# An abstract characterization of noncommutative projective lines

## TL;DR

This paper characterizes when a $k$-linear abelian category can be viewed as a noncommutative projective line, providing conditions and applications to specific noncommutative $bP^1$-bundles.

## Contribution

It offers necessary and sufficient conditions for a category to be a noncommutative projective line and applies this to Piontkovski's noncommutative projective lines.

## Key findings

- Characterization of noncommutative projective lines
- Conditions for a category to be a noncommutative $bP^1$-bundle
- Identification of $bP^1_n$ as a noncommutative projectivization

## Abstract

Let $k$ be a field. We describe necessary and sufficient conditions for a $k$-linear abelian category to be a noncommutative projective line, i.e. a noncommutative $\mathbb{P}^{1}$-bundle over a pair of division rings over $k$. As an application, we prove that $\mathbb{P}^{1}_{n}$, Piontkovski's $n$th noncommutative projective line, is the noncommutative projectivization of an $n$-dimensional vector space.