An abstract characterization of noncommutative projective lines

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 ${\mathbb{P}}_n^1$, Piontkovski's $n$th noncommutative projective line, is the noncommutative projectivization of an $n$-dimensional vector space.

Numerous minor changes throughout, including title. Lemma 2.2 significantly strengthened. Main results are identical to previous version

References in corpus (3)

• A categorical characterization of quantum projective spaces
• Witt's theorem for noncommutative conics
• A representation theoretic study of noncommutative symmetric algebras