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- W1970867887 abstract "In 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 ) = 0̸ . 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 ) = 0̸ and hence G ( Γ ) is Cohen–Macaulay. We consider a class of numerical semigroups Γ = ∑ i = 0 3 N m 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 non-decreasing. 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." @default.
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- W1970867887 date "2011-07-01" @default.
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- W1970867887 title "CM defect and Hilbert functions of monomial curves" @default.
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- W1970867887 doi "https://doi.org/10.1016/j.jpaa.2010.08.007" @default.
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