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
$\Gamma_{i} \rightarrow \Gamma_{i-1}$ 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
${\mathbb{P}}_{X^{2}}(\mathcal{E})$.
Readership
Graduate students and research mathematicians interested in
algebraic geometry.