Date of Award
Master of Science (MS)
School of Mathematical and Statistical Sciences
Free resolutions for an ideal are constructions that tell us useful information about the structure of the ideal. Every ideal has one minimal free resolution which tells us significantly more about the structure of the ideal. In this thesis, we consider a specific type of resolution, the Lyubeznik resolution, for a monomial ideal I, which is constructed using a total order on the minimal generating set G(I). An ideal is called Lyubeznik if some total order on G(I) produces a minimal Lyubeznik resolution for I. We investigate the problem of characterizing whether an ideal I is Lyubeznik by using covers of generators of I, a construction due to Guo, Wu, and Yu  that is distinct from its Lyubeznik resolution. For monomial ideals of a polynomial ring, we characterize all Lyubeznik ideals that are minimally generated by four or fewer generators, and provide the total order that produces the minimal Lyubeznik resolution for all such ideals.
Fontes, Nathan S., "Lyubeznik Ideals Minimally Generated by Four or Fewer Elements" (2022). All Theses. 3852.