Journal of Algebraic Combinatorics

, Volume 37, Issue 2, pp 289–312 | Cite as

The stable set of associated prime ideals of a polymatroidal ideal

Article

Abstract

The associated prime ideals of powers of polymatroidal ideals are studied, including the stable set of associated prime ideals of this class of ideals. It is shown that polymatroidal ideals have the persistence property and for transversal polymatroids and polymatroidal ideals of Veronese type the index of stability and the stable set of associated ideals is determined explicitly.

Keywords

Associated prime ideals Polymatroidal ideals Analytic spread 

Notes

Acknowledgements

This paper was partially written during the visit of the second and third author at the Universität Duisburg-Essen, Campus Essen. The second author wants to thank The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy for supporting her. The third author was supported by a Romanian grant awarded by UEFISCDI, project number 83/2010, PNII-RU code TE_46/2010, program Human Resources, “Algebraic modeling of some combinatorial objects and computational applications”.

References

  1. 1.
    Borna, K.: On linear resolution of powers of an ideal. Osaka J. Math. 46, 1047–1058 (2009) MathSciNetMATHGoogle Scholar
  2. 2.
    Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Camb. Philos. Soc. 86, 35–39 (1979) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brodmann, M.: Asymptotic stability of \(\operatorname{Ass}(M/I^{n}M)\). Proc. Am. Math. Soc. 74, 16–18 (1979) MathSciNetGoogle Scholar
  4. 4.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Revised edn. Cambridge University Press, Cambridge (1996) Google Scholar
  5. 5.
    Chen, J., Morey, S., Sung, A.: The stable set of associated primes of the ideal of a graph. Rocky Mt. J. Math. 32, 71–89 (2002) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    CoCoATeam: CoCoA: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it
  7. 7.
    Conca, A., Herzog, J.: Castelnuovo–Mumford regularity of products of ideals. Collect. Math. 54, 137–152 (2003) MathSciNetMATHGoogle Scholar
  8. 8.
    Eisenbud, D., Huneke, C.: Cohen–Macaulay Rees algebras and their specialization. J. Algebra 81, 202–224 (1983) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Francisco, C., Tai Ha, H., Van Tuyl, A.: Coloring of hypergraphs, perfect graphs and associated primes of powers of monomial ideals. J. Algebra 331, 224–242 (2011) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gitler, I., Reyes, E., Villarreal, R.H.: Blowup algebras of squarefree monomial ideals and some links to combinatorial optimization problems. Rocky Mt. J. Math. 39, 71–102 (2009) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Greuel, G.M., Pfister, G., Schönemann, H.: Singular 2.0. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de
  12. 12.
    Herzog, J., Hibi, T.: Discrete polymatroids. J. Algebr. Comb. 16, 239–268 (2002) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Herzog, J., Hibi, T.: Monomial Ideals. GTM, vol. 260. Springer, Berlin (2010) Google Scholar
  15. 15.
    Herzog, J., Simis, A., Vasconcelos, W.: Arithmetic of normal Rees algebras. J. Algebra 143, 269–294 (1991) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Herzog, J., Hibi, T., Trung, N.V., Zheng, X.: Standard graded vertex cover algebras, cycles and leaves. Trans. Am. Math. Soc. 360, 6231–6249 (2008) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Huneke, C.: On the associated graded ring of an ideal. Ill. J. Math. 26, 121–137 (1982) MathSciNetMATHGoogle Scholar
  18. 18.
    Martinez-Bernal, J., Morey, S., Villarreal, R.H.: Associated primes of powers of edge ideals. arXiv:1103.0992v3
  19. 19.
    Morey, S., Villarreal, R.H.: Edge ideals: algebraic and combinatorial properties. arXiv:1012.5329v3
  20. 20.
    Villarreal, R.H.: Rees cones and monomial rings of matroids. Linear Algebra Appl. 428, 2933–2940 (2008) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Vladoiu, M.: Equidimensional and unmixed ideals of Veronese type. Commun. Algebra 36, 3378–3392 (2008) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität Duisburg-Essen, Campus EssenEssenGermany
  2. 2.The Abdus Salam International Centre for Theoretical Physics (ICTP)TriesteItaly
  3. 3.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

Personalised recommendations