherian ring. The topological space of a Noetherian scheme $X$ is a Noetherian space, and the local rings $\mathcal O_{X,x}$ are Noetherian. If every point of a scheme has an open affine Noetherian neighbourhood, the scheme is called locally Noetherian. A quasi-compact locally Noetherian scheme is a Noetherian scheme. An example of a Noetherian scheme is a scheme of finite type over a field (an algebraic variety) or over any Noetherian ring.


Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Noetherian scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_scheme&oldid=32136
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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