Betti Cones of Stanley-Reisner Ideals

The aim of this thesis is to investigate the Betti diagrams of squarefree monomial ideals in polynomial rings. Betti diagrams encode information about the minimal free graded resolutions of these ideals, and are therefore important algebraic invariants.

Computing resolutions is a difficult task in general, but in our case there are tools we can use to simplify it. Most immediately, the Stanley-Reisner Correspondence assigns a unique simplicial complex to every squarefree monomial ideal, and Hochster’s Formula allows us to compute the Betti diagrams of these ideals from combinatorial properties of their corresponding complexes. This reduces the algebraic problem of computing resolutions to the (often easier) combinatorial problem of computing homologies. As such, most of our work is combinatorial in nature.

The other key tool we use in studying these diagrams is Boij-Soderberg Theory. This theory views Betti diagrams as vectors in a rational vector space, and investigates them by considering the convex cone they generate. This technique has proven very instructive: it has allowed us to classify all Betti diagrams up to integer multiplication. This thesis applies the theory more narrowly, to the cones generated by diagrams of squarefree monomial ideals.

In Chapter 2 we introduce all of these concepts, along with some preliminary results in both algebra and combinatorics we will need going forward. Chapter 3 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.