Namespaces
Variants
Actions

Difference between revisions of "Noetherian scheme"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
Line 1: Line 1:
−
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066860/n0668601.png" /> is a Noetherian topological space, and the local rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066860/n0668602.png" /> 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.
+
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.


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
Retrieved from "https://encyclopediaofmath.org/index.php?title=Noetherian_scheme&oldid=32136"