On the Hilbert function of certain rings of monomial curves

Molinelli, Silvia; Tamone, Grazia

doi:10.1016/0022-4049(94)00058-q

Cited by 24 publications

(18 citation statements)

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“…Note that since the explicit description of the standard basis of the numerical semigroup Γ generated by an almost arithmetic progression in known (for example, see [17,18]), one can use the above theorem to obtain the proofs of the results proved in [13,12] much easily than their original proofs.…”

Section: Lemma 13mentioning

confidence: 97%

“…This proves once again 1.1(3) for a semigroup ring R: The degree deg(h R ) of the h-polynomial is the reduction exponent r 0 , of R. Further, if G(Γ ) is Cohen-Macaulay, then h R (Z) = ∑ r 0 r=0 |S(r)| · Z r by Theorem 1.6 and 1.9(3). If Γ is generated by an arithmetic progression, then G(Γ ) is always Cohen-Macaulay and the explicit computation of the hpolynomial is done in [13]. If Γ is generated by an almost arithmetic progression, then a characterization for the CohenMacaulayness of G(Γ ) is given (in most cases) in [12] and the explicit computation of the h-polynomial is done in [20].…”

Section: Lemma 13mentioning

confidence: 99%

“…If either ν ≤ 3 or ν ≤ e ≤ ν + 2, then (see [4,5]) the Hilbert function H R of R is nondecreasing. However, if either ν ≥ 4 or e ≥ ν + 3, then H R can be locally decreasing (see for example, [16,22,6,13]). Moreover, if R is the semigroup ring K [[Γ ]] of a numerical semigroup Γ ⊆ N generated by an arithmetic sequence, then the associated graded ring G(Γ ) := gr m (R) := ⊕ n∈N m n /m n+1 is always Cohen-Macaulay and hence the Hilbert function of R is non-decreasing (see [13]).…”

Section: Introductionmentioning

confidence: 97%

“…However, if either ν ≥ 4 or e ≥ ν + 3, then H R can be locally decreasing (see for example, [16,22,6,13]). Moreover, if R is the semigroup ring K [[Γ ]] of a numerical semigroup Γ ⊆ N generated by an arithmetic sequence, then the associated graded ring G(Γ ) := gr m (R) := ⊕ n∈N m n /m n+1 is always Cohen-Macaulay and hence the Hilbert function of R is non-decreasing (see [13]). Furthermore, if Γ is generated by an almost arithmetic sequence, then a characterization (in most cases) for the Cohen-Macaulayness of G(Γ ) is given in [12] and the Hilbert function of R is non-decreasing (see [26]).…”

Section: Introductionmentioning

confidence: 97%

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CM defect and Hilbert functions of monomial curves

Patil

Tamone

2011

Journal of Pure and Applied Algebra

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a b s t r a c tIn this article we consider a semigroup ring R = K [[Γ ]] of a numerical semigroup Γ and study the Cohen-Macaulayness of the associated graded ring G(Γ ) := gr m (R) := ⊕ n∈N m n /m n+1 and the behaviour of the Hilbert function H R of R. We define a certain (finite) subset B(Γ ) ⊆ Γ and prove that G(Γ ) is Cohen-Macaulay if and only if B(Γ ) = ∅. Therefore the subset B(Γ ) is called the Cohen-Macaulay defect of G(Γ ). Further, we prove that if the degree sequence of elements of the standard basis of Γ is non-decreasing, then B(Γ ) = ∅ and hence G(Γ ) is Cohen-Macaulay. We consider a class of numerical semigroups Γ = ∑ 3 i=0 Nm i generated by 4 elements m 0 , m 1 , m 2 , m 3 such that m 1 + m 2 = m 0 + m 3 -so called ''balanced semigroups''. We study the structure of the Cohen-Macaulay defect B(Γ ) of Γ and particularly we give an estimate on the cardinality |B(Γ , r)| for every r ∈ N. We use these estimates to prove that the Hilbert function of R is nondecreasing. Further, we prove that every balanced ''unitary'' semigroup Γ is ''2-good'' and is not ''1-good'', in particular, in this case, G(Γ ) is not Cohen-Macaulay. We consider a certain special subclass of balanced semigroups Γ . For this subclass we try to determine the Cohen-Macaulay defect B(Γ ) using the explicit description of the standard basis of Γ ; in particular, we prove that these balanced semigroups are 2-good and determine when exactly G(Γ ) is Cohen-Macaulay.

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Section: Lemma 13mentioning

confidence: 97%

Section: Lemma 13mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

CM defect and Hilbert functions of monomial curves

Patil

Tamone

2011

Journal of Pure and Applied Algebra

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“…We recall that G is Cohen-Macaulay (Proposition 1.1 in [11]) and, if a 0 = (t − 1)n + r, t ≥ 2 and 1 ≤ r ≤ n, then the h-polynomial of G is…”

Section: Preliminariesmentioning

confidence: 99%

Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

Sharifan

Zaare-Nahandi

2009

Journal of Pure and Applied Algebra

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a b s t r a c tLet A(C) be the coordinate ring of a monomial curve C ⊆ A n corresponding to the numerical semigroup S minimally generated by a sequence a 0 , . . . , a n . In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr m (A) with respect to the maximal ideal m of A = A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr m (A) in the case a 0 , . . . , a n is a generalized arithmetic sequence.

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Numerical semigroup algebras

Barucci

Multiplicative Ideal Theory in Commutative Algebra

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On the Hilbert function of certain rings of monomial curves

Cited by 24 publications

References 2 publications

CM defect and Hilbert functions of monomial curves

CM defect and Hilbert functions of monomial curves

Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

Numerical semigroup algebras

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