The tree of quadratic transforms of a regular local ring of dimension two

TL;DR

This paper investigates the topology of the quadratic tree of 2-dimensional regular local overrings of a local ring and studies the structure of rings formed as intersections within this tree, highlighting differences in Noetherian properties.

Contribution

It provides a detailed analysis of the topological structure of the quadratic tree and characterizes the rings obtained as intersections, including non-Noetherian cases.

Findings

Finite intersections in the quadratic tree are Noetherian.

The structure of intersection rings is well understood for finite cases.

Some intersection rings are not Noetherian.

Abstract

Let $D$ be a 2-dimensional regular local ring and let $Q(D)$ denote the quadratic tree of 2-dimensional regular local overrings of $D$. We explore the topology of the tree $Q(D)$ and the family ${\mathcal{R}}(D)$ of rings obtained as intersections of rings in $Q(D)$. If $A$ is a finite intersection of rings in $ Q(D)$, then $A$ is Noetherian and the structure of $A$ is well understood. However, other rings in ${\mathcal{R}}(D)$ need not be Noetherian. The two main goals of this paper are to examine topological properties of the quadratic tree $Q(D)$, and to examine the structure of rings in the set ${\mathcal{R}}(D)$.