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, $mathcal{E}$ is a coherent ${mathcal{O}}_{X}$-bimodule and $mathcal{I} subset T(mathcal{E})$ is a graded ideal then we develop a compatibility theory on adjoint squares in order to construct the functor $Gamma_{n}$ of flat families of truncated $T(mathcal{E})/mathcal{I}$-point modules of length $n+1$. For $n geq 1$ we represent $Gamma_{n}$ as a closed subscheme of ${mathbb{P}}_{X^{2}}({mathcal{E}}^{otimes n})$. The representing scheme is defined in terms of both ${mathcal{I}}_{n}$ 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 $Gamma_{i} ightarrow Gamma_{i-1}$ which makes the set ${Gamma_{n}}$ an inverse system. In order for the point modules of $T(mathcal{E})/mathcal{I}$ 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 ${mathsf{Proj}} T(mathcal{E})/mathcal{I}$ is a quantum ruled surface. In this case, we show the point modules over $T(mathcal{E})/mathcal{I}$ are parameterized by the closed points of ${mathbb{P}}_{X^{2}}(mathcal{E})$.