Nagoya Mathematical Journal

Componentwise linear ideals

Jürgen Herzog and Takayuki Hibi

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A componentwise linear ideal is a graded ideal $I$ of a polynomial ring such that, for each degree $q$, the ideal generated by all homogeneous polynomials of degree $q$ belonging to $I$ has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal $I_{\Delta}$ arising from a simplicial complex $\Delta$ is componentwise linear if and only if the Alexander dual of $\Delta$ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

Article information

Nagoya Math. J., Volume 153 (1999), 141-153.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D25
Secondary: 13D02: Syzygies, resolutions, complexes 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]


Herzog, Jürgen; Hibi, Takayuki. Componentwise linear ideals. Nagoya Math. J. 153 (1999), 141--153.

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  • A. Aramova and J. Herzog, Koszul cycles and Eliahou–Kervaire type resolutions , J. Algebra, 181 (1996), 347–370.
  • A. Aramova, J. Herzog and T. Hibi, Squarefree lexsegment ideals , Math. Z., 228 (1998), 353–378.
  • A. Aramova, J. Herzog and T. Hibi, Gotzmann theorems for exterior algebras and combinatorics , J. Algebra, 191 (1997), 174–211.
  • A. Aramova, J. Herzog and T. Hibi, Weakly stable ideals , Osaka J. Math., 34 (1997), 745–755.
  • A. M. Bigatti, Upper bounds for the Betti numbers of a given Hilbert function , Comm. Algebra, 21 (1993), 2317 – 2334.
  • W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge University Press, Cambridge, New York, Sydney (1993).
  • W. Bruns and J. Herzog, Semigroup rings and simplicial complexes , J. Pure and Appl. Algebra, 122 (1997), 185–208.
  • H. Charalambous and E. G. Evans, Resolutions obtained by iterated mapping cones , J. Algebra, 176 (1995), 750–754.
  • J. A. Eagon and V. Reiner, Resolutions of Stanley–Reisner rings and Alexander duality , J. Pure and Appl. Algebra, 130 (1998), 265–275.
  • D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer–Verlag (1995).
  • S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals , J. Algebra, 129 (1990), 1–25.
  • G. Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes , Math. Z, 158 (1978), 61–70.
  • M. Green, Restrictions of linear series to hyperplanes, and some results of macaulay and Gotzmann, in “Algebraic Curves and Projective Geometry” , Springer Lect. Notes in Math., No. 1389, 1988, pp. 76–89.
  • T. Hibi, Level rings and algebras with straightening laws , J. Algebra, 117 (1988), 343–362.
  • T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, Glebe, N.S.W., Australia (1992).
  • M. Hochster, Cohen–Macaulay rings, combinatorics, and simplicial complexes, in “Ring Theory II” (B. R. McDonald and R. Morris, Eds.) , Lect. Notes in Pure and Appl. Math., No. 26, Dekker, New York, 1977, pp. 171–223.
  • H. A. Hulett, Maximum Betti Numbers for a given Hilbert function , Comm. Algebra, 21 (1993), 2335–2350.
  • K. Pardue, Nonstandard Borel-fixed ideals , Ph. D. thesis, Brandeis University, 1994.
  • R. P. Stanley, Combinatorics and Commutative Algebra, Second Ed., Birkhäuser, Boston, Basel, Stuttgart (1996).
  • R. P. Stanley, A monotonicity property of $h$-vectors and $h^*$-vectors , Europ. J. Combin., 14 (1993), 251–258.
  • N. Terai and T. Hibi, Computation of Betti numbers of monomial ideals associated with cyclic polytopes , Discrete and Comput. Geom., 15 (1996), 287–295.
  • N. Terai and T. Hibi, Computation of Betti numbers of monomial ideals associated with stacked polytopes , Manuscripta Math., 92 (1997), 447–453.
  • N. Terai and T. Hibi, Alexander duality theorem and second Betti numbers of Stanley–Reisner rings , Adv. in Math., 124 (1996), 332–333.
  • A. M. Duval, Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes , Electronic J. Combinatorics, 3 (1996), #R 21.