Ph.D. (2008) Cornell University
Abstract: We study the minimal free resolutions of monomial ideals in two different settings.
In the first part, we consider quadratic monomial ideals and focus on the squarefree case, namely that of an edge ideal. We study the minimal free resolution of such an ideal over a polynomial ring in the case where the resolution is linear. In particular, we provide an explicit minimal free resolution under the assumption that the graph associated with the edge ideal satisfies specific combinatorial conditions. In addition, we construct a regular cellular structure on the resolution.
In the second part, we consider free resoutions over toric rings. We describe the structure of the minimal free resolution of a monomial ideal over a toric ring of a rational normal curve. We also prove that Macaulay's Theorem holds for toric rings of rational normal curves.