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Journal of Algebraic Combinatorics

, Volume 49, Issue 1, pp 1–20 | Cite as

Depth and regularity modulo a principal ideal

  • Giulio Caviglia
  • Huy Tài Hà
  • Jürgen Herzog
  • Manoj Kummini
  • Naoki Terai
  • Ngo Viet TrungEmail author
Article

Abstract

We study the relationship between depth and regularity of a homogeneous ideal I and those of (If) and I : f, where f is a linear form or a monomial. Our results have several interesting consequences on depth and regularity of edge ideals of hypergraphs and of powers of ideals.

Keywords

Depth Regularity Monomial ideal Powers of an ideal Edge ideal Very well-covered graph Chordal graph 

Mathematics Subject Classification

Primary 13C20 Secondary 13D45 14B05 05C65 

Notes

Acknowledgements

Giulio Caviglia is partially supported by the Simons Foundation (Grant #209661). Huy Tài Hà is partially supported by the Simons Foundation (Grant #279786). Ngo Viet Trung is partially supported by Vietnam National Foundation for Science and Technology Development (Grant #101.04-2017.19) and the project VAST.HTQT.NHAT.1/16-18. The main part of this work was done during research stays of the authors at the American Institute of Mathematics in the SQuaRE program “Ordinary powers and symbolic powers” during the period 2012–2014. The authors are grateful to S. A. Seyed Fakhari for pointing out a mistake of Theorem 5.1 in the first version of this paper and to an anonymous referee for mentioning that Lemma 4.1(i) and (ii) were already proved in [31] and [32], respectively.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Giulio Caviglia
    • 1
  • Huy Tài Hà
    • 2
  • Jürgen Herzog
    • 3
  • Manoj Kummini
    • 4
  • Naoki Terai
    • 5
  • Ngo Viet Trung
    • 6
    Email author
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsTulane UniversityNew OrleansUSA
  3. 3.Fachbereich 6, MathematikUniversität Duisburg-EssenEssenGermany
  4. 4.Chennai Mathematical InstituteSiruseriIndia
  5. 5.Faculty of Culture and EducationSaga UniversitySagaJapan
  6. 6.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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