Difference between revisions of "Noetherian scheme"
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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 | + | 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. |
Revision as of 20:25, 3 May 2014
A scheme admitting a finite open covering by spectra of Noetherian rings (cf. 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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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
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