Author ORCID Identifier
Date Available
5-15-2024
Year of Publication
2024
Degree Name
Doctor of Philosophy (PhD)
Document Type
Doctoral Dissertation
College
Arts and Sciences
Department/School/Program
Mathematics
First Advisor
Dr. Uwe Nagel
Abstract
Computational commutative algebra has become an increasingly popular area of research. Central to the theory is the notion of a Gröbner basis, which may be thought of as a nonlinear generalization of Gaussian elimination. In 2019, Nagel and Römer introduced FI- and OI-modules over FI- and OI-algebras, which provide a framework for studying sequences of related modules defined over sequences of related polynomial rings. In particular, they laid the foundations of a theory of Gröbner bases for certain classes of OI-modules. In this dissertation we develop an OI-analog of Buchberger's algorithm in order to compute such Gröbner bases, as well as an OI-analog of Schreyer's theorem to compute their modules of syzygies. We also give an application of our results to the computation of free OI-resolutions, and showcase our Macaulay2 package "OIGroebnerBases.m2'' which implements these constructions. Lastly, we show how our results can be tweaked to compute free FI-resolutions.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2024.189
Recommended Citation
Morrow, Michael, "Computational Methods for OI-Modules" (2024). Theses and Dissertations--Mathematics. 114.
https://uknowledge.uky.edu/math_etds/114
