UC Berkeley

Boij-Soederberg theory represents a breakthrough in our understanding of free resolutions. Boij and Soederberg proposed the radical perspective that the numerics of graded free resolutions are best understood "up to scalar multiplication." The conjectures of Boij and Soederberg were then proven in a series of papers. Further, these papers describes the cone of cohomology tables of vector bundles on projective n-space by illustrating a duality with the cone of free resolutions. We use the phrase Boij-Soederberg theory to refer to the study of these two cones and the corresponding decomposition theorems.

Applications of graded free resolutions appear throughout algebraic geometry, commutative algebra, topology, combinatorics, and more; since Boij-Soederberg theory provides a structure theorem about the shapes of graded free resolutions, we might hope to apply the theory widely. One such application arose instantly, leading to a proof of the Herzog-Huneke-Srinivasan Multiplicity Conjecture, which had been open for decades.

However, Boij-Soederberg theory is such a radical departure from usual approaches to free resolutions that it is not immediately applicable to many of the situations where free resolutions arise. Boij-Soederberg theory is almost orthogonal to all previous approaches to understanding graded free resolutions, as the theory uses the combinatorial structure of Betti diagrams to group modules into families, whereas more traditional approaches use flatness to understand families of modules. The overarching goal of this thesis is thus to connect Boij-Soederberg theory with some of the previous avenues of research where graded free resolutions have arisen, and we pursue this theme in several directions.

Chapter 2 builds a framework for overcoming the limitation of working "up to scalar multiplication'". Namely, we apply Boij-Soederberg theoretic results about the cone of Betti diagrams in order to investigate the integral structure of the semigroup of Betti diagrams. Our main results show that this semigroup is locally finitely generated, but that it can otherwise be quite pathological. In addition, we construct a number of nontrivial obstructions which prevent a diagram in the cone of Betti diagrams from belonging to the semigroup of Betti diagrams.

In Chapter 3, we consider the question of whether the Boij-Soederberg decomposition of a Betti diagram of a module is related to a flat deformation of the module. This is an essential mystery raised by Boij-Soederberg theory, and we provide the first results in this direction by producing large families where the Boij-Soederberg decomposition of a Betti diagram closely reflects a special filtration of the module. These results suggest that Boij-Soederberg theory might have deeper, as yet undiscovered, consequences for free resolutions. In addition, we provide applications to the classification of very singular spaces of matrices. This application is related to questions about classifying vector spaces of low rank matrices and torsion-free sheaves on projective spaces. This chapter is based on joint work with David Eisenbud and Frank-Olaf Schreyer.

In Chapter 4, we apply Boij-Soederberg theory to prove a special case of a famous conjecture in commutative algebra: the Buchsbaum-Eisenbud-Horrocks Rank Conjecture. Whereas the Multiplicity Conjecture is about the possible shapes of graded free resolutions, the Buchsbaum-Eisenbud-Horrocks Rank Conjecture is about the possible sizes of graded free resolutions. The conjecture is closely related to a topological conjecture of Carlsson about certain finite group actions on products of spheres. We prove a broad new case of the rank conjecture, and our method of proof--which involves a combination of Boij-Soederberg theory and optimization--is essentially unrelated to any previous work on the conjecture.

Finally, in Chapter 5, we apply Boij-Soederberg theory to the study of the asymptotics of free resolutions. Namely, we provide a lower bound for the Betti numbers of I^t when I is an ideal generated in a single degree. This builds on recent studies of asymptotic Castelnuovo-Mumford regularity, by Cutkosky, Herzog, Kodiyalam, Trung, and Wang.