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Componentwise linear ideals

Published online by Cambridge University Press:  22 January 2016

Jürgen Herzog  and
Takayuki Hibi
Jürgen Herzog
Affiliation:
FB 6 Mathematik und Informatik, Universität-GHS-Essen, Essen 45117, Germany, juergen.herzog@uni-essen.de
Takayuki Hibi
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan, hibi@math.sci.osaka-u.ac.jp
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Abstract

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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 IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ 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.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 153 , 1999 , pp. 141 - 153
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

References

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