Species and non-commutative P^1's over non-algebraic bimodules

TL;DR

This paper explores non-commutative projective lines over non-algebraic bimodules, providing a classification of modules and a geometric interpretation, extending previous results by Ringel.

Contribution

It offers a complete description of coherent sheaves on these non-commutative lines and establishes their derived equivalence to bimodule species, generalizing prior work.

Findings

Categories of coherent sheaves are fully described.

Non-commutative projective lines are derived equivalent to bimodule species.

Provides a geometric interpretation for module classification.

Abstract

We study non-commutative projective lines over not necessarily algebraic bimodules. In particular, we give a complete description of their categories of coherent sheaves and show they are derived equivalent to certain bimodule species. This allows us to classify modules over these species and thus generalize, and give a geometric interpretation for, results of C. Ringel.