A constructive approach to one-dimensional Gorenstein k-algebras
aa r X i v : . [ m a t h . A C ] J a n A CONSTRUCTIVE APPROACH TO ONE-DIMENSIONALGORENSTEIN k-ALGEBRAS
J. ELIAS ∗ AND M. E. ROSSI ∗∗ Abstract.
Let R be the power series ring or the polynomial ring over a field k and let I be an ideal of R. Macaulay proved that the Artinian Gorenstein k -algebras R/I arein one-to-one correspondence with the cyclic R -submodules of the divided power seriesring Γ . The result is effective in the sense that any polynomial of degree s producesan Artinian Gorenstein k -algebra of socle degree s. In a recent paper, the authors ex-tended Macaulay’s correspondence characterizing the R -submodules of Γ in one-to-onecorrespondence with Gorenstein d-dimensional k -algebras. However, these submodulesin positive dimension are not finitely generated. Our goal is to give constructive andfinite procedures for the construction of Gorenstein k -algebras of dimension one and anycodimension. This has been achieved through a deep analysis of the G -admissible sub-modules of Γ . Applications to the Gorenstein linkage of zero-dimensional schemes andto Gorenstein affine semigroup rings are discussed. Introduction
Gorenstein rings were introduced by A. Grothendieck and they are a generalizationof complete intersections, indeed the two notions coincide in codimension two by a wellknown result by Serre. Codimension three Gorenstein rings are completely described byBuchsbaum and Eisenbud’s structure theorem, [2], but despite many attempts the gen-eral construction of Gorenstein rings remains an open problem in higher codimension. A.Kustin and M. Miller, in a series of papers, studied the structure of Gorenstein ideals ofcodimension 4, see [24]. More recently, M. Reid studied their projective resolution aimingto extend the result of D. Buchsbaum and D. Eisenbud, see [30]. Gorenstein rings areof great interest in many areas of mathematics and they have appeared as an importantcomponent in a significant number of problems with applications to commutative algebra,singularity theory, number theory and more recently to combinatorics, among other areas.The lack of a general structure of Gorenstein rings is the main obstacle in several prob-lems, among them the Gorenstein linkage. For a complete and interesting presentation ofGorenstein rings, see H. Bass’s and C. Huneke’s papers, [1] and [19].Let R denote the power series ring k [[ z , . . . , z n ]] or the polynomial ring k [ z , . . . , z n ]over a field k and let I ⊂ R be an ideal (homogeneous if R is a polynomial ring). Wedenote by M the maximal ideal of R generated by z , · · · , z n .As an effective consequence of Matlis duality, it is known that an Artinian ring R/I isa Gorenstein k -algebra if and only if I is the set of solutions of a system of polynomial Date : January 20, 2021. ∗ Partially supported by MTM2016-78881-P. ∗∗ Partially supported by PRIN 2010-11 “Geometria delle varieta’ algebriche”.2010 MSC: Primary 13H10; Secondary 13H15; 14C05. differential operators with constant coefficients. Macaulay, at the beginning of the 20thcentury, proved that the Artinian Gorenstein k -algebras are in correspondence with thecyclic R -submodules of the divided powers ring Γ = k DP [ Z , · · · , Z n ] where the elementsof R act as derivatives (or contraction) on Γ, see [13], [21].Thanks to the effective construction of the Artinian Gorenstein algebras, in the lasttwenty years several authors have applied this device to several problems, among others:Waring’s problem [15], the n-factorial conjecture in combinatorics and geometry [17], thecactus rank [29], the geometry of the punctual Hilbert scheme of Gorenstein schemes [21],the classification up to analytic isomorphism of Artinian Gorenstein rings [12], and theKoszulness of k -algebras [4].Recently in [10] the authors extended Macaulay’s correspondence characterizing thesubmodules of Γ , called G -admissible submodules, in one-to-one correspondence withGorenstein d-dimensional k -algebras (Theorem 2.3). These submodules in positive di-mension are not finitely generated. Clearly this is an obstacle to the effective constructionof Gorenstein algebras of positive dimension.In this paper we give a constructive and finite procedure for producing Gorenstein one-dimensional k -algebras of any codimension. Thanks to a deep analysis of the structureof the G -admissible R -submodules of Γ , it is possible to write an algorithmic procedurefor constructing step-by-step a finite subset of Γ , called a G -admissible set, as a goodcandidate for being extended to a G -admissible submodule. In the graded case, similarly tothe Artinian case, a suitable DP-polynomial H ∈ Γ uniquely determines a one-dimensionalGorenstein ring and hence a G -admissible R -submodule, see Proposition 2.11, Remark3.2. This is no longer true in the local case where more sophisticated techniques willbe necessary. By using the theory of the standard bases (or Hironaka bases), Theorem3.4 gives necessary and sufficient effective conditions to build up the Gorenstein one-dimensional k -algebras from a finite subset of Γ . In particular, starting from a suitable finite G -admissible set H , we can construct all the ideals I of R such that R/I is aGorenstein one-dimensional k -algebra and the dual contains H . Example 3.7 shows thattwo Gorenstein rings sharing the same H in general are not analytically isomorphic, evenif they share the same associated graded ring. In the last two sections we apply the resultsto two classes of examples: the construction of the inverse system of d distinct Gorensteinpoints of P k and the inverse system of a class of Gorenstein semigroup rings in A k . Thecomputations are performed by using the computer algebra system Singular [5] and theSingular library INVERSE-SYST.lib [9].2. The structure of the Inverse system
Let V be a vector space of dimension n over a field k where, unless specifically statedotherwise, k is an infinite field of any characteristic. If V denotes the k -vector space h z , . . . , z n i , then we denote by V ∗ = h Z , . . . , Z n i the dual basis. Let P = Sym k · V = ⊕ i ≥ Sym k i V be the standard graded polynomial ring in n variables over k and Γ = D k · ( V ∗ ) = ⊕ i ≥ D k i ( V ∗ ) = ⊕ i ≥ Hom k ( P i , k ) be the graded P -module of graded k -linearhomomorphisms from P to k , hence Γ ≃ k DP [ Z , . . . , Z n ] the divided power ring. Inparticular Γ j = h{ Z [ L ] | | L | = j }i is the span of the dual generators to z L = z l · · · z l n where L denotes the multi-index L = ( l , . . . , l n ) ∈ N n of length | L | = P i l i . If L ∈ Z n then we set Z [ L ] = 0 if any component of L is negative. The monomials Z [ L ] are called NVERSE SYSTEMS OF ONE-DIMENSIONAL GORENSTEIN RINGS 3 divided power monomials (DP-monomials) and the elements F = P L b L Z [ L ] of Γ thedivided power polynomials (DP-polynomials).We recall that Γ is an R -module with R acting on Γ by contraction as follows. Thisaction is also called apolarity. Definition 2.1. If h = P M a M z M ∈ R and F = P L b L Z [ L ] ∈ Γ , then the contraction of F by h is defined as h ◦ F = X M,L a M b L Z [ L − M ] For short, from now on we write Z L instead of Z [ L ] .Recall that the injective hull E R ( k ) of k as an R -module is isomorphic as an R -moduleto the divided power ring Γ (see [14], [28]). For detailed information see [6], [13], [21],Appendix A. If the characteristic of the field k is zero, then there is a natural isomorphismof R -algebras between (Γ , ◦ ), where ◦ is the contraction already defined, and the usualpolynomial ring P replacing contraction with taking partial derivatives. In this paper wealways consider Γ as an R -module by contraction.If I ⊂ R is an ideal of R then ( R/I ) ∨ = Hom R ( R/I, E R ( k )) is the R -submodule of Γ I ⊥ = { g ∈ Γ | I ◦ g = 0 } . This submodule of Γ is called
Macaulay’s inverse system of I . Given an R -submodule W of Γ , then the dual W ∨ = Hom R ( W, E R ( k )) is the ring R/ Ann R ( W ) whereAnn R ( W ) = { f ∈ R | f ◦ g = 0 for all g ∈ W } is an ideal of R . If I is a homogeneous ideal of R (resp. W is generated by homogeneouspolynomials) then I ⊥ is generated by homogeneous polynomials of Γ (resp. Ann R ( W ) isan homogeneous ideal of R ) and I ⊥ = ⊕ I ⊥ j where I ⊥ j = { F ∈ Γ j | h ◦ F = 0 for all h ∈ I j } . Notice that the Hilbert function of
R/I can be computed by I ⊥ , see for instance[10] Section 2. Macaulay in [25, IV] proved a particular case of Matlis duality, calledMacaulay’s correspondence, between the ideals I ⊆ R such that R/I is an Artinianlocal ring and R -submodules W = I ⊥ of Γ which are finitely generated. Macaulay’scorrespondence is an effective method for computing Gorenstein Artinian rings, see [3],Section 1, [20], [15] and [21]. We summarize with the statement Artinian Gorenstein k -algebras A = R/I of socle degree s correspond to cyclic R -submodules of Γ generated bya DP-polynomial F = 0 of degree s . The authors extended Macaulay’s correspondence characterizing the d -dimensional lo-cal Gorenstein k -algebras in terms of suitable submodules of Γ, see [10]. This result inthe one-dimensional case will be our starting point and for completeness we include herethe statement. Definition 2.2. An R -submodule M of Γ is called G -admissible if it admits a countablesystem of generators { H l } l ∈ N + satisfying the following conditions (1) There exists a linear form z ∈ R such that for all l ∈ N + z ◦ H l = (cid:26) H l − if l > otherwise. (2) Ann R ( h H l i ) ◦ H l +1 = h H i for all l ∈ N + . J. ELIAS AND M. E. ROSSI
If this is the case, we say that M = h H l , l ∈ N + i is a G -admissible R -submodule of Γ withrespect to the linear form z ∈ R . Notice that if
R/I has positive depth, then there always exists a linear form z ∈ R which is regular modulo I because k is infinite. Theorem 2.3.
There is a one-to-one correspondence C between the following sets:(i) one-dimensional Gorenstein k -algebras A = R/I ,(ii) non-zero G -admissible R -submodules M = h H l , l ∈ N + i of Γ .In particular, given an ideal I ⊂ R with A = R/I satisfying ( i ) and z a linear regularelement modulo I, then C ( A ) = I ⊥ = h H l , l ∈ N + i ⊂ S with h H l i = ( I + ( z l )) ⊥ is G -admissible. Conversely, given an R -submodule M of Γ satisfying (ii), then C − ( M ) = R/I with I = Ann R ( M ) = \ l ∈ N + Ann R ( h H l i ) . The main goal of this paper is to construct G -admissible R -submodules of Γ . Notice that if we fix any DP-polynomial H ∈ Γ such that z ◦ H = 0 , then M = h Z l H , l ∈ N i is G -admissible with respect to z, but the corresponding one-dimensionalGorenstein ring is not of great interest because it is a cone over the Gorenstein Artinianring corresponding to the DP-polynomial H , see [10], Proposition 4.1. In the next remarkwe observe that the choice of H is very important in the construction of a G -admissiblemodule, because it encodes much information on the corresponding Gorenstein ring. Remark 2.4.
Let A = R/I be a 1-dimensional Gorenstein ring and let z be a linearregular element modulo I. Let I ⊥ = h H l , l ∈ N + i be the corresponding G -admissible dualmodule with respect to z. We recall that by Theorem 2.3, we have h H i = ( I +( z )) ⊥ . Henceby Macaulay’s correspondence, deg( H ) coincides with the socle degree of the Artinianreduction A/zA.
On the other hand we have the following inequality on the multiplicity e ( A ) of A : e ( A ) ≤ Length R ( A/zA ) = dim k ( h H i ) . In particular the equality holds if z is a superficial element of A. If A is a standardgraded k -algebra which is Gorenstein, then the above equality holds for every regularelement z modulo I. Hence e ( A ) = dim k ( h H i ) (the dimension as a k -vector space ofthe R -module generated by H ) and reg ( A ) = deg H where reg ( A ) is the Castelnuovo-Mumford regularity of A. As a consequence, in the graded case, important geometric information such as the mul-tiplicity, the arithmetic genus, or more generally, the Hilbert polynomial of the Gorenstein k -algebra can be controlled by the choice of H , the first step in the construction of a G -admissible dual module.Recall that if I is an homogeneous ideal, then the dual R -submodule I ⊥ of Γ canbe generated by homogeneous (in the usual meaning) DP-polynomials and converselyif the R -submodule of Γ is homogeneous, then the ideal I is homogeneous. Hence thecorrespondence will be between one-dimensional Gorenstein standard graded k -algebrasand G -admissible homogeneous R -submodules of Γ . The following result refines Theorem2.3 in the case of graded k -algebras. NVERSE SYSTEMS OF ONE-DIMENSIONAL GORENSTEIN RINGS 5
Theorem 2.5.
There is a one-to-one correspondence C between the following sets:(i) one-dimensional Gorenstein standard graded k-algebras A = R/I of multiplicity e = e ( A ) (resp. Castelnuovo-Mumford regularity r = reg ( A ) )(ii) non-zero G -admissible homogeneous R -submodules M = h H l , l ∈ N + i of Γ suchthat dim k h H i = e (resp. deg H = r ) In the construction of G -admissible R -modules, the following remark suggests to us toproceed step by step starting from H . Remark 2.6.
Given two DP-polynomials
H, G ∈ Γ , we say that G is a primitive of H with respect to z ∈ R if z ◦ G = H . From the definition of the contraction ◦ , we will get G = ZH + C for some C ∈ Γ such that z ◦ C = 0 . We remark that ZH denotes the usual multiplicationin a polynomial ring and we do not use the internal multiplication in Γ as DP-polynomials.Hence, according to Definition 2.2, part (1), given a G -admissible module generated by { H l } l ∈ N + , we have that H l +1 is a primitive of H l with respect to z for every positiveinteger l. In particular H is a primitive of H and so on. Hence there exist C , . . . , C l inΓ (depending on condition (2) of Definition 2.2) such that z ◦ C i = 0 and for all positive l (1) H l +1 = z l H + z l − C + · · · + zC l − + C l In the effective construction of a Gorenstein ring
R/I, note that we may assume Γ = k DP [ Z , . . . , Z n ] and take z = z . This presentation has the advantage that we may assume H , C , . . . , C l in k DP [ Z , . . . , Z n ] . According to Remark 2.6, starting from a DP-polynomial H , in each step we have tochoose the “constants” C i imposing condition (2) of Definition 2.2. We have implementeda Singular routine to determine the possible DP-polynomials H l +1 from H l , if they exist[5].Guided by the construction of one-dimensional Gorenstein k -algebras R/I in a finiteand effective number of steps, we state the following questions:
Question 2.7.
When can a finite set { H , . . . , H t +1 } of DP-polynomials of Γ verifyingconditions (1) and (2) of Definition 2.2 be completed to a system of generators of a G -admissible R -module M = h H l , l ∈ N + i ? Question 2.8.
Does there exist an integer t such that the finite subset { H , . . . , H t +1 } ofa G -admissible R -module M = h H l , l ∈ N + i determines (uniquely?) the Gorenstein ideal I = Ann R ( M ) ? According to the definition of a G -admissible R -submodule of Γ , see Definition 2.2, wesay: Definition 2.9. A finite set of DP-polynomials H = { H l | < l ≤ t } is a G-admissible set with respect to a linear element z ∈ R if it satisfies conditions (1) and(2) of Definition 2.2 with respect to z ∈ R. Note that an algorithm in Singular has been implemented for computing G -admissiblesets with respect to z ∈ R starting from a DP -polynomial H such that z ◦ H = 0 . J. ELIAS AND M. E. ROSSI
Remark 2.10.
Notice that in H = { H l ; 0 < l ≤ t } the last polynomial H t determinesthe full sequence H , · · · , H t − by contraction with respect to z . Thus H is identifiedby the DP-polynomial H t . But H t is not an arbitrary polynomial simply of the shapedescribed in Remark 2.6. In fact, in Definition 2.2, condition (1) does not imply condition(2) as the following example shows. Let us consider the finite set: H = { H = X , H = X Y, H = X + X Y , H = X Y + X Y , H = X Y + X Y } . This set satisfiescondition (1) with respect to y ; and it satisfies condition (2) except for the last polynomialsince Ann R ( h H i ) ◦ H % h H i . However, if we replace H with H + X we get a G-admissible set with respect to y. From now on if J is an ideal of R (not necessarily homogeneous), we denote by J ≤ s R the ideal of R generated by all the elements of J in M s \ M s +1 if any, otherwise (0) . Inthe homogeneous case, it means generated by forms of J of degree at most s. The nextresult gives a first positive answer to Question 2.8 in the homogeneous case.
Proposition 2.11.
Let t be a positive integer and let H = { H l | < l ≤ t } be ahomogeneous G-admissible set with respect to z ∈ R. Let deg H = r and assume t ≥ r + 2 . If H can be extended to a G -admissible R -module, then the corresponding gradedGorenstein k -algebra A = R/I is uniquely determined and I = Ann R ( H r +2 ) ≤ r +1 R. The previous result was proved in [10], Proposition 4.2. It depends on the fact thatthe maximum degree of a minimal system of generators of I is at most reg ( A ) + 1 and itcoincides with the socle degree socdeg( A/zA ) + 1 = r + 1 . Clearly it is also true that I = Ann R ( H t ) ≤ r +1 R. The previous result can be improved if we know the maximum degree t of the generatorsof I in which case one can replace r = deg H by t − ≤ r ) , hence I = Ann R ( H t +1 ) ≤ t R. Remark 2.12.
As in the Artinian case, the previous result tells us that there is a one-to-one correspondence between Gorenstein graded k -algebras of dimension one and socledegree r and suitable homogeneous DP-polynomials H r +2 of degree 2 r + 1 . We present here an example.
Example 2.13.
We consider the following G -admissible set of DP-polynomials in Γ = k P D [ X, Y ] with respect to y of R = k [[ x, y ]]: H = { H = X , H = X Y, H = X + X Y , H = X Y + X Y } It is a subset of the G -admissible R -submodule M of Γ generated by { H t = i ≤ t − X i =0 X i Y t − − i | t ≥ } In this case r = deg H = 2 , hence according to Proposition 2.11 I = Ann R ( H ) ≤ R. NVERSE SYSTEMS OF ONE-DIMENSIONAL GORENSTEIN RINGS 7
In fact, I = Ann R ( H ) ≤ R = ( x − xy ) is a Gorenstein ideal, y is a non-zero divisor in A = R/I and I ⊥ = M. We remark that in particular H (actually H ) determines theGorenstein ideal I and hence the G -admissible R -module I ⊥ . We end this section with a very partial answer (for the moment) to Question 2.7 in thehomogeneous case. We will need the following proposition.
Proposition 2.14.
Let H = { H l ; 0 < l ≤ t } be a finite G-admissible set with respect to z ∈ R . Then Ann R ( H t +1 ) + ( z t ) = Ann R ( H t ) for all t = 1 , · · · , t − . In particular Ann R ( H t +1 ) + ( z ) = Ann R ( H ) . Proof.
Since z ◦ H t +1 = H t we have h H t i ⊂ h H t +1 i , and then Ann R ( H t +1 ) ⊂ Ann R ( H t ).From condition (1) we deduce that z t ◦ H t = 0 so ( z t ) ⊂ Ann R ( H t ).Let now a ∈ Ann R ( H t ), from conditions (2) and (1) there exists b ∈ R such that a ◦ H t +1 = b ◦ H = b ◦ ( z t ◦ H t +1 ) . Hence a − bz t ∈ Ann R ( H t +1 ), so a ∈ Ann R ( H t +1 ) + ( z t ). (cid:3) Corollary 2.15.
Assume dim R = 2 . Let t be a positive integer and let H = { H l ; 0 Denote by ( x, z ) a minimal system of generators of the maximal ideal of R. Because z ◦ H = 0 and r = deg H , we may write H = x r and hence Ann( H ) = ( x r +1 , z ) . ByProposition 2.14 we have Ann( H t +1 ) + ( z ) = Ann( H ) for every t ≥ , hence Ann( H t +1 ) ⊆ ( x r +1 , z ) . We also know that Ann( H t +1 ) is generated by a regular sequence, say ( F t , G t ) . Hence we may assume F t = x r +1 + zM t for some form M t of degree r and G t = zN t forsome form N t ∈ R. We prove deg G t ≥ t +1 for every t = 0 , . . . , t . We proceed inductivelyon t. If t = 0 , then M = 0 and G = z. Assume Ann( H j +1 ) = ( x r +1 + zM j , zN j ) withdeg N j ≥ j and prove that Ann( H j +2 ) = ( x r +1 + zM j +1 , zN j +1 ) with deg N j +1 ≥ j + 1 . Now zN j +1 ◦ H j +2 = 0 implies N j +1 ◦ H j +1 = 0 . Hence N j +1 ∈ ( x r +1 + zM j , zN j ) and weconclude. Hence Ann R ( H r +2 ) ≤ r +1 R = ( x r +1 + zM r +1 ) = I. (cid:3) Unfortunately, Proposition 2.11 and Proposition 2.15 cannot be extended to the non-homogeneous case, even in the algebraic case. In fact the information on the multiplicityis not enough to determine the singularity. The following example is a counterexample inthe non-graded case to both the above results. Example 2.16. For all n ≥ A n = k [[ x, y ]] / ( f n )with f n = y − x n . Notice that, for all n ≥ , A n is an algebraic Gorenstein ring of mul-tiplicity e ( A n ) = 2 . On the other hand A n /xA n = k [ y ] / ( y ) is an Artinian reduction of A n , so H = Y and hence deg H = 1 . If n ≥ , it is clear that we cannot obtain the ideal( f n ) after deg H + 2 = 3 steps as Proposition 2.11 and Corollary 2.15 suggest. J. ELIAS AND M. E. ROSSI One dimensional Gorenstein local or graded rings The aim of this section is to give constructive answers to Questions 2.8 and 2.7 in thelocal and graded case. It will be useful to recall some well known facts concerning thestandard bases in the local case. Let R be the ring of formal power series with coefficientsin k with maximal ideal M and let P be the corresponding polynomial ring. Notice that P = gr M ( R ) is the associated graded ring of R . For every element f ∈ R \ { } we canwrite f = f v + f v +1 + · · · , where f v is not zero and f j is a homogeneous polynomial ofdegree j in P for every j ≥ v. We say that v ( f ) is the order of f (or M -valuation),denote f v by f ∗ and call it the initial form of f. If f = 0 we agree that its order is ∞ . Wedenote by I ∗ the ideal of P generated by all the initial forms of the elements in I. A set ofgenerators (not necessarily minimal) of I, say { f , . . . , f r } , is called a standard basis (orHironaka’s basis) of I if their initial forms generate I ∗ . Notice that the associated gradedring of A = R/I with respect to the maximal ideal is isomorphic to P/I ∗ . It is known thatthe Krull dimension of A and of gr M ( R/I ) = P/I ∗ coincide, as well, by definition, theirHilbert function. Unfortunately, even if A is Gorenstein, then in general gr M ( R/I ) is nolonger Gorenstein nor even Cohen-Macaulay, see for instance the examples in Section 5.Notice that if I is a homogeneous ideal in P, then any homogeneous system of generatorsof I is a standard basis. A set of generators of I is called a minimal standard basis of I if the initial forms minimally generate I ∗ . Notice that the orders of the elementsin a minimal system of generators of I are not uniquely determined, instead the ordersof the elements of a minimal standard basis of I (and the number of generators) areuniquely determined by the first graded Betti numbers of I ∗ . If { f , . . . , f r } is a minimalstandard basis of I, then v ( f i ) = deg( f ∗ i ) and r is the minimal number of generators of thehomogeneous ideal I ∗ . We recall that an element z ∈ M \ M is superficial for A if andonly if z ∗ is a homogeneous filter-regular element of degree one in G = gr M ( R/I ) = P/I ∗ or equivalently (0 : G z ∗ ) j = 0 for all large degree j. If depth A > , then a superficialelement is also a regular element modulo I, see for instance [31], Section 1.2.The next Theorem has a central role in this paper. In particular it extends and itrefines Proposition 2.11 in the local case. Theorem 3.1. Let A = R/I be a one-dimensional Gorenstein ring and let I ⊥ = h H i | i ∈ N + i be the correspondent G -admissible module with respect to z (a regular linear elementmodulo I ). Given t ≥ e = dim k h H i , let { h , . . . , h r } be the elements of a minimalstandard basis of Ann R ( H t +1 ) such that v ( h i ) ≤ t . Then (i) (Ann R ( H t +1 ) ∗ ) ≤ t = ( I ∗ ) ≤ t . (ii) There exist α , . . . , α r ∈ R such that { h + α z t +1 , . . . , h r + α r z t +1 } is a minimalstandard basis of I . (iii) Ann R ( H t +1 ) = I + ( z t +1 ) = ( h , . . . , h r ) + ( z t +1 ) .Assume that z is a superficial element of A and denote by J the ideal generated by { h , . . . , h r } and B = R/J . (iv) B is a one-dimensional Gorenstein ring and z is a regular element modulo J. Inparticular J ∗ = I ∗ and hence the Hilbert functions of A and B agree. (v) Assume that { h , . . . , h s } is a minimal system of generators of J , then { h + α z t +1 , . . . , h s + α s z t +1 } is a minimal system of generators of I . NVERSE SYSTEMS OF ONE-DIMENSIONAL GORENSTEIN RINGS 9 Proof. ( i ) Since I + ( z t +1 ) = Ann R ( H t +1 ) we get that I ∗ ⊂ Ann R ( H t +1 ) ∗ . Conversely,let F ∈ Ann R ( H t +1 ) ∗≤ t be a homogeneous form. Then there are f ∈ I and β ∈ R suchthat ( f + βz t +1 ) ∗ = F . Since the degree of F is less than or equal to t, we deduce that F = f ∗ ∈ I ∗≤ t .( ii ) Let g , . . . , g p be a minimal standard basis of I . Since t ≥ e = dim k h H i ≥ e ( R/I )and I ∗ is minimally generated by forms of degree less than or equal to e ( R/I ), [8], wehave that deg g ∗ i ≤ t , i = 1 , · · · , p .Let { h , . . . , h r } be the elements of a minimal standard basis of Ann R ( H t +1 ) such thatdeg h ∗ i ≤ t. From ( i ) we get( g ∗ , . . . , g ∗ p ) ≤ t = I ∗≤ t = Ann R ( H t +1 ) ∗≤ t = ( h ∗ , . . . , h ∗ r ) ≤ t and then(2) I ∗ = ( g ∗ , . . . , g ∗ p ) = ( h ∗ , . . . , h ∗ r ) . Thus h ∗ , . . . , h ∗ r is a minimal system of generators of I ∗ .For all h i ∈ I + ( z t +1 ) we can write, i = 1 , . . . , r , h i = γ i − α i z t +1 with γ i ∈ I and α i ∈ R . From this identity we have h i + α i z t +1 = γ i ∈ I , so h ∗ i = ( h i + α i z t +1 ) ∗ ∈ I ∗ for i = 1 , . . . , r . Hence { h i + α i z t +1 } i =1 ,...,r is a minimal standard basis of I .( iii ) It is a consequence of ( ii ).( iv ) We consider the morphism ππ : X = Spec (cid:18) R [ w ]( h + w t +1 α z t +1 , . . . , h r + w t +1 α r z t +1 ) (cid:19) −→ Spec( k [ w ])Notice that π − (0) is B and that for all a = 0 ∈ k we have π − ( a ) ∼ = A .First we prove that B is one-dimensional.We assume that z is a superficial element of degree one of A , so e is the multiplicity of A . We denote by m (resp. n ) the maximal ideal of A (resp. B ).From the identity I + ( z t +1 ) = J + ( z t +1 ), t ≥ e , we deduce I + M e +1 = J + M e +1 , so m j / m j +1 = n j / n j +1 for j = e − , e . Since the multiplication by z induces an isomorphism m e − m e .z −→ m e m e +1 we get an isomorphism n e − n e .z −→ n e n e +1 By Nakayama’s lemma we deduce n e = z n e − , so n j = z n j − for all j ≥ e .From this we get that dim( B ) ≤ 1. Assume that dim( B ) = 0. From the upper semi-continuity of the dimension over the ground field k of the fibers of the morphism π weget dim( A ) = 0 which is a contradiction with the hypothesis dim( A ) = 1.From the identity I + ( z t +1 ) = J + ( z t +1 ) we get I + ( z ) = J + ( z ), so the multiplicity e ′ of B satisfies e ′ ≤ dim k ( R/J + ( z )) = dim k ( R/I + ( z )) = e. We denote by e ′ T − e ′ (resp. eT − e ) the Hilbert Polynomial of B (resp. A ). Assume e ′ < e and let l be an integer such that e ′ l − e ′ < el − e . Consider now the morphism π l : X l = Spec (cid:18) R [ w ]( h + w t +1 α z t +1 , . . . , h r + w t +1 α r z t +1 ) + M l +1 R [ w ] (cid:19) −→ Spec( k [ w ])We have dim k ( π − l (0)) = e ′ l − e ′ and that dim k ( π − l ( a )) = el − e for all a = 0 ∈ k .By upper semi-continuity of the dimension over the ground field k of the fibers of themorphism π l we get a contradiction, so e ′ = e .We know that n j = z n j − for all j ≥ e = e ′ so n j − n j .z −→ n j n j +1 is an isomorphism for all j ≥ e ′ = e . Hence z is a superficial degree one element of B .Since dim k ( B/ ( z )) = e ′ we get that z is a nonzero divisor of B by [31], Proposition 1.2(3).Since A and B are one-dimensional Cohen-Macaulay local rings of multiplicity e , I ∗ and J ∗ are minimally generated by forms of degree less than or equal to e . On the otherhand J ∗≤ e = ( J + ( z t +1 )) ∗≤ e = ( I + ( z t +1 )) ∗≤ e = ( h ∗ , · · · , h ∗ r ) ≤ e = I ∗≤ e Hence I ∗ = J ∗ and the Hilbert functions of A and B agree.( v ) Assume that { h , · · · , h s } , s ≤ r , is a minimal system of generators of the ideal J .Assume now that s < r and let s < j ≤ r . Then there exist a ij ∈ R, i = 1 , · · · , s , suchthat h j = s X i =1 a ij h i Thus h j + α j z t +1 − s X i =1 a ij ( h i + α i z t +1 ) = z t +1 ( α j − s X i =1 a ij α i ) ∈ I Since z is a non-zero divisor modulo I we deduce α j − s X i =1 a ij α i = β j ∈ I and then h j + α j z t +1 = s X i =1 a ij ( h i + α i z t +1 ) + β j z t +1 for j = s + 1 , · · · , r . Since β j z t +1 ∈ M I , by Nakayama’s Lemma I is generated by { h i + α i z t +1 } i =1 , ··· ,s .Assume that { h i + α i z t +1 } i =1 , ··· ,s is not a minimal system of generators, for instance if h s + α s z t +1 = s − X i =1 a ij ( h i + α i z t +1 ) NVERSE SYSTEMS OF ONE-DIMENSIONAL GORENSTEIN RINGS 11 for some a ij ∈ R, i = 1 , · · · , s , then h s − s − X i =1 a ij h i = z t +1 ( − α s + s − X i =1 a ij α i )Since z is a non-zero divisor modulo the ideal J we get α s − s − X i =1 a ij α i = β j ∈ J and then h s − s − X i =1 a ij h i ∈ m J which is a contradiction with the assumption that { h , · · · , h s } is a minimal system ofgenerators of J . (cid:3) Notice that in Theorem 3.1, the ideal Ann R ( H t +1 ) = I + ( z t +1 ) has generators ofvaluation v ≤ t because t ≥ e ( R/I ) . Hence the assumptions of this result are consistent. Remark 3.2. If the ideal I is homogeneous, then Ann R ( H t +1 ) = Ann R ( H t +1 ) ∗ and I = I ∗ . In this case, in the previous result, we have necessarily α i = 0 for every i. Hence H e +1 determines the ideal I where e = e ( R/I ) . This confirms Proposition 2.11 because e ≥ r + 1 where r = deg H . We remark that in the homogeneous case, H r +2 determinesthe Gorenstein ideal I, while instead in the non homogeneous case, the DP-polynomial H e +1 determines a Gorenstein ideal I (but not uniquely). We will see that H , . . . , H e +1 can be completed in different ways to a G -admissible module, see Example 3.6.The next result gives a sufficient condition for obtaining a Gorenstein 1-dimensionalring from a finite sequence H only requiring that the DP-polynomials are primitive ofeach other (in the sense of Remark 2.6), i.e. we only require that they satisfy condition(1) of Definition 2.2. Proposition 3.3. Let H = { H , . . . , H t +1 } be a finite set of DP-polynomials satisfyingcondition (1) of Definition 2.2 with respect to a linear form z of R. Then z i ∈ Ann R ( H i ) for all i = 1 , . . . , t + 1 . Let I be an ideal of R such that Ann R ( H t +1 ) = I + ( z t +1 ) . Assume z regular modulo I ,then Ann R ( H i ) = I + ( z i ) for all i = 1 , . . . , t + 1 . In particular R/I is a one-dimensional Gorenstein k -algebra and I ⊥ contains { H , . . . , H t +1 } . Proof. First of all, z i ∈ Ann R H i for all i = 1 , . . . , t + 1 since z i ◦ H i = 0 by assumption.We proceed now by descending recurrence. Given a ∈ I + ( z i ), i ≤ t , we have az ∈ I + ( z i +1 ) = Ann R ( H i +1 ). Hence( az ) ◦ H i +1 = a ◦ ( z ◦ H i +1 ) = a ◦ H i = 0so a ∈ Ann R ( H i ). For all a ∈ Ann R ( H i ) we have 0 = a ◦ H i = ( az ) ◦ H i +1 . Hence az ∈ Ann R ( H i +1 ) = I + ( z i +1 ). Since z is a non-zero divisor modulo I we get a ∈ I + ( z i ). It follows that R/I is one-dimensional Gorenstein k -algebra because I + ( z ) = Ann R H is Artinian Gorenstein and z is I -regular. We conclude by Theorem 2.3. (cid:3) We give an answer to Questions 2.7 and 2.8 and we describe all the Gorenstein ideals I of R of multiplicity e = e ( R/I ) such that I ⊥ contains a G -admissible set H = { H , · · · , H t +1 } with t ≥ e = dim k h H i . In particular this allows us to find a system of generators of theGorenstein ideal I by a finite effective procedure. We note that a procedure has beenimplemented in Singular using the next result. Theorem 3.4. Let H = { H , · · · , H t +1 } be a G -admissible set with respect to a linearform z with t ≥ e = dim k h H i . Then the following conditions are equivalent: ( i ) There exists a G -admissible R -submodule of Γ with respect to z ∈ R, say M H , obtainedby completion of H . In particular if I = Ann R ( M H ) , then A = R/I is one-dimensionaland Gorenstein, z is regular modulo I, and Ann( H t +1 ) = I + ( z t +1 ) . ( ii ) Ann R ( H t +1 ) = ( h , . . . , h r ) + ( z t +1 ) where h , . . . , h r are the elements of a minimalstandard basis of Ann R ( H t +1 ) with v ( h i ) ≤ t and there exist α , · · · , α r ∈ R such that z is regular modulo ( h + α z t +1 , . . . , h r + α r z t +1 ) . If this is the case, then I = ( h + α z t +1 , . . . , h r + α r z t +1 ) and I ⊥ = M H . Proof. Assume ( i ) , then there exists a G -admissible R -submodule of Γ with respect to z ∈ R, say M H , obtained by completion of H . Let I = Ann R ( M H ) be the correspondingideal. By Theorem 2.3, R/I is one-dimensional and Gorenstein, z is regular modulo I, and Ann R ( H t +1 ) = I + ( z t +1 ) . By Theorem 3.1, given h , . . . , h r the elements of aminimal standard basis of Ann R ( H t +1 ) with v ( h i ) ≤ t, we know that there is a family ofelements α , · · · , α r ∈ R such that { h + α z t +1 , . . . , h r + α r z t +1 } is a minimal standardbasis of I and hence z is a non-zero divisor modulo ( h + α z t +1 , . . . , h r + α r z t +1 ) = I. Conversely, if we assume ( ii ) and consider the ideal I = ( h + α z t +1 , . . . , h r + α r z t +1 ) , then Ann R ( H t +1 ) = ( h , . . . , h r ) + ( z t +1 ) = I + ( z t +1 ) . Because z is regular modulo I, thenby Lemma 3.3, R/I is Gorenstein one-dimensional, hence I ⊥ is G -admissible with respectto z and it contains H . Hence I ⊥ = M H . (cid:3) Remark 3.5. We remark that the previous result is effective. Given a finite set H = { H , · · · , H t +1 } with t ≥ e = dim k h H i , by using [9] we can check if H is G-admissiblewith respect to a linear form z . We can compute Ann R ( H t +1 ) and determine the elements h , . . . , h r of a minimal standard basis of Ann R ( H t +1 ) with v ( h i ) ≤ t . Let { h , · · · , h s } be a minimal system of generators of J = ( h , . . . , h r ) . Let α , . . . , α s be elements of R such that z is a regular superficial element modulo I α = ( h + α z t +1 , . . . , h s + α s z t +1 ),then R/I α and R/J are Gorenstein one-dimensional rings with I ∗ α = J ∗ sharing the sameHilbert function. In fact R/I α is Gorenstein one dimensional by Proposition 3.3 and weconclude concerning R/J by Theorem 3.1. Example 3.6. Consider R = k [[ x, y, z ]] . According to Remark 3.5, we wish to constructthe ideals I in R such that R/I are Gorenstein local rings of dimension 1 and multiplicity5 sharing the same G -admissible set with respect to x ∈ R. Consider a polynomial H ∈ Γ = k P D [ Y, Z ] such that dim k h H i = 5 . Let H = Z + Y . One can verify that H = { H , H = XH , H = X H , H = X H + Y Z + Y Z , H = XH , H = XH } NVERSE SYSTEMS OF ONE-DIMENSIONAL GORENSTEIN RINGS 13 is a finite G-admissible set with respect to x. The elements of a minimal standard basis of Ann R ( H ) of valuation ≤ e = 5 are h = yz − x , h = z − y , h = y − x z. We verify that x is regular modulo the ideal J = ( h , h , h ) = ( h , h ) , hence by Theorem 3.4 R/J is one-dimensional Gorenstein.Notice that the ideal J = ( h , h , h ) = ( h , h ) is the defining ideal in R of the semigroupring k [[ t , t , t ]] . Consider now the ideal I α = ( h + α x , h + α x ) with α , α ∈ R. Since x is regular modulo I α for every α , α ∈ R, then I α describes all the ideals of R such that R/I α is a one-dimensional Gorenstein ring of multiplicity 5 and the dual is a completionof H . In particular the family of Gorenstein ideals I α share the same associated gradedring, hence the same Hilbert function, because they have the same tangent cone.In general, it is hard to prove that two ideals are non analytically isomorphic. In thenext example we give two deformations I , I of J , see Remark 3.5, such that I , I , J arepairwise non-analytically isomorphic and they come from the same G -admissible set. Example 3.7. Consider R = k [[ x, y, z ]] and the G -admissible set with respect to x ∈ R H = { H = Y + Z , H i = X i − H , i = 2 , · · · , } . Hence H = X Y + X Z and Ann R ( H ) = ( yz − x , z − y , x ) . Following the notations ofRemark 3.5, J = ( h , h ) with h = yz − x , h = z − y . Notice that J is the defining idealof the Gorenstein monomial curve k [[ t , t , t ]] . Consider the following two deformationsof J : I = ( h + x , h ) and I = ( h , h + x ), that is respectively ( α , α ) = ( x , α , α ) = (0 , x ) . Both ideals define reduced one-dimensional Gorenstein rings. Byusing Singular we compute the multiplicity sequences of the above ideals: { , , , , ... } for J , { , , , , , ... } for I , and I defines an ordinary singularity with 5 non-singularbranches. Hence J, I , I are pairwise non-analytically isomorphic.4. 0 -dimensional Gorenstein schemes As an application of the previous results, in this section we will present some examplesand discussions that we hope will be useful in the theory of Gorenstein linkage (G-linkage).Note that the lack of a general structure of homogeneous Gorenstein ideals of higher codi-mension is the main obstacle to extending the Gorenstein liaison theory in codimension atleast three; the codimension two Gorenstein liaison case is well understood, see [23]. See,for instance, [27], [24] and [22] for some constructions of particular families of Gorensteinalgebras.We say that two schemes X and Y in P n k that are reduced and without common com-ponents are directly Gorenstein-linked if their union is arithmetically Gorenstein. Moregenerally, if we call I ( G ) the Gorenstein ideal in R = k [ x , . . . , x n ], then I ( G ) : I ( X ) = I ( Y ) and I ( G ) : I ( Y ) = I ( X ) . Equivalence classes in Gorenstein linkage are determined by the equivalence relation gen-erated by direct Gorenstein linkage. In terms of the inverse system, it is easy to verifythat I ( G ) : I ( X ) = I ( Y ) is equivalent to I ( X ) ⊥ = I ( Y ) ◦ I ( G ) ⊥ . We say X is glicci if it is in the Gorenstein linkage equivalence class of a completeintersection. In analogy to the codimension 2 theory (where Gorenstein is complete in-tersection), one hopes that every arithmetically Cohen-Macaulay scheme is glicci and,in particular, every finite set of points in P should be glicci, see for instance [18], [23].This was verified by R. Hartshorne in [18] and by D. Eisenbud, R. Hartshorne and F.O.Schreyer in [7] if d = | X | ≤ , d = 37 , . Usually it is very difficult to construct a set of distinct points G whose defining ideal isGorenstein. General points are very far in general from being Gorenstein. The problembecomes even more difficult if the set G must contain a given set X of points as thelinkage theory requires. We take advantage of the results of this paper for giving explicitexamples of reduced one dimensional Gorenstein k -algebras A = k [ x , . . . , x n ] /I ( G ) withgiven Hilbert function. Our approach aims to be constructive, but for the moment wecannot deduce a theoretic approach.Our goal is the following. Given the integers d ; e and an appropriate Hilbert functionof degree d+e admissible for a Gorenstein graded k -algebra of dimension one, we willconstruct some examples of reduced Gorenstein schemes G in P k with the given Hilbertfunction. We verify that G contains a degree d subscheme X with generic Hilbert functionwhose complement Y of degree e also has generic Hilbert function. It turns out thatthere is only a finite number of possibilities for the Hilbert function of such Gorensteinschemes. We recall that in our construction the Hilbert function is determined by thesuitable choice of H . In fact the entries of the h -vector are determined by dim k h H i j . Wesay that a 0-dimensional scheme in P n k has generic Hilbert function if it is maximal, that isHF X ( j ) = Max { deg X, (cid:0) n + jj (cid:1) } . In general I ( X ) is level, hence also the structure of I ( X ) ⊥ is known by [26]. Different methods for constructing points which are glicci in special casesare known. Our hope is that our computations could suggest some theoretical methodsto use more abstractly.In this section let R = k [ x, y, z, w ] and hence Γ = k DP [ X, Y, Z, W ] . We assume theground field k is of characteristic zero. We may assume without loss of generality thatthe points we want to construct do not lie on w = 0 . Hence we want to find a G -admissibleset H = { H , . . . , H t +1 } in Γ with respect to w such that dim k h H i = d + e. Moreoverif t = maximum degree of the generators of I ( G ) or t = deg H , then by the previousresults: I ( G ) = Ann R ( H t +1 ) ≤ t R. If X is glicci, then there exists a G -admissible sequence H = { H , . . . , H t +1 } such that( I ( X ) + ( w i )) ⊥ ⊆ h H i i for all i = 1 , . . . , t + 1 . Example 4.1. (5 Gorenstein points) We know that a set of 4 points in P k is glicci, infact they are linked to a complete intersection by a Gorenstein scheme of length 5 . Thisis a very special case, in fact if we consider a set X of 4 points in linear general positionand we take a sufficiently general point Y , then G = X ∩ Y consists of 5 points whichare Gorenstein (but not a complete intersection). If we consider H = X + Y + Z , then dim k h H i = dim k h , X, Y, Z, H i = 5 and we can take the G -admissible set H = { H , H = W H + X + Y + Z , H = W H + X + Y + Z } with respect to w ∈ R. NVERSE SYSTEMS OF ONE-DIMENSIONAL GORENSTEIN RINGS 15 In this case, then I = Ann R ( H ) ≤ R is radical and it is the defining ideal of a Gorensteinset of 5 points containing 4 points in linear general position. Example 4.2. (Gorenstein set of 14 points) It is known that a set of 10 points in P k isglicci, in fact they are linked to a set of 4 points by a Gorenstein scheme of length 14 . In this example we use the inverse system to find a Gorenstein zero-scheme consisting of d + e = 14 points which are the union of two reduced subschemes X and Y respectivelyof d = 10 and e = 4 points with maximal Hilbert function.Consider H = X Y + Y Z + XZ ∈ k DP [ X, Y, Z ] and we observe that dim k h H i = 14with Hilbert function h i = dim k h H i i : { , , , , } . Recall that Ann R ( H ) = I ( G )+( w ) . We want to construct a 1-dimensional Gorenstein graded algebra R/I ( G ) from a G -admissible set H = { H , H , H , H } . Notice that in Theorem 3.4, we can choose t = 3because by the Hilbert function we know that the ideal is generated in degree 3 . By usingSingular’s library [9], given H j with j = 1 , . . . , t, it is possible to find the admissibleelements C j ∈ k DP [ X, Y, Z ] such that H j +1 = W H j + C j . In the first step we provethat we can choose C any homogeneous form of degree 5 in the variables X, Y, Z. Take C = XY Z . Consider H = W H + XY Z . We realize computationally that this choice of C fixesthe successive elements C = − X Y − XY Z + 2 Z and C = − XY − X Y Z − X Y Z − X Z , then H = W H + W C + W C + C . Hence I = Ann R ( H ) ≤ R = ( x − y z + z w, x y − z + 2 xw ,xy − xzw + 2 yw , y − xz + yzw, x z + 2 w , xyz − y w + zw , yz − x w )is Gorenstein and dim R/I = 1 . Using Singular, we can verify that the ideal is radical,hence it is the defining ideal of a Gorenstein set of 14 distinct points. We can also provethat it is the union of two subsets of points comprised of 10 and 4 points with maximalHilbert function.It would be more interesting to have a theoretic argument proving that starting froma given subscheme X of d = 10 distinct points with maximal Hilbert function, then itis always possible to find e = 4 distinct points, say Y, such that the union defines a0-dimensional Gorenstein scheme of length d + e = 14 . In our case I ( Y ) is generated by 6 quadrics q = x + wL , q = xy + wL , . . . , q = z + wL with L i suitable linear forms in R and I ( X ) = Ann R ( q ◦ H , . . . , q ◦ H ) ≤ R where H = { H , H , H , H } is a G -admissible set. Our approach suggests to find J =( q , . . . , q ) = I (4) imposing the following conditions: ( I ( X ) + ( w i )) ⊥ = J ◦ H i , i =1 , · · · , 4, with H = { H , H , H , H } G -admissible.In our particular case the system has a solution and we have: L = 3 x + 12 y + 3 z +7 w, L = 6 x + 11 y + 9 z + 11 w, L = 12 x + 10 y + 11 z + 7 w, L = 9 x + 7 y + 5 z + 11 w, L = x + 10 y + 4 z + 3 w, L = 10 x + 2 y + 10 z + 2 w and I ( X ) = ( − x + 3 x y + 3 y − y z +3 xz + 3 z + xyw, − x + 6 x y − y − y z − xz + yz + 6 z + x w, − x − x y − y − y z − xz − z + z w, − x − x y + 5 y − y z + 6 xz − z + yzw, − x + x y + 4 y + xyz − y z + 4 xz + z + y w x − x y + xy − y + 2 y z − xz − z + xzw ) . Example 4.3. (Gorenstein sets of 30 and 55 points) This case was considered by Hartshornein [18] to be a possible counterexample to the “conjecture”. But Eisenbud, Hartshorneand Schreyer in [7] proved that a set of 20 points in P k is glicci, in fact they are linkedto a set of 10 points by a Gorenstein scheme of length 30 . In this example we use the inverse system to find a set G of d + e = 30 Gorenstein points which are the union oftwo reduced subschemes X and Y respectively of d = 20 and e = 10 points with maximalHilbert function. We may proceed as in the previous example.In this case deg H = 6 because the socle degree of R/I ( G ) is 6 . We consider H = X + Y + Z + X Y + Y Z + XZ ∈ k DP [ X, Y, Z ] because dim k h H i = 30 and theHilbert function is: { , , , , , , } . Recall that Ann R ( H ) = I ( G ) + ( w ) . We wantto construct a 1-dimensional Gorenstein graded algebra R/I ( G ) from a G -admissible set H = { H , . . . , H } . Notice that in Theorem 3.4, we can choose t = 4 because by theHilbert function we know that the ideal is generated in degree 4 . By using Singular’slibrary [9] we can find C , . . . , C and we get H = W H + W C + · · · + W C + C . Atthe first step we choose C = X Y Z , then as before we realized that all the successiveforms are uniquely determined.In the same paper, Eisenbud, Hartshorne and Schreyer stated that the degree of thesmallest collection of general points in P k not yet known to be glicci is 34 . As before, whatwe are able to construct, with our methods, is a set G of 55 Gorenstein points containinga subscheme of 34 distinct points with generic Hilbert function, whose complementary 21points has also generic Hilbert function.In this case deg H = 8 because the socle degree of R/I ( G ) is 8 . We consider H = X + Y + Z + X Y Z + X Y Z + X Y Z ∈ k DP [ X, Y, Z ] because dim k h H i = 55 andits Hilbert function is: { , , , , , , , , } . Recall that Ann R ( H ) = I ( G ) + ( w ) . Wewant to construct a 1-dimensional Gorenstein graded algebra R/I ( G ) from a G -admissibleset H = { H , . . . , H } . Notice that in Theorem 3.4, we can choose t = 5 because by theHilbert function we know that the ideal is generated in degree ≤ . By using Singular’slibrary [9] we can find C , . . . , C and we get H = W H + W C + · · · + W C + C . Wechoose C = XY Z , then by the algorithm we realize that, as in the previous examples,all the successive “constants” C i are uniquely determined. For the moment we cannotunderstand the reason for this constraint. We think that this could be the key of aconstructive general argument.5. Gorenstein Semigroup rings Semigroup rings are a broad class of one-dimensional domains and they are a test casefor many open problems. Hence we end this paper with the construction of the inversesystem of the defining ideal of a family of semigroup rings. In particular we study acorresponding G -admissible set in Γ according to Theorem 3.4. Let b ≥ A = k [[ t b , t b +1 , t b +3 ]] . It is easy to see that A = k [[ x, y, z ]] /I where I = ( xz − y , z b − x b +1 ) . Thus A is aone-dimensional Gorenstein local domain of type (2 , b ) and multiplicity e = 3 b, see [16],Example 5.5. It is an interesting class of ideals because the associated graded ring is notCohen-Macaulay even if the ideal is a complete intersection. The Hilbert function of A NVERSE SYSTEMS OF ONE-DIMENSIONAL GORENSTEIN RINGS 17 has a very special shape. For every b ≥ A has b − A ( t ) = t = 0 , t + 2 t = 1 , · · · , b − , b t = b, b + 1 t = b + 1 , b + k t = b + 2 k, k = 1 , · · · , b − , b + k + 1 t = b + 2 k + 1 , k = 1 , · · · , b − , b t ≥ b − . We compute the inverse system I ⊥ = h H t , t ∈ N + i of I with respect to x . Notice that theideal I is homogeneous in the weighted space defined by deg b = (3 b, b + 1 , b + 3), i.e.deg b ( x i y j z k ) = 3 bi + (3 b + 1) j + (6 b + 3) k. Because the ideal I is homogeneous with respect to deg b , the DP-polynomial H t is homoge-neous as well. Notice that e = 3 b = dim k h H i . We determine that H = { H , · · · , H b +1 } is a G-admissible set in Γ = k DP [ X, Y, Z ] with respect to x which determines I. Claim : H = Y Z b − and deg b ( H t ) = 6 b + 3 bt − t ≥ H is a generator of ( I + ( x )) ⊥ . Since I + ( x ) = ( x, y , z b ), an easycomputation give us H = y z b − . Recall that H t = x ◦ H t +1 . From this we get the secondpart of the claim.We write δ ( b, t ) = 6 b + 3 bt − 1, for b ≥ t ≥ 1; we denote by M b,t the set ofhomogeneous monomials of degree δ ( b, t ) in Γ = k DP [ X, Y, Z ]. For all t ≥ δ ( b, t ) H t = X m ∈M b , t m . Claim : Ann R ( H b +1 ) = I + ( x b +1 ).Notice that we only have to prove that (1) x b +1 ◦ H b +1 = 0(2) ( xz − y ) ◦ H b +1 = 0(3) ( z b − x b +1 ) ◦ H b +1 = 0We set δ = δ ( b, b + 1) = 15 b + 3 b − . Recall that deg( H b +1 ) = δ .(1) For all monomials m = X i Y j Z k of degree δ we have to prove that i < b + 1. Assumethat i ≥ b + 1. The Diophantine equation(3 b + 1) j + (6 b + 3) k = δ − bi has the following solutions:( j, k ) = ( − , δ − bi ) + λ (6 b + 3 , − (3 b + 1))for all λ ∈ Z . Since j, k ≥ δ − bi ≥ A = 2( δ − bi )6 b + 3 ≤ λ ≤ B = δ − bi b + 1 . In particular i < b + 1 and then 3 b + 1 ≤ i < b + 1. We can perform the correspondingEuclidean divisions A = 5 b − i + r , B = 5 b − i + r with 0 < r = i − b − b +3 < r = i − b − b +1 < 1. Hence there are no integers λ such that A ≤ λ ≤ B . Hence i < b + 1.(2) We prove that xz ◦ M b, b +1 = y ◦ M b, b +1 . Given m = X i Y j Z k ∈ M b , + wehave that xz ◦ m = 0 if i = 0 or k = 0. Hence xz ◦ M b, b +1 is the set of monomials m ′ = X i − Y j Z k − such that m = X i Y j Z k ∈ M b, b +1 and i, k ≥ 1. Since m ′ = y ◦ ( Y m ′ )and y m ′ ∈ M b, b +1 we get xz ◦ M b, b +1 ⊂ y ◦ M b, b +1 . 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Joan EliasDepartament de Matem`atiques i Inform`aticaUniversitat de Barcelona (UB)Gran Via 585, 08007 Barcelona, Spain Email address : [email protected] Maria Evelina RossiDipartimento di MatematicaUniversit`a di genovaVia Dodecaneso 35, 16146 Genova, Italy Email address ::