Author(s): Richie Sheng, Deborah Wooton
Mentor(s): Tim Tribone, Trung Chau
Institution U of U
A central topic of interest in the field of combinatorial commutative algebra is to understand the structure of a special class of an algebraic object called monomial ideals. One tool to do so is called a minimal free resolution, which encodes various invariants about the monomial ideal to which it corresponds. However, while finding a free resolution for a given monomial ideal is generally straightforward, such constructions are rarely minimal. Other constructions always give a minimal complex (a more general class of objects, to which free resolutions belong), but these complexes may not be resolutions. For example, given a monomial ideal, we can find the ideal’s Scarf complex via a relatively straightforward procedure. If the Scarf complex is a resolution, then it will always be the minimal free resolution for the ideal. However, Scarf complexes are not necessarily free resolutions. Two specific types of monomial ideals are path ideals and connected ideals, which come from mathematical objects called graphs. Graphs encapsulate pairwise relationships (represented as edges) between a set of objects (represented as vertices). Path ideals and connected ideals correspond to different paths within and different connected subsets of the graph, respectively. We fully characterize which graphs have connected ideals that have minimal free resolutions coinciding with the Scarf complex construction and partially classify which graphs have path ideals with this property. Given a graph, we can therefore know whether the Scarf complex is an appropriate tool to find the minimal free resolution of the graph’s path ideal or connected ideal and thereby obtain the structural information contained in the minimal free resolution.