# Reverse lexicographic and lexicographic shifting

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## Abstract

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δ_{lex}—an operation that transforms a monomial ideal of S = **K**[*x*_{i}: 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 *I* ⊂ *S* 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*.

## Keywords

Shifting Reverse lexicographic## References

- 1.A. Aramova and J. Herzog, “Almost regular sequences and Betti numbers,”
*American J. Math.***122**(2000), 689–719.MathSciNetGoogle Scholar - 2.A. Aramova, J. Herzog, and T. Hibi, “Squarefree lexsegment ideals,”
*Math. Z.***228**(1998), 353–378.CrossRefMathSciNetGoogle Scholar - 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.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.A.M. Bigatti, A. Conca, and L. Robbiano, “Generic initial ideals and distractions,”
*Comm. Algebra***33**(6) (2005), 1709–1732.Google Scholar - 6.A. Björner and G. Kalai, “An extended Euler-Poincare theorem,”
*Acta Math.***161**(1988), 279–303.MathSciNetGoogle Scholar - 7.A. Conca, “Reduction numbers and initial ideals,”
*Proc. Amer. Math. Soc.***131**(2003), 1015–1020.CrossRefzbMATHMathSciNetGoogle Scholar - 8.D. Eisenbud,
*Commutative Algebra with a View Toward Algebraic Geometry*, Springer-Verlag GTM, New York, 1995.Google Scholar - 9.D. Eisenbud,
*The Geometry of Syzygies*, Springer-Verlag GTM, 2004, to appear.Google Scholar - 10.S. Eliahou and M. Kervaire, “Minimal resolutions of some monomial ideals,”
*J. Alg.***129**(1990), 1–25.CrossRefMathSciNetGoogle Scholar - 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.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.
- 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.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.E. Nevo, “Algebraic shifting and basic constructions on simplicial complexes,” preprint (2002).Google Scholar
- 17.K. Pardue, “Deformation classes of graded modules and maximal Betti numbers,”
*Illinois J. Math***40**(1996), 564–585.zbMATHMathSciNetGoogle Scholar - 18.R. Stanley,
*Combinatorics and Commutative Algebra*, Second Edition, Birkhäuser, Boston, 1996.Google Scholar