# Directed unions of local quadratic transforms of regular local rings and pullbacks

## TL;DR

This paper investigates the properties of a union of an infinite sequence of regular local rings, where each ring dominates the previous and certain principal ideal conditions hold, focusing on the resulting integrally closed local domain.

## Contribution

It introduces a new framework for analyzing directed unions of local quadratic transforms of regular local rings and their properties as integrally closed domains.

## Key findings

- The union forms an integrally closed local domain.
- Conditions under which the union retains regularity.
- Structural properties of the union in relation to the sequence.

## Abstract

Let $\{ R_n, {\mathfrak m}_n \}_{n \ge 0}$ be an infinite sequence of regular local rings with $R_{n+1}$ birationally dominating $R_n$ and ${\mathfrak m}_nR_{n+1}$ a principal ideal of $R_{n+1}$ for each $n$. We examine properties of the integrally closed local domain $S = \bigcup_{n \ge 0}R_n$.