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Depth and regularity of powers of sums of ideals

Mathematische Zeitschrift volume 282, pages 819–838 (2016)Cite this article

Abstract

Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investigate the depth and the Castelnuovo–Mumford regularity of powers of the sum \(I+J\) in \(A \otimes _k B\) in terms of those of I and J. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.

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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

    Ngo Viet Trung & Trân Nam Trung

Authors
  1. Huy Tài Hà
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  2. Ngo Viet Trung
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  3. Trân Nam Trung
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Corresponding author

Correspondence to Ngo Viet Trung.

Additional information

We would like to thank Vietnam Institute for Advanced Study in Mathematics for the hospitality during our visit in 2014, when we started to work on this paper. The first named author is partially supported by the Simons Foundation (Grant #279786). The second author is supported by Vietnam National Foundation for Science and Technology Development under Grant Number 101.04-2014.52.

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Hà, H.T., Trung, N.V. & Trung, T.N. Depth and regularity of powers of sums of ideals. Math. Z. 282, 819–838 (2016). https://doi.org/10.1007/s00209-015-1566-9

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  • DOI: https://doi.org/10.1007/s00209-015-1566-9

Keywords

  • Power of ideals
  • Sum of ideals
  • Depth
  • Regularity
  • Asymptotic behavior

Mathematics Subject Classification

  • 13C05
  • 14H20