## Graduation Semester and Year

Spring 2024

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

## First Advisor

Michaela Vancliff

## Abstract

During the past 36 years, some research in noncommutative algebra has been driven by attempts to classify AS-regular algebras of global dimension four. Such algebras are often considered to be noncommutative analogues of polynomial rings. In the 1980s, Artin, Tate, and Van den Bergh introduced a projective scheme that parametrizes the point modules over a graded algebra generated by elements of degree one. In 2002, Shelton and Vancliff introduced the concept of line scheme, which is a projective scheme that parametrizes line modules.

This dissertation is in two parts. In the first part, we consider a 1-parameter family of quadratic AS-regular algebras of global dimension four that have a finite point scheme and a line scheme that is a union of three distinct lines, with multiplicities 8, 6, and 6, respectively.

In the second part, we discuss a certain family of quadratic AS-regular algebras $A$ of global dimension $\geq 2$, where $A$ is an Ore extension of a twist, by an automorphism, of the polynomial ring on $n$ variables, where $1 \leq n < \infty$.

## Keywords

line modules, point modules, projective spaces, jose, lozano, michaela, vancliff, quadratic, quantum, zero locus

## Disciplines

Algebra | Algebraic Geometry

## License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License

## Recommended Citation

Lozano, Jose E., "POINT MODULES AND LINE MODULES OF CERTAIN QUADRATIC QUANTUM PROJECTIVE SPACES" (2024). *Mathematics Dissertations*. 2.

https://mavmatrix.uta.edu/math_dissertations/2