Betti Numbers of Syzygies and Cohomology of Coherent Sheaves

Abstract

The Betti numbers of a graded module over the polynomial ring form a table of numerical invariants that refines the Hilbert polynomial. A sequence of papers sparked by conjectures of Boij and Söderberg have led to the characterization of the possible Betti tables up to rational multiples---that is, to the rational cone generated by the Betti tables. We will summarize this work by describing the cone and the closely related cone of cohomology tables of vector bundles on projective space, and we will give new, simpler proofs of some of the main results. We also explain some of the applications of the theory, including the one that originally motivated the conjectures of Boij and Söderberg, a proof of the Multiplicity Conjecture of Herzog, Huneke and Srinivasan.

References in corpus (7)

• Graded Betti numbers of Cohen-Macaulay modules and the Multiplicity conjecture
• The Existence of Pure Free Resolutions
• Pieri resolutions for classical groups
• Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case
• A special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture
• Filtering free resolutions
• Gorenstein Modules of Finite Length

Cited by in corpus (10)

• Tensor complexes: Multilinear free resolutions constructed from higher tensors
• Categorified duality in Boij-Söderberg Theory and invariants of free complexes
• Filtering free resolutions
• Shapes of free resolutions over a local ring
• Triplets of pure free squarefree complexes
• Three flavors of extremal Betti tables
• Questions about Boij-Söderberg theory
• Non-Ulrich representation type
• The reduction number and degree bound of projective subschemes
• Bounds for the Multiplicity of Gorenstein algebras