Advertisement

Collectanea Mathematica

, Volume 63, Issue 3, pp 361–374 | Cite as

Associated primes of powers of edge ideals

  • José Martínez-Bernal
  • Susan Morey
  • Rafael H. Villarreal
Article

Abstract

Let G be a graph and let I be its edge ideal. Our main result shows that the sets of associated primes of the powers of I form an ascending chain. It is known that the sets of associated primes of I i and \({\overline{I^i}}\) stabilize for large i. We show that their corresponding stable sets are equal. To show our main result we use a classical result of Berge from matching theory and certain notions from combinatorial optimization.

Keywords

Associated primes Edge-ideal Integral closure Perfect matching Analytic spread 

Mathematics Subject Classification (2000)

Primary 13C13 Secondary 13A30 13F55 05C25 05C75 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brodmann M.: Asymptotic stability of Ass(M/I n M). Proc. Amer. Math. Soc. 74, 16–18 (1979)MathSciNetGoogle Scholar
  2. 2.
    Bruns, W., Ichim, B.: Normaliz 2.0, Computing normalizations of affine semigroups. Available from http://www.math.uos.de/normaliz (2008)
  3. 3.
    Chen J., Morey S., Sung A.: The stable set of associated primes of the ideal of a graph. Rocky Mountain J. Math. 32, 71–89 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Diestel R.: Graph Theory, Graduate Texts in Mathematics, vol. 173, 2nd edn. Springer-Verlag, New York (2000)Google Scholar
  5. 5.
    Dupont L.A., Villarreal R.H.: Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones. Algebra Discrete Math. 10(2), 64–86 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dupont L.A., Reyes E., Villarreal R.H.: Cohen-Macaulay clutters with combinatorial optimization properties and parallelizations of normal edge ideals. São Paulo J. Math. Sci. 3(1), 61–75 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Francisco C.A., Hà H.T., Van Tuyl A.: A conjecture on critical graphs and connections to the persistence of associated primes. Discrete Math. 310, 2176–2182 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Francisco C., Hà H.T., Van Tuyl A.: Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals. J. Algebra 331(1), 224–242 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Golumbic, M.C.: Algorithmic graph theory and perfect graphs, 2nd edn, Annals of Discrete Mathematics, vol. 57. Elsevier Science B.V., Amsterdam (2004)Google Scholar
  10. 10.
    Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/ (1996)
  11. 11.
    Hà H.T., Morey S.: Embedded associated primes of powers of square-free monomial ideals. J. Pure Appl. Algebra 214(4), 301–308 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Herzog J., Hibi T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Huneke C., Swanson I.: Integral Closure of Ideals Rings, and Modules, London Math. Soc., Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)Google Scholar
  14. 14.
    Lovász L., Plummer M.D.: Matching Theory, Annals of Discrete Mathematics, vol. 29. Elsevier Science B.V., Amsterdam (1986)Google Scholar
  15. 15.
    Martínez-Bernal, J., Renterí a, C., Villarreal, R.H.: Combinatorics of symbolic Rees algebras of edge ideals of clutters, Contemp. Math (in press)Google Scholar
  16. 16.
    Matsumura H.: Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)Google Scholar
  17. 17.
    McAdam S.: Asymptotic prime divisors and analytic spreads. Proc. Amer. Math. Soc. 80, 555–559 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    McAdam S.: Asymptotic Prime Divisors, Lecture Notes in Mathematics, vol. 103. Springer-Verlag, New York (1983)Google Scholar
  19. 19.
    Morey, S., Villarreal, R.H.: Edge ideals: algebraic and combinatorial properties, Progress in Commutative Algebra: Ring Theory, Homology, and Decompositions, De Gruyter. Preprint, \({{\tt arXiv:1012.5329v2 [math.AC]}}\) 2010, (in press)Google Scholar
  20. 20.
    Ratliff L.J. Jr.: On prime divisors of I n, n large. Michigan Math. J. 23(4), 337–352 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Ratliff L.J. Jr.: On asymptotic prime divisors. Pacific J. Math. 111(2), 395–413 (1984)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Schrijver A.: Combinatorial Optimization, Algorithms and Combinatorics 24. Springer-Verlag, Berlin (2003)Google Scholar
  23. 23.
    Simis A., Vasconcelos W.V., Villarreal R.H.: On the ideal theory of graphs. J. Algebra 167, 389–416 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Vasconcelos W.V.: Integral Closure. Springer Monographs in Mathematics. Springer-Verlag, New York (2005)Google Scholar
  25. 25.
    Villarreal R.H.: Cohen-Macaulay graphs. Manuscripta Math. 66, 277–293 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Villarreal R.H.: Monomial Algebras. Dekker, New York (2001)zbMATHGoogle Scholar
  27. 27.
    Villarreal R.H.: Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs. J. Algebraic Combin. 27(3), 293–305 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Universitat de Barcelona 2011

Authors and Affiliations

  • José Martínez-Bernal
    • 1
  • Susan Morey
    • 2
  • Rafael H. Villarreal
    • 1
  1. 1.Departamento de MatemáticasCentro de Investigación y de Estudios Avanzados del IPNMexico CityMexico
  2. 2.Department of MathematicsTexas State UniversitySan MarcosUSA

Personalised recommendations