CM defect and Hilbert functions of monomial curves

Abstract

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.

On semigroup rings with decreasing Hilbert function

“…(n i , n j ) = (1,4) : S 1 =< 13, 14, 17, 29, 32, 33, 35, 36, 37, 38 > k ′ = 1 c = 26 (n i , n j ) = (2,8) : S 2 =< 13, 15, 21, 32, 38, 40, 44, 46, 48, 50 > k ′ = 1 c = 38 (n i , n j ) = (3, 12), : S 3 =< 13, 16, 25, 35, 44, 47, 53, 56, 59, 62 > k ′ = 1 c = 50 (n i , n j ) = (4,3) : S 4 =< 13, 17, 16, 35, 36, 37, 38, 40, 41, 44 > k ′ = 0 c = 32 (n i , n j ) = (5,7) : S 5 =< 13, 18, 20, 41, 43, 45, 47, 48, 50, 55 > k ′ = 0 c = 43 (n i , n j ) = (6, 11) : S 6 =< 13, 19, 24, 44, 49, 54, 55, 59, 60, 66 > k ′ = 0 c = 54 (n i , n j ) = (7,2) : S 7 =< 13, 20, 28, 47, 55, 62, 63, 70, 71, 77 > k ′ = 0 c = 65 (n i , n j ) = (8,6) : S 8 =< 13, 21, 19, 44, 46, 48, 50, 54, 56, 62 > k ′ = −1 c = 50 (n i , n j ) = (9, 10) : S 9 =< 13, 22, 23, 53, 54, 55, 56, 63, 64, 73 > k ′ = −1 c = 61 (n i , n j ) = (10, 1) : S 10 =< 13, 23, 27, 56, 60, 64, 68, 70, 74, 84 > k ′ = −1 c = 72 (n i , n j ) = (11, 5) : S 11 =< 13, 24, 31, 59, 66, 73, 77, 80, 84, 95 > k ′ = −1 c = 83 (n i , n j ) = (12,9)…”

“…If R = k[[S]] is a semigroup ring, many authors proved that H R is non-decreasing in several cases: • S is generated by an almost arithmetic sequence (if the sequence is arithmetic, then G is Cohen-Macaulay) [17], [13] • S belongs to particular subclasses of four-generated semigroups, which are symmetric [1], or which have Buchsbaum tangent cone [3] • S is balanced [12], [3] • S is obtained by techniques of gluing numerical semigroups [2], [9] • S satisfies certain conditions on particular subsets of S (see below) [ [14] (here recalled in Example 1.6). When G is not Cohen-Macaulay, a useful method to describe H R can be found in some recent papers (see [12], [3], [5]): it is based on the study of certain subsets of S, called D k and C k , (k ∈ N).…”

“…: S12 =< 13, 25, 22, 53, 56, 59, 62, 68, 71, 80 > k ′ = −2 c = 68 Case v = e − With notation 1.1 and 1.2, the Hilbert functionof R ′ = R/t e R, is H R ′ (z) = [1, v−1, a, b, c, d] with a+b+c+d = 4.3. There exist n i , n j ∈ Ap 1 , distinct elements, such that3n i ∈ Ap 3 2n i + n j n i + 2n j 3n j , D h + e =            4n i hn i + n j (h − 1)n i + 2n j · · · (h + 1)n j , (h = 2 or h = 3).Proof.…”

One-dimensional Gorenstein local rings with decreasing Hilbert function

“…• In [22] D.P. Patil and the third author, for the rings associated with balanced numerical semigroups with embedding dimension 4;…”

On the Cohen–Macaulayness of tangent cones of monomial curves inA4(K)

“…Consider n 1 = 627, n 2 = 1546, n 3 = 1662 and n 4 = 3377. The toric ideal I(C) is minimally generated by the set G = {x 18 1 − x 3 2 x 4 3 , x 25 2 − x 7 3 x 8 4 , x 11 3 − x 13 1 x 3 4 , x 11 4 − x 5 1 x 22 2 , x 5 1 x 7 3 − x 3 2 x 3 4 }. Here a 24 = 8, a 34 = 3, a 12 = 3, a 1 = 18, a 2 = 25 and a 13 = 4 < 7 = a 23 .…”