Zariski–Riemann space
Concept in algebraic geometry / From Wikipedia, the free encyclopedia
In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve.
Zariski–Riemann spaces were introduced by Zariski (1940, 1944) who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by Nagata (1962) who used them to show that algebraic varieties can be embedded in complete ones.
Local uniformization (proved in characteristic 0 by Zariski) can be interpreted as saying that the Zariski–Riemann space of a variety is nonsingular in some sense, so is a sort of rather weak resolution of singularities. This does not solve the problem of resolution of singularities because in dimensions greater than 1 the Zariski–Riemann space is not locally affine and in particular is not a scheme.