Ideal theory of infinite directed unions of local quadratic transforms

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 $S^{\ast }$ of $S$ has the form $S^{\ast }=W\cap T$, where $W$ is the rank $1$ valuation overring of $V$.

Final version, to appear in J. of Algebra

Cited by in corpus (1)

• Asymptotic properties of infinite directed unions of local quadratic transforms