Tangent cones of monomial curves obtained by numerical duplication

  • Marco D’AnnaEmail author
  • Raheleh Jafari
  • Francesco Strazzanti


Given a numerical semigroup ring \(R=k\llbracket S\rrbracket \), an ideal E of S and an odd element \(b \in S\), the numerical duplication \(S \bowtie ^b E\) is a numerical semigroup, whose associated ring \(k\llbracket S \bowtie ^b E\rrbracket \) shares many properties with the Nagata’s idealization and the amalgamated duplication of R along the monomial ideal \(I=(t^e \mid e\in E)\). In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen–Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when \(\mathrm{gr}_{\mathfrak {m}}(I)\) is Cohen–Macaulay and when \(\mathrm{gr}_{\mathfrak {m}}(\omega _R)\) is a canonical module of \(\mathrm{gr}_{\mathfrak {m}}(R)\) in terms of numerical semigroup’s properties, where \(\omega _R\) is a canonical module of R.


Numerical semigroups Numerical duplication Associated graded ring Cohen–Macaulay rings Gorenstein rings Homogeneous numerical semigroups 

Mathematics Subject Classification

13A30 13H10 20M14 20M25 



  1. 1.
    Anderson, D.D., Winders, M.: Idealization of a module. J. Commut. Algebra 1(1), 3–56 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Barucci, V., D’Anna, M., Strazzanti, F.: A family of quotients of the Rees algebra. Commun. Algebra 43(1), 130–142 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barucci, V., D’Anna, M., Strazzanti, F.: Families of Gorenstein and almost Gorenstein rings. Ark. Mat. 54(2), 321–338 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barucci, V., Dobbs, D.E., Fontana, M.: Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domain. Memoirs of the American Mathematical Society, vol. 125(598). American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  5. 5.
    Barucci, V., Fröberg, R.: Associated graded rings of one dimensional analytically irreducible rings. J. Algebra 304(1), 349–358 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bryant, L.: Goto numbers of a numerical semigroup ring and the Gorensteiness of associated graded rings. Commun. Algebra 38(6), 2092–2128 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    D’Anna, M.: A construction of Gorenstein rings. J. Algebra 306(2), 507–519 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    D’Anna, M., Fontana, M.: An amalgamated duplication of a ring along an ideal: basic properties. J. Algebra Appl. 6(3), 443–459 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    D’Anna, M., Micale, V., Sammartano, A.: When the associated graded ring of a semigroup ring is complete intersection. J. Pure Appl. Algebra 217(6), 1007–1017 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    D’Anna, M., Strazzanti, F.: The numerical duplication of a numerical semigroup. Semigroup Forum 87(1), 149–160 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Delgado, M., García-Sánchez, P.A., Morais, J.: “NumericalSgps”—a GAP package, Version 1.1.5 (2017) Accessed 25 Sept 2017
  12. 12.
    Fossum, R.M., Griffith, P.A., Reiten, I.: Trivial Extensions of Abelian Categories Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory. Lecture Notes in Mathematics, vol. 456. Springer, Berlin (1975)zbMATHGoogle Scholar
  13. 13.
    The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.8.4 (2016). Accessed 4 June 2016
  14. 14.
    García, A.: Cohen–Macaulayness of the associated graded of a semigroup ring. Commun. Algebra 10, 393–415 (1982)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Jäger, J.: Längenberechnung und kanonische ideale in eindimensionalen ringen. Arch. Math. 29, 504–512 (1997)zbMATHGoogle Scholar
  16. 16.
    Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Accessed 19 July 2018
  17. 17.
    Herzog, J., Rossi, M.E., Valla, G.: On the depth of the symmetric algebra. Trans. Am. Math. Soc. 296(2), 577–606 (1986)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Huang, I.-C.: Residual complex on the tangent cone of a numerical semigroup ring. Acta Math. Vietnam. 40(1), 149–160 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Jafari, R., Armengou, S.Zarzuela: Homogeneous numerical semigroups. Semigroup Forum 97(2), 278–306 (2018)MathSciNetGoogle Scholar
  20. 20.
    Lipman, J.: Stable ideals and ARF rings. Am. J. Math. 93, 649–685 (1971)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Oneto, A., Strazzanti, F., Tamone, G.: One-dimensional Gorenstein local rings with decreasing Hilbert function. J. Algebra 489, 91–114 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ooishi, A.: On the associated graded modules of canonical modules. J. Algebra 141, 143–157 (1991)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups. Springer Developements in Mathematics, vol. 20. Springer, Berlin (2009)Google Scholar
  24. 24.
    Şahin, M.: Extensions of toric varieties. Electron. J. Comb. 18(1), 1 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Stamate, D.I.: Betti numbers for numerical semigroup rings. In: Ene, V., Miller, E. (eds.) Multigraded Algebra and Applications. NSA 2016. Springer Proceedings in Mathematics and Statistics, vol. 238. Springer, Cham (2018)Google Scholar

Copyright information

© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di CataniaCataniaItaly
  2. 2.Mosaheb Institute of MathematicsKharazmi UniversityTehranIran
  3. 3.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  4. 4.Institut de Matemàtica Universitat de BarcelonaBarcelonaSpain

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