Points on Quantum Projectivizations (eBook)

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 ${/mathbb{P}}_{X^{2}}(/mathcal{E})$.