# Points on Quantum Projectivizations

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*Adam Nyman*

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

#### Table of Contents

# Table of Contents

## Points on Quantum Projectivizations

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. Compatibilities on Squares 916
- Chapter 3. Construction of the Functor Γ[sub(n)] 3542
- Chapter 4. Compatibility with Descent 5764
- Chapter 5. The Representation of Γ[sub(n)] for Low n 7986
- Chapter 6. The Bimodule Segre Embedding 101108
- Chapter 7. The Representation of Γ[sub(n)] for High n 129136
- Bibliography 139146
- Index 141148 free