|
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 who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They 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. 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. ==Definition== The Zariski–Riemann space of a field ''K'' over a base field ''k'' is a locally ringed space whose points are the valuation rings containing ''k'' and contained in ''K''. Sometimes the valuation ring ''K'' itself is excluded, and sometimes the points are restricted to the zero-dimensional valuation rings (those whose residue field has transcendence degree zero over ''k''). 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. It is made into a locally ringed space by assigning to any open subset the intersection of the valuation rings of the points of the subset. The local ring at any point is the corresponding valuation ring. The Zariski–Riemann space of a function field can also be constructed as the inverse limit of all complete (or projective) models of the function field. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Zariski–Riemann space」の詳細全文を読む スポンサード リンク
|