Connectedness and integrally closed local overrings of two-dimensional regular local rings

TL;DR

This paper establishes a correspondence between certain connected sets of valuation overrings and integrally closed local overrings in two-dimensional regular local rings, deepening understanding of their structural relationships.

Contribution

It introduces a novel one-to-one correspondence linking closed connected sets of valuation overrings with specific integrally closed local overrings in two-dimensional regular local rings.

Findings

Correspondence between connected valuation sets and integrally closed overrings

Characterization of non-essential and non-divisorial valuation rings

Enhanced understanding of the structure of overrings in regular local rings

Abstract

Let $D$ be a two-dimensional regular local ring. We prove there is a one-to-one correspondence between closed connected sets in the space of valuation overrings of $D$ that dominate $D$ and the integrally closed local overrings of $D$ that are not essential valuation rings or divisorial valuation rings of $D$.