The geometry of points on quantum projectivizations

TL;DR

This paper investigates the geometric structure of point modules over certain graded algebras associated with schemes, showing they are parameterized by a projectivized bundle, and constructs resolutions in special cases.

Contribution

It characterizes the parameter space of point modules over noncommutative graded algebras using geometric methods and constructs explicit resolutions for modules over projective lines.

Findings

Point modules are parameterized by the closed points of a projectivized bundle.

The functor of flat families of point modules is studied and characterized.

Explicit bimodule resolutions are constructed for the case when the base scheme is projective line.

Abstract

Suppose $S$ 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. We study the geometry of the functor $\Gamma_{n}$ of flat families of truncated $\mathcal{B}=T(\mathcal{E})/\mathcal{I}$-point modules of length $n+1$. We then use the results of our study to show that the point modules over $\mathcal{B}$ are parameterized by the closed points of ${\mathbb{P}}_{X^{2}}(\mathcal{E})$. When $X={\mathbb{P}}^{1}$, we construct, for any $\mathcal{B}$-point module, a graded ${\mathcal{O}}_{X}-\mathcal{B}$-bimodule resolution.