Zariski topology
Let denote the affine space over a field . The Zariski topology on is defined to be the topology whose closed sets are the sets
where is any ideal in the polynomial ring . For any affine variety , the Zariski topology on is defined to be the subspace topology induced on as a subset of .
Let denote –dimensional projective space over . The Zariski topology on is defined to be the topology whose closed sets are the sets
where is any homogeneous ideal in the graded –algebra . For any projective variety , the Zariski topology on is defined to be the subspace topology induced on as a subset of .
The Zariski topology is the predominant topology used in the study of algebraic geometry. Every regular morphism of varieties is continuous in the Zariski topology (but not every continuous map in the Zariski topology is a regular morphism). In fact, the Zariski topology is the weakest topology on varieties making points in closed and regular morphisms continuous.
Title | Zariski topology |
---|---|
Canonical name | ZariskiTopology |
Date of creation | 2013-03-22 12:38:11 |
Last modified on | 2013-03-22 12:38:11 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 4 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 14A10 |
Related topic | PrimeSpectrum |