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

Betti Numbers of Graded Modules and Cohomology of Vector Bundles