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
HTML articles powered by AMS MathViewer

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

PDF | Request permission

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

David A. Buchsbaum and David Eisenbud, Remarks on ideals and resolutions, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971 & Convegno di Geometria, INDAM, Rome, 1972) Academic Press, London-New York, 1973, pp. 193–204. MR 337946
David A. Buchsbaum and David Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259–268. MR 314819, DOI 10.1016/0021-8693(73)90044-6
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
David Eisenbud, Gunnar Fløystad, and Frank-Olaf Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4397–4426. MR 1990756, DOI 10.1090/S0002-9947-03-03291-4
David Eisenbud and Frank-Olaf Schreyer, Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc. 16 (2003), no. 3, 537–579. With an appendix by Jerzy Weyman. MR 1969204, DOI 10.1090/S0894-0347-03-00423-5
Christopher A. Francisco and Hema Srinivasan, Multiplicity conjectures, Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math., vol. 254, Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 145–178. MR 2309929, DOI 10.1201/9781420050912.ch5

http://www.math.uiuc.edu/Macaulay2/

Robin Hartshorne and André Hirschowitz, Cohomology of a general instanton bundle, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 365–390. MR 683638
J. Herzog and M. Kühl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 (1984), no. 13-14, 1627–1646. MR 743307, DOI 10.1080/00927878408823070
Jürgen Herzog and Hema Srinivasan, Bounds for multiplicities, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2879–2902. MR 1458304, DOI 10.1090/S0002-9947-98-02096-0
D. Kirby, A sequence of complexes associated with a matrix, J. London Math. Soc. (2) 7 (1974), 523–530. MR 337939, DOI 10.1112/jlms/s2-7.3.523
Shigeru Mukai, Curves and symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 1, 7–10. MR 1158012

http://www.kurims.

kyoto-u.ac.jp/preprint/preprint$\_$y2003.html

C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119 (French). MR 374130
Frank-Olaf Schreyer, Syzygies of canonical curves and special linear series, Math. Ann. 275 (1986), no. 1, 105–137. MR 849058, DOI 10.1007/BF01458587
Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR 1988690, DOI 10.1017/CBO9780511546556