Dictionary

Zariski topology en · NOUN

Etymology

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

Meanings

1. 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.
   • If Y#92;subseteqKⁿ is an affine variety, then by definition the Zariski topology on Y has the subvarieties of Y as closed sets. 2011, Gregor Kemper, A Course in Commutative Algebra, Springer, page 34:
   • 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 X#92;timesY of two affine varieties is stronger than the product of the Zariski topologies on X and Y. 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:
   • 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.
   • 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. 2008, Gert-Martin Greuel, Gerhard Pfister, A Singular Introduction to Commutative Algebra, 2nd edition, Springer, page 523:

Forms

Relateds

Zariski

Translations (3)

ru топология Зарисского (Translations) · pl topologia Zariskiego (Translations) · de Zariski-Topologie (Translations)

wikipedia: Oscar Zariski