Acta Mathematica Vietnamica

, Volume 43, Issue 1, pp 67–81 | Cite as

Stability of Depth and Cohen-Macaulayness of Integral Closures of Powers of Monomial Ideals

  • Le Tuan HoaEmail author
  • Tran Nam Trung


Let I be a monomial ideal in a polynomial ring \(R = k[x_{1},\dots ,x_{r}]\). In this paper, we give an upper bound on \(\overline {\text {dstab}} (I)\) in terms of r and the maximal generating degree d(I) of I such that \(\text {depth} R/\overline {I^{n}}\) is constant for all \(n\geqslant \overline {\text {dstab}}(I)\). As an application, we classify the class of monomial ideals I such that \(\overline {I^{n}}\) is Cohen-Macaulay for some integer n ≫ 0.


Depth Monomial ideal Simplicial complex Integral closure 

Mathematics Subject Classification (2010)

13D45 05C90 



This work is partially supported by NAFOSTED (Vietnam) under the grant number 101.04-2015.02.


  1. 1.
    Bivia-Ausina, C.: The analytic spread of monomial ideals. Comm. Algebra 31, 3487–3496 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brodmann, M.P.: The asymptotic nature of the analytic spread. Math. Proc. Cambridge Philos Soc. 86, 35–39 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Eisenbud, D., Huneke, C.: Cohen-Macaulay Rees algebras and their specializations. J. Algebra 81, 202–224 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Giang, D.H., Hoa, L.T.: On local cohomology of a tetrahedral curve. Acta Math. Vietnam. 35, 229–241 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ha, H.T., Nguyen, D.H., Trung, N.V., Trung, T.N.: Symbolic powers of sums of ideals. Preprint ArXiv:1702.01766
  6. 6.
    Herrmann, M., Ikeda, S., Orbanz, U.: Equimultiplicity and Blowing up. Springer-Verlag (1988)Google Scholar
  7. 7.
    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Herzog, J., Qureshi, A.A.: Persistence and stability properties of powers of ideals. J. Pure Appl. Algebra 219, 530–542 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Herzog, J., Takayama, Y., Terai, N.: On the radical of a monomial ideal. Arch. Math. 85, 397–408 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hoa, L.T., Kimura, K., Terai, N., Trung, T.N.: Stability of depths of symbolic powers of Stanley-Reisner ideals. J. Algebra 473, 307–323 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hoa, L.T., Trung, T.N.: Partial Castelnuovo-Mumford regularities of sums and intersections of powers of monomial ideals. Math. Proc. Cambridge Philos Soc. 149, 1–18 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    McAdam, S., Eakin, P.: The asymptotic Ass. J. Algebra 61, 71–81 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Minh, N.C, Trung, N.V.: Cohen-Macaulayness of powers of two-dimensional square-free monomial ideals. J. Algebra 322, 4219–4227 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Reid, L., Roberts, L.G., Vitulli, M.A.: Some results on normal homogeneous ideals. Comm. Algebra 31, 4485–4506 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley (1998)Google Scholar
  16. 16.
    Takayama, Y.: Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48, 327–344 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Terai, N., Trung, N.V.: Cohen-Macaulayness of large powers of Stanley-Reisner ideals. Adv. Math. 229(2), 711–730 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Terai, N., Yoshida, K.: Locally complete intersection Stanley-Reisner ideals. Illinois J. Math. 53(2), 413–429 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Trung, T.N.: Stability of associated primes of integral closures of monomial ideals. J. Combin. Ser. A. 116, 44–54 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Trung, T.N.: Stability of depths of powers of edge ideals. J. Algebra 452, 157–187 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vasconcelos, W.: Integral Closure: Rees Algebras, Multiplicities, Algorithms. Springer Monographs in Mathematics. Springer, New York (2005)zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Institute of MathematicsVASTHanoiVietnam

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