Betti Numbers of Syzygies and Cohomology of Coherent Sheaves

TL;DR

This paper reviews the characterization of Betti tables and cohomology tables of vector bundles, providing new proofs and applications, including a proof of the Multiplicity Conjecture, advancing the understanding of algebraic invariants.

Contribution

It offers a simplified proof of the Boij-S"oderberg theory results and explores applications like the proof of the Multiplicity Conjecture.

Findings

Characterization of the cone of Betti tables

Description of the cone of cohomology tables

Proof of the Multiplicity Conjecture

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 S\"oderberg 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 S\"oderberg, a proof of the Multiplicity Conjecture of Herzog, Huneke and Srinivasan.