Betti Numbers of Graded Modules and Cohomology of Vector Bundles

TL;DR

This paper proves a strengthened form of Boij and Soederberg's conjecture on Betti tables of Cohen-Macaulay modules, with applications to the Multiplicity Conjecture and cohomology of vector bundles.

Contribution

It establishes a strengthened conjecture on Betti tables and characterizes the cone of cohomology tables of vector bundles on projective spaces.

Findings

Proof of the strengthened Boij-Soderberg conjecture

Verification of the Multiplicity Conjecture for Cohen-Macaulay modules

Characterization of cohomology tables of vector bundles

Abstract

Mats Boij and Jonas Soederberg (math.AC/0611081) have conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring can be decomposed in a certain way as a positive linear combination of Betti tables of modules with pure resolutions. We prove, over any field, a strengthened form of their conjecture. Applications include a proof of the Multiplicity Conjecture of Huneke and Srinivasan and a proof of the convexity of a fan naturally associated to the Young lattice. We also characterize the rational cone of all cohomology tables of vector bundles on projective spaces in terms of the cohomology tables of "supernatural" bundles. This characterization is dual, in a certain sense, to our characterization of Betti tables.