Tangent cones of monomial curves obtained by numerical duplication

Abstract

Given a numerical semigroup ring $R=k[[S]]$, 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 $k[[S{\bowtie }^{b}E]]$ 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 $R$.

References in corpus (1)

• When the associated graded ring of a semigroup ring is Complete Intersection

Cited by in corpus (1)

• The Apéry Set of a Good Semigroup