Categorified duality in Boij-Söderberg Theory and invariants of free complexes

Abstract

We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-Söderberg 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.

33 pages

References in corpus (3)

• Betti Numbers of Syzygies and Cohomology of Coherent Sheaves
• The cone of Betti diagrams over a hypersurface ring of low embedding dimension
• The cone of Betti tables over a rational normal curve

Cited by in corpus (10)

• The cone of Betti tables over a rational normal curve
• The cone of Betti tables over three non-collinear points in the plane
• Triplets of pure free squarefree complexes
• Towards Boij-Söderberg theory for Grassmannians: the case of square matrices
• Zipping Tate resolutions and exterior coalgebras
• Foundations of Boij-Söderberg Theory for Grassmannians
• Three flavors of extremal Betti tables
• Questions about Boij-Söderberg theory
• Small projective spaces and Stillman uniformity for sheaves
• Syzygies of ${\mathbb{P}}^1\times {\mathbb{P}}^1$: data and conjectures