Zariski space
Encyclopedia
In algebraic geometry
, a Zariski space, named for Oscar Zariski
, has several different meanings:
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a Zariski space, named for Oscar Zariski
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
, has several different meanings:
- A topological space that is Noetherian (every open set is quasicompact)
- A topological space that is Noetherian and also soberSober spaceIn mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...
(every nonempty closed irreducible subset is the closure of a unique point). The spectrum of any commutative Noetherian ring is a Zariski space in this sense . - A Zariski–Riemann spaceZariski–Riemann spaceIn algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a space whose points are valuation rings containing k and properly contained in K...
of valuations of a field.