Mathematische Zeitschrift

, Volume 274, Issue 3–4, pp 805–819 | Cite as

Poset embeddings of Hilbert functions

Article

Abstract

For a standard graded algebra \(R\), we consider embeddings of the poset of Hilbert functions of \(R\)-ideals into the poset of \(R\)-ideals, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements–Lindström is proved. We exhibit a condition on the embedding that ensures that the classification of Hilbert functions is obtained with images of lexicographic segment ideals.

Notes

Acknowledgments

We thank A. Conca and the referee for helpful comments. The computer algebra system Macaulay2 [13] provided valuable assistance in studying examples.

References

  1. 1.
    Bigatti, A.M., Conca, A., Robbiano, L.: Generic initial ideals and distractions. Comm. Algebra 33(6), 1709–1732 (2005)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  3. 3.
    Bigatti, A.M.: Upper bounds for the Betti numbers of a given Hilbert function. Comm. Algebra 21(7), 2317–2334 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Conca, A., Herzog, J., Hibi, T.: Rigid resolutions and big Betti numbers. Comment. Math. Helv. 79(4), 826–839 (2004)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Clements, G.F., Lindström, B.: A generalization of a combinatorial theorem of Macaulay. J. Comb. Theory 7, 230–238 (1969)MATHCrossRefGoogle Scholar
  6. 6.
    Conca, A.: Koszul homology and extremal properties of Gin and Lex. Trans. Am. Math. Soc. 356(7), 2945–2961 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Conca, A., Rossi, M.E., Valla, G.: Gröbner flags and Gorenstein algebras. Compos. Math. 129(1), 95–121 (2001)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Eisenbud, D., Green, M., Harris, J.: Cayley–Bacharach theorems and conjectures. Bull. Am. Math. Soc. (N.S.) 33(3), 295–324 (1996)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Eisenbud, D.: Commutative Algebra, Graduate Texts in Mathematics, vol. 150, Springer, New York. With a view toward algebraic geometry (1995)Google Scholar
  10. 10.
    Francisco, C.A.,Richert, B. P.: Lex-Plus-Powers Ideals, Syzygies and Hilbert Functions, pp. 113–144 (2007)Google Scholar
  11. 11.
    Gasharov, V., Horwitz, N., Peeva, I.: Hilbert functions over toric rings. Michigan Math. J. 57, 339–357 (2008). (Special volume in honor of Melvin Hochster)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gasharov, V., Murai, S., Peeva, I.: Hilbert schemes and maximal Betti numbers over Veronese rings. Math. Z. 267(1–2), 155–172 (2011)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Grayson D.R., Stillman, M. E.: Macaulay 2, A Software System for Research in Algebraic Geometry. Available at http://www.math.uiuc.edu/Macaulay2/ (2006)
  14. 14.
    Hulett, H.A.: Maximum Betti numbers of homogeneous ideals with a given Hilbert function. Comm. Algebra 21(7), 2335–2350 (1993)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Katona G.: A Theorem of Finite Sets, Theory of Graphs (Proc. Colloq., Tihany, 1966), pp. 187–207 (1968)Google Scholar
  16. 16.
    Kruskal, J.B.: The Number of Simplices in a Complex, Mathematical Optimization, Techniques, pp. 251–278 (1963)Google Scholar
  17. 17.
    Mermin, J.: Lexlike sequences. J. Algebra 303(1), 295–308 (2006)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Mermin, J.: Monomial regular sequences. Proc. Am. Math. Soc. 138(6), 1983–1988 (2010). (electronic)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Mermin, J., Murai, S.: Betti numbers of lex ideals over some macaulay–lex rings. J. Algebraic Combin. 31(2), 299–318 (2010)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Mermin, J., Murai, S.: The lex-plus-powers conjecture holds for pure powers. Adv. Math. 226(4), 3511–3539 (2011)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Mermin, J., Peeva, I.: Lexifying ideals. Math. Res. Lett. 13(2–3), 409–422 (2006)MathSciNetMATHGoogle Scholar
  22. 22.
    Miller, E., Sturmfels, B.: Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)Google Scholar
  23. 23.
    Pardue, K.: Deformation classes of graded modules and maximal Betti numbers. Illinois J. Math. 40(4), 564–585 (1996)MathSciNetMATHGoogle Scholar
  24. 24.
    Sbarra, E.: Upper bounds for local cohomology for rings with given Hilbert function. Comm. Algebra 29(12), 5383–5409 (2001)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Stanley, R.P.: Enumerative Combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge (1997). (With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Chennai Mathematical InstituteSiruseriIndia

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