Asymptotic properties of infinite directed unions of local quadratic transforms

TL;DR

This paper investigates the asymptotic behavior of order valuation rings arising from infinite sequences of local quadratic transforms of regular local rings, linking these properties to the ring-theoretic nature of their union.

Contribution

It provides a detailed analysis of the asymptotic properties of order valuations and their relation to the structure of the union ring, including explicit descriptions and examples.

Findings

Existence of a unique limit point V for the family of order valuation rings.

Connection between asymptotic valuation behavior and whether the union ring is archimedean or integrally closed.

Examples of union rings that are archimedean and integrally closed but not valuation domains.

Abstract

We consider infinite sequences {R_n} of successive local quadratic transforms of a regular local ring. Let S denote the directed union of the sequence of regular local rings R_n. We previously showed the existence of a unique limit point V of the family of order valuation rings of the sequence. In this paper, we examine asymptotic properties of this family of order valuations. We link this asymptotic behavior to ring-theoretic properties of S, namely whether S is archimedean and whether S is completely integrally closed. We construct examples of such S that are archimedean and completely integrally closed but not valuation domains. We give an explicit description of V, where the description depends on whether S is archimedean or non-archimedean.