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Regularity and Koszul property of symbolic powers of monomial ideals

Mathematische Zeitschrift volume 298, pages 1487–1522 (2021)Cite this article

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.

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Notes

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

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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.

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Authors and Affiliations

  1. Faculty of Natural Sciences, Hong Duc University, No. 565 Quang Trung, Dong Ve, Thanh Hoa, Vietnam

    Le Xuan Dung & Truong Thi Hien

  2. Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Hop D. Nguyen & Tran Nam Trung

  3. TIMAS, Thang Long University, Hanoi, Vietnam

    Tran Nam Trung

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  1. Le Xuan Dung
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  2. Truong Thi Hien
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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

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  • 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