Betti Cones of Stanley-Reisner Ideals
David Carey
TL;DR
This work studies Betti diagrams of squarefree monomial ideals by translating algebraic data into combinatorial information through the Stanley–Reisner correspondence and Hochster’s formula, and by organizing Betti diagrams within Boij–Söderberg cones. It provides precise dimensional descriptions for cones generated by diagrams of squarefree monomial ideals (D_n) and edge ideals (C_n), including height-restricted subcones (D_n^h, C_n^h), and develops a framework of PR complexes to realize pure Betti diagrams of any degree type. The results yield both upper and lower bounds that match, giving exact dimensions, and establish a pathway to generate pure diagrams of arbitrary degree type, advancing the Boij–Söderberg program in the squarefree/Stanley–Reisner setting. The work also introduces and analyzes PR complexes, linking homology index sets to degree-type data and clarifying the relationship with Cohen–Macaulay theory and duality.
Abstract
The aim of this thesis is to investigate the Betti diagrams of squarefree monomial ideals in polynomial rings. We use two key tools to help us study these diagrams. The first is the Stanley-Reisner Correspondence, which assigns a unique simplicial complex to every squarefree monomial ideal, and thus allows us to compute the Betti diagrams of these ideals from combinatorial properties of their corresponding complexes. As such, most of our work is combinatorial in nature. The second tool is Boij-Soderberg Theory, which views Betti diagrams as vectors in a rational vector space, and investigates them by considering the convex cone they generate. This thesis applies the theory to the cones generated by diagrams of squarefree monomial ideals. We begin by introducing all of these concepts, along with some preliminary results in both algebra and combinatorics. Chapter 3 then presents the dimensions of our cones, along with the vector spaces they span. Chapters 4 and 5 are devoted to the pure Betti diagrams in these cones, and the combinatorial properties of their associated complexes. Finally, Chapter 6 builds on these results to prove a partial analogue of the first Boij-Soderberg conjecture for squarefree monomial ideals, by detailing an algorithm for generating pure Betti diagrams of squarefree monomial ideals of any degree type.