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''on an affine space $A^n$''
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{{MSC|14}}
 +
{{TEX|done}}
  
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 $A^n$''
 +
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 [[Affine algebraic set|Affine algebraic set]]) in $A^n$, the [[induced topology]] on $X$ is also known as the Zariski topology. In a similar
[[Affine algebraic set|Affine algebraic set]]) in $A^n$, the induced
+
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
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  
 
are all the sets  
 
$$V(\mathfrak l) = \{{\mathfrak p}\in {\rm Spec A} \mid {\mathfrak p} \supset {\mathfrak l}\},$$
 
$$V(\mathfrak l) = \{{\mathfrak p}\in {\rm Spec A} \mid {\mathfrak p} \supset {\mathfrak l}\},$$
Line 13: Line 12:
  
 
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
+
{{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
+
{{Cite|Se}}. An affine scheme endowed with the Zariski
topology is quasi-compact.
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topology is a [[quasi-compact space]].
  
The topology most naturally defined on an arbitrary
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The topology most naturally defined on an arbitrary [[scheme]] is also called the Zariski topology in order to distinguish between it and the [[Etale topology|étale topology]], or, if the variety $X$ is defined
[[Scheme|scheme]] is also called the Zariski topology in order to
+
over the field ${\mathbb C}$, between it and the topology of an analytic space on the set of complex-valued points of $X({\mathbb C})$.
distinguish between it and the
 
[[Etale topology|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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD
+
{|
valign="top"> O. Zariski, "The compactness of the Riemann manifold of
+
|-
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
+
|-
valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, , ''Fibre spaces
+
|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;
+
|-
translated from French)</TD></TR></table>
+
|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>
 

Latest revision as of 20:50, 15 March 2023

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 a quasi-compact space.

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=30760
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article