Skip to main content

Symbolic powers of sums of ideals


Let I and J be nonzero ideals in two Noetherian algebras A and B over a field k. Let \(I+J\) denote the ideal generated by I and J in \(A\otimes _k B\). We prove the following expansion for the symbolic powers:

$$\begin{aligned} (I+J)^{(n)} = \sum _{i+j = n} I^{(i)} J^{(j)}. \end{aligned}$$

If A and B are polynomial rings and if \({{\,\mathrm{char}\,}}(k) = 0\) or if I and J are monomial ideals, we give exact formulas for the depth and the Castelnuovo–Mumford regularity of \((I+J)^{(n)}\), which depend on the interplay between the symbolic powers of I and J. The proof involves a result of independent interest which states that the induced map \({{\,\mathrm{Tor}\,}}_i^A(k,I^{(n)}) \rightarrow {{\,\mathrm{Tor}\,}}_i^R(k,I^{(n-1)})\) is zero for any homogeneous ideal I and \(i \ge 0\), \(n \ge 0\). We also investigate other properties and invariants of \((I+J)^{(n)}\) such as the equality between ordinary and symbolic powers, the Waldschmidt constant and the Cohen–Macaulayness.

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


  1. Ahangari Maleki, R.: The Golod property of powers of ideals and Koszul ideals. J. Pure Appl. Algebra 223(no. 2), 605–618 (2019)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  3. Bocci, C., Cooper, S., Guardo, E., Harbourne, B., Janssen, M., Nagel, U., Seceleanu, A., Van Tuyl, A., Vu, T.: The Waldschmidt constant for squarefree monomial ideals. J. Algebra Comb. 44(4), 875–904 (2016)

    MathSciNet  Article  Google Scholar 

  4. Brodmann, M.: Asymptotic stability of \(\text{ Ass }(M/I^nM)\). Proc. Am. Math. Soc. 74, 16–18 (1979)

    MATH  Google Scholar 

  5. Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  6. Chudnovsky, G.V.: Singular points on complex hypersurfaces and multidimensional Schwarz Lemma, In: Seminaire de Theorie des Nombres, Paris 1979–80, Progress in Math., vol. 12, 29–69, Birkhäuser, (1981)

  7. Cooper, S.M., Embree, R.J.D., Hà, H.T., Hoefel, A.H.: Symbolic powers of monomial ideals. Proc. Edinburgh Math. Soc. 27, 39–55 (2017)

    MathSciNet  Article  Google Scholar 

  8. Cutkosky, S.D., Herzog, J., Srinivasan, H.: Asymptotic growth of algebras associated to powers of ideals. Math. Proc. Camb. Philos. Soc. 148, 55–72 (2010)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  10. Eisenbud, D.: Commutative Algebra: with a View Toward Algebraic Geometry. Springer, New York (1995)

    Book  Google Scholar 

  11. Eisenbud, D., Hochster, M.: A Nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions. J. Algebra 58(1), 157–161 (1979)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Eliahou, S., Kervaire, M.: Minimal resolutions of some monomial ideals. J. Algebra 129, 1–25 (1990)

    MathSciNet  Article  Google Scholar 

  14. Esnault, H., Viehweg, E.: Sur une minoration du degré d’hypersurfaces s’annulant en certains points. Ann. Math. 263, 75–86 (1983)

    MathSciNet  Article  Google Scholar 

  15. Francisco, C., Hà, H.T., Van Tuyl, A.: Splittings of monomial ideals. Proc. Am. Math. Soc. 137, 3271–3282 (2009)

    MathSciNet  Article  Google Scholar 

  16. Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry.

  17. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique IV, Étude locale des schémas et des morphismes de schémas, Seconde partie, Publications Mathématiques de l‘IHÉS 24, (1965)

  18. Hà, H.T., Nguyen, H.D., Trung, N.V., Trung, T.N.: The depth function of polynomial ideals, In preparation

  19. Hà, H.T., Trung, N.V., Trung, T.N.: Depth and regularity of powers of sums of ideals. Math. Z. 282, 819–838 (2016)

    MathSciNet  Article  Google Scholar 

  20. Harbourne, B., Huneke, C.: Are symbolic powers highly evolved? J. Ramanujan Math. Soc. 28A, 247–266 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Herzog, J.: Algebraic and homological properties of powers and symbolic powers of ideals, Lect. Notes, CIMPA School on Combinatorial and Computational Aspects of Commutative Algebra, Lahore, (2009)

  22. 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 

  23. 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 

  24. 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 

  25. Hoa, L.T., Trung, T.N.: Castelnuovo-Mumford regularity of symbolic powers of two-dimensional square-free monomial ideals. J. Commut. Algebra 8(1), 77–88 (2016)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  27. McCullough, J., Peeva, I.: Counterexamples to the Eisenbud-Goto regularity conjecture. J. Am. Math. Soc. 31, 473–496 (2018)

    MathSciNet  Article  Google Scholar 

  28. 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 

  29. Nguyen, H.D., Vu, T.: Powers of sums and their homological invariants. J. Pure Appl. Algebra 223(7), 3081–3111 (2019)

    MathSciNet  Article  Google Scholar 

  30. O’Carroll, L., Qureshi, M.A.: Primary rings and tensor products of algebras. Math. Proc. Cambridge Philos. Soc. 92, 41–48 (1982)

    MathSciNet  Article  Google Scholar 

  31. Sabzrou, H., Tousi, M., Yassemi, S.: Simplicial join via tensor products. Manuscripta Math. 126, 255–272 (2008)

    MathSciNet  Article  Google Scholar 

  32. Seidenberg, A.: The prime ideals of a polynomial ideal under extension of the base field. Annali di Matematica Pura ed Applicata 102, 57–59 (1975)

    MathSciNet  Article  Google Scholar 

  33. Sullivant, S.: Combinatorial symbolic powers. J. Algebra 319, 115–142 (2008)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  35. Trung, N.V., Tuan, T.M.: Equality of ordinary and symbolic powers of Stanley–Reisner ideals. J. Algebra 328, 77–93 (2011)

    MathSciNet  Article  Google Scholar 

  36. Vamos, P.: On the minimal prime ideals of a tensor product of two fields. Math. Proc. Cambridge Phil. Soc. 84, 25–35 (1978)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  38. Waldschmidt, M.: Propriétés arithmétiques de fonctions de plusieurs variables II, In Séminaire P. Lelong (Analyse), 1975/76, Lecture Notes Math. 578, Springer, 108–135, (1977)

  39. Zariski, O., Samuel, P.: Commutative Algebra, vol. I. Springer, New York (1975)

    MATH  Google Scholar 

Download references


H.T. Hà is partially supported by the Simons Foundation (Grant #279786) and Louisiana Board of Regents (Grant #LEQSF(2017-19)-ENH-TR-25). H.D. Nguyen is supported by a Marie Curie fellowship of the Istituto Nazionale di Alta Matematica. H.D. Nguyen and T.N. Trung are partially supported by Project ICRTM01\(\_\)2019.01 of the International Centre for Research and Postgraduate Training in Mathematics (ICRTM), Institute of Mathematics, VAST. T.N. Trung is partially supported by Vietnam National Foundation for Science and Technology Development (Grant #101.04-2018.307). Part of this work was done during a research stay of the authors at Vietnam Institute for Advanced Study in Mathematics.

Author information

Authors and Affiliations


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

Hà, H.T., Nguyen, H.D., Trung, N.V. et al. Symbolic powers of sums of ideals. Math. Z. 294, 1499–1520 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Symbolic power
  • Sum of ideals
  • Associated prime
  • Tensor product
  • Binomial expansion
  • Depth
  • Castelnuovo–Mumford regularity
  • Tor-vanishing
  • Depth function

Mathematics Subject Classification

  • Primary 13C15
  • 14B05
  • Secondary 13D07
  • 18G15