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Symbolic powers of sums of ideals

Mathematische Zeitschrift volume 294, pages 1499–1520 (2020)Cite this article

Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA, 70118, USA

    Huy Tài Hà

  2. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Hop Dang Nguyen, Ngo Viet Trung & Tran Nam Trung

  3. International Centre for Research and Postgraduate Training, Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Ngo Viet Trung

  4. TIMAS, Thang Long University, Nghiem Xuan Yem Road, Hoang Mai district, Hanoi, Vietnam

    Tran Nam Trung

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  1. Huy Tài Hà
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  2. Hop Dang Nguyen
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Correspondence to Ngo Viet Trung.

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Hà, H.T., Nguyen, H.D., Trung, N.V. et al. Symbolic powers of sums of ideals. Math. Z. 294, 1499–1520 (2020). https://doi.org/10.1007/s00209-019-02323-8

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  • DOI: https://doi.org/10.1007/s00209-019-02323-8

Keywords

  • 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