Places of algebraic function fields in arbitrary characteristic

  title={Places of algebraic function fields in arbitrary characteristic},
  author={Franz-Viktor Kuhlmann},
  journal={Advances in Mathematics},
  • F. Kuhlmann
  • Published 10 November 2004
  • Mathematics
  • Advances in Mathematics

Kronecker Function Rings of Transcendental Field Extensions

We consider the ring Kr(F/D), where D is a subring of a field F, that is the intersection of the trivial extensions to F(X) of the valuation rings of the Zariski–Riemann space consisting of all

Density of composite places in function fields and applications to real holomorphy rings

Given an algebraic function field F |K and a place ℘ on K, we prove that the places that are composite with extensions of ℘ to finite extensions of K lie dense in the space of all places of F , in a

The constructible topology on spaces of valuation domains

We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an arbitrary spectral space and we observe that this topology coincides with the

On the geometry of Prüfer intersections of valuation rings

Let F be a field, let D be a subring of F and let Z be an irreducible subspace of the space of all valuation rings between D and F that have quotient field F. Then Z is a locally ringed space whose

Witt equivalence of function fields over global fields

In our work we investigate Witt equivalence of general function fields over global fields. It is proven that for any two such fields K and L the Witt equivalence induces a canonical bijection between

The structure of spaces of valuations and the local uniformization problem

The problem of resolution of singularities is a major problem in algebraic geometry. Local uniformization can be seen as its local version. For varieties over fields of characteristic zero, local


It is proved that in an extremal valued field of finite p-degree, the images of all additive polynomials have the optimal approximation property and can be used to improve the axiom system for the elementary theory of Laurent series fields over finite fields.



On local uniformization in arbitrary characteristic, I

We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K

Embedding problems over large fields

In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on

Value groups, residue fields, and bad places of rational function fields

We classify all possible extensions of a valuation from a ground field K to a rational function field in one or several variables over K. We determine which value groups and residue fields can

On places of algebraic function fields.

In this paper we study the space of all places of a function field in n variables. We denote by F/k an (algebraic) function field, i.e. F is a fmitely generated extension of k of transcendence degree

Local Uniformization on Algebraic Varieties

1. In [10] (p. 650) we have proved a uniformization theorem for zero-dimensional valuations on an algebraic surface, over an algebraically closed ground field K (of characteristic zero). In the

On valuation spectra

If K is an ordered field then every convex subring of K is a valuation ring of K. This easy but fundamental observation has made valuation theory a very natural and important tool in real algebraic

Prime ideal structure in commutative rings

0. Introduction. Let ' be the category of commutative rings with unit, and regard Spec (as in [1]) as a contravariant functor from ' to g$7 the category of topological spaces and continuous maps. The

Model theoretic methods in the theory of topological fields.

Suppose β is a base of τ. Then clearly the sentences (1) — (4) do not change their meaning when the variables U and F r nge over β instead of τ. The reason is that 3 F is used only in front of

Resolution of singularities : a research textbook in tribute to Oscar Zariski : based on the courses given at the working week in Obergurgl, Austria, September 7-14, 1997

Oscar Zariski 1899-1986.- Resolution of Singularities 1860-1999.- 1: Classes of the Working Week.- Alterations and resolution of singularities.- Reduction of singularities for differential