Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



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
Published electronically: October 27, 2008
MathSciNet review: 2505303
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [2006] M. Boij and J. Söderberg. Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture. math.AC/0611081.
  • [2008] M. Boij and J. Söderberg. Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case. Preprint: arXiv:0803.1645.
  • [1973a] D. Buchsbaum and D. Eisenbud. Remarks on ideals and resolutions. Symposia Math. XI (1973) 193-204. MR 0337946 (49:2715)
  • [1973b] D. A. Buchsbaum and D. Eisenbud. What makes a complex exact? J. Algebra 25 (1973) 259-268. MR 0314819 (47:3369)
  • [1995] D. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Math. 150, Springer-Verlag, New York (1995). MR 1322960 (97a:13001)
  • [2003] D. Eisenbud, G. Fløystad and F.-O. Schreyer. Sheaf cohomology and free resolutions over exterior algebras. Trans. Amer. Math. Soc. 355 (2003) 4397-4426. MR 1990756 (2004f:14031)
  • [2007] D. Eisenbud, G. Fløystad and J. Weyman. The existence of pure free resolutions. arXiv:0709.1529.
  • [2003] D. Eisenbud and F.-O. Schreyer. Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16 (2003) 537-579. MR 1969204 (2004j:14067)
  • [2007] D. Erman. The Semigroup of Betti Diagrams. arXiv:0806.4401
  • [2007] C.A Francisco and H. Srinivasan. Multiplicity conjectures. In Syzygies and Hilbert Functions, ed. I. Peeva, Lect. Notes in Pure and Appl. Math., Chapman and Hall, NY, 2007. MR 2309929 (2008f:13044)
  • [M2] D. R. Grayson and M. E. Stillman. Macaulay 2, a software system for research in algebraic geometry. Available at
  • [1982] R. Hartshorne and A. Hirschowitz. Cohomology of a general instanton bundle. Ann. Sci. de l'École Normale Sup. 15 (1982) 365-390. MR 683638 (84c:14011)
  • [1984] J. Herzog and M. Kühl. On the Betti numbers of finite pure and linear resolutions. Comm. in Alg. 12 (13) (1984) 1627-1646. MR 743307 (85e:13021)
  • [1998] J. Herzog and H. Srinivasan. Bounds for multiplicities. Trans. Am. Math. Soc. 350 (1998) 2879-2902. MR 1458304 (99g:13033)
  • [1974] D. Kirby. A sequence of complexes associated to a matrix. J. London Math. Soc. 7 (1974) 523-530. MR 0337939 (49:2708)
  • [2008] M. Kunte. Gorenstein modules of finite length. Thesis, Uni. des Saarlandes (2008).
  • [1992] S. Mukai. Curves and symmetric spaces. Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 1, 7-10. MR 1158012 (93d:14042)
  • [2003] S. Mukai. Curves and Symmetric Spaces II. RIMS preprint 2003, http://www.kurims.
  • [1973] C. Peskine and L. Szpiro. Dimension projective finie et cohomologie locale. Applications la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck. Inst. Hautes Etudes Sci. Publ. Math. 42 (1973) 47-119. MR 0374130 (51:10330)
  • [1986] F.-O. Schreyer. Syzygies of canonical curves and special linear series. Math. Ann. 275 (1986), no. 1, 105-137. MR 849058 (87j:14052)
  • [2006] J. Söderberg. Graded Betti numbers and $ h$-vectors of level modules. Preprint, arxiv:math.AC/0612047.
  • [2003] J. Weyman. Cohomology of Vector Bundles and Syzygies. Cambridge Tracts in Math. 149 (2003). MR 1988690 (2004d:13020)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14F05, 13D02, 13D25, 14N99

Retrieve articles in all journals with MSC (2000): 14F05, 13D02, 13D25, 14N99

Additional Information

David Eisenbud
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720

Frank-Olaf Schreyer
Affiliation: Mathematik und Informatik, Universität des Saarlandes, Campus E2 4, D-66123 Saarbrücken, Germany

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.

American Mathematical Society