Directed unions of local monoidal transforms and GCD domains

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.