Let O be a monomial curve in the affine algebraic e-space over a field K and P be the relation ideal of O. If O is defined by a sequence of e positive integers some e -1 of which form an arithmetic sequence then we construct a minimal set of generators for P and write an explicit formula for ~(P ).
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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