On local uniformization of orderings

Abstract

The main application of the results of this paper is to prove the existence of real valuation rings of the quotient field K of an excellent domain A having prescribed centers, ranks, rational ranks and residue dimensions. The major part of the paper develops the machinery necessary for this task. Let α be a point in the real spectrum of the ring A. Much of the paper is devoted to showing that α can be uniformized by means of quadratic transforms. Assume now that A is local. Write Gα for the topological space of all generations of α in the quotient field K, write Uα for the subset of elements βGα which uniformize α, and write Bα for the set of βGα which blow up α. The two main theorems state that the interior of Uα is dense in Gα and that the set Bα is dense in Gα.