manuscripta mathematica

, Volume 112, Issue 4, pp 489–509 | Cite as

Symmetric and Rees algebras of Koszul cycles and their Gröbner bases

  • Jürgen HerzogEmail author
  • Zhongming Tang
  • Santiago Zarzuela


In this paper we study the symmetric algebra S(E i ) and Rees algebra R(E i ) of the modules E i of i-cycles of the Koszul complex associated with the sequence of indeterminates \({{x_1,\ldots, x_n}}\) of a polynomial ring \({{K[x_1,\ldots, x_n]}}\). For i=2 and i=n−2 we show that \({{x_1,\ldots, x_n}}\) is a d-sequence on S(E i ) and R(E i ) and we determine Gröbner bases and Sagbi bases related to these algebras.


Polynomial Ring Symmetric Algebra Koszul Complex Sagbi Base 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bruns, W., Herzog, J.: Cohen-Macaulay rings, Cambridge: Cambridge University Press, 1993Google Scholar
  2. 2.
    Conca, A., Herzog, J., Valla, G.: Sagbi bases with applications to blow up algebras. J. Reine Angew. Math. 474, 113–138 (1996)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Goto, S., Hayasaka, F., Kurano, K., Nakamura, Y.: Algebras generated by the entries of a certain submatrix and the maximal minors. Preprint, 2002Google Scholar
  4. 4.
    Herzog, J., Restuccia, G., Tang, Z.: L s-sequences and symmetric algebras. Manuscripta Math. 104 (4), 479–501 (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Huneke, C.: Symbolic powers of prime ideals and special graded algebras. Comm. Alg. 9 (4), 339–366 (1981)zbMATHGoogle Scholar
  6. 6.
    Huneke, C.: The theory of d-sequences and Powers of ideals. Adv. Math. 46, 249–279 (1982)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Huneke, C., Ulrich, B.: Divisor class groups and deformations. Am. J. Math. 107 (6), 1265–1303 (1985)zbMATHGoogle Scholar
  8. 8.
    Möller, H. M., Mora, F.: New constructive methods in classical ideal theory. J. Alg. 100, 138–178 (1986)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Simis, A., Ulrich, B., Vasconcelos, W.V.: Jacobian dual fibrations. Am. J. Math. 115 (1), 47–75 (1993)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jürgen Herzog
    • 1
    Email author
  • Zhongming Tang
    • 2
  • Santiago Zarzuela
    • 3
  1. 1.FB6 Mathematik und InformatikUniversität-GHS-EssenEssenGermany
  2. 2.Department of MathematicsSuzhou UniversitySuzhouP.R. China
  3. 3.Departament d’Àlgebra i GeometriaUniversitat de BarcelonaBarcelonaSpain

Personalised recommendations