## 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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## Bibliographic Information

**David Eisenbud**- Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
- MR Author ID: 62330
- ORCID: 0000-0002-5418-5579
- Email: eisenbud@math.berkeley.edu
**Frank-Olaf Schreyer**- Affiliation: Mathematik und Informatik, Universität des Saarlandes, Campus E2 4, D-66123 Saarbrücken, Germany
- MR Author ID: 156975
- Email: schreyer@math.uni-sb.de
- Received by editor(s): January 17, 2008
- Published electronically: October 27, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**22**(2009), 859-888 - MSC (2000): Primary 14F05, 13D02; Secondary 13D25, 14N99
- DOI: https://doi.org/10.1090/S0894-0347-08-00620-6
- MathSciNet review: 2505303

Dedicated: Dedicated to Mark Green, whose work connecting Algebraic Geometry and Free Resolutions has inspired us for a quarter of a century, on the occasion of his sixtieth birthday