Projective Geometry Associated to some Quadratic, Regular Algebras of Global Dimension Four

Author

Derek C. Tomlin

Graduation Semester and Year

2018

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Michaela Vancliff

Abstract

The attempted classification of regular algebras of global dimension four, so-called quantum P³s, has been a driving force for modern research in noncommutative algebra. Inspired by the work of Artin, Tate, and Van den Bergh, geometric methods via schemes of d-linear modules have been developed by various researchers to further their classification. In this thesis, we compute and analyze the line scheme of two families of algebras -- for both families, almost every algebra can be considered a candidate for a generic quadratic quantum P³. For the first family of algebras, we find that, viewed as a closed subscheme of P⁵, the generic member has a one-dimensional line scheme consisting of eight irreducible curves: one nonplanar elliptic curve in a P³, one nonplanar rational curve with a unique singular point, two planar elliptic curves, and two subschemes, each consisting of the union of a nonsingular conic and a line. For the second family of algebras, we find that, viewed as a closed subscheme of P⁵, the generic member has a one-dimensional line scheme consisting of seven irreducible curves: three nonplanar elliptic curves in a P³ and four planar elliptic curves. Additionally, regarding the first family of algebras, we relate distinguished points of the line scheme to distinguished elements in the algebras. In particular, we explore a connection between certain right ideals of the algebras and how they intersect with a particular family of normalizing sequences.

Keywords

Projective algebraic geometry, Noncommutative algebra

Disciplines

Mathematics | Physical Sciences and Mathematics

License

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

Comments

Degree granted by The University of Texas at Arlington

Recommended Citation

Tomlin, Derek C., "Projective Geometry Associated to some Quadratic, Regular Algebras of Global Dimension Four" (2018). Mathematics Dissertations. 201.
https://mavmatrix.uta.edu/math_dissertations/201