Advertisement

On the depth and Stanley depth of the integral closure of powers of monomial ideals

  • S. A. Seyed FakhariEmail author
Article
  • 4 Downloads

Abstract

Let \({\mathbb {K}}\) be a field and \(S={\mathbb {K}}[x_1,\ldots ,x_n]\) be the polynomial ring in n variables over \({\mathbb {K}}\). For any monomial ideal I, we denote its integral closure by \({\overline{I}}\). Assume that G is a graph with edge ideal I(G). We prove that the modules \(S/\overline{I(G)^k}\) and \(\overline{I(G)^k}/\overline{I(G)^{k+1}}\) satisfy Stanley’s inequality for every integer \(k\gg 0\). If G is a non-bipartite graph, we show that the ideals \(\overline{I(G)^k}\) satisfy Stanley’s inequality for all \(k\gg 0\). For every connected bipartite graph G (with at least one edge), we prove that \(\mathrm{sdepth}(I(G)^k)\ge 2\), for any positive integer \(k\le \mathrm{girth}(G)/2+1\). This result partially answers a question asked in Seyed Fakhari (J Algebra 489:463–474, 2017). For any proper monomial ideal I of S, it is shown that the sequence \(\{\mathrm{depth}(\overline{I^k}/\overline{I^{k+1}})\}_{k=0}^{\infty }\) is convergent and \(\lim _{k\rightarrow \infty }\mathrm{depth}(\overline{I^k}/\overline{I^{k+1}})=n-\ell (I)\), where \(\ell (I)\) denotes the analytic spread of I. Furthermore, it is proved that for any monomial ideal I, there exists an integer s such that
$$\begin{aligned} \mathrm{depth} (S/I^{sm}) \le \mathrm{depth} (S/{\overline{I}}), \end{aligned}$$
for every integer \(m\ge 1\). We also determine a value s for which the above inequality holds. If I is an integrally closed ideal, we show that \(\mathrm{depth}(S/I^m)\le \mathrm{depth}(S/I)\), for every integer \(m\ge 1\). As a consequence, we obtain that for any integrally closed monomial ideal I and any integer \(m\ge 1\), we have \(\mathrm{Ass}(S/I)\subseteq \mathrm{Ass}(S/I^m)\).

Keywords

Depth Edge ideal Integral closure Stanley depth Stanley’s inequality 

Mathematics Subject Classification

13C15 05E40 13B22 

Notes

Acknowledgements

The author thanks Irena Swanson for suggesting determinantal trick in the proof of Lemma 4.4. He also thanks the referee for careful reading of the paper and for valuable comments.

References

  1. 1.
    Alilooee, A., Banerjee, A.: Powers of edge ideals of regularity three bipartite graphs. J. Commut. Algebra 9, 441–454 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alipour, A., Seyed Fakhari, S.A., Yassemi, S.: Stanley depth of factor of polymatroidal ideals and edge ideal of forests. Arch. Math. (Basel) 105, 323–332 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Camb. Philos. Soc. 86(1), 35–39 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  5. 5.
    Burch, L.: Codimension and analytic spread. Proc. Camb. Philos. Soc. 72, 369–373 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Duval, A.M., Goeckner, B., Klivans, C.J., Martin, J.L.: A non-partitionable Cohen–Macaulay simplicial complex. Adv. Math. 299, 381–395 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Herzog, J.: A survey on Stanley depth. In: Bigatti, A., Giménez, P., Sáenz-de-Cabezón, E. (eds.) Monomial Ideals, Computations and Applications. Proceedings of MONICA 2011. Lecture Notes in Mathematics, vol. 2083. Springer (2013)Google Scholar
  8. 8.
    Herzog, J., Hibi, T.: Monomial Ideals. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291(2), 325–650 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Herzog, J., Rauf, A., Vladoiu, M.: The stable set of associated prime ideals of a polymatroidal ideal. J. Algebraic Combin. 37, 289–312 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Herzog, J., Takayama, Y., Terai, N.: On the radical of a monomial ideal. Arch. Math. (Basel) 85(5), 397–408 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hoa, L.T., Trung, T.N.: Stability of depth and Cohen–Macaulayness of integral closures of powers of monomial ideals. Acta Math. Vietnam 43, 67–81 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huneke, C., Swanson, I.: Integral Closure of Ideals Rings, and Modules, London Mathematical Society, Lecture Note Series vol. 336. Cambridge University Press, Cambridge (2006)Google Scholar
  14. 14.
    Popescu, D.: Bounds of Stanley depth. An. St. Univ. Ovidius. Constanta 19(2), 187–194 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Pournaki, M.R., Seyed Fakhari, S.A., Tousi, M., Yassemi, S.: What is \(\ldots \) Stanley depth? Not. Am. Math. Soc. 56(9), 1106–1108 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pournaki, M.R., Seyed Fakhari, S.A., Yassemi, S.: Stanley depth of powers of the edge ideal of a forest. Proc. Am. Math. Soc. 141(10), 3327–3336 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Seyed Fakhari, S.A.: Stanley depth of the integral closure of monomial ideals. Collect. Math. 64, 351–362 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Seyed Fakhari, S.A.: Stanley depth of weakly polymatroidal ideals and squarefree monomial ideals. Ill. J. Math. 57(3), 871–881 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Seyed Fakhari, S.A.: Depth, Stanley depth and regularity of ideals associated to graphs. Arch. Math. (Basel) 107, 461–471 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Seyed Fakhari, S.A.: On the Stanley depth of powers of edge ideals. J. Algebra 489, 463–474 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Seyed Fakhari, S.A.: Depth and Stanley depth of symbolic powers of cover ideals of graphs. J. Algebra 492, 402–413 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Seyed Fakhari, S.A.: Stanley depth and symbolic powers of monomial ideals. Math. Scand. 120, 5–16 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Singla, P.: Minimal monomial reductions and the reduced fiber ring of an extremal ideal. Ill. J. Math. 51(4), 1085–1102 (2007)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Stanley, R.P.: Linear Diophantine equations and local cohomology. Invent. Math. 68(2), 175–193 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Takayama, Y.: Combinatorial characterizations of generalized Cohen–Macaulay monomial ideals. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 48, 327–344 (2005)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Vasconcelos, W.: Integral Closure, Rees Algebras, Multiplicities, Algorithms. Springer Monographs in Mathematics. Springer, Berlin (2005)zbMATHGoogle Scholar
  27. 27.
    Villarreal, R.H.: Monomial Algebras. Dekker, New York (2001)zbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran

Personalised recommendations