Journal of Algebraic Combinatorics

, Volume 23, Issue 2, pp 107–123 | Cite as

Reverse lexicographic and lexicographic shifting

  • Eric BabsonEmail author
  • Isabella Novik
  • Rekha Thomas


A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δlex—an operation that transforms a monomial ideal of S = K[xi: i ∈ ℕ] that is finitely generated in each degree into a squarefree strongly stable ideal—is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal IS is fixed by lexicographic shifting if and only if I is a universal squarefree lexsegment ideal (abbreviated USLI) of S. Moreover, in the case when I is finitely generated and is not a USLI, it is verified that all the ideals in the sequence \(\{ \Delta_{\rm lex}^{i} (I) \}_{i=0}^{\infty}\)} are distinct. The limit ideal \(\bar{\Delta}(I) = {\rm lim}_{i \rightarrow \infty} \Delta_{\rm lex}^{i} (I)\) is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I.


Shifting Reverse lexicographic 


  1. 1.
    A. Aramova and J. Herzog, “Almost regular sequences and Betti numbers,” American J. Math. 122 (2000), 689–719.MathSciNetGoogle Scholar
  2. 2.
    A. Aramova, J. Herzog, and T. Hibi, “Squarefree lexsegment ideals,” Math. Z. 228 (1998), 353–378.CrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Aramova, J. Herzog, and T. Hibi, “Shifting operations and graded Betti numbers,” J. Algebraic Combin. 12(3), (2000), 207–222.CrossRefMathSciNetGoogle Scholar
  4. 4.
    D. Bayer, H. Charalambous, and S. Popescu, “Extremal Betti numbers and applications to monomial ideals,” J. Algebra 221(2), (1999), 497–512.CrossRefMathSciNetGoogle Scholar
  5. 5.
    A.M. Bigatti, A. Conca, and L. Robbiano, “Generic initial ideals and distractions,” Comm. Algebra 33(6) (2005), 1709–1732.Google Scholar
  6. 6.
    A. Björner and G. Kalai, “An extended Euler-Poincare theorem,” Acta Math. 161 (1988), 279–303.MathSciNetGoogle Scholar
  7. 7.
    A. Conca, “Reduction numbers and initial ideals,” Proc. Amer. Math. Soc. 131 (2003), 1015–1020.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag GTM, New York, 1995.Google Scholar
  9. 9.
    D. Eisenbud, The Geometry of Syzygies, Springer-Verlag GTM, 2004, to appear.Google Scholar
  10. 10.
    S. Eliahou and M. Kervaire, “Minimal resolutions of some monomial ideals,” J. Alg. 129 (1990), 1–25.CrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Herzog, “Generic initial ideals and graded Betti numbers,” Computational Commutative Algebra and Combinatorics, (ed. T. Hibi), Vol. 33 of Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, 2002, pp. 75–120.Google Scholar
  12. 12.
    M. Hochster, “Cohen-Macaulay rings, combinatorics and simplicial complexes,” Proc. Ring Theory II, Lect. Notes in Pure and Applied Math., 26, Dekker, New York, 1977, pp. 171–223.Google Scholar
  13. 13.
    G. Kalai, “Symmetric matroids,” J. Combin. Theory Ser. B 50 (1990), 54–64.Google Scholar
  14. 14.
    G. Kalai, “The diameter of graphs of convex polytopes and f-vector theory,” Applied Geometry and Discrete Mathematics – The Victor Klee Festschrift (eds: P. Gritzmann and B. Sturmfels), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 387–411.Google Scholar
  15. 15.
    G. Kalai, “Algebraic shifting,” in Computational Commutative Algebra and Combinatorics, T. Hibi (ed.), Vol. 33 of Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, 2002, pp. 121–163.Google Scholar
  16. 16.
    E. Nevo, “Algebraic shifting and basic constructions on simplicial complexes,” preprint (2002).Google Scholar
  17. 17.
    K. Pardue, “Deformation classes of graded modules and maximal Betti numbers,” Illinois J. Math 40 (1996), 564–585.zbMATHMathSciNetGoogle Scholar
  18. 18.
    R. Stanley, Combinatorics and Commutative Algebra, Second Edition, Birkhäuser, Boston, 1996.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattle

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