Graduation Semester and Year

Spring 2024



Document Type


Degree Name

Doctor of Philosophy in Mathematics



First Advisor

Michaela Vancliff


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 $1 \leq n < \infty$.


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


Algebra | Algebraic Geometry



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