Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three
aa r X i v : . [ m a t h . A C ] J un TANGENT CONE OF NUMERICAL SEMIGROUP RINGS OFEMBEDDING DIMENSION THREE
YI-HUANG SHEN
Abstract.
In this paper, we give new characterizations of the Buchsbaumand Cohen-Macaulay properties of the tangent cone gr m ( R ), where ( R, m ) is anumerical semigroup ring of embedding dimension 3. In particular, we confirmthe conjectures raised by Sapko on the Buchsbaumness of gr m ( R ). Introduction
Throughout this paper let N denote the set { , , , · · · } . A numerical semigroup G generated by n , . . . , n d ∈ N is the set { P ni =1 a i n i | a i ∈ N } . It is a subsemigroupof N . For simplicity, we always assume that G is minimally generated by thesegenerators with n < · · · < n d and gcd( n , . . . , n d ) = 1, unless stated otherwise.Let K be a field and t an indeterminate over K . As a subring of the power seriesring V = K [[ t ]], the ring R = K [[ t n , . . . , t n d ]] is the numerical semigroup ring associated to G with m = ( t n , . . . , t n d ) R being the unique maximal ideal. In thiscase, d is the embedding dimension of R and n is the Hilbert-Samuel multiplicityof R with respect to m .The numerical semigroup ring R is a natural homomorphic image of the powerseries ring S = K [[ x , . . . , x d ]]. The kernel I of this surjection is a binomial ideal(cf. Gilmer [9, Corollary 7.3]) and it will be referred to as the defining ideal of R .Let C be the monomial curve having the parametrization x = t n , x = t n , . . . , x d = t n d . To study the tangent cone of C at the origin, it is very natural to take a closer lookat the initial form ideal I ∗ of the defining ideal I , which is the kernel of the naturalhomomorphism between the associated graded ringsgr n ( S ) = ∞ M i =0 n i / n i +1 ։ gr m ( R ) = ∞ M i =0 m i / m i +1 . Here n and m are the maximal ideals of S and R respectively. The initial formideal I ∗ can be computed from I , for instance, by using the method in Eisenbud[7, Section 15.10.3]. We will also refer to gr m ( R ) as the tangent cone of R .Every two-generated numerical semigroup has a principal defining ideal. Hence,the first non-trivial example arises in the case when the embedding dimension d =3. In this situation, the defining ideal I is always three-generated (cf. Herzog[14]). Furthermore, Herzog [15] and Robbiano and Valla [17] independently provedthat gr m ( R ) is Cohen-Macaulay if and only if the minimal number of generators Mathematics Subject Classification.
Primary 13A30, Secondary 13P10, 13H10.
Key words and phrases.
Numerical Semigroup Rings; Tangent Cone; Cohen-Macaulayness;Buchsbuamness. µ ( I ∗ ) ≤
3. We are able to provide an additional equivalent characterization in termsof reduction number and index of nilpotency. Recall that the ideal Q = ( t n ) R isa principal reduction of the maximal ideal m , with reduction number r Q ( m ) =min (cid:8) r | Q m r = m r +1 (cid:9) and index of nilpotency s Q ( m ) = min (cid:8) s | m s +1 ⊆ Q (cid:9) . Itfollows easily from the definition that r Q ( m ) ≥ s Q ( m ), with equality when gr m ( R )is Cohen-Macaulay (cf. Valabrega and Valla [21, Corollary 2.7]). On the other hand,it is not very difficult to see that r Q ( m ) = s Q ( m ) in general will not lead to theCohen-Macaulayness of gr m ( R ). However, three-generated numerical semigroupsturn out to be very special. We will prove in Theorem 3.3 that when R is anumerical semigroup ring of embedding dimension 3, gr m ( R ) is Cohen-Macaulay ifand only if r Q ( m ) = s Q ( m ).We also study the Buchsbaum and 2-Buchsbaum properties in terms of the 0-thlocal cohomology modules. Let M = L ∞ i =1 n i / n i +1 be the homogeneous maximalideal of gr n ( S ). Then gr m ( R ) is said to be k -Buchsbaum if M k · H M (gr m ( R )) = 0.Normally, 1-Buchsbaum will simply be referred to as Buchsbaum. It is evident thatif length(H M (gr m ( R )) ≤
1, then gr m ( R ) is Buchsbaum, and if length(H M (gr m ( R )) ≤
2, then gr m ( R ) is 2-Buchsbaum. Interestingly enough, the converses are also true inthis case. We prove in Theorem 3.9 and 3.10, that when R is a numerical semigroupring of embedding dimension 3, the associated graded ring gr m ( R ) is Buchsbaumif and only if length(H M (gr m ( R ))) ≤
1, and gr m ( R ) is 2-Buchsbaum if and onlyif length(H M (gr m ( R ))) ≤
2, respectively. In particular, the three conjectures con-cerning the Buchsbaumness of the tangent cone gr m ( R ), raised by Sapko [19], areconfirmed.In Barucci and Fr¨oberg [3] and D’Anna et al. [6], the authors introduced severalinvariants for the numerical semigroup G . Using these invariants, they gave varioussufficient and/or necessary conditions for the tangent cone gr m ( R ) to be Cohen-Macaulay or Buchsbaum. As an application of our treatment, we will show inTheorem 3.20, that the sufficient condition in D’Anna et al. [6, Theorem 3.8] isalso necessary for the tangent cone gr m ( R ) to be Buchsbaum, under the furtherassumption that the embedding dimension d = 3.The main technique in this paper is to manipulate the standard basis of thedefining ideal I . The standard basis algorithm can generate a standard basis froma binomial minimal generating set of I . When the embedding dimension d is small,it is possible to carry out the standard basis algorithm by hand. This approachturns out to be very useful for the investigation of the tangent cone when theembedding dimension is three. We will go over briefly related theory in the nextsection. 2. Preliminaries
Let us begin by explaining several key ingredients of the numerical semigroup G = h n , . . . , n d i . Recall that we assume gcd( n , . . . , n d ) = 1. Hence for everyinteger g ≫
0, we have g ∈ G . The integer f = max { z ∈ Z | z G } is called the Frobenius number of G . Let e be a nonzero element in G . The Ap´ery set of G with respect to e is Ap( G, e ) = { w , · · · , w e − } , where w i is the smallest elementin G congruent to i modulo e . Sometimes we write the elements of Ap( G, e ) inincreasing order: e w = 0 < e w < · · · < e w e − = e + f . The following lemma givesan important characterization of Gorenstein numerical semigroup rings. ANGENT CONE OF NUMERICAL SEMIGROUP RINGS 3
Lemma 2.1 ([1, 4, 16]) . Let G = h n , . . . , n d i be a numerical semigroup and R be itsassociated numerical semigroup ring. Furthermore, let e = n be the multiplicity of R and f be the Frobenius number of G . Then the following conditions are equivalent. (a) The numerical semigroup ring R is Gorenstein. (b) The numerical semigroup G is symmetric in the sense that for every z ∈ Z , z ∈ G if and only if f − z G . (c) e w i + e w e − − i = e w e − for every integer i such that ≤ i ≤ e − . Recall that N = { , , , . . . } . For every z ∈ G , we have z = P i a i n i for some a i ∈ N . We will frequently refer to such a linear combination as a representation of z with respect to G . The integer P i a i is called the length of this representation.It is obvious that ord m ( t z ) = max { P a i | P a i n i = z, a i ∈ N } . When there is noconfusion, we also write this number as ord G ( z ) and similarly define min-ord G ( z )to be min { P a i | P a i n i = z, a i ∈ N } . The ratio ord G ( z )min-ord G ( z ) is called the elasticityof z with respect to G . We say z = P i a i n i is a maximal representation of z withrespect to G if P a i = ord G ( z ).The semigroup G can be equipped with a partial order ≦ G : for a, b ∈ G , wewrite a ≦ G b if b − a ∈ G . This order relation was considered, for instance,in Rosales and Garc´ıa-S´anchez [18]. Another important partial order is (cid:22) G : for a, b ∈ G , we write a (cid:22) G b if there exists an element c in G such that a + c = b and ord G ( a ) + ord G ( c ) = ord G ( b ). The partial order (cid:22) G in this formulation wassuggested by Lance Bryant. Lemma 2.2 (Bryant [5, Corollary 3.20]) . Let ( R, m ) be a Gorenstein numericalsemigroup ring associated to a semigroup G = h n , . . . , n d i , and assume that theassociated graded ring gr m ( R ) is Cohen-Macaulay. Then gr m ( R ) is Gorenstein ifand only if the following condition holds for e = n : w i (cid:22) G w e − for all w i ∈ Ap(
G, n ) . ( † ) Remark . In the previous lemma, if the numerical semigroup G is symmetricand the elasticity of w e − with respect to G is 1, then every representation of w e − is maximal. Hence the condition ( † ) holds automatically.The following remark to Lemma 2.2 will be useful for the proof of Theorem 3.3. Remark . Let G = h n , . . . , n d i be a symmetric numerical semigroup and R itsassociated numerical semigroup ring. Suppose m is the maximal ideal of R with aprincipal reduction Q = ( t n ) R . If the condition ( † ) holds and s Q ( m ) = r Q ( m ), thengr m ( R ) is Gorenstein. We do not need to assume that gr m ( R ) is Cohen-Macaulayin advance. For the proof, see Bryant [5, Theorem 3.14].For a Gorenstein numerical semigroup ring, the index of nilpotency s Q ( m ) canbe computed by using the m -adic order of w e − . Lemma 2.5.
Let ( R, m ) be a Gorenstein numerical semigroup ring associated tothe semigroup G = h n , . . . , n d i , and let f denote the Frobenius number of G . Thenfor the principal reduction Q = ( t n ) R of m , s Q ( m ) = ord G ( f + n ) .Proof. We always haves Q ( m ) = max { ord G ( w ) | = w ∈ Ap(
G, n ) } . When G is symmetric, the maximum is obviously achieved at ord G ( f + n ). (cid:3) YI-HUANG SHEN
By convention, let S = K [[ x , . . . , x d ]] be a power series ring over a field K and n be its maximal ideal. This ring maps naturally onto the numerical semigroupring R = K [[ t n , . . . , t n d ]]. For each nonzero element x ∈ S , let o = ord n ( x ) < ∞ be the n -adic order of x . We denote by x ∗ the residue class of x in n o / n o +1 and callit the initial form of x . The initial form ideal I ∗ ⊆ gr n ( S ) is generated by x ∗ forall x ∈ I , and gr m ( R ) ∼ = gr n ( S ) /I ∗ canonically. For our numerical semigroup ring R , the radical of the initial ideal I ∗ is very simple. Lemma 2.6. √ I ∗ = h x ∗ , . . . , x ∗ d i gr n ( S ) .Proof. Consider the binomials f i := x n i − x n i ∈ I for 2 ≤ i ≤ d . Since n Let T = K [ X ] = K [ x , . . . , x d ] be a polynomial ring over a field K . (a) A total order > τ on the set of monomials (cid:8) X α | α ∈ N d (cid:9) ⊆ T is a monomialorder if X α > τ X β = ⇒ X α + γ > τ X β + γ for any α, β, γ ∈ N d .(b) A monomial order > τ is a local order if 1 > τ X α for all α = 0 ∈ N d ; it is a global order or term order if 1 < τ X α for all α = 0 ∈ N d .(c) A local monomial order > τ is degree compatible if deg( X α ) < deg( X β ) = ⇒ X α > τ X β for any α, β ∈ N d .The negative degree reverse lexicographic order that we shall introduce here isa very useful monomial order. It is local and degree compatible. Definition 2.8. A monomial order > ds such that X α > ds X β ⇐⇒ n deg( X α ) < deg( X β ) or (cid:16) deg( X α ) = deg( X β ) and ∃ ≤ i ≤ d : α ( n ) = β ( n ) , . . . , α ( i + 1) = β ( i + 1) , α ( i ) < β ( i ) (cid:17)o , is called a negative degree reverse lexicographic order on T = K [ x , . . . , x d ].Fix a monomial order > τ on T . For a nonzero polynomial f = P c α X α , the leading monomial of f is LM( f ) := max > τ { X α | c α = 0 } . When LM( f ) = X α , wecall LC( f ) := c α the leading coefficient of f and LT( f ) := c α X α the leading term of f . For two nonzero polynomials f and g in T , the s -polynomial is defined asfollows: spoly( f, g ) := lcm(LM( f ) , LM( g ))LT( f ) f − lcm(LM( f ) , LM( g ))LT( g ) g. A finite set B ⊆ I is a standard basis of an ideal I ⊆ T if for any nonzero f ∈ I ,there exists an element g ∈ B satisfying LM( g ) | LM( f ). The famous Buchberger’scriterion (cf. Greuel and Pfister [11, Theorem 1.7.3]) says that a generating set B = { g , . . . , g t } of I is a standard basis if and only there exist c ijk ∈ T such that ANGENT CONE OF NUMERICAL SEMIGROUP RINGS 5 for all i and j , s ( g i , g j ) = P k c ijk g k and LM( c ijk g k ) < τ LM( s ( g i , g j )) when c ijk = 0.With a global monomial order, a standard basis B can always be generated froma generating set B = { g , . . . , g t ′ } of I by applying the standard basis algorithm(cf. Greuel and Pfister 11, Section 1.7). Roughly speaking, this algorithm extendsthe generating set B to the standard basis B by successively adding nonzero s -polynomials spoly( g i , g j ). Since new generators will also be needed for calculatingthe s -polynomials, when working with a local monomial order, the standard basisalgorithm might not terminate in finite steps. Nevertheless, this is not a problemwhen dealing with the defining ideal of a numerical semigroup ring. The finitenessis guaranteed by the algorithm described in Section 15.10.3 of Eisenbud [7] witha global monomial order. The algorithm in Eisenbud [7] uses a homogenizationtechnique and has been implemented in the package TangentCone of Macaulay2 [10]. Furthermore, one can remove the redundancies in the standard basis obtainedhere and arrives at a reduced standard basis which is uniquely determined, seeDefinition 1.6.2 and Exercise 1.6.1 of Greuel and Pfister [11] for clarity.From now on, fix a numerical semigroup G = h n , . . . , n d i . Definition 2.9. For distinct α = ( α (1) , . . . , α ( d )) and β = ( β (1) , . . . , β ( d )) ∈ N d ,the binomial f = X α − X β ∈ T is weakly balanced with respect to G if P i α ( i ) n i = P i β ( i ) n i . The binomial f is called balanced if it is weakly balanced, deg( X α ) =deg( X β ), and X α and X β are coprime.For the numerical semigroup ring R , the defining ideal I is generated by weaklybalanced binomials (cf. Gilmer [9, Corollary 7.3]). If we fix a degree compatiblelocal monomial order and apply the standard basis algorithm (cf. Greuel and Pfister[11, Section 1.7]) to this generating set, we are able to obtain a reduced standardbasis { f , . . . , f s } . In this case, the initial form ideal I ∗ is minimally generated bythe corresponding initial forms: I ∗ = h f ∗ , . . . , f ∗ s i gr n ( S ) . Since each f i is also a weakly balanced binomial, f ∗ i is either a monomial or abalanced binomial. In the latter case, roughly speaking, f i = f ∗ i . In the rest ofthis paper, when we say that g is a minimal generator of I ∗ , it is understood that g ∈ { f ∗ , . . . , f ∗ s } when the minimal binomial generating set of I and the monomialorder is clear.Our next task is to choose a suitable monomial order > τ for K [ x , . . . , x d ]. Definition 2.10. A local monomial order > τ is nice in the variable x i if thefollowing holds: n deg( X α ) < deg( X β ) or (cid:16) X α − X β is balanced, β ( i ) > (cid:17)o = ⇒ X α > τ X β . Being nice is really a mild condition. For instance, the following monomial orderis nice in x : X α > X β def ⇐⇒ n deg( X α ) < deg( X β ) or (cid:16) deg( X α ) = deg( X β ) and ∃ ≤ i ≤ d : α (1) = β (1) , . . . , α ( i − 1) = β ( i − , α ( i ) < β ( i ) (cid:17)o . When d = 3, the negative degree reverse lexicographic order is also nice in x . Definition 2.11. Let K [ x , . . . , x d ] be a polynomial ring with a degree compatiblelocal monomial order > τ and G = h n , . . . , n d i the underlying numerical semigroup. YI-HUANG SHEN An ideal J ⊆ K [ x , . . . , x d ] is called almost balanced if it satisfies the following twoconditions:(a) √ J = ( x , . . . , x d );(b) there is a reduced standard basis { f , . . . , f t } of J such that f i is either amonomial or a balanced binomial.For a numerical semigroup ring ( R, m ) and its defining ideal I , it is clear thatthe initial form ideal I ∗ is almost balanced. The following lemma is crucial whendiscussing the Cohen-Macaulayness of gr m ( R ). Lemma 2.12. Let > τ be a local monomial order for T = K [ x , . . . , x d ] that isnice in x , and J an almost balanced T -ideal. Suppose { f , . . . , f t } forms a reducedstandard basis of I as in the previous definition. Then T /J is Cohen-Macaulay ifand only if for any f i , either f i is binomial or x does not divide f i .Proof. Observe that J is homogeneous. Hence for the homogeneous maximal ideal m = h x , . . . , x d i of T , T /J is Cohen-Macaulay if and only if ( T /J ) m is Cohen-Macaulay, if and only if x is a regular element on ( T /J ) m .If some f i = x X α , then X α J since { f , . . . , f s } is a reduced standard basis.Therefore J is not a perfect ideal.Conversely, suppose that x f ∈ J m and 0 = f J m . By multiplying suitableunit element in T m , we may assume that f ∈ T . Notice that x LM( f ) = LM( x f )is divisible by some LM( f i ). But LM( f ) is not divisible by this LM( f i ), henceLM( f i ) is divisible by x . Since the monomial order > τ is nice in x , f i cannot bea balanced binomial. Hence it is a monomial. (cid:3) Example 2.13. Let K be a field, R = K [[ t , t , t ]] and m = ( t , t , t ) R . Thenthe defining ideal is I = ( x x − x , x − x x , x x − x ) ⊆ K [[ x , x , x ]] . With respect to the negative degree reverse lexicographic order, the set (cid:8) x x − x , x − x x , x x − x , x − x (cid:9) forms a reduced standard basis of I . Thus the initial form ideal is I ∗ = ( x x , x , x x , x , ) ⊆ K [ x , x , x ] . Since the generator x x is divisible by x , gr m ( R ) is not Cohen-Macaulay. Thisnon-Cohen-Macaulay property also follows immediately from the fact that I ∗ isgenerated by more than 3 elements.The α -invariants of the numerical semigroup G will also be needed in our inves-tigation. Definition 2.14. For the numerical semigroup G = h n , · · · , n d i , define α i = min { α ∈ N | αn i ∈ h n , . . . , b n i , . . . , n d i , α = 0 } for 1 ≤ i ≤ d . Here for the “truncated” semigroup h n , . . . , b n i , . . . , n d i , we do notrequire that gcd { n j | j = i } = 1.In Example 2.13, we have α = 5, α = 3 and α = 2. ANGENT CONE OF NUMERICAL SEMIGROUP RINGS 7 When the Embedding Dimension d = 3In this section, we will always use the negative degree reverse lexicographic orderon gr n ( S ) ∼ = K [ x , x , x ]. Hence if f = x b − x a x c is a balanced binomial withrespect to the numerical semigroup G = h n , n , n i , then the leading monomial of f is x b .The basis of the initial form ideal I ∗ will be constructed as in Section 2 fromthe binomial basis given in the following fundamental theorem for three-generatednumerical semigroups.3.1. Fundamental Theorem.Theorem 3.1 ([14]) . Let R be a numerical semigroup ring corresponding to thenumerical semigroup G = h n , n , n i . Then for the α -invariants α i as defined inDefinition 2.14, and suitable numbers α ij ∈ N where ≤ i, j ≤ , the followingconditions hold. (a) If R is Gorenstein, then, after a permutation ( i, j, k ) of (1 , , , the definingideal is I = ( x α i i − x α j j , x α k k − x α ki i x α kj j ) , and the Frobenius number of G is f = ( α i − n i + ( α k − n k − n j . (b) If R is not Gorenstein, then I = ( x α − x α x α , x α − x α x α , x α − x α x α ) , where α i = α ji + α ki for all permutation ( i, j, k ) of (1 , , . Furthermore,each α ij > for ≤ i = j ≤ . The invariants appeared in this theorem have been studied extensively, for in-stance, in Fel [8] and Rosales and Garc´ıa-S´anchez [18]. Now, applying Lemma2.12 to Theorem 3.1, one can quickly give arithmetic conditions for gr m ( R ) to beCohen-Macaulay. Corollary 3.2. Resume the notation from Theorem 3.1. (a) If I = ( x α − x α , x α − x α x α ) , then gr m ( R ) is a complete intersectionand I ∗ is generated by { x α , x α } . (b) If I = ( x α − x α , x α − x α x α ) , then gr m ( R ) is Cohen-Macaulay if andonly if α ≤ α + α . When gr m ( R ) is Cohen-Macaulay, I ∗ is generatedby { x α , ( x α − x α x α ) ∗ } . (c) If I = ( f := x α − x α , f := x α − x α x α ) , we can always assumethat α < α . Then gr m ( R ) is Cohen-Macaulay if and only if α + α ≤ α + α − α . Set f := x α + α − x α x α − α , the s -polynomial of f and f .When gr m ( R ) is Cohen-Macaulay, I ∗ is generated by { x α , x α x α , f ∗ } . (d) If I = ( x α − x α x α , x α − x α x α , x α − x α x α ) , then gr m ( R ) isCohen-Macaulay if and only if α ≤ α + α . When gr m ( R ) is Cohen-Macaulay, I ∗ is generated by { x α x α , ( x α − x α x α ) ∗ , x α } . YI-HUANG SHEN Cohen-Macaulayness. The purpose of this subsection is to establish a newcharacterization for the Cohen-Macaulayness of gr m ( R ) when the embedding di-mension d = 3. Notice that the ideal Q = ( t n ) R is a principal reduction ofthe maximal ideal m . We want to connect the Cohen-Macaulay property withthe reduction number r Q ( m ) = min (cid:8) r | Q m r = m r +1 (cid:9) and the index of nilpo-tency s Q ( m ) = min (cid:8) s | m s +1 ⊆ Q (cid:9) . It follows easily from the definition thatr Q ( m ) ≥ s Q ( m ). When gr m ( R ) is Cohen-Macaulay, a result of Valabrega andValla [21, Corollary 2.7] implies that r Q ( m ) = s Q ( m ). On the other hand, it isnot very difficult to see that r Q ( m ) = s Q ( m ) in general will not lead to the Cohen-Macaulayness of gr m ( R ). However, it is different for a numerical semigroup ring ofembedding dimension 3. Theorem 3.3. Suppose ( R, m ) is a numerical semigroup ring of embedding dimen-sion 3, and Q = ( t n ) R is a principal reduction of the maximal ideal m . The tangentcone gr m ( R ) is Cohen-Macaulay if and only if the index of nilpotency s Q ( m ) equalsthe reduction number r Q ( m ) .Proof. Let G = h n , n , n i be the associated numerical semigroup. Since the “onlyif” part is clear, we may assume that s Q ( m ) = r Q ( m ) and proceed to show thatgr m ( R ) is Cohen-Macaulay. For every x ∈ S = K [[ x , x , x ]], we will write x forits image in R = S/I , where I is the defining ideal. The integer e will be themultiplicity n .(1) First, we study the case when G is symmetric, i.e., the numerical semigroupring R is a complete intersection. Now, for the Ap´ery set element w e − = f + n where f is the Frobenius number of G , ord G ( w e − ) = s Q ( m ) byLemma 2.5. Using the same notation as in Theorem 3.1, we have threecases.(i) When ( i, j, k ) = (1 , , i, j, k ) = (1 , , w e − = ( α − n + ( α − n by part (a) of Theorem 3.1. This is obviously the unique representa-tion of w e − with respect to G . It follows from Remark 2.3 that thecondition ( † ) holds. When r Q ( m ) = s Q ( m ), gr m ( R ) is Gorenstein byRemark 2.4.(iii) Suppose ( i, j, k ) = (2 , , α < α . Nowthe Frobenius number can be written as f = ( α + α − n + ( α − n − n , therefore w e − = ( α + α − n + ( α − n . This gives the maximal representation of w e − with respect to G , ands Q ( m ) = ord m ( w e − ) = ( α + α − 1) + ( α − 1) by Lemma 2.5. Inthis case, the tangent cone gr m ( R ) is Cohen-Macaulay if and only if α + α ≤ α + α − α by Corollary 3.2(c). For this subcase, wewant to prove that the following conditions are equivalent:(a) gr m ( R ) is Cohen-Macaulay. ANGENT CONE OF NUMERICAL SEMIGROUP RINGS 9 (b) r Q ( m ) = s Q ( m ).(c) r Q ( m ) ≤ α + α − ⇒ (b): It is clear.(b) ⇒ (c): For r = r Q ( m ), x α + α − x α − ∈ m r . So x α + α x α − ∈ m r +1 = Q m r . But x α + α x α − = x α x α + α − = x α x α − , and x is a regular element in the domain R . Hence x α − x α − ∈ m r .We want to show that ord m ( x α − x α − ) = ( α − 1) + ( α − α − n + ( α − n is theunique representation of this element with respect to G . Suppose not,then ( α − n + ( α − n = an + bn + cn , with a, b, c ∈ N and b > 0. By the minimality of α and α , one musthave a ≤ α − c ≤ α − 1. Now( α − − a ) n + ( α − − c ) n = bn Since b > b ≥ α by the minimality of α . Hence( α − − a ) n + ( α − − c ) n = ( b − α ) n + α n , thus ( α − − a ) n = ( b − α ) n + ( c + 1) n , which contradicts the minimality of α . This shows that (b) implies(c).(c) ⇒ (a): We have α + α + α − Q ( m ) ≤ r Q ( m ) ≤ α + α − α + α ≤ α + α − α . Hence gr Q ( m )is Cohen-Macaulay and (a) holds.(2) Next, we consider the case when the semigroup group G is not symmetric.Recall that the defining ideal is I = ( f := x α − x α x α , f := x α − x α x α , f := x α − x α x α ) . Our aim is to show that if s Q ( m ) = r Q ( m ), then α ≤ α + α . First ofall, with the partial order ≦ G we introduced in Section 2, we havemax ≦ G Ap( G, n ) = { ( α − n + ( α − n , ( α − n + ( α − n } , from Rosales and Garc´ıa-S´anchez [18, Lemma 4]. Therefore, the index ofnilpotency iss Q ( m ) = max { α + α − , α + α − } , by a proof similar to that of Lemma 2.5. Now we are ready to completethe proof.(i) The case when s Q ( m ) = α + α − α + α − r = r Q ( m ). Then x α x α − ∈ m r +1 = Q m r . Notice that x α x α − = x α x α − . Hence x α − x α − ∈ m r .Similar to the Gorenstein case, one can show that the representation z = ( α − n +( α − n ∈ G is unique, hence ord m ( x α − x α − ) = α + α − ≥ r = s = α + α − 2. Thus α ≤ α + α − α = α + α , and gr m ( R ) is Cohen-Macaulay.(ii) If s Q ( m ) > α + α − Q ( m ) = s Q ( m ), then let δ := α − α > α + α − r − δ . Now ord m ( x α − x α − ) ≥ r − δ , hence x α + δ x α − ∈ m r +1 = Q m r . It follows easily that x α − x δ x α − ∈ m r . Suppose to the contrary that gr m ( R ) is not Cohen-Macaulay, then α = α + α > α + α . Hence δ = α − α < α − α < α .We claim that the representation P : z = ( α − n + δn + ( α − n ∈ G is maximal. Let Q : z = an + bn + cn be a distinct representation of z . Then we have 6 cases when compar-ing the coefficients of P and Q . The proof of the claim is straightfor-ward and easy. To avoid unnecessary repetition, we just consider theexemplifying case where α − ≥ a , δ < b and α − ≥ c . Whence( b − δ ) n = ( α − − a ) n + ( α − − c ) n . By the choice of α , b − δ ≥ α , hence( b − δ − α ) n = ( α − − a − α ) n + ( α − − c − α ) n . Or equivalently( a + 1) n + ( b − δ − α ) n = ( α − − c ) n . This implies that 0 < α − − c < α , which is against the choice of α .The argument for other cases is similar. Now ord m ( x α − x δ x α − ) = δ + α + α − ≥ r = s = α + α − δ . Hence α ≤ α + α − α = α + α , and gr m ( R ) is again Cohen-Macaulay. (cid:3) We thank Lance Bryant for the helpful comments regarding Theorem 3.3. Example 3.4. Let K be a field, R = K [[ t , t , t ]]. Then Q = ( t ) R is a principalreduction of the maximal ideal m = ( t , t , t ) R . We have r Q ( m ) = s Q ( m ) = 5,hence gr m ( R ) is Cohen-Macaulay by Theorem 3.3.The statement of Theorem 3.3 fails if the embedding dimension is 4. Example 3.5. Let R = K [[ t , t , t , t ]], Q = ( t ) R and m = ( t , t , t , t ) R .Then R is Gorenstein and we have s Q ( m ) = r Q ( m ) = 4. But t = t t ∈ m (( m : R m ) ∩ m ) \ m . Thus, by D’Anna et al. [6, Corollary 2.3 and Remark 2.7], gr m ( R ) is not evenBuchsbaum.If the 1-dimensional local ring R is not associated to any numerical semigroup,then the theorem might still fail, even when R has embedding dimension 3. Theprototype of the following example is due to Lance Bryant. ANGENT CONE OF NUMERICAL SEMIGROUP RINGS 11 Example 3.6. The computer algebra system Singular [12] suggests that the ideal I = ( a + c + b , a b + ac + b ) is a prime ideal in the polynomial ring Q [ a, b, c ].Let R = Q [[ a, b, c ]] /IR . The initial form ideal I ∗ = ( b c + ac , abc , a c , a b, a ),hence Q [ a, b, c ] /I ∗ is not Cohen-Macaulay. On the other hand, Q = ( b − c ) R is aprincipal reduction of the maximal ideal m = ( a, b, c ) R . It is not difficult to seethat r Q ( m ) = s Q ( m ) = 6.3.3. Buchsbaumness and -Buchsbaumness. Recall that for a one-dimensionalstandard graded ring A with the unique homogeneous maximal ideal M , a finitelygenerated A -module M is called k -Buchsbaum if M k · H M ( M ) = 0. The 1-Buchsbaum condition is simply called Buchsbaum , and 0-Buchsbaum modules areprecisely the Cohen-Macaulay modules.In this subsection, we will mainly investigate the Buchsbaum and 2-Buchsbaumproperty of gr m ( R ), where ( R, m ) is a numerical semigroup ring of embedding di-mension 3. Denote the homogeneous maximal ideal of gr n ( S ) ∼ = K [ x , x , x ] by M . Since gr m ( R ) = gr n ( S ) /I ∗ , we will write the image of f ∈ gr n ( S ) in gr m ( R )as f . For the local cohomology module H M (gr m ( R )), we can also replace M by the homogeneous maximal ideal of gr m ( R ). Let r be the reduction numberof m . Then it is not very difficult to show that H M (gr m ( R )) = (0 : gr m ( R ) M r )(cf. D’Anna et al. [6, Lemma 2.2]). Therefore, gr m ( R ) is Buchsbaum if and only H M (gr m ( R )) = (0 : gr m ( R ) M ).Sapko investigated the tangent cone of numerical semigroup rings and made thefollowing conjectures regarding the Buchsbaumness. Conjecture 3.7 ([19]) . Let ( R, m ) be a numerical semigroup ring of embeddingdimension . (a) If gr m ( R ) is Buchsbaum, then the initial form ideal I ∗ of I is generated by4 elements, and for some integer k ≥ , (0 : gr m ( R ) M ) = ( x k ) gr m ( R ) . (b) gr m ( R ) is Buchsbaum if and only if length( H M (gr m ( R ))) ≤ . The main theme of this subsection is to confirm the above conjectures, and provesimilar results when the tangent cone is 2-Buchsbaum. Lemma 3.8. Let ( R, m ) be a numerical semigroup ring of embedding dimension . If gr m ( R ) is not Cohen-Macaulay and M is the homogeneous maximal idealof gr m ( R ) , then the 0-th local cohomology module H M (gr m ( R )) is principal and isgenerated by x γ for suitable γ ∈ N .Proof. Recall that given a degree-compatible local monomial order like the > ds , theinitial form ideal I ∗ is generated by the initial forms of a binomial standard basisof I . Since n < n < n , I ∗ is generated by forms of the following 4 types, withall visible exponents strictly positive:(a) x α ,(b) x γ or a balanced binomial x γ − x γ x γ ,(c) x a x c ,(d) x b x c .For any minimal generating set, there is exactly one generator of type (a). Thesame is true for generators of type (b). To see this, it suffices to notice that if x γ − x γ x γ is balanced, then x γ is its leading monomial. On the other hand,there might be more than one generators of type (c) or (d).It follows from Lemma 2.12 that I ∗ is Cohen-Macaulay if and only if generatorsof type (c) do not exist. If gr m ( R ) is not Cohen-Macaulay, then H M (gr m ( R )) = 0.We claim that this local cohomology module is generated by x γ where γ = min { c | x a x c is a generator of I ∗ of type (c) for some nonzero a, c ∈ N } . Since √ I ∗ = ( x , x ), x γ ∈ H M (gr m ( R )). On the other hand, I ∗ + ( x γ ) is (notnecessarily minimally) generated by x γ together with the remaining generators of I ∗ of type (b) or (d). This last ideal is Cohen-Macaulay by Lemma 2.12. Hencethe local cohomology module H M (gr m ( R )) is generated by x γ . (cid:3) For a one-dimensional Cohen-Macaulay local ring ( R, m ), when length( H M (gr m ( R ))) ≤ 1, one sees immediately that the associated graded ring gr m ( R ) is Buchsbaum.When R is a numerical semigroup ring of embedding dimension 3, the previouslemma implies that the converse is also true. Theorem 3.9. Let ( R, m ) be a numerical semigroup ring of embedding dimension . Then gr m ( R ) is Buchsbaum if and only if length( H M (gr m ( R ))) ≤ . Next, we study the 2-Buchsbaumness of the tangent cone. When length( H M (gr m ( R ))) ≤ 2, the associated graded ring gr m ( R ) is clearly 2-Buchsbaum. We found that theconverse is also true for a numerical semigroup ring of embedding dimension 3. Theorem 3.10. Let ( R, m ) be a numerical semigroup ring of embedding dimension . Then gr m ( R ) is 2-Buchsbaum if and only if length( H M (gr m ( R ))) ≤ . Proof. It suffices to assume that gr m ( R ) is 2-Buchsbaum, not Cohen-Macaulay, andinvestigate the length of the local cohomology module. Lemma 3.8 guarantees amonomial minimal generator x a x c in I ∗ . Since x x c , x c ∈ I ∗ , we have 1 ≤ a ≤ α − ≤ c ≤ α − I ∗ having theform x a x c . It is easy to see that this could fail only when both x x α − and x x α − are minimal generators of I ∗ . Since they are minimal, there exist β , β ∈ N suchthat both x β − x x α − and x β − x x α − are weakly balanced binomials in I .Because n > n > n , we must have β > β and x β − β x = x . Hence G istwo-generated, contradicting our assumption of d = 3.Meanwhile, we notice that x x c ∈ I ∗ . Hence either α = 2 or this monomial isdivisible by the leading monomial x b x c ′ of a minimal generator of I ∗ with 1 ≤ b ≤ ≤ c ′ ≤ c . If α = 2, then Corollary 3.2 implies that I ∗ is Cohen-Macaulay.Hence α > I ∗ having the form x b x c ′ with 1 ≤ b ≤ ≤ c ′ ≤ c .Now we are ready to show that length( H M (gr m ( R ))) ≤ x a x c = x x α − . Notice that x x α − , x x α − ∈ I ∗ . Eachof them has to be divisible by some monomial minimal generator of I ∗ ofthe form x b x c ′ with 1 ≤ b ≤ 2, 1 ≤ c ′ ≤ c . But there is at most one suchgenerator. Hence this generator must divide the gcd( x x α − , x x α − ) = x x α − . In particular, x x α − ∈ I ∗ . Consequently the vector space H M (gr m ( R )) = ( x α − ) gr m ( R ) is generated by (cid:8) x α − , x α − (cid:9) . ANGENT CONE OF NUMERICAL SEMIGROUP RINGS 13 (b) The case x a x c = x x α − can never happen. Notice that the image of x α − generates the local cohomology module. Hence x x · x α − ∈ I ∗ .We know that there cannot exist two distinct minimal generators of theform x α x γ in I ∗ . Since x x α − is assumed to be a minimal generator, ithas to divide x x α − , which is impossible.(c) Assume that x a x c = x x α − . Notice that x α ∈ I ∗ . Hence the local coho-mology module is generated as a vector space by (cid:8) x α − (cid:9) or (cid:8) x α − , x x α − (cid:9) .(d) Assume that x a x c = x x α − . We have x x x α − ∈ I ∗ by the 2-Buchsbaumness.Since x x x α − is not a minimal generator, either x x α − ∈ I ∗ or x x α − ∈ I ∗ . Because x x α − is a minimal generator, the first option cannot hap-pen. Hence x x α − ∈ I ∗ and the local cohomology module is generated asa vector space by (cid:8) x α − , x x α − (cid:9) . (cid:3) Lemma 3.11. Suppose ( R, m ) is a Gorenstein numerical semigroup ring with em-bedding dimension d = 3 and gr m ( R ) is -Buchsbaum, then gr m ( R ) is indeed Cohen-Macaulay.Proof. Suppose to the contrary that gr m ( R ) is not Cohen-Macaulay. Then α ≥ I ∗ has exactly one minimal generatorof the form x γ x γ with γ , γ > 0. Furthermore, γ = 1 or 2, and α = γ + 1 or γ + 2. Since x x α − ∈ I ∗ , there exists a generator f = x β x γ − x α belonging tothe binomial reduced standard basis of I with β ≤ γ ≤ α − 1. Since γ < α and α ≥ 3, we have β > f is not a new generator generated fromthe standard basis algorithm. Instead, it has to be one of the minimal binomialgenerators of I .By Theorem 3.1, when R is Gorenstein, the defining ideal, after a permutation( i, j, k ) of (1 , , I = ( x α i i − x α j j , x α k k − x α ki i x α ki j ) . By symmetry, we can always assume that i < j . Now one can characterize whenthe associated graded ring is 2-Buchsbaum in terms of these α ’s. By our discussionfor x β x γ , we only need the check the case where ( i, j, k ) = (2 , , I = ( f := x α − x α , f := x α − x α x α ) . It is evident that f ∗ = x α and we can assume that 0 ≤ α < α , hence α > f ∗ = − x α x α and it is non-comparable with f ∗ . Applying the standardbasis algorithm, we get f := spoly( f , f ) = − x α + α + x α x α − α , which mustbelong to the reduced standard basis of I . Notice that I ∗ is perfect if and onlyif α + α ≤ α + α − α . Since we have assumed that I ∗ is not perfect, f ∗ = x α x α − α . Now by our discussion above, α ≤ 2. But if G is minimallygenerated by 3 elements, then α > 2, and this is a contradiction. Thus, gr m ( R ) isCohen-Macaulay. (cid:3) Proposition 3.12. Suppose ( R, m ) is a numerical semigroup ring of embedding di-mension and gr m ( R ) is 2-Buchsbaum, then the initial form ideal I ∗ is -generated.Proof. We may assume that gr m ( R ) is not Cohen-Macaulay. Hence by the proofsof Theorem 3.10 and Lemma 3.11, we have α ≥ R is not Gorenstein. Nowit suffices to show that I has exactly one more standard basis element in additionto its 3 minimal binomial generators. (a) Suppose that x x α − is a minimal generator for I ∗ , then I = ( f := x α − x x , f := x α − x x α − , f := x α − x α − x α − )by case (a) of the proof for 3.10 together with Theorem 3.1. We observethat spoly( f , f ) and spoly( f , f ) do not contribute to the standard basis.Since gr m ( R ) is not Cohen-Macaulay, α > α − α − ≥ 1. If α = 3, then x generates the local cohomology module and x x ∈ I ∗ . Thus there isa generator f = x β x − x γ ∈ I in the standard basis with β = 1 or 2.Observe that f := spoly( f , f ) = − x α +12 x + x α +11 . Since n > n and α ≥ > β , this will imply that γ < α , which contradicts the choice of α .Hence α ≥ f := spoly( f , f ) = x α +12 − x α +11 x α − . By the 2-Buchsbaumness, x α − is not the generator for the local cohomology moduleand x α +12 has to be the leading monomial. The standard basis algorithmwill stop at this step.(b) Suppose that x x α − is a minimal generator for I ∗ . Then I = ( f := x α − x α x , f := x α − x x α − , f := x α − x α − x α − α ) , with α = 1 or 2. The standard basis algorithm generates f := spoly( f , f ) = x α + α − x α +11 x α − . If the tangent cone is 2-Buchsbaum, then α + α ≤ α + α − 1. And then the algorithm stops at this step.(c) If x x α − is a minimal generate for I ∗ , then by the proof for Theorem3.10, α = 1 and the defining ideal is I = ( f := x α − x x , f := x α − x x α − , f := x α − x α − x α − ) . Similar to the previous case, the standard basis algorithm will only con-tribute an additional basis element f := spoly( f , f ) = x α +12 − x α +21 x α − . (cid:3) Example 3.13. Let K be a field, R = K [[ t , t , t ]] and m = ( t , t , t ) R . For thisnumerical semigroup ring, the defining ideal I = ( x − x x , x − x x , x − x x ) ⊆ K [[ x , x , x ]] and the initial form ideal I ∗ = ( x x , x x , x , x ) ⊆ K [ x , x , x ],which is 4-generated. Let M be the homogeneous maximal ideal of gr m ( R ). Thenthe local cohomology module H M (gr m ( R )) is generated by the element x in gr m ( R )as an Artinian R -module. It is also generated by the elements x and x x as a K -vector space. Therefore, gr m ( R ) is 2-Buchsbaum, but not Buchsbaum. Proposition 3.14. Suppose ( R, m ) is a numerical semigroup ring of embedding di-mension . If gr m ( R ) is Buchsbaum, but not Cohen-Macaulay, then for the principalreduction ideal Q = ( t n ) R of m and the α -invariant α , r Q ( m ) = α = s Q ( m ) + 1 .Proof. Let r = r Q ( R ) be the reduction number and s = s Q ( R ) the index of nilpo-tency. Since gr m ( R ) is Buchsbaum, by Lemma 3.8, H M (gr m ( R )) is generated by x α − , α = α = 1 and α = α − 1. Now by Theorem 3.1, α = 1, α = α − α = α − 1. Thus the defining ideal I has the following standard basis:(a) f := x α − x x with α ≥ f := x α − x x α − with α ≥ α + 1,(c) f := x α − x α − x α − with α ≤ α + α − f := x α +12 − x α +11 x α − with α ≤ α + α − ANGENT CONE OF NUMERICAL SEMIGROUP RINGS 15 The inequality in case (d) follows from the fact that I has only 4 standard basis,hence the standard basis algorithm has to stop after generating f . Since α ≥ α + 1, by using the binomials f , f and f , it is straightforward to show that( x , x ) α +1 ⊆ x m α . Hence m α +1 = x m α and r ≤ α . On the other hand, itfollows from the definition of α that x α − ( x ). Hence s ≥ α − 1. Since gr m ( R )is not Cohen-Macaulay, Theorem 3.3 implies that r > s . These three inequalitieslead to r = α = s + 1. (cid:3) Example 3.15. Let K be a field, R = K [[ t , t , t ]], Q = ( t ) R and m =( t , t , t ) R . The defining ideal is I = ( x − x x , x − x x , x − x x ) ⊆ K [[ x , x , x ]] . The initial form ideal is I ∗ = ( x , x , x x , x x ) ⊆ K [ x , x , x ]. For the homo-geneous maximal ideal M of gr m ( R ), the local cohomology module H M (gr m ( R ))is generated by the element x in gr m ( R ) as a K -vector space. Hence gr m ( R ) isBuchsbaum, but not Cohen-Macaulay. We have r Q ( m ) = α = 4 and s Q ( m ) = 3. Example 3.16. The converse of Proposition 3.14 is not true. In Example 3.13, Q = ( t ) R is a principal reduction of the maximal ideal m = ( t , t , t ) R , satisfyingr Q ( m ) = α = 4 and s Q ( m ) = 3. However, the tangent cone gr m ( R ) is 2-Buchsbaum,but not Buchsbaum.At the end of this subsection, we give another characterization of Buchsbaumnessof the tangent cone gr m ( R ), with a different flavor from that of Theorem 3.9.Let G = h n , . . . , n d i be a general numerical semigroup with multiplicity e = n .If the associated semigroup ring is ( R, m ) with Quot( R ) being the total quotientring and r being the reduction number of m , then the numerical semigroup G ′ of theblowup ring R ′ := S n ≥ ( m n : Quot( R ) m n ) = ( m r : Quot( R ) m r ) is h n , n − n , n − n , . . . , n d − n i (cf. Barucci [2, Section 3.3]).Let Ap( G, e ) = { w , . . . , w e − } be the Ap´ery set of G with respect to e , where w i is the smallest element in G congruent to i modulo e . Similarly, let Ap( G ′ , e ) = (cid:8) w ′ , . . . , w ′ e − (cid:9) . Furthermore, let M = G \ { } be the maximal ideal of the semi-group G . In Barucci and Fr¨oberg [3] and D’Anna et al. [6] the following invari-ants for G were defined. For each i = 0 , , . . . , e − 1, let a i = ( w i − w ′ i ) /e , b i =max { n | w i ∈ nM } , c i = min { n | w ′ i ∈ nM − ne } and d i = min { n | w ′ i ∈ nM − nM } .All these invariants are non-negative integers. Theorem 3.17 ([3, Theorem 2.6]) . The tangent cone gr m ( R ) is Cohen-Macaulayif and only if a i = b i for each i = 0 , , . . . , e − . Proposition 3.18 ([6, Proposition 3.5]) . We always have b i ≤ a i ≤ c i ≤ d i ≤ r ,where r is the reduction number of the maximal ideal. Moreover, b i < a i if andonly if a i < c i . Theorem 3.19 ([6, Theorem 3.8]) . Suppose d i = a i + 1 for every i such that a i > b i . Then gr m ( R ) is Buchsbaum. We want to show that the condition in Theorem 3.19 is also necessary when theembedding dimension d = 3. Theorem 3.20. Let ( R, m ) be a numerical semigroup ring of embedding dimension . Then the associated graded ring gr m ( R ) is Buchsbaum if and only if d i = a i + 1 for every i such that a i > b i . Proof. By Theorems 3.17 and 3.19, we may assume that gr m ( R ) is Buchsbaum, butnot Cohen-Macaulay, and prove that d i = a i + 1 for every i such that a i > b i .Let M be the maximal ideal of the semigroup G , e = n the multiplicity of R and r the reduction number of m . From the discussion in D’Anna et al. [6, Section3] we know that the blowup semigroup G ′ = rM − re . Furthermore, Remark 3.3of D’Anna et al. [6] says that a i > b i if and only if there exists s ′ ≡ i (mod e )in G ′ such that s ′ + ( h + 1) e ∈ hM \ ( h + 1) M for some non-negative integer h . Since s ′ ∈ G ′ , s ′ + rM ⊆ rM . Hence if s ′ + ( h + 1) e ∈ hM \ ( h + 1) M ,then s ′ + ( h + 1) e + rM ⊆ ( h + 1 + r ) M , thus t s ′ +( h +1) e := t s ′ +( h +1) e + m h +1 ∈ H M (gr m ( R )).Recall that α = min { α ∈ N | αn ∈ h n , n i , α = 0 } . Since G is three-generatedand gr m ( R ) is Buchsbaum, Lemma 3.8 shows that H M (gr m ( R )) is the R/ m -vectorspace generated by x α − . Whence s ′ + ( h + 1) e = ( α − n . For this reason,there exists a unique s ′ ∈ G ′ such that s ′ + ( h + 1) e ∈ hM \ ( h + 1) M for some h ∈ N , and if a i > b i , then s ′ ≡ i (mod e ). Fix this i . By virtue of Proposition3.18, now it suffices to show that a i + 1 = r .Let α be the invariant that is defined similarly to α . Proposition 3.14 showsthat r = α . By the definition of α , t ( α − n ( t n ) R , hence the Ap´ery element w i equals ( α − n . Notice that a i < r . Hence, in order to show that a i = r − 1, it suffices to show that w i − ( r − e ∈ G ′ = rM − re , or equivalently,( α − n + e ∈ α M . But this follows from the binomial f = x α − x x α − inthe proof of Proposition 3.14. (cid:3) Example 3.21. Assume the notation in Example 3.15. We have already knownthat r( m ) = 4 and gr m ( R ) is Buchsbaum. For the semigroup G = h , , i ,the Ap´ery set is Ap( G, 5) = { , , , , } . The blowup semigroup is G ′ = { , , →} , hence Ap( G ′ , 5) = { , , , , } . The invariants are a = { , , , , } , b = { , , , , } , c = { , , , , } and d = { , , , , } . Notice that a i > b i if andonly if i = 4, and d = a + 1. Remark . The numerical semigroup G = h , , , i given by D’Anna et al.[6, Remark 3.9] shows that Theorem 3.20 fails if we allow the embedding dimensionto be 4.We conclude this paper by an additional remark. Remark . The standard basis method in this paper turns out to be less fruitfulif the embedding dimension d ≥ 4. However, when d = 4 and the tangent conegr m ( R ) is Gorenstein, we are able to provide further insights with the help of linkagetheory. For instance, the initial form ideal I ∗ satisfies µ ( I ∗ ) ≤ 5. This echoes aresult of Bresinsky [4]: for every Gorenstein numerical semigroup ring ( R, m ) ofembedding dimension 4, the defining ideal I satisfies µ ( I ) ≤ 5. Detailed discussionis available in Shen [20]. Acknowledgement This paper is part of my Ph.D. thesis at Purdue University, which was writtenunder the supervision of Professor Bernd Ulrich. I want to express my sincere grat-itude to Professor Bernd Ulrich for the advising, encouragement and inspiration. Iam also grateful to Dr. Lance Bryant and Professor William Heinzer for the stim-ulating comments during the preparation of this work. I want to acknowledge the ANGENT CONE OF NUMERICAL SEMIGROUP RINGS 17 support provided by GAP [13] and Singular [12]. In addition, I thank Dr. LanceBryant for bringing to my attention the research of V. Sapko, and for his Singular library that facilitates the computations of initial form ideals. Finally, I thank thereferee for the careful reading and valuable comments and suggestions. References [1] Ap´ery, R. (1946). Sur les branches superlin´eaires des courbes alg´ebriques. C.R. Acad. Sci. Paris Multiplicative ideal theory in commutative algebra . 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Generators and relations of abelian semigroups and semi-group rings. Manuscripta Math. Nagoya Math.J. Proc. Amer. Math. Soc. Math.Proc. Cambridge Philos. Soc. Arch. Math. (Basel) Comm. Algebra [20] Shen, Y.-H. (2009). Monomial curves, Gorenstein ideals and Stanley decompo-sitions . PhD thesis, Purdue University, West Lafayette, Indiana, 2009.[21] Valabrega, P., Valla, G. (1978). Form rings and regular sequences. NagoyaMath. J. Department of Mathematics, University of Science and Technology of China, Hefei,Anhui, 230026, China E-mail address ::