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Tangent cones of monomial curves obtained by numerical duplication

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

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.

Keywords

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

Mathematics Subject Classification

13A30 13H10 20M14 20M25 

Notes

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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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