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Difference between revisions of "Zariski topology"

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''on an affine space $A^n$''
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The topology defined on $A^n$ by taking the closed sets to be the
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The ''Zariski topology on an affine space $A^n$''
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is the topology defined on $A^n$ by taking the closed sets to be the
 
algebraic subvarieties of $A^n$. If $X$ is an affine algebraic variety
 
algebraic subvarieties of $A^n$. If $X$ is an affine algebraic variety
 
(see
 
(see
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The Zariski topology was first introduced by O. Zariski
 
The Zariski topology was first introduced by O. Zariski
[[#References|[1]]], as a topology on the set of valuations of an
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{{Cite|Za}}, as a topology on the set of valuations of an
 
algebraic function field. Though, in general, the Zariski topology is
 
algebraic function field. Though, in general, the Zariski topology is
 
not separable, many constructions of algebraic topology carry over to
 
not separable, many constructions of algebraic topology carry over to
 
it
 
it
[[#References|[2]]]. An affine scheme endowed with the Zariski
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{{Cite|Se}}. An affine scheme endowed with the Zariski
 
topology is quasi-compact.
 
topology is quasi-compact.
  
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD
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{|
valign="top"> O. Zariski, "The compactness of the Riemann manifold of
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|-
an abstract field of algebraic functions" ''Bull. Amer. Math. Soc.'' ,
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|valign="top"|{{Ref|Ha}}||valign="top"| R. Hartshorne, "Algebraic geometry", Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}}
'''50''' : 10 (1944) pp. 683–691 {{MR|0011573}} {{ZBL|0063.08390}} </TD></TR><TR><TD
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|-
valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, , ''Fibre spaces
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|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre,, ''Fibre spaces and their applications'', Moscow (1958) pp. 372–450 (In Russian; translated from French)
and their applications'' , Moscow (1958) pp. 372–450 (In Russian;
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|-
translated from French)</TD></TR></table>
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|valign="top"|{{Ref|Za}}||valign="top"| O. Zariski, "The compactness of the Riemann manifold of an abstract field of algebraic functions" ''Bull. Amer. Math. Soc.'', '''50''' : 10 (1944) pp. 683–691 {{MR|0011573}} {{ZBL|0063.08390}}
 
+
|-
 
+
|}
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD
 
valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977)
 
pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 

Revision as of 12:02, 24 November 2013

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

The Zariski topology on an affine space $A^n$ is the topology defined on $A^n$ by taking the closed sets to be the algebraic subvarieties of $A^n$. If $X$ is an affine algebraic variety (see Affine algebraic set) in $A^n$, the induced topology on $X$ is also known as the Zariski topology. In a similar manner one defines the Zariski topology of the affine scheme ${\rm Spec}\; A$ of a ring $A$ (sometimes called the spectral topology) — the closed sets are all the sets $$V(\mathfrak l) = \{{\mathfrak p}\in {\rm Spec A} \mid {\mathfrak p} \supset {\mathfrak l}\},$$ where ${\mathfrak l}$ is an ideal of $A$.

The Zariski topology was first introduced by O. Zariski [Za], as a topology on the set of valuations of an algebraic function field. Though, in general, the Zariski topology is not separable, many constructions of algebraic topology carry over to it [Se]. An affine scheme endowed with the Zariski topology is quasi-compact.

The topology most naturally defined on an arbitrary scheme is also called the Zariski topology in order to distinguish between it and the étale topology, or, if the variety $X$ is defined over the field ${\mathbb C}$, between it and the topology of an analytic space on the set of complex-valued points of $X({\mathbb C})$.

References

[Ha] R. Hartshorne, "Algebraic geometry", Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
[Se] J.-P. Serre,, Fibre spaces and their applications, Moscow (1958) pp. 372–450 (In Russian; translated from French)
[Za] O. Zariski, "The compactness of the Riemann manifold of an abstract field of algebraic functions" Bull. Amer. Math. Soc., 50 : 10 (1944) pp. 683–691 MR0011573 Zbl 0063.08390
How to Cite This Entry:
Zariski topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zariski_topology&oldid=30761
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article