Zariski–Riemann space
Encyclopedia
In algebraic geometry
, a Zariski–Riemann space or Zariski space of a subring
k of a field
K is a space whose points are valuation ring
s containing k and properly contained in K. When k is the ring of complex numbers and K the field of rational functions of a complex algebraic curve, then the Zariski–Riemann space has essentially the same underlying set as the usual Riemann surface
, a compact nonsingular model of the curve. Zariski–Riemann spaces were introduced by who called them Riemann manifolds or (confusingly) Riemann surfaces, and were named Zariski–Riemann spaces after Oscar Zariski
and Bernhard Riemann
by who used them to show that algebraic varieties can be embedded in complete ones.
If S is the Zariski–Riemann space of a subring k of a field K, it has a topology defined by taking a basis of open sets to be the valuation rings containing a given finite subset of K. The space S is quasi-compact.
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–Riemann space or Zariski space of a subring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
k of a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K is a space whose points are valuation ring
Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D....
s containing k and properly contained in K. When k is the ring of complex numbers and K the field of rational functions of a complex algebraic curve, then the Zariski–Riemann space has essentially the same underlying set as the usual Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
, a compact nonsingular model of the curve. Zariski–Riemann spaces were introduced by who called them Riemann manifolds or (confusingly) Riemann surfaces, and were named Zariski–Riemann spaces after Oscar Zariski
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
and Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
by who used them to show that algebraic varieties can be embedded in complete ones.
If S is the Zariski–Riemann space of a subring k of a field K, it has a topology defined by taking a basis of open sets to be the valuation rings containing a given finite subset of K. The space S is quasi-compact.