Narrow Results

The purpose of this paper is to examine the freeness of nonzero reflexive modules M and N over a regular local ring R, under the condition is zero.

Let $(R,𝔪)$ be a regular local ring of dimension $d\ge 2$. A local monoidal transform of $R$ is a ring of the form $R_1=R{[\frac{𝔭}{x}]}_{{𝔪}_1}$, where $x\in 𝔭$ is a regular parameter, $𝔭$ is a regular prime ideal of $R$ and ${𝔪}_1$ is a maximal ideal of $R[\frac{𝔭}{x}]$ lying over $𝔪.$ In this paper, 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.

The following sections are included:

• Introduction
• Basics
• Valuations and blowing ups
• A Complete Set of Key Polynomials
• The order relation on $\mathcal{V}$
• Nonmetric Tree Structure on $\mathcal{V}$