Ideal theory of infinite directed unions of local quadratic transforms

TL;DR

This paper investigates the ideal-theoretic structure of infinite unions of local quadratic transforms of a regular local ring, revealing unique minimal overring decompositions and properties of integral closures.

Contribution

It establishes the existence of a unique minimal Noetherian overring and describes the integral closure structure of infinite unions of local quadratic transforms.

Findings

Existence of a unique minimal proper Noetherian overring T of S.

Decomposition of S as T intersect V, where V is a limit valuation overring.

Complete integral closure of S expressed as W intersect T, with W a rank 1 valuation overring.

Abstract

Let $R$ be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating $R$, there exists a unique sequence $\{R_n\}$ of local quadratic transforms of $R$ along this valuation domain. We consider the situation where the sequence $\{ R_n \}_{n \ge 0}$ is infinite, and examine ideal-theoretic properties of the integrally closed local domain $S = \bigcup_{n \ge 0} R_n$. Among the set of valuation overrings of $R$, there exists a unique limit point $V$ for the sequence of order valuation rings of the $R_n$. We prove the existence of a unique minimal proper Noetherian overring $T$ of $S$, and establish the decomposition $S = T \cap V$. If $S$ is archimedian, then the complete integral closure $S^{*}$ of $S$ has the form $S^{*} = W \cap T$, where $W$ is the rank $1$ valuation overring of $V$.