The geometry of arithmetic noncommutative projective lines

TL;DR

This paper investigates the geometric properties of arithmetic noncommutative projective lines, classifying their vector bundles, isomorphisms, and automorphism groups to understand their structure.

Contribution

It provides a comprehensive classification of arithmetic noncommutative projective lines, including their isomorphisms and automorphisms, advancing the understanding of their geometric structure.

Findings

Proved these spaces are integral.

Classified vector bundles over them.

Determined their automorphism groups.

Abstract

Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space equal to the projectivization of the noncommutative symmetric algebra of a k-central two -sided vector space of rank two over K. We study the geometry of these spaces. More precisely, we prove they are integral, we classify vector bundles over them, we classify them up to isomorphism, and we classify isomorphisms between them. Using the classification of isomorphisms, we compute the automorphism group of an arithmetic noncommutative projective line.