Boij-Söderberg Decompositions, Cellular Resolutions, and Polytopes

Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. In chapter 2 using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules. These results are published in the Journal of Algebra, see [25]. In chapter 3 we provide some further results about Boij-Söderberg decompositions. We show how truncation of a pure diagram impacts the decomposition. We also prove constructively that every integer multiple of a pure diagram of codimension 2 can be realized as the Betti table of a module. In chapter 4 we introduce the idea of a c-polar self-dual polytope. We prove that in dimension 2 only the odd n-gons have an embedding which is polar self-dual. We also define the family of Ferrers polytopes. We prove that the Ferrers polytope in dimension d is d-polar self-dual hence establishing a nontrivial example of a polar self-dual polytope in all dimension. Finally we prove that the Ferrers polytope in dimension d supports a cellular resolution of the Stanley-Reisner ring of the (d+3)-gon