Graduation Semester and Year
Spring 2025
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics
First Advisor
David A. Jorgensen
Abstract
Tor-persistence is the claim that Tor of a module with itself is only zero if the module has finite projective dimension. Work done by Avramov, Iyengar, Nasseh, Sather-Wagstaff, and various other authors have proved Tor-persistence of modules over certain rings. In this work, we will prove Tor-persistence for certain modules over determinantal rings, specifically for the hypersurface defined by the determinant of a generic matrix. We will then give an explicit proof that Tor^R_2(M,M) is never zero, that Tor^R_1(M,M)=0, and due to the periodicity of the given free resolution, our result can be extended to the entire complex, showing that all even-degree Tor modules are non-zero and all odd-degree Tor modules are zero. Alongside these results about the generic square matrix, we examine results about Tor for non-square matrices, specifically for matrices that have fewer rows than columns. We conclude the discussion by specializing our generic matrix to provide a host of instances of Tor-persistence over many other rings.
Keywords
Algebra, Homological algebra, Tor, Tor-persistence, Determinantal rings
Disciplines
Algebra | Mathematics | Physical Sciences and Mathematics
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Ajani, Tatheer F., "Self-Tor Persistence of Modules over Determinantal Rings" (2025). Mathematics Dissertations. 266.
https://mavmatrix.uta.edu/math_dissertations/266