Tangent cones of numerical semigroup rings

Benítez, Teresa Cortadellas; Armengou, Santiago Zarzuela

doi:10.1090/conm/502/09856

Cited by 5 publications

(7 citation statements)

References 7 publications

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“…It has been shown by two of the authors of the present paper in [7] that this Apéry table provides precise information about the structure of G(S) as a graded module over the fiber cone of the ideal generated by the minimal reduction x = t n1 , which is a polynomial ring in one variable over k. This structure is in principle weaker than the structure of G(S) as a ring itself, but as it was observed in the more general context of the study of the fiber cone of ideals with analytic spread one [6], it provides enough information to determine several invariants and properties of the tangent cone, such as the regularity or the Cohen-Macaulay property. Moreover, in [8] the family of invariants given by this structure was explicitly related to other families of invariants of one-dimensional local rings like the microinvariants defined by J. Elias [15], or the Apéry invariants defined by Barucci-Fröberg in [3], see also [4].…”

Section: Introductionsupporting

confidence: 57%

“…The following fact proved by Lemma 2.1] in the more general setting of one-dimensional equicharacteristic analytically irreducible and residually rational domains, will be crucial for our results (see also [7,Lemma 2.1] for a proof in the case of numerical semigroup rings): let I be a fractional ideal of A and f 0 , . .…”

Section: The Apéry Table Of a Monomial Curvementioning

confidence: 99%

“…In order to study the structure of G(S) more in detail we recall the considerations made in [7], coming from an idea in [3]. If we put, for n ≥ 0, Ap(nM ) = {ω n,0 = ne, .…”

Section: The Apéry Table Of a Monomial Curvementioning

confidence: 99%

“…Before showing how to read the structure of G(S) as F (t e )-module in the Apéry table, we recall the following notation introduced in [7].…”

Section: The Apéry Table Of a Monomial Curvementioning

confidence: 99%

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On the Apéry sets of monomial curves

2012

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In this paper, we use the Apéry table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In particular, we precise the shape of the Apéry table of a numerical semigroup of embedding dimension 3, when the tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new proofs for two conjectures raised by V.

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Section: Introductionsupporting

confidence: 57%

Section: The Apéry Table Of a Monomial Curvementioning

confidence: 99%

“…In order to study the structure of G(S) more in detail we recall the considerations made in [7], coming from an idea in [3]. If we put, for n ≥ 0, Ap(nM ) = {ω n,0 = ne, .…”

Section: The Apéry Table Of a Monomial Curvementioning

confidence: 99%

“…Before showing how to read the structure of G(S) as F (t e )-module in the Apéry table, we recall the following notation introduced in [7].…”

Section: The Apéry Table Of a Monomial Curvementioning

confidence: 99%

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On the Apéry sets of monomial curves

2012

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“…(3) A is a finitely generated A-module, and hence is a semi-local, one-dimensional Cohen-Macaulay ring; (4) x is a regular element of A ;…”

Section: Lemma 5 With the Notations Above Introduced The Following Imentioning

confidence: 99%

Apery and micro-invariants of a one-dimensional Cohen–Macaulay local ring and invariants of its tangent cone

Cortadellas¹,

Zarzuela²

2011

Journal of Algebra

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Given a one-dimensional equicharacteristic Cohen-Macaulay local ring A, Juan Elias introduced in 2001 the set of micro-invariants of A in terms of the first neighborhood ring. On the other hand, if A is a one-dimensional complete equicharacteristic and residually rational domain, Valentina Barucci and Ralf Fröberg defined in 2006 a new set of invariants in terms of the Apery set of the value semigroup of A. We give a new interpretation for these sets of invariants that allow to extend their definition to any onedimensional Cohen-Macaulay ring. We compare these two sets of invariants with the one introduced by the authors for the tangent cone of a one-dimensional Cohen-Macaulay local ring and give explicit formulas relating them. We show that, in fact, they coincide if and only if the tangent cone G( A) is Cohen-Macaulay. Some explicit computations will also be given.

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On monomial curves obtained by gluing

Jafari

Armengou

2013

Semigroup Forum

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We study arithmetic properties of tangent cones associated to affine monomial curves, using the concept of gluing. In particular we characterize the Cohen-Macaulay and Gorenstein properties of tangent cones of some families of monomial curves obtained by gluing. Moreover, we provide new families of monomial curves with non-decreasing Hilbert functions. introductionA monomial curve C in the affine space A d k over a field k consists on the set of points defined parametrically by X 1 = t m1 , . . . , X d = t m d , for some positive integers m 1 < · · · < m d . In order to be sure that different parameterizations give rise to different monomial curves, we may assume that gcd(m 1 , . . . , m d ) = 1.In fact, it is known that the set C is an affine variety whose coordinate ring issee for instance E. Reyes, R. H. Villarreal and L. Zárate [17]. The set S = {r 1 m 1 + · · · + r d m d ; r i ≥ 0} is a subset of the non-negative integers N ∪ {0} which is closed under addition, and the condition gcd(m 1 , . . . , m d ) = 1 is equivalent to the property # N \ S < ∞. In other words, S =< m 1 , . . . , m d > is a numerical semigroup minimally generated by the unique minimal system of generators {m 1 , . . . , m d }. The coordinate ring R is called the numerical semigroup ring associated to S and it is denoted by k[S]. Since we are interested in the arithmetical properties at the origin, which is the only singular point of the curve C, we will consider the ring R = k[[t m1 , . . . , t m d ]] = k[[S]]. Note that R is a complete one-dimensional local domain with maximal ideal m = (t m1 , . . . , t m d ).We also consider the tangent cone associated to k[[S]]; that is the graded ring G(S) := n≥0 m n /m n+1 .

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Tangent cones of numerical semigroup rings

Cited by 5 publications

References 7 publications

On the Apéry sets of monomial curves

On the Apéry sets of monomial curves

Apery and micro-invariants of a one-dimensional Cohen–Macaulay local ring and invariants of its tangent cone

On monomial curves obtained by gluing

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