Zariski topology

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Zariski topology

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Etymology

After Russian-born American mathematician Oscar Zariski (1899—1986), regarded as one of the most influential algebraic geometers of the 20th century.

Noun

Zariski topology (plural Zariski topologies)

1. (algebraic geometry) Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring.

   The Zariski topology allows tools from topology to be used in the study of algebraic varieties, even when the underlying field is not a topological field.

   • 1994, V. I. Danilov, “II. Algebraic Varieties and Schemes”, in I. R. Shafarevich, editor, Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes, Springer, page 184:

     Thus the Zariski open subsets are 'very big'; in particular, the Zariski topology is highly non-Hausdorff.
     A further difference with the classical topology is that the Zariski topology on the product ${\displaystyle X\times Y}$ of two affine varieties is stronger than the product of the Zariski topologies on ${\displaystyle X}$ and ${\displaystyle Y}$.

   • 2008, Gert-Martin Greuel, Gerhard Pfister, A Singular Introduction to Commutative Algebra, 2nd edition, Springer, page 523:

     In particular, “singular” is a local notion, where “local” so far was mainly considered with respect to the Zariski topology. However, since the Zariski topology is so coarse, small neighbourhoods in the Zariski topology might not be local enough.

   • 2011, Gregor Kemper, A Course in Commutative Algebra, Springer, page 34:

     If ${\displaystyle Y\subseteq K^{n}}$ is an affine variety, then by definition the Zariski topology on ${\displaystyle Y}$ has the subvarieties of ${\displaystyle Y}$ as closed sets.

Usage notes

The application of Zariski topology to algebraic varieties is a central idea of scheme theory, a branch of algebraic geometry.

Related terms

• Zariski (attributive form)

Translations

Translations

German: Zariski-Topologie f Russian: топология Зарисского f (topologija Zarisskovo)

Further reading

• Zariski topology on nLab
• Zariski Topology on MathWorld
• Scheme theory on Wikipedia.Wikipedia
• Spectrum of a ring on Wikipedia.Wikipedia
• Zariski geometry on Wikipedia.Wikipedia