Depth functions of symbolic powers of homogeneous ideals

Abstract

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function \({{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R -{{\,\mathrm{pd}\,}}I^{(t)} - 1\), where \(I^{(t)}\) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and \({{\,\mathrm{pd}\,}}\) denotes the projective dimension. It has been an open question whether the function \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is non-increasing if I is a squarefree monomial ideal. We show that \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is almost non-increasing in the sense that \({{\,\mathrm{depth}\,}}R/I^{(s)} \ge {{\,\mathrm{depth}\,}}R/I^{(t)}\) for all \(s \ge 1\) and \(t \in E(s)\), where

$$\begin{aligned} E(s) = \bigcup _{i \ge 1}\{t \in {\mathbb {N}}|\ i(s-1)+1 \le t \le is\} \end{aligned}$$

(which contains all integers \(t \ge (s-1)^2+1\)). The range E(s) is the best possible since we can find squarefree monomial ideals I such that \({{\,\mathrm{depth}\,}}R/I^{(s)} < {{\,\mathrm{depth}\,}}R/I^{(t)}\) for \(t \not \in E(s)\), which gives a negative answer to the above question. Another open question asks whether the function \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is always constant for \(t \gg 0\). We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that \(I^{(t)}\) is integrally closed for \(t \gg 0\) (e.g. if I is a squarefree monomial ideal), then \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is constant for \(t \gg 0\) with

$$\begin{aligned} \lim _{t \rightarrow \infty }{{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R - \dim \oplus _{t \ge 0}I^{(t)}/{\mathfrak {m}}I^{(t)}. \end{aligned}$$

Our last result (which is the main contribution of this paper) shows that for any positive numerical function \(\phi (t)\) which is periodic for \(t \gg 0\), there exist a polynomial ring R and a homogeneous ideal I such that \({{\,\mathrm{depth}\,}}R/I^{(t)} = \phi (t)\) for all \(t \ge 1\). As a consequence, for any non-negative numerical function \(\psi (t)\) which is periodic for \(t \gg 0\), there is a homogeneous ideal I and a number c such that \({{\,\mathrm{pd}\,}}I^{(t)} = \psi (t) + c\) for all \(t \ge 1\).

This is a preview of subscription content, log in to check access.

Change history

  • 11 October 2019

    The original proof of Theorem 3.3 incorrect. The correction concerns only this proof and does not affect any result of the paper.

Notes

  1. 1.

    The notation of \(P_F\) in [16] is different.

  2. 2.

    There is a typo in [37, Lemma 1.5]. In the formula for \(M_1\) one has to replace \(\text {Assm}(R)\setminus {\mathcal V}(I_F)\) by \(\text {Assm}(R)\cap {\mathcal V}(I_F)\).

  3. 3.

    The proof for Proposition 3.2(ii) and (iii) of [30] has errors, which can be corrected as follows. For Proposition 3.2(ii), we consider the exact consequence \(0 \rightarrow M \cap N \rightarrow L \rightarrow (L/M) \oplus (L/N)\) and apply Corollary 2.5(i) and Lemma 3.1. Proposition 3.2(iii) follows from the exact sequence \(M \oplus N \rightarrow L \rightarrow L/(M+N) \rightarrow 0\) by applying Corollary 2.5(ii) and Lemma 3.1.

References

  1. 1.

    Bandari, S., Herzog, J., Hibi, T.: Monomial ideals whose depth function has any given number of strict local maxima. Ark. Mat. 52, 11–19 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Björner, A.: Topological methods. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, pp. 1819–1872. Birkhäuser, North-Holland (1995)

    Google Scholar 

  3. 3.

    Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Camb. Philos. Soc. 86(1), 35–39 (1979)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  5. 5.

    Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes in Mathematics, vol. 1327. Springer, Berlin (1980)

    Google Scholar 

  6. 6.

    Constantinescu, A., Pournaki, M.R., Seyed Fakhari, S.A., Terai, N., Yassemi, S.: Cohen–Macaulayness and limit behavior of depth for powers of cover ideals. Commun. Algebra 43, 143–157 (2015)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cutkosky, S.D.: Symbolic algebras of monomial primes. J. Reine Angew. Math. 416, 71–89 (1991)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Ein, L., Lazarsfeld, R., Smith, K.: Uniform bounds and symbolic powers on smooth varieties. Invent. Math. 144, 241–252 (2001)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Eisenbud, D., Hochster, M.: A Nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions. J. Algebra 58, 157–161 (1979)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Eisenbud, D., Mazur, B.: Evolutions, symbolic squares, and fitting ideals. J. Reine Angew. Math. 488, 189–201 (1997)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Hà, H.T., Nguyen, H.D., Trung, N.V., Trung, T.N.: Depth functions of powers of homogeneous ideals, Preprint, arXiv:1904.07587

  12. 12.

    Hà, H.T., Trung, N.V.: Membership criteria and containments of powers of monomial ideals. Acta Math. Vietnam 44, 117–139 (2019)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Herzog, J., Hibi, T., Trung, N.V.: Symbolic powers of monomial ideals and vertex power algebras. Adv. Math. 210, 304–322 (2007)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Herzog, J., Qureshi, A.A.: Persistence and stability properties of powers of ideals. J. Pure Appl. Algebra 229, 530–542 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hien, H.T.T., Lam, H.M., Trung, N.V.: Saturation and associated primes of powers of edge ideals. J. Algebra 439, 225–244 (2015)

    MathSciNet  Article  Google Scholar 

  17. 17.

    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)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hoa, L.T., Tam, N.D.: On some invariants of a mixed product of ideals. Arch. Math. 94, 327–337 (2010)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hoa, L.T., Trung, T.N.: Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals. Math. Proc. Camb. Philos. Soc. 149, 1–18 (2010)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hochster, M.: Rings of invariants of Tori, Cohen–Macaulay rings generated by monomials, and polytopes. Ann. Math. 96, 318–337 (1972)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hochster, M., Huneke, C.: Comparison of symbolic and ordinary powers of ideals. Invent. Math. 147, 349–369 (2002)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Huneke, C.: On the finite generation of symbolic blow-ups. Math. Z. 179, 465–472 (1982)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings and Modules. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  24. 24.

    Kaiser, T., Stehlik, M., Skrekovski, R.: Replication in critical graphs and the persistence of monomial ideals. J. Combin. Theory Ser. A 123, 239–251 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kimura, K., Terai, N., Yassemi, S.: The projective dimension of symbolic powers of the edge ideal of a very well-covered graph. Nagoya Math. J. 230, 160–179 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Matsuda, K., Suzuki, T., Tsuchiya, A.: Nonincreasing depth functions of monomial ideals. Glasgow Math. J. 60, 505–511 (2018)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Minh, N.C., Trung, N.V.: Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals. Adv. Math. 226, 1285–1306 (2011)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Morey, S., Villarreal, R.H.: Edge ideals: algebraic and combinatorial properties. In: Francisco, C., Klinger, L.C., Sather-Wastaff, S., Vassilev, J.C. (eds.) Progress in Commutative Algebra, Combinatorics and Homology, vol. 1, pp. 85–126. De Gruyter, Berlin (2012)

    Google Scholar 

  29. 29.

    Nhi, D.V.: Specializations of direct limits and of local cohomology modules. Proc. Edinb. Math. Soc. 50, 459–475 (2007)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Nhi, D.V., Trung, N.V.: Specialization of modules. Commun. Algebra 27, 2959–2978 (1999)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Roberts, P.: A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian. Proc. Am. Math. Soc. 94, 589–592 (1985)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Schrijver, A.: The Theory of Linear and Integer Programming. Wiley, New York (1999)

    Google Scholar 

  33. 33.

    Seidenberg, A.: The hyperplane sections of normal varieties. Trans. Am. Math. Soc. 69, 357–386 (1950)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Takayama, Y.: Combinatorial characterizations of generalized Cohen–Macaulay monomial ideals. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48, 327–344 (2005)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Terai, N., Trung, N.V.: On the associated primes and the depth of the second power of squarefree monomial ideals. J. Pure Appl. Algebra 218, 1117–1129 (2014)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Terai, N., Trung, N.V.: Cohen–Macaulayness of large powers of Stanley–Reisner ideals. Adv. Math. 229, 711–730 (2012)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Trung, N.V.: Über die Übertragung der Ringeigenschaften zwischen \(R\) und \(R[u]/(F)\). Math. Nachr. 92, 215–224 (1978)

    Article  Google Scholar 

  38. 38.

    Trung, N.V.: Squarefree monomial ideals and hypergraphs, Lectures Report to AIM, Workshop on Integral Closure, Adjoint Ideals and Cores, Palo Alto (2006). https://aimath.org/WWN/integralclosure/Trung.pdf

  39. 39.

    Trung, N.V.: Hypergraphs, polyhedra and monomial ideals. In: Proceedings of the 5th Joint Japan–Vietnam on Commutative Algebra, Institute of Mathematics, Hanoi, pp 15–29 (2010)

  40. 40.

    Trung, N.V., Ikeda, S.: When is the Rees algebra Cohen–Macaulay? Commun. Algebra 17, 2893–2922 (1997)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Varbaro, M.: Symbolic powers and matroids. Proc. Am. Math. Soc. 139, 2357–2366 (2011)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Hop Dang Nguyen is partially supported by Project CT 0000.03/19-21 of Vietnam Academy of Science and Technology. Ngo Viet Trung is partially supported by Vietnam National Foundation for Science and Technology Development. Part of this work was done during research stays of the authors at Vietnam Institute for Advanced Study in Mathematics. The authors would like to thank Huy Tài Hà and Tran Nam Trung for their collaboration on the joint paper [11] which initiated this work. They are also grateful to the referee for many suggestions which help improve the presentation of the paper. After the revision of this paper, the authors have been informed that Theorem 2.7 has been recently obtained in a modified form by different methods by J. Montano and L. Nunez-Betancourt (arXiv:1809.02308) and S. A. Seyed Fakhari (arXiv:1812.03742).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ngo Viet Trung.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nguyen, H.D., Trung, N.V. Depth functions of symbolic powers of homogeneous ideals. Invent. math. 218, 779–827 (2019). https://doi.org/10.1007/s00222-019-00897-y

Download citation

Keywords

  • Symbolic power
  • Projective dimension
  • Depth
  • Asymptotic behavior
  • Monomial ideal
  • Integrally closed ideal
  • Degree complex
  • Local cohomology
  • Bertini-type theorem
  • System of linear diophantine inequalities

Mathematics Subject Classification

  • 13C15
  • 14B05