Points on Quantum Projectivizations

The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizataions, a significant class of examples in non-commutative algebraic geometry. More precisely, if S is an affine, noetherian scheme, X is a separated, noetherian S-scheme, E is a coherent OX -bimodule and I ⊂ T (E) is a graded ideal then we develop a compatibility theory on adjoint squares in order to construct the functor Γn of flat families of truncated T (E)/I-point modules of length n + 1. For n ≥ 1 we represent Γn as a closed subscheme of PX2(E ). The representing scheme is defined in terms of both In and the bimodule Segre embedding, which we construct. Truncating a truncated family of point modules of length i+ 1 by taking its first i components defines a morphism Γi → Γi−1 which makes the set {Γn} an inverse system. In order for the point modules of T (E)/I to be parameterizable by a scheme, this system must be eventually constant. In [20], we give sufficient conditions for this system to be constant and show that these conditions are satisfied when ProjT (E)/I is a quantum ruled surface. In this case, we show the point modules over T (E)/I are parameterized by the closed points of PX2(E).