Complements of the point schemes of noncommutative projective lines

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 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 for an analogous open subscheme of the noncommutative quadrics of Artin, Tate and Van den Bergh.