Date of Award
8-2022
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair/Advisor
Dr. Keri Sather-Wagstaff
Committee Member
Dr. Michael Burr
Committee Member
Dr. James Coykendall
Committee Member
Dr. Felice Manganiello
Abstract
We investigate algebra structures on resolutions of a special class of Cohen-Macaulay simplicial complexes. Given a simplicial complex, we define a pure simplicial complex called the purification. These complexes arise as a generalization of certain independence complexes and the resultant Stanley-Reisner rings have numerous desirable properties, e.g., they are Cohen-Macaulay. By realizing the purification in the context of work of D'alì, et al., we obtain a multi-graded, minimal free resolution of the Alexander dual ideal of the Stanley-Reisner ideal. We augment this in a standard way to obtain a resolution of the quotient ring, which is likewise minimal and multi-graded. Ultimately, we propose an explicit product on the resolution and prove that, if associative, this product imparts a differential graded (DG) algebra structure on the minimal resolution.
Recommended Citation
Morra, Todd Anthony, "Minimal Differential Graded Algebra Resolutions Related to Certain Stanley-Reisner Rings" (2022). All Dissertations. 3113.
https://tigerprints.clemson.edu/all_dissertations/3113