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Zariski topology

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on an affine space $A^n$

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 $$V(\mathfrak l) = \{{\mathfrak p}\in {\rm Spec A} | {\mathfrak p} \supset {\mathfrak l}\},$$

where ${\mathfrak l}$ is an ideal of $A$.

The Zariski topology was first introduced by O. Zariski [1], 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 [2]. 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

[1] 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
[2] J.-P. Serre, , Fibre spaces

and their applications , Moscow (1958) pp. 372–450 (In Russian;

translated from French)


Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2
How to Cite This Entry:
Zariski topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zariski_topology&oldid=19569
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article