Categorified duality in Boij-S\"oderberg Theory and invariants of free complexes

TL;DR

This paper develops a categorical framework for duality in Boij-S"oderberg theory, enabling a deeper understanding of invariants of free complexes and extending the theory to new contexts like toric varieties.

Contribution

It introduces a categorical pairing that categorifies Eisenbud and Schreyer's functionals, broadening the scope of Boij-S"oderberg theory to more general settings.

Findings

Constructed a pairing between derived categories that categorifies key functionals.

Described the cone of Betti tables for complexes with specified homology.

Extended the theory to toric varieties, creating multigraded analogues.

Abstract

We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-S\"oderberg conjectures describing numerical invariants of syzygies. The new foundation allows us to extend the reach of the theory substantially. More explicitly, we construct a pairing between derived categories that simultaneously categorifies all the functionals used by Eisenbud and Schreyer. With this new tool, we describe the cone of Betti tables of finite, minimal free complexes having homology modules of specified dimensions over a polynomial ring, and we treat many examples beyond polynomial rings. We also construct an analogue of our pairing between derived categories on a toric variety, yielding toric/multigraded analogues of the Eisenbud-Schreyer functionals.