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Difference between revisions of "Noetherian space"

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A [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668701.png" /> in which every strictly decreasing chain of closed subspaces breaks off. An equivalent condition is: Any non-empty family of closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668702.png" /> ordered by inclusion has a minimal element. Every subspace of a Noetherian space is itself Noetherian. If a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668703.png" /> has a finite covering by Noetherian subspaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668704.png" /> is itself Noetherian. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668705.png" /> is Noetherian if and only if every open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668706.png" /> is quasi-compact. A Noetherian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668707.png" /> is the union of finitely many irreducible components.
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Examples of Noetherian spaces are some spectra of commutative rings (cf. [[Spectrum of a ring|Spectrum of a ring]]). For a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668708.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n0668709.png" /> (the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n06687010.png" />) is Noetherian if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n06687011.png" /> is a [[Noetherian ring|Noetherian ring]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n06687012.png" /> is the nil radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066870/n06687013.png" />.
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A [[topological space]] $X$ in which every strictly decreasing chain of closed subspaces breaks off. An equivalent condition is: Any non-empty family of closed subsets of $X$ ordered by inclusion has a minimal element. Every subspace of a Noetherian space is itself Noetherian. If a space $X$ has a finite covering by Noetherian subspaces, then $X$ is itself Noetherian. A space $X$ is Noetherian if and only if every open subset of $X$ is quasi-compact. A Noetherian space $X$ is the union of finitely many irreducible components.
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Examples of Noetherian spaces are some spectra of commutative rings (cf. [[Spectrum of a ring]]). For a ring $A$ the space $\mathrm{Spec}(A)$ (the spectrum of $A$) is Noetherian if and only if $A/J$ is a [[Noetherian ring|Noetherian ring]], where $J$ is the nil radical of $A$.
  
 
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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD></TR>
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Revision as of 20:19, 30 November 2014


A topological space $X$ in which every strictly decreasing chain of closed subspaces breaks off. An equivalent condition is: Any non-empty family of closed subsets of $X$ ordered by inclusion has a minimal element. Every subspace of a Noetherian space is itself Noetherian. If a space $X$ has a finite covering by Noetherian subspaces, then $X$ is itself Noetherian. A space $X$ is Noetherian if and only if every open subset of $X$ is quasi-compact. A Noetherian space $X$ is the union of finitely many irreducible components.

Examples of Noetherian spaces are some spectra of commutative rings (cf. Spectrum of a ring). For a ring $A$ the space $\mathrm{Spec}(A)$ (the spectrum of $A$) is Noetherian if and only if $A/J$ is a Noetherian ring, where $J$ is the nil radical of $A$.

References

[1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001


Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Noetherian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_space&oldid=35225
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article