Canonical trace ideal and residue for numerical semigroup rings
aa r X i v : . [ m a t h . A C ] A ug CANONICAL TRACE IDEAL AND RESIDUE FOR NUMERICALSEMIGROUP RINGS
J ¨URGEN HERZOG, TAKAYUKI HIBI AND DUMITRU I. STAMATE
Abstract.
For a numerical semigroup ring K [ H ] we study the trace of its canon-ical ideal. The colength of this ideal is called the residue of H . This invariantmeasures how far is H from being symmetric, i.e. K [ H ] from being a Gorensteinring. We remark that the canonical trace ideal contains the conductor ideal, andwe study bounds for the residue.For 3-generated numerical semigroups we give explicit formulas for the canonicaltrace ideal and the residue of H . Thus, in this setting we can classify those whoseresidue is at most one (the nearly-Gorenstein ones), and we show the eventualperiodic behaviour of the residue in a shifted family. Introduction
Let ( R, m , K ) be a local ring (or a positively graded K -algebra) which is Cohen-Macaulay and possesses a canonical module ω R . In [13] the trace ideal of ω R isused as a tool to stratify the Cohen-Macaulay rings and to define the class of nearlyGorenstein rings. We recall that if N is any R -module, its trace is the ideal tr( N ) = P ϕ ∈ Hom R ( N,R ) ϕ ( N ) in R .The relevance of tr( ω R ) (also called the canonical trace ideal of R ) stems fromthe fact that it describes the non-Gorenstein locus of the ring R . Namely, by [13,Lemma 2.1], for any p ∈ Spec( R ), p ⊇ tr( ω R ) if and only if R p is not a Gorensteinring. Thus tr( ω R ) = R if and only if R is a Gorenstein ring. In [13], the ring R iscalled nearly Gorenstein when tr( ω R ) ⊇ m . Also, the residue of R , denoted res( R )is defined as the length of the module R/ tr( ω R ). Several other invariants for suchrings are surveyed in [2].In this paper we study bounds, and in small codimension we give exact formulas,for res( R ) when R is the semigroup ring K [ H ] associated to the numerical semigroup H and the field K . This allows to determine the nearly Gorenstein property in somefamilies of semigroups.We outline the structure of the paper. First, in Section 1 we transfer the terminol-ogy and notations from rings to the setting of numerical semigroups. A numericalsemigroup H is a subsemigroup of N containing 0 such that the number of gaps g ( H ) = | N \ H | is finite. The largest gap (i.e. positive integer not in H ) is the Mathematics Subject Classification.
Primary 13H10, 20M10, 20M25; Secondary 13D02,05E40.
Key words and phrases. numerical semigroup, residue, canonical module, trace ideal, nearlyGorenstein, conductor, shifted family. robenius number F( H ). In Proposition 1.4 we show that if H is generated by anarithmetic sequence, then K [ H ] is nearly Gorenstein.As a measure of how far is K [ H ] from being Gorenstein (equivalently, that H issymmetric, cf. [17]), we introduce the residue of H defined asres( H ) = dim K K [ H ] / tr( ω K [ H ] ) . Clearly, res( H ) = 0 when H is symmetric, and res( H ) ≤ K [ H ] isnearly Gorenstein. The exponents of the monomials in tr( ω K [ H ] ) form a semigroupideal tr( H ) ⊆ H . We note in Proposition 1.1 that if H is not symmetric, then C H ⊆ tr( H ) ⊆ H \ { } , where C H is the semigroup ideal generated by the elementsof H larger than F( H ).This observation gives a first estimateres( H ) ≤ n ( H ) := |{ x ∈ H : x < F( H ) }| in Corollary 1.2. Examples computed with the NumericalSgps package [6] in GAP[9] indicate (Question 1.3) that another bound might also hold:(1) res( H ) ≤ n ( H ) − g ( H ) . This bound is proved to be correct if K [ H ] is nearly Gorenstein, and also if H is3-generated, cf. Proposition 2.2.When H is 3-generated and not symmetric, the relation ideal I H ⊂ K [ x , x , x ]of K [ H ] is given by the maximal minors of the structure matrix of H , which is ofthe form A = (cid:18) x a x a x a x b x b x b (cid:19) . (2)With this notation we derive in Proposition 2.1 thatres( H ) = Y i =1 min { a i , b i } . Working with the structure matrix of H allows us to parametrize explicitly thenon-symmetric 3-generated semigroups H whose trace is at either end of the interval[ C H , H \ { } ], see Theorem 2.3 and Proposition 2.5.Example 1.5 shows that res( H ) may take any nonnegative integer value, even ifwe fix the number of generators of H . Still, once we fix n < · · · < n e , the residueof the semigroups in the shifted family {h n + j, . . . , n e + j i} j ≥ seem to changeperiodically with j , for j ≫
0. This goes in the same direction as a recent numberof other results about eventually periodic properties in this shifted family, see [16],[32], [14], [30], [4], [24]. Using [29], we prove in Theorem 3.2 that given n < n < n and letting H j = h n + j, n + j, n + j i we have res( H j ) = res( H j +( n − n ) ) for all j ≫
0. In this setup, in Corollary 3.4 we obtain another upper bound for res( H j )when j ≫
0, depending on n − n .In the Appendix we prove the inclusion of the conductor ideal in any trace ideal,and we characterize when the equality holds. This is made in the more generalcontext of extensions of local rings R ⊆ e R with isomorphic residue fields, e R adiscrete valuation ring in Q ( R ) and a finite R -module.. . The canonical trace ideal of a semigroup, or Rings to semigroupstransition
A numerical semigroup H is a submonoid of N , and unless stated otherwise weassume | N \ H | < ∞ . Say H is minimally generated by n < n < . . . < n e with e >
1. We write H = h n , . . . , n e i . The number e is called the embedding dimension of H and the number n the multiplicity of H . One always has n ≤ e . We say that H has minimal multiplicity if n = e . In this case, one also says that H has maximalembedding dimension , cf. [28].The elements in the set G ( H ) = N \ H are called the gaps of H . As | G ( H ) | < ∞ ,there exists a largest integer F( H ), called the Frobenius number of H , such thatF( H ) H .We denote by M the subset H \ { } . The elements f ∈ G ( H ) with f + M ∈ H are called pseudo-Frobenius numbers . The set of pseudo-Frobenius numbers willbe denoted by PF( H ). The cardinality of PF( H ) is called the type of H , denotedtype( H ).We fix a field K . The positively graded K -subalgebra K [ H ] = K [ t n , . . . , t n e ] of K [ t ] is the semigroup ring of H . Its graded maximal ideal is m = ( t n , . . . , t n e ).The embedding dimension (resp. multiplicity) of H is also the embedding dimension(resp. multiplicity) of K [ H ] in the algebraic sense. The polynomial ring K [ t ] is afinite module over K [ H ] and is the integral closure of K [ H ] in its quotient field Q ( K [ H ]) = K ( t ). The module K [ t ] /K [ H ] has finite length and a K -basis given bythe residue classes of { t a : a ∈ G ( H ) } .The canonical module ω K [ H ] of K [ H ] is the fractionary K [ H ]-ideal generated bythe elements t − f with f ∈ PF( H ), see [7, Exercise 21.11]. Therefore, the Cohen-Macaulay type of K [ H ] is equal to type( H ). In particular, K [ H ] is Gorenstein if andonly if PF( H ) = { F( H ) } . Kunz [17] showed that K [ H ] is Gorenstein if and only if H is symmetric , i.e. for all x ∈ Z either x ∈ H , or F( H ) − x ∈ H . The anti-canonicalideal of K [ H ] is the fractionary ideal ω − K [ H ] = { x ∈ Q ( K [ H ]) : x · ω K [ H ] ⊆ K [ H ] } .Since K [ H ] is a domain, by [13, Lemma 1.1] one has tr( ω K [ H ] ) = ω K [ H ] · ω − K [ H ] .We mention that the almost Gorenstein numerical semigroup rings (as definedby Barucci and Fr¨oberg in [1], see also [11]) are a proper subclass of the nearlyGorenstein ones, by [13, Proposition 6.1]. For our purposes, we will take as definitionfor almost Gorensteinness Nari’s characterization which we explain next.Let PF( H ) = { f , . . . , f τ − , F( H ) } , with f i < f i +1 for 1 ≤ i < τ −
2. It is knownby Nari [21] that K [ H ] is almost Gorenstein, if and only if(3) f i + f τ − i = F( H ) for i = 1 , . . . , ⌊ τ / ⌋ . The semigroup H is called almost symmetric if K [ H ] is almost Gorenstein, and H is called nearly Gorenstein , if K [ H ] is nearly Gorenstein. These two classes ofsemigroups have been recently considered in [20].A subset I ⊂ Z is called a relative ideal of H if I + H ⊆ I and h + I ⊆ H forsome h ∈ H . If moreover I ⊆ H , then I is called an ideal of H .Let Ω H and Ω − H be the set of exponents of the monomials in ω K [ H ] , and in ω − K [ H ] respectively. Then Ω H and Ω − H are relative ideals of H called the canonical, espectively the anti-canonical ideal of H . We define the trace of H as tr( H ) =Ω H + Ω − H . It is clear that tr( H ) is an ideal in H consisting of the exponents of themonomials in tr( K [ H ]).In this notation, H is nearly Gorenstein if and only if M ⊆ tr( H ).The semigroup ring K [ H ] is 1-dimensional, so its canonical trace ideal is eitherthe whole ring, or it is an m -primary ideal. Equivalently, K [ H ] / tr( ω K [ H ] ) is a finitedimensional vector space with a K -basis given by { t h : h ∈ H \ tr( H ) } . We definethe residue of H as the residue of K [ H ], namely(4) res( H ) = dim K K [ H ] / tr( ω K [ H ] ) = | H \ tr( H ) | . Thus res( H ) = 0 means that H is symmetric, and res( H ) ≤ H isnearly Gorenstein.The conductor of the extension K [ H ] ⊆ K [ t ] is the ideal C K [ t ] /K [ H ] = ( t h : h > F( H )) K [ H ] , which explains why the quantity c ( H ) := F( H ) + 1 is named the conductor of H .We denote C H = { h : t h ∈ C K [ t ] /K [ H ] } , which is an ideal in H minimally generated by c ( H ) , c ( H ) + 1 , . . . , c ( H ) + n −
1. An important observation is that tr( H ) containsthe conductor ideal C H . We skip the proof for now, since it follows from PropositionA.1 in the Appendix, where we consider a more general situation. Proposition 1.1.
For any numerical semigroup H one has C H ⊆ tr( H ) ⊆ H. If H is not symmetric then C H ⊆ tr( H ) ⊆ M . As a corollary we obtain an upper bound for res( H ).We define the set of non-gaps of H to be N G ( H ) = { x ∈ H : x < F( H ) } and wedenote n ( H ) = | N G ( H ) | . Corollary 1.2.
For any numerical semigroup H one has res( H ) ≤ n ( H ) , withequality if and only if tr( H ) = C H .Proof. The desired inequality follows from the observation that n ( H ) = | N G ( H ) | = | H \ C H | ≥ | H \ tr( H ) | = res( H ) . (cid:3) The map ρ : N G ( H ) → G ( H ) given by ρ ( x ) = F( H ) − x for all x in N G ( H ) iswell defined and injective. Also, | N G ( H ) | + | G ( H ) | = F( H ) + 1, hence denoting g ( H ) = | G ( H ) | we have n ( H ) ≤ g ( H ).Numerical experiments with GAP ([9]) indicate that another bound for res( H )might also hold. We formulate the following question. Question 1.3.
Given a numerical semigroup H , is it true that res( H ) ≤ g ( H ) − n ( H )?This question has a positive answer for symmetric semigroups: by [8, Lemma1(f)] H is symmetric if and only if n ( H ) = g ( H ). In Proposition 2.2 we also confirmQuestion 1.3, when H is 3-generated. or any integer a > H = h a, a +1 , . . . , a − i is nearly Gorensteinand not symmetric (see Proposition 1.4), and it has res( H ) = 1 = n ( H ) < g ( H ) − n ( H ) = a −
2. This shows that the bound in Question 1.3 is not always smallerthan the one given by Corollary 1.2.The second list of inclusions in Proposition 1.1 are sharp, as confirmed by Propo-sition 1.4 and Example 1.5 below.The following result shows that a numerical semigroup generated by an arithmeticsequence is nearly Gorenstein. We also characterize when such semigroups are al-most symmetric, taking into account that the symmetric case was known from workof Gimenez, Sengupta and Srinivasan in [10].
Proposition 1.4.
Let e > , and H = h a, a + d, . . . , a + ( e − d i with a, d coprimenonnegative integers and e ≤ a . Then (a) H is nearly Gorenstein; (b ) H is symmetric if and only if a ≡ e − ; (c) H is almost symmetric if and only if a = e or a ≡ e − .Proof. It is known from [10, Theorem 4.7] that τ = type( H ) is the unique integer1 ≤ τ ≤ e − a = k ( e −
1) + τ + 1 with k integer. Equivalently, k = ⌊ a − e − ⌋ .Tripathi [31, Theorem on page 3] shows thatPF( H ) = (cid:26) a (cid:22) x − e − (cid:23) + dx : a − τ ≤ x ≤ a − (cid:27) . For a − τ ≤ x ≤ a − k ( e − ≤ x − ≤ k ( e −
1) + ( τ − ⌊ x − e − ⌋ = k .This implies that F( H ) = ak + d ( a −
1) and(5) PF( H ) = { F( H ) − ( τ − d, . . . , F( H ) − d, F( H ) } , hence the canonical ideal Ω H is generated by W = {− F( H ) , − F( H ) + d, . . . , − F( H ) + ( τ − d } . For part (a) we consider the set W ′ = { F( H ) + a, F( H ) + a + d, . . . , F( H ) + a + ( e − τ ) d } ⊂ H. An element in W + W ′ is of the form a +( i + j ) d with 0 ≤ i ≤ τ − ≤ j ≤ e − τ .This way we obtain the generators of H : a, a + d, . . . , a + ( e − d , which shows that W ′ ⊂ Ω − H and Ω H + Ω − H ⊇ M . Equivalently, H is nearly Gorenstein.Part (b) is known and may be traced back to [10, Theorem 2.2] or (less explicitlyin) [25]. The statement is an immediate consequence of the fact that K [ H ] isGorenstein if and only if τ = 1.For part (c), using (b), it is enough to treat the case of H being almost symmetric,but not symmetric. This is equivalent (using (3) and (5)) to(F( H ) − ( τ − d ) + (F( H ) − d ) = F( H ) , which is equivalent toF( H ) = τ d. fter we substitute the values of F( H ) and τ in the previous equation, we get ak + d ( a −
1) = ( a − − k ( e − d,k ( a + d ( e − ,k = 0 . Note that e ≤ a and by the way k was defined, we may express k = (cid:4) a − e − (cid:5) .Therefore k = 0 if and only if a = e . (cid:3) Next, we present a family of numerical semigroups H such that C H = tr( H ). Example 1.5.
For the integers m > q > H = h m, qm + 1 , qm + 2 , . . . , qm + m − i . This is a semigroup with minimal multiplicity, hence its pseudo-Frobenius numbersare obtained by subtracting m from the rest of the minimal generators. This givesPF( H ) = { ( q − m +1 , ( q − m +2 , . . . , qm − } , a list of m − x ∈ Ω − H , i.e. − PF( H )+ x ⊂ H . This can happen only if x − F( H ) = x − qm +1 ≥ qm , equivalently x ≥ qm −
1. Consequently, tr( H ) = { x : x ≥ qm } = C H , andres( H ) = |{ , m, . . . , ( q − m }| = q .2. The case of -generated numerical semigroups When the numerical semigroup H is 3-generated, the results in [13, Section 3] canbe applied to obtain a simple formula of res( H ) from the defining ideal of K [ H ].Assume H is minimally generated by n , n , n , not necessarily listed increasingly.Let ϕ : S = K [ x , x , x ] → K [ H ] the algebra map given by ϕ ( x i ) = t n i for i =1 , . . . ,
3. Then ker( ϕ ) = I H , the defining ideal of K [ H ].It is proven in [12] that H is symmetric, equivalently K [ H ] is a complete intersec-tion, if and only if, up to a permutation, d = gcd( n , n ) > n ∈ h n /d, n /d i .Assume H is not symmetric. We recall from [12] how to compute the ideal I H in this case. We find the positive integers c , c , c minimal with the property thatthere exist nonnegative integers a i , b i , i = 1 , . . . , c n = b n + a n ,c n = a n + b n , (6) c n = b n + a n . Such a i , b i are positive, unique, and c i = a i + b i for i = 1 , . . . ,
3. In this notation,the ideal I H is the ideal of maximal minors of the matrix A = (cid:18) x a x a x a x b x b x b (cid:19) , (7)that we call the structure matrix of the semigroup H .It is noticed in [22, page 69] that one can recover n , n , n from the matrix A bycomputing the K -vector space dimension for the isomorphic rings K [ H ] / ( t n ) ∼ = S/ ( x , I H ) ∼ = K [ x , x ] / ( x a + b , x b x a , x a + b ) , nd the other two cases, see [28, Lemma 10.23] for a different approach. Namely,we get n = a a + b a + b b ,n = a a + a b + b b , (8) n = a a + b a + b b . It follows from the Hilbert-Burch theorem ([3, Theorem 1.4.17]) that the transpose A T is the relation matrix of I H , i.e. the sequence0 → S A T −→ S → I H → R = K [ H ] is 2, hence by [13, Corollary 3.4] we gettr( ω R ) = I ( ¯ A T ) = ( t n i a i , t n i b i : i = 1 , . . . , , where ¯ A T is obtained by applying ϕ on the entries of A T . We may formulate thefollowing result. Proposition 2.1.
Assume H is a non-symmetric -generated numerical semigroupand let R = K [ H ] . With notation as in (6) , we set d i = min { a i , b i } for ≤ i ≤ .Then tr( ω R ) = ( t d n , t d n , t d n ) R, and res( H ) = d d d . Proof.
The first part is clear from the discussion above. Since R/ tr( ω R ) ∼ = S/ ( I H , x d , x d , x d ) ∼ = S/ ( x d , x d , x d )we obtain that res( H ) = dim K R/ tr( ω R ) = d d d . (cid:3) We may now give a positive answer to Question 1.3, in embedding dimension 3.
Proposition 2.2.
For any -generated numerical semigroup H one has res( H ) ≤ g ( H ) − n ( H ) . Proof. If H is symmetric we actually have equality 0 = res( H ) = g ( H ) − n ( H ), asnoted in [8, Lemma 1(f)]. Assume H is not symmetric and that it has a structurematrix A denoted as in (7). Nari et al. prove in [22, Theorem 3.2] that2 g ( H ) − (F( H ) + 1) ∈ { a a a , b b b } . Using Proposition 2.1, we obtainres( H ) ≤ min { a a a , b b b } ≤ g ( H ) − (F( H ) + 1) = g ( H ) − n ( H ) . (cid:3) As an application of Proposition 2.1 we will characterize the 3-generated numericalsemigroups such that their trace is at either end of the interval [ C H , M ]. Theorem 2.3.
Let H be a -generated numerical semigroup. Then tr( H ) = M ifand only if one of the following cases occurs: (i) H = h ab + b + 1 , b + c + 1 , ac + a + c i where a, b, c are positive integers with gcd( b + c − , ab − c ) = 1 , or (ii) H = h bc + b + 1 , ca + c + 1 , ab + a + 1 i , where a, b, c are positive integers with gcd( bc + b + 1 , ca + c + 1) = 1 . n case (i) , F( H ) = abc + bc − b − { , ab − c } , and in case (ii) , F( H ) = 2 abc − .Proof. Assume H = h n , n , n i such that tr( H ) = M . By [13, Corollary 3.5], thatis equivalent to I ( A ) = ( x , x , x ), where A is the matrix attached to H as in (7).Clearly, H is not symmetric, hence up to a permutation of the variables, there areessentially two (overlapping) cases to consider.Case 1: A = (cid:18) x x a x b x x x c (cid:19) , with a, b, c > . Using (8) we get n = ab + b +1 , n = b + c +1 , n = ac + a + c , as desired. It is easyto check that gcd( n , n ) = gcd( n , n ) = gcd( n , n ), hence 1 = gcd( n , n , n ) =gcd( n , n − n ) = gcd( b + c − , ab − c ).Conversely, let n = ab + b + 1 , n = b + c + 1 , n = ac + a + c for some positiveintegers a, b, c such that gcd( b + c − , ab − c ) = 1. Arguing as above we see thatthe generators of H are pairwise coprime, hence H is a numerical semigroup whichis not symmetric. It is easy to verify the following equations:(1 + c ) n = n + bn , (1 + a ) n = n + n , (9) (1 + b ) n = cn + an . We claim that these are the minimal relations (6) among n , n , n .Since a , b in (6) are positive, unique and (1 + a ) n = n + n , we may identify c = 1 + a and a = b = 1.After substituting n = (1 + a ) n − n into c n = b n + a n , we get c ((1 + a ) n − n ) = b n + a n , hence( c (1 + a ) − b ) n = ( c + a ) n . Since n and n are coprime, there exists a positive integer ℓ so that c + a = ℓn .Thus, c + a ≥ b + c + 1.On the other hand, comparing (9) and (6) we obtain that c ≤ c and a = c − b ≤ ( b +1) − b , hence c + a ≤ c + b . This implies that c + a = b + c +1,and moreover c = c + 1 and a = b . We can now identify the rest of the coefficientsin (6): c = 1 + b, b = c, a = a , which shows that the matrix A has the desiredentries.Case 2: A = (cid:18) x x x x b x c x a (cid:19) , with a, b, c > . Using (8) we get n = bc + b + 1 , n = ca + c + 1 , n = ab + a + 1. It is easy to seethat gcd( n , n ) = gcd( n , n ) = gcd( n , n ), hence the desired description for H .Conversely, let a, b, c be positive integers with gcd( bc + b +1 , ca + c +1) = 1. It nowfollows from [27, Theorem 14] (and its proof) that H = h bc + b +1 , ca + c +1 , ab + a +1 i is a pseudo-symmetric numerical semigroup whose matrix A is the one we startedwith this case. t is shown in [26, Theorem 2.2.3] and [18, Exercise 5, pp. 145] that for anynon-symmetric numerical semigroup H = h n , n , n i one hasF( H ) = max { c n + b n , c n + a n } , where c , c , a , b are as in (6). It is now an easy exercise to derive the announcedformulas for F( H ), when H belongs to either one of the two families consideredabove. (cid:3) Remark 2.4.
As noticed by Nari, Numata and Keiichi Watanabe in [22, Corol-lary 3.3] (see also [23, Corollary 2.9]), the format of the matrix A in case (ii) ofTheorem 2.3 corresponds to H being pseudo-symmetric, which is equivalent in em-bedding dimension 3 to H being almost symmetric and not symmetric, see [23,Proposition 2.3]. The complete parametrization of 3-generated pseudo-symmetricnumerical semigroups was obtained by Rosales and Garc´ıa-S´anchez in [27]. Proposition 2.5.
Assume H is a non-symmetric -generated numerical semigroup.Then tr( H ) = C H if and only if H = h , a + 1 , a + 2 i for some positive integer a .Proof. Assume tr( H ) = C H .It follows from Proposition 2.1 that µ (tr( H )) ≤
3. If µ (tr( H )) = 2, then e ( H ) = µ ( C H ) = 2 and H is symmetric, a contradiction. Therefore, µ (tr( H )) = 3, hence H has multiplicity 3. (We can get the same thing by applying Corollary A.5.) Listingits generators increasingly we have that either H = h , a + 1 , b + 2 i with 0 < a ≤ b ,or H = h , b + 2 , a + 1 i with a > b > a ≤ b . Then 3 b + 2 / ∈ h , a + 1 i , hence 3 b + 2 ≤ F( h , a + 1 i ) = 6 a − b < a . It is easy to check that the structure matrix (7) is A = (cid:18) x a − b x x x x x b − a +11 (cid:19) . Hence res( H ) = 2 a − b by Proposition 2.1. Note that 0 , , , . . . , a −
1) are not in C H ,hence 2 a − b = res( H ) = | H \ C H | ≥ a . This gives a = b and H = h , a + 1 , a + 2 i .If a > b then arguing as in the previous case we obtain 3 a +1 ≤ F( h , b +2 i ) = 6 b +1, and a < b . Clearly 0 , , , . . . , b are not in C H , hence res( H ) = | H \ C H | ≥ b + 1.On the other hand, the structure matrix of H is A = (cid:18) x b − a +11 x x x x x a − b (cid:19) , and Proposition 2.1 gives res( H ) = 2 b − a + 1. Thus 2 b − a + 1 ≥ b + 1, and b ≥ a ,a contradiction.Example 1.5 confirms that for any a > H = h , a + 1 , a + 2 i satisfies tr( H ) = C H . (cid:3) Remark 2.6.
From the proof of Proposition 2.5 we see that for any a > h , a + 1 , a + 2 i ) = a . . The residue in shifted families of semigroups
Remark 2.6 indicates that the residue of a 3-generated numerical semigroup H may be as large as possible. However, this is not the case in a shifted family ofsemigroups, as we verify below.Firstly, we extend the definition of residue from (4) to arbitrary affine subsemi-groups of N . In this sense, for any semigroup H ⊂ N containing 0 we letres( H ) = res (cid:18) d H (cid:19) , where d = gcd( h : h ∈ H ) . Given the sequence of integers a : a < · · · < a e , for any j we denote a + j : a + j, . . . , a e + j . The shifted family of a is the family { a + j } j ≥ . It has been provedthat for large enough shifts several properties occur periodically in the shifted familyof semigroups {h a + j i} j ≥ and their semigroup rings { K [ h a + j i ] } j ≥ , see [16], [32],[14], [29]. For instance, Jayanthan and Srinivasan [16] showed that for j ≫ K [ h a + j i ] is complete intersection (CI) ⇐⇒ K [ h a + j + ( a e − a ) i ] is CI . More generally, Vu ([32, Theorem 1.1]) showed thatfor j ≫ , β i ( K [ h a + j i ]) = β i ( K [ h a + j + ( a e − a ) i ]) for all i. In particular, for j ≫ K [ h a + j i ] and K [ h a + j + ( a e − a ) i ] areGorenstein at the same time. This implies that the semigroups h a + j i and h a + j +( a e − a ) i are symmetric at the same time. Equivalently,for j ≫ , res( h a + j i ) = 0 ⇐⇒ res( h a + j + ( a e − a ) i ) = 0 . It is natural to ask the following.
Question 3.1.
Given the list of integers a : a < · · · < a e , is it true thatfor j ≫ , res( h a + j i ) = res( h a + j + ( a e − a ) i )?We remark that a positive answer to Question 3.1 implies that for numericalsemigroups with bounded width, their residue is also bounded. We recall that thewidth of a numerical semigroup H is defined in [14] as the difference between thelargest and the smallest minimal generator of H .Numerical experiments with GAP ([9], [6]) indicate that Question 3.1 might havea positive answer. Next we confirm it in case e ≤
3. If e = 2, then h a + j, a + j i is symmetric for all j , and we are done. The case e = 3 is proved in the followingtheorem. We first note that when studying asymptotic properties in a shifted family { a + j } j , we may assume a = 0. Theorem 3.2.
Given the integers < a < b , let D = gcd( a, b ) and k a,b =max { b ( b − aD − , baD } . For any integer j we denote H j = h j, j + a, j + b i .Then res( H j ) = res( H j + b ) for all j > k a,b . Before giving the proof of Theorem 3.2 we recall a result from [29] (extendingJayanthan and Srinivasan’s [16, Theorem 1.4]) about the occurence of symmetricsemigroups in a shifted family {h j, a + j, b + j i} j ≥ . emma 3.3. ( [29, Theorem 3.1] ) With notation as in Theorem 3.2, let (10) T = Y p prime, ν p ( a ) <ν p ( b ) p ν p ( b ) , where for any integer n we denote ν p ( n ) = max { i : p i divides n } . Then for j > k a,b the semigroup H j is symmetric if and only if j is a multiple of T . In particular, inthe family of semigroups { H j } j>k a,b the symmetric property occurs periodically withprincipal period T .Proof. (of Theorem 3.2).We start with j > k a,b . By Lemma 3.3, if H j is symmetric then H j + b is symmetric,too, hence res( H j ) = res( H j + b ) = 0.Assume H j is not symmetric. By Lemma 3.3, H j + ℓb is not symmetric for all ℓ ≥ A j + ℓb the structure matrix (7) of the non-symmetric semigroup H j + ℓb .For ( n , n , n ) = (0 , a, b ) + j + ℓb , it is proved in [29, Theorem 2.2] that for any ℓ ≥ bD n = b − aD n + aD n . This implies that A j = x ( b − a ) /D x a x a x b x a/D x b ! , where a , a , b , b are positive integers (depending on j ) such that a + b = b/D ,by (11).Let e = gcd( a, b ) / gcd( j, a, b ). Proposition 4.2 in [29] explains how the equations(6) change when we shift up by b . According to this result, only the last column of A j changes and we obtain A j + b = x ( b − a ) /D x a x a + e x b x a/D x b + e ! . Iterating this, we have that A j + ℓb = x ( b − a ) /D x a x a + ℓe x b x a/D x b + ℓe ! , for ℓ ≥ . Proposition 2.1 gives(12) res( H j + ℓb ) = min { ( b − a ) /D, b + ℓe } · min { a/D, a + ℓe } · min { a , b } . For ℓ ≥ max { b − aD − , aD } = b k a,b it is easy to see that b + ℓe ≥ ( b − a ) /D and a + ℓe ≥ a/D . Hence, (12) becomes(13) res( H j + ℓb ) = min { a , b } · a ( b − a ) /D , which is a formula not involving ℓ .The argument above shows that for any j > k a,b we have that res( H j ) =res( H j + b ). This concludes the proof. (cid:3) orollary 3.4. With notation as in Theorem 3.2, for j > k a,b the residue of H j isan integer divisible by ( b − a ) a/D and res( H j ) < b / D . Proof. If H j is symmetric, the inequality to prove is clear. Assume H j is not sym-metric and j > k a,b . By (13), we have res( H j ) = min { a , b } · a ( b − a ) /D , with a , b positive integers such that a + b = b/D . This shows the first part of theclaim. The second part is obtained from the following chain of inequalitiesres( H j ) ≤ a ( b − a ) D (cid:18) bD − (cid:19) < ab ( b − a ) D ≤ (cid:18) b (cid:19) · D = 8 b D , where for the last inequality we used the known fact that √ xyz ≤ ( x + y + z ) / x, y, z > (cid:3) Corollary 3.5.
With notation as in Theorem 3.2, for j > k a,b the semigroup H j is nearly Gorenstein if and only if H j + b is nearly Gorenstein. We make a comment about the frequency of occurences of the symmetric, almostsymmetric and nearly Gorenstein property in a shifted family.
Remark 3.6.
Let 0 < a < b . For j ≥ H j = h j, j + a, j + b i . We usethe constant k a,b introduced in Theorem 3.2.Lemma 3.3 shows that we find symmetric semigroups for arbitrarily large shifts j . From the formula (10) for the principal period T we infer that T >
1, otherwise a = b , which is false. This means that there is no j such that H j is symmetric forall j > j .When b = 2 a , the semigroup H j is generated by an arithmetic sequence. UsingProposition 1.4 we get that H j is nearly Gorenstein for all j >
0. On the otherhand, by Lemma 3.3, when j > b we have that H j is symmetric if and only if j isdivisible by 2 ν ( b ) .It is however possible that in the shifted family { H j } j ≥ , for large j the onlynearly Gorenstein semigroups are the symmetric ones. Indeed, if a, b are coprimeand a >
1, by Corollary 3.4 we have that for j > k a,b , either res( H ) = 0, orres( H ) ≥ a ( b − a ) > { H j } j ≥ . This can also beseen as follows. According to Nari et al. [22], the structure matrix A j for an almostsymmetric semigroup H j must have one row consisting of linear forms. However,it is proven in [29, Proposition 4.2] that for j > k a,b the matrix A j + b is obtainedfrom A j by increasing the exponents of the last column by gcd( a, b ) / gcd( a, b, j ).This shows that for j > k a,b + b the semigroup H j is not almost symmetric. Theanalysis of the occurence of infinitely many almost symmetric semigroups in theshifted family of a 4-generated numerical semigroup is made in [15, Section 7]. Appendix A. Trace ideals as conductor ideals in local rings
Some statements in this paper can be formulated for extensions of 1-dimensionallocal domains where the conductor is defined. In this appendix we would like to nderstand the case when the trace of an ideal coincides with the conductor ideal,and some consequences this draws.Hereafter, ( R, m ) is a 1-dimensional Cohen-Macaulay local ring with canonicalmodule ω R . Assume now that ( R, m ) ⊂ ( e R, n ) is an extension of local rings, where e R is a finite R -module and a discrete valuation ring such that e R ⊆ Q ( R ). We alsoassume that the inclusion map R → e R induces an isomorphism R/ m → e R/ n . Theset C e R/R = { x ∈ R : x e R ⊆ R } is called the conductor of this extension and it is an ideal of both R and e R . Withthe notation introduced we have Proposition A.1.
For any ideal I ⊂ R one has C e R/R ⊆ tr( I ) . In particular, C e R/R ⊆ tr( ω R ) .Proof. There exists f ∈ I such that I e R = ( f ) e R . Now let g ∈ C e R/R . Then ( g/f ) I ⊆ ( g/f ) I e R = g e R ⊆ R . Thus g/f ∈ I − . Since g = ( g/f ) f with f ∈ I , it follows that g ∈ I − · I = tr( I ), where for the last equation we used [13, Lemma 1.1]. (cid:3) Corollary A.2. R is nearly Gorenstein, if C e R/R = m . For the rest of this section, the ideal quotients are computed in Q ( R ) = Q ( e R ).The following lemma will be used in the proof of Proposition A.4. Lemma A.3. e R = R : C e R/R
Proof.
It is clear from the definition of C e R/R that e R ⊆ R : C e R/R .For the reverse inclusion, let f ∈ R : C e R/R . If f / ∈ e R , then since Q ( R ) = Q ( e R )we may write f = εt − a with ε invertible in e R , a positive integer and t a generatorof the maximal ideal of e R . Since C e R/R is an ideal in e R , there exists b > C e R/R = ( t b ) e R .The property f ∈ R : C e R/R now reads as εt − a C e R/R ⊆ R , or equivalently, that C e R/R ⊆ t a R . This implies that b ≥ a . Since we may write t b − = f · g where g = ε − t a + b − is clearly in C e R/R , it follows that t b − ∈ R .We claim that t b − e R ⊆ R . This will then lead to a contradiction, since t b −
6∈ C e R/R .It is clear that t b − n = C e R/R ⊆ R . Thus is suffices to show that if ν ∈ e R is a unit,then νt b − ∈ R . To prove this, we use our assumption that R/ m → e R/ n is anisomorphism. In order to simplify notation we may assume that R/ m = e R/ n . Thenthis implies that there exists h ∈ R such that ν − h ∈ n . Thus, ν = h + h with h ∈ R and h ∈ n , and therefore νt b − = ht b − + h t b − . Since both summands onthe right hand side of this equation belong to R , the claim follows.This concludes the proof of the inclusion R : C e R/R ⊆ e R . (cid:3) Our next result gives several characterizations of the situation when tr( I ) = C e R/R ,in terms of the fractionary ideal I − . We first recall that for any fractionary ideal J f R , the ideal quotient J : J may be identified with the endomorphism ring End( J )of J , see [19, Lemma 3.14]. Proposition A.4.
Let I ⊂ R be an ideal. The following conditions are equivalent: (i) tr( I ) = C e R/R . (ii) End( I − ) = e R . (iii) I − ∼ = e R .Proof. (i) ⇒ (ii): If tr( I ) = C e R/R , then using Lemma A.3 and the remark followingit, we get e R = R : C e R/R = R : ( I · I − ) = ( R : I ) : I − = End( I − ) . (ii) ⇒ (i): Suppose that tr( I ) = C e R/R . Then C e R/R is properly contained in tr( I ).It follows that C e R/R is properly contained in tr( I ) e R . Let t be generator of themaximal ideal of e R . Then there exist integers a < b such that tr( I ) e R = t a e R and C e R/R = t b e R . Clearly, t b ∈ C e R/R e R = C e R/R . Since tr( I ) e R is a principal ideal, thereexists f ∈ tr( I ) such that tr( I ) e R = ( f ) e R . Therefore, there exists u invertible in e R such that t a = u · f .Since by assumption e R = End( I − ) (which is R : tr( I )), we obtain that t a ∈ R .We may write t b − = t b − a − · t a , where t b − a − ∈ e R and t a ∈ tr( I ). Using again that e R = R : tr( I ), it follows that t b − ∈ R . Arguing as in the proof of Lemma A.3 weobtain t b − ∈ C e R/R , which is a contradiction.(ii) ⇒ (iii): If End( I − ) = e R , then I − e R = I − . Since any nonzero e R -ideal isisomorphic to e R , the assertion follows.(iii) ⇒ (ii) is obvious. (cid:3) For an R -module M we let e ( M ) denote its multiplicity. Corollary A.5.
Let R be as before, and assume in addition that R is of the form R = S/J with ( S, n ) regular local ring of dimension and J ⊆ n .If tr( ω R ) = C e R/R then e ( R ) ≤ µ ( J ) . Moreover, if R is an almost complete inter-section, then R has minimal multiplicity.Proof. Since R is a 1-dimensional domain, it it Cohen-Macaulay and proj dim S ( R ) =1. Thus J has a minimal presentation 0 → S g − → S g → J , where g = µ ( J ).Now Proposition A.4(iii) together with tr( ω R ) = C e R/R imply that ω − R and e R areisomorphic R -modules, and they must have the same number of minimal generatorsover R . Hence dim R/ m ( e R/ m e R ) = µ ( ω − R ) ≤ g , by [13, Corollary 3.4].As e R is a discrete valuation ring, there exists f ∈ m such that m e R = f e R . Since e R is a finitely generated R -module of rank 1, it follows that e ( R ) = e ( e R ), see [3,Corollary 4.7.9]. Now e ( e R ) = dim R/ m m k e R/ m k +1 e R for k ≫
0. Since m k e R/ m k +1 e R = f k e R/f k +1 e R ∼ = e R/f e R ∼ = e R/ m e R , we conclude from the above considerations that e ( R ) ≤ g , as desired. f R is an almost complete intersection, then µ ( J ) = height( J ) + 1 = dim S ,hence e ( R ) ≤
3. On the other hand, a celebrated inequality of Abhyankar gives3 = emb dim R ≤ e ( R ) + dim( R ) − e ( R ). Thus e ( R ) = 3, as desired. (cid:3) Acknowledgement . We gratefully acknowledge the use of SINGULAR ([5]) andof the numericalsgps package ([6]) in GAP ([9]) for our computations.Dumitru Stamate was partly supported by a fellowship at the Research Instituteof the University of Bucharest (ICUB) and by the University of Bucharest, Facultyof Mathematics and Computer Science through the 2019 Mobility Fund.
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J¨urgen Herzog, Fachbereich Mathematik, Universit¨at Duisburg-Essen, CampusEssen, 45117 Essen, Germany
E-mail address : [email protected] Takayuki Hibi, Department of Pure and Applied Mathematics, Graduate Schoolof Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
E-mail address : [email protected] Dumitru I. Stamate, ICUB/Faculty of Mathematics and Computer Science, Uni-versity of Bucharest, Str. Academiei 14, Bucharest – 010014, Romania
E-mail address : [email protected]@fmi.unibuc.ro