Intersection Theory on Non-commutative Surfaces

Consider a non-commutative algebraic surface, X, and an effective divisor Y on X, as defined by Van den Bergh. We show that the Riemann-Roch theorem, the genus formula, and the self intersection formula from classical algebraic geometry generalize to this setting. We also apply our theory to some special cases, including the blow up of X in a point, and show that the self intersection of the exceptional divisor is −1. This is used to give an example of a non-commutative surface with a commutative P1 which cannot be blown down, because its self intersection is +1 rather than −1. We also get some results on Hilbert polynomials of modules on X.