Graduation Semester and Year
Summer 2025
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics
First Advisor
David A. Jorgensen
Second Advisor
Dimitar Grantcharov
Third Advisor
Michaela Vancliff
Fourth Advisor
Theresa Jorgensen
Fifth Advisor
Minerva Cordero
Abstract
The Koszul homology of a local ring is a powerful tool in commutative algebra as it provides information on the structure and properties of the ring. In this research, we explore the relationship between quotients of regular local rings and their Koszul homology algebra. One such relationship is detailed by the Tate-Assmus theorem, which asserts, in part, that a ring is a complete intersection if and only if the Koszul homology is generated by its degree 1 homology elements. An objective of this research is to examine and identify the properties of a minimal intersection and its Koszul homology algebra. In particular, we show that a local ring is a minimal intersection if and only if the Koszul homology algebra of the ring localized at any prime ideal decomposes as the tensor product of two subalgebras.
Keywords
Commutative Algebra, Homological Algebra, Koszul Complex, Koszul Homology Algebra, Minimal Intersections, Local Rings
Disciplines
Algebra
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Grebel, Kathryn A., "Algebra Structures for the Koszul Homology of Minimal Intersections" (2025). Mathematics Dissertations. 269.
https://mavmatrix.uta.edu/math_dissertations/269