Complements of the point schemes of noncommutative projective lines

TL;DR

This paper investigates the structure of the complement of the point scheme in noncommutative projective lines, showing it forms a noncommutative Dedekind domain of Gelfand-Kirillov dimension 1 and analyzing its simplicity.

Contribution

It characterizes the localized coordinate ring as a noncommutative Dedekind domain and explores conditions for its simplicity, extending understanding of noncommutative projective geometry.

Findings

The complement of the point scheme is a noncommutative Dedekind domain.

The localized ring has Gelfand-Kirillov dimension 1.

Simplicity of these Dedekind domains follows a dichotomy similar to noncommutative quadrics.

Abstract

Recently, Chan and Nyman constructed noncommutative projective lines via a noncommutative symmetric algebra for a bimodule $V$ over a pair of fields. These noncommutative projective lines of contain a canonical closed subscheme (the point scheme) determined by a normal family of elements in the noncommutative symmetric algebra. We study the complement of this subscheme when $V$ is simple, the coordinate ring of which is obtained by inverting said normal family. We show that this localised ring is a noncommutative Dedekind domain of Gelfand-Kirillov dimension 1. Furthermore, the question of simplicity of these Dedekind domains is answered by a similar dichotomy to an analogous open subscheme of the noncommutative quadrics of Artin, Tate and Van den Bergh.