Directed unions of local monoidal transforms and GCD domains

TL;DR

This paper investigates the properties of rings formed by infinite unions of local monoidal transforms of regular local rings, focusing on conditions under which these unions are GCD domains, and extends results to more general GCD domain unions.

Contribution

It introduces a detailed study of infinite directed unions of local monoidal transforms and characterizes when these unions are GCD domains, extending to broader GCD domain contexts.

Findings

Identifies conditions for unions to be GCD domains.

Provides structural insights into local monoidal transforms.

Extends results to general GCD domain unions.

Abstract

Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d \geq 2$. A local monoidal transform of $R$ is a ring of the form $R_1= R[\frac{\mathfrak{p}}{x}]_{\mathfrak{m}_1}$ where $x \in \mathfrak{p}$ is a regular parameter, $\mathfrak{p}$ is a regular prime ideal of $R$ and $ \mathfrak{m}_1 $ is a maximal ideal of $ R[\frac{\mathfrak{p}}{x}] $ lying over $ \mathfrak{m}. $ In this article we study some features of the rings $ S= \cup_{n \geq 0}^{\infty} R_n $ obtained as infinite directed union of iterated local monoidal transforms of $R$. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.