Directed unions of local monoidal transforms and GCD domains

Abstract

Let $(R,\mathfrak{m})$ be a regular local ring of dimension $d\ge 2$. 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 $S={\cup }_{n\ge 0}^{\infty }{R}_n$ 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.

21 pages, comments are welcome