Betti numbers of graded modules and cohomology of vector bundles

Eisenbud, David; Schreyer, Frank-Olaf

doi:10.1090/S0894-0347-08-00620-6

Betti numbers of graded modules and cohomology of vector bundles
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by David Eisenbud and Frank-Olaf Schreyer

J. Amer. Math. Soc. 22 (2009), 859-888

DOI: https://doi.org/10.1090/S0894-0347-08-00620-6

Published electronically: October 27, 2008

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Abstract:

In the remarkable paper Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture, Mats Boij and Jonas Söderberg conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring is a positive linear combination of Betti tables of modules with pure resolutions. We prove a strengthened form of their conjectures. 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.

With the same tools we show that the cohomology table of any vector bundle on projective space is a positive rational linear combination of the cohomology tables of what we call supernatural vector bundles. Using this result we give new bounds on the slope of a vector bundle in terms of its cohomology.

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