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

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.