Journal of Algebraic Combinatorics

, Volume 12, Issue 3, pp 207–222 | Cite as

Shifting Operations and Graded Betti Numbers

  • Annetta Aramova
  • Jürgen Herzog
  • Takayuki Hibi


The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a squarefree monomial ideal are bounded by those of the corresponding squarefree lexsegment ideal.

algebraic shifting shifted complexes generic initial ideals extremal Betti numbers 


  1. 1.
    I. Anderson, Combinatorics of Finite Sets, Oxford Science Publications, 1987.Google Scholar
  2. 2.
    A. Aramova and J. Herzog, “Almost regular sequences and Betti numbers,” Amer. J. Math. 122 (2000), 689–719.Google Scholar
  3. 3.
    A. Aramova, J. Herzog, and T. Hibi, “Squarefree lexsegment ideals,” Math. Z. 228 (1998), 353–378.Google Scholar
  4. 4.
    A. Aramova, J. Herzog, and T. Hibi, “Ideals with stable Betti numbers,” Adv. Math. 152 (2000), 72–77.Google Scholar
  5. 5.
    A. Aramova, J. Herzog, and T. Hibi, “Gotzmann theorems for exterior algebras combinatorics,” J. Alg. 191 (1997), 174–211.Google Scholar
  6. 6.
    D. Bayer, H. Charalambous, and S. Popescu, “Extremal Betti numbers and Applications to Monomial Ideals,” J. Alg. 221 (1999), 497–512.Google Scholar
  7. 7.
    W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised edition, Cambridge, 1996.Google Scholar
  8. 8.
    A. Björner and G. Kalai, “An extended Euler-Poincaré theorem,” Acta Math. 161 (1988), 279–303.Google Scholar
  9. 9.
    D. Eisenbud, Commutative Algebra, with a View Towards Algebraic Geometry, Graduate Texts Math., Springer, 1995.Google Scholar
  10. 10.
    S. Eliahou and M. Kervaire, “Minimal resolutions of some monomial ideals,” J. Alg. 129 (1990), 1–25.Google Scholar
  11. 11.
    M. Green, “Generic initial ideals,” in Proc. CRM-96, Six Lectures on Commutative Algebra, Barcelona, Spain, Vol. 166, pp. 119–186, Birkhäuser, 1998.Google Scholar
  12. 12.
    T. Hibi, Algebraic Combinatorics on Convex Polytopes, CarslawPublications, Glebe, N.S.W., Australia, 1992.Google Scholar
  13. 13.
    J. Herzog and N. Terai, “Stable properties of algebraic shifting,” Results Math. 35 (1999), 260–265.Google Scholar
  14. 14.
    M. Hochster, “Cohen-Macaulay rings, combinatorics, and simplicial complexes,” in Proc. Ring Theory II, Lect. Notes in Pure and Appl Math., Vol. 26, pp. 171–223, Deeker, New York, 1977.Google Scholar
  15. 15.
    G. Kalai, “Algebraic shifting,” Unpublished manuscript, 1993.Google Scholar
  16. 16.
    G. Kalai, “The diameter of graphs of convex polytopes and f-vector theory,” in Proc. Applied Geometry and Discrete Mathematics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 4, pp. 387–411, Amer. Math. Soc., 1991.Google Scholar
  17. 17.
    R.P. Stanley, Combinatorics and Commutative Algebra, Birkhäuser, 1983.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Annetta Aramova
  • Jürgen Herzog
  • Takayuki Hibi

There are no affiliations available

Personalised recommendations