Skip to main content

Regularity and Koszul property of symbolic powers of monomial ideals

Abstract

Let I be a homogeneous ideal in a polynomial ring over a field. Let \(I^{(n)}\) be the n-th symbolic power of I. Motivated by results about ordinary powers of I, we study the asymptotic behavior of the regularity function \({{\,\mathrm{reg}\,}}(I^{(n)})\) and the maximal generating degree function \(\omega (I^{(n)})\), when I is a monomial ideal. It is known that both functions are eventually quasi-linear. We show that, in addition, the sequences \(\{{{\,\mathrm{reg}\,}}I^{(n)}/n\}_n\) and \(\{\omega (I^{(n)})/n\}_n\) converge to the same limit, which can be described combinatorially. We construct an example of an equidimensional, height two squarefree monomial ideal I for which \(\omega (I^{(n)})\) and \({{\,\mathrm{reg}\,}}(I^{(n)})\) are not eventually linear functions. For the last goal, we introduce a new method for establishing the componentwise linearity of ideals. This method allows us to identify a new class of monomial ideals whose symbolic powers are componentwise linear.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. This result can be proved quickly using Corollary 5.6.

References

  1. Ahangari Maleki, R., Rossi, M.E.: Regularity and linearity defect of modules over local rings. J. Commut. Algebra 4, 485–504 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Bahiano, C.E.N.: Symbolic powers of edge ideals. J. Algebra 273(2), 517–537 (2004)

    MathSciNet  Article  Google Scholar 

  3. Catalisano, M.V., Trung, N.V., Valla, G.: A sharp bound for the regularity index of fat points in general position. Proc. Am. Math. Soc. 118(3), 717–724 (1993)

    MathSciNet  Article  Google Scholar 

  4. Cooper, S., Embree, R.J.D., Hà, H.T., Hoefel, A.H.: Symbolic powers of monomial ideals. Proc. Edinb. Math. Soc. (2) 60(1), 39–55 (2017)

    MathSciNet  Article  Google Scholar 

  5. Crispin, V., Emtander, E.: Componentwise linearity of ideals arising from graphs. Matematiche (Catania) 63(2), 185–189 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Cutkosky, S.D.: Irrational asymptotic behaviour of Castelnuovo–Mumford regularity. J. Reine Angew. Math. 522, 93–103 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Cutkosky, S.D., Herzog, J., Trung, N.V.: Asymptotic behaviour of the Castelnuovo–Mumford regularity. Compos. Math. 118, 243–261 (1999)

    MathSciNet  Article  Google Scholar 

  8. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150. Springer, New York (1995)

    Google Scholar 

  9. Francisco, C., Hà, H.T.: Whiskers and sequentially Cohen–Macaulay graphs. J. Combin. Theory Ser. A 115(2), 304–316 (2008)

    MathSciNet  Article  Google Scholar 

  10. Francisco, C.A., Van Tuyl, A.: Sequentially Cohen–Macaulay edge ideals. Proc. Am. Math. Soc. 135(8), 2327–2337 (2007)

    MathSciNet  Article  Google Scholar 

  11. Goto, S., Nishida, K., Watanabe, K.-I.: Non-Cohen–Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question. Proc. Am. Math. Soc. 120(2), 383–392 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2

  13. Gu, Y., Hà, H.T., O’Rourke, J.L., Skelton, J.W.: Symbolic powers of edge ideals of graphs. Algebra, Commun (2020). https://doi.org/10.1080/00927872.2020.1745221

    Book  MATH  Google Scholar 

  14. Herzog, J., Hibi, T.: Componentwise linear ideals. Nagoya Math. J. 153, 141–153 (1999)

    MathSciNet  Article  Google Scholar 

  15. Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, Berlin (2011)

    Book  Google Scholar 

  16. Herzog, J., Hibi, T., Ohsugi, H.: Powers of Componentwise Linear Ideals, Abel Symposium, vol. 6. Springer, Berlin (2011)

    MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  18. Herzog, J., Hoa, L.T., Trung, N.V.: Asymptotic linear bounds for the Castelnuovo–Mumford regularity. Trans. Am. Math. Soc. 354(5), 1793–1809 (2002)

    MathSciNet  Article  Google Scholar 

  19. Herzog, J., Iyengar, S.B.: Koszul modules. J. Pure Appl. Algebra 201, 154–188 (2005)

    MathSciNet  Article  Google Scholar 

  20. 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 

  21. 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 

  22. Iyengar, S.B., Römer, T.: Linearity defects of modules over commutative rings. J. Algebra 322, 3212–3237 (2009)

    MathSciNet  Article  Google Scholar 

  23. Kodiyalam, V.: Asymptotic behaviour of Castelnuovo–Mumford regularity. Proc. Am. Math. Soc. 128(2), 407–411 (2000)

    MathSciNet  Article  Google Scholar 

  24. Lyubeznik, G.: On the arithmetical rank of monomial ideals. J. Algebra 112(1), 86–89 (1988)

    MathSciNet  Article  Google Scholar 

  25. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Springer, Berlin (2005)

    MATH  Google Scholar 

  26. Minh, N.C., Trung, T.N.: Regularity of symbolic powers and arboricity of matroids. Forum Math. 31(2), 465–477 (2019)

    MathSciNet  Article  Google Scholar 

  27. Nguyen, H.D.: Notes on the linearity defect and applications. Ill. J. Math. 59(3), 637–662 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Nguyen, H.D., Trung, N.V.: Depth functions of symbolic powers of homogeneous ideals. Invent. Math. 218, 779–827 (2019)

    MathSciNet  Article  Google Scholar 

  29. Nguyen, H.D., Vu, T.: Linearity defect of edge ideals and Fröberg’s theorem. J. Algebr. Combin. 44(1), 165–199 (2016)

    MathSciNet  Article  Google Scholar 

  30. Nguyen, H.D., Vu, T.: Powers of sums and their homological invariants. J. Pure Appl. Algebra 223, 3081–3111 (2019)

    MathSciNet  Article  Google Scholar 

  31. Reid, L., Roberts, L.G., Vitulli, M.A.: Some results on normal homogeneous ideals. Commun. Algebra 31(9), 4485–4506 (2003)

    MathSciNet  Article  Google Scholar 

  32. 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 

  33. Römer, T.: On minimal graded free resolutions. Ph.D. dissertation, University of Essen, Germany (2001)

  34. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1998)

    MATH  Google Scholar 

  35. Şega, L.: Homological properties of powers of the maximal ideal of a local ring. J. Algebra 241(2), 827–858 (2001)

    MathSciNet  Article  Google Scholar 

  36. Seyed Fakhari, S.A.: Symbolic powers of cover ideal of very well-covered and bipartite graphs. Proc. Am. Math. Soc. 146(no. 1), 97–110 (2018)

    MathSciNet  Article  Google Scholar 

  37. Seyed Fakhari, S.A.: Regularity of symbolic powers of cover ideals of graphs. Collect. Math. 70(no. 2), 187–195 (2019)

    MathSciNet  Article  Google Scholar 

  38. Trung, N.V., Wang, H.: On the asymptotic behavior of Castelnuovo–Mumford regularity. J. Pure Appl. Algebra 201(1–3), 42–48 (2005)

    MathSciNet  Article  Google Scholar 

  39. Trung, T.N.: Stability of associated primes of integral closures of monomial ideals. J. Combin. Ser. A. 116, 44–54 (2009)

    MathSciNet  Article  Google Scholar 

  40. Vasconcelos, W.: Integral Closure: Rees Algebras, Multiplicities, Algorithms. Springer Monographs in Mathematics. Springer, Berlin (2005)

    MATH  Google Scholar 

  41. Villarreal, R.: Cohen–Macaulay graphs. Manuscr. Math. 66(3), 277–293 (1990)

    MathSciNet  Article  Google Scholar 

  42. Villarreal, R.: Monomial Algebras. Second Edition. Monographs and Research Notes in Mathematics. Chapman and Hall, New York (2015)

    Google Scholar 

  43. Yanagawa, K.: Alexander duality for Stanley–Reisner rings and squarefree \({\mathbb{N}}^n\)-graded modules. J. Algebra 225, 630–645 (2000)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

L.X. Dung, T.T. Hien and T.N. Trung are partially supported by NAFOSTED (Vietnam) under the grant number 101.04-2018.307. H.D. Nguyen is partially supported by International Centre for Research and Postgraduate Training in Mathematics (ICRTM) under Grant number ICRTM01\(\_\)2020.05. H.D. Nguyen and T.N. Trung are also grateful to the support of Project CT 0000.03/19-21 of the Vietnam Academy of Science and Technology. Part of this work was done during our stay at the Vietnam Institute for Advanced Study in Mathematics. Finally, the authors would like to thank the anonymous referee for useful comments which have helped them to improve the quality of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tran Nam 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

Dung, L.X., Hien, T.T., Nguyen, H.D. et al. Regularity and Koszul property of symbolic powers of monomial ideals. Math. Z. 298, 1487–1522 (2021). https://doi.org/10.1007/s00209-020-02657-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-020-02657-8

Keywords

  • Castelnuovo–Mumford regularity
  • Symbolic power
  • Componentwise linear
  • Koszul module
  • Cover ideal

Mathematics Subject Classification

  • 13D02
  • 05C90
  • 05E40
  • 05E45