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Symbolic powers of sums of ideals

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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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