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Betti numbers of graded modules and cohomology of vector bundles


Authors: David Eisenbud and Frank-Olaf Schreyer
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
Published electronically: October 27, 2008
MathSciNet review: 2505303
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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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Additional Information

David Eisenbud
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
Email: eisenbud@math.berkeley.edu

Frank-Olaf Schreyer
Affiliation: Mathematik und Informatik, Universität des Saarlandes, Campus E2 4, D-66123 Saarbrücken, Germany
Email: schreyer@math.uni-sb.de

DOI: https://doi.org/10.1090/S0894-0347-08-00620-6
Received by editor(s): January 17, 2008
Published electronically: October 27, 2008
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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