Zariski
Speech parts
1. Zariski - Adjective
2. Zariski - Adverb
3. Zariski - Proper noun
Meaning
Zariski
A surname.
Zariski (not comparable)
attributive form of Zariski topology.
Zariski closure — Zariski decomposition — Zariski density
Zariski (not comparable)
attributive form of Zariski topology.
Zariski closed — Zariski open — Zariski dense
Examples
A basis for the Zariski topology can be constructed as follows.
This formulation is analogous to the construction of the cotangent space to define the Zariski tangent space in algebraic geometry.
The Zariski topology on Sch is generated by the pretopology whose covering families are jointly surjective families of scheme-theoretic open immersions.
If one only considers the points of A, i.e. the maximal ideals in R, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets).
In both categories, a covering family is defined be a family which is a cover on Zariski open subsets.
The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes.