On the structure of Stanley-Reisner rings associated to cyclic polytopes
aa r X i v : . [ m a t h . A C ] O c t ON THE STRUCTURE OF STANLEY-REISNER RINGSASSOCIATED TO CYCLIC POLYTOPES
JANKO B ¨OHM AND STAVROS ARGYRIOS PAPADAKIS
Abstract.
We study the structure of Stanley–Reisner rings associatedto cyclic polytopes, using ideas from unprojection theory. Considerthe boundary simplicial complex ∆( d, m ) of the d -dimensional cyclicpolytope with m vertices. We show how to express the Stanley-Reisnerring of ∆( d, m + 1) in terms of the Stanley–Reisner rings of ∆( d, m ) and∆( d − , m − Introduction
Gorenstein commutative rings form an important class of commutativerings. For example, they appear in algebraic geometry as canonical ringsof regular surfaces and anticanonical rings of Fano n -folds and in algebraiccombinatorics as Stanley–Reisner rings of sphere triangulations. In codi-mensions 1 and 2 they are complete intersections and in codimension 3 theyare Pfaffians [2], but, to our knowledge, no structure theorems are knownfor higher codimensions.Unprojection theory [11], which analyzes and constructs complicated com-mutative rings in terms of simpler ones, began with the aim of partly fillingthis gap. The first kind of unprojection which appeared in the literature isthat of type Kustin–Miller, studied originally by Kustin and Miller [8] andlater by Reid and the second author [9, 10]. Starting from a codimension1 ideal J of a Gorenstein ring R such that the quotient R/J is Gorenstein,Kustin–Miller unprojection uses the information contained in Hom R ( J, R ) toconstruct a new Gorenstein ring S which is birational to R and corresponds Mathematics Subject Classification.
Primary 13F55 ; Secondary 13H10, 13D02,05E99.J. B. supported by DFG (German Research Foundation) through Grant BO3330/1-1. S. P. was supported by the Portuguese Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia through Grant SFRH/BPD/22846/2005 of POCI2010/FEDER and through ProjectPTDC/MAT/099275/2008. to the contraction of V ( J ) ⊂ Spec R . See Subsection 2.2 for a precise defini-tion of Kustin–Miller unprojection and the introduction of [3] for referencesto applications.In the paper [3], the authors proved that on the algebraic level of Stanley–Reisner rings, stellar subdivisions of Gorenstein* simplicial complexes cor-respond to Kustin–Miller unprojections and gave applications to Stanley-Reisner rings associated to stacked polytopes. In the present paper, we useunprojection theory to study the structure of Stanley–Reisner rings associ-ated to cyclic polytopes. This setting is different from the one studied in [3]since here, except for some easy subcases, stellar subdivisions do not appearand the unprojection ideals are more complicated.Our main result, which is stated precisely in Theorems 3.3 and 4.4, canbe described as follows. Assume d ≥ d + 1 < m . Consider the cyclicpolytope which has m vertices and dimension d , and denote by ∆( d, m ) itsboundary simplicial complex. We show how to express the Stanley-Reisnerring of ∆( d, m + 1) in terms of the Stanley–Reisner rings of ∆( d, m ) and∆( d − , m −
1) via Kustin–Miller unprojection. Moreover, a similar result isalso true for the remaining cases d = 2 , m = d + 1, see Subsections 3.1,3.2, 4.1 and 4.2. In Section 5 we give a combinatorial interpretation of ourconstruction.As an application, in Section 6 we inductively identify the minimal gradedfree resolutions of the Stanley–Reisner rings k [∆( d, m )]. We use this iden-tification in Proposition 6.6 to calculate the graded Betti numbers of theserings, recovering results originally due to Schenzel [12] for d even and Teraiand Hibi [13] for d odd. Our derivation is more algebraic than the one in[13], and does not use Hochster’s formula or Alexander duality. Finally,Subsection 6.2 contains examples and a link to related computer algebracode.An interesting open question is whether there are other families of Goren-stein Stanley–Reisner rings related by unprojections in a similar way ascyclic polytopes, compare also the discussion in [3, Section 6].2. Preliminaries
Assume k is a field, and m a positive integer. An (abstract) simpli-cial complex on the vertex set { , . . . , m } is a collection ∆ of subsets of { , . . . , m } such that (i) all singletons { i } with i ∈ { , . . . , m } belong to ∆and (ii) σ ⊂ τ ∈ ∆ implies σ ∈ ∆. The elements of ∆ are called faces andthose maximal with respect to inclusion are called facets . The dimension ofa face σ is defined as one less than the cardinality of σ . The dimension of∆ is the maximum dimension of a face. Any abstract simplicial complex ∆has a geometric realization, which is unique up to linear homeomorphism. TANLEY-REISNER RINGS ASSOCIATED TO CYCLIC POLYTOPES 3
For any subset W of { , . . . , m } , we denote by x W the square-free mono-mial in the polynomial ring k [ x , . . . , x m ] with support W , in other words x W is the product of x t for t ∈ W . The ideal I ∆ of k [ x , . . . , x m ] whichis generated by the square-free monomials x W with W / ∈ ∆ is called the Stanley-Reisner ideal of ∆. The face ring , or
Stanley-Reisner ring , of ∆over k , denoted k [∆], is defined as the quotient ring of k [ x , . . . , x m ] by theideal I ∆ .Assume R = k [ x , . . . , x m ] is a polynomial ring over a field k with thedegrees of all variables x i positive, and denote by m = ( x , . . . , x m ) themaximal homogeneous ideal of R . Assume M is a finitely generated graded R -module. Denote by0 → F g → F g − → · · · → F → F → M → M as R -module, and write F i = ⊕ j R ( − j ) b ij . The integer b ij is called the ij -th graded Betti number of M and is alsodenoted by b ij ( M ). For fixed i we set b i ( M ) = P j b ij ( M ). The integer b i ( M ) is the rank of the free R -module F i in the category of (ungraded) R -modules, and(2.1) b i ( M ) = dim R/m
Tor Ri ( R/m, M ) , cf. [7, Proposition 1.7]. For more details about free resolutions and Bettinumbers see, for example, [6, Sections 19, 20].Assume R is a ring. An element r ∈ R will be called R -regular if themultiplication by r map R → R, u ru is injective. A sequence r , . . . , r n of elements of R will be called a regular R -sequence if r is R -regular, and,for 2 ≤ i ≤ n , we have that r i is R/ ( r , . . . , r i − )-regular.Assume k is a field, and a, m, n three positive integers with m < n and2 a ≤ n − m + 2. We define the ideal I a,m,n ⊂ k [ x m , x m +1 , . . . , x n ] by I a,m,n = ( x t x t . . . x t a (cid:12)(cid:12) m ≤ t , t a ≤ n, t j + 2 ≤ t j +1 for 1 ≤ j ≤ a − . The assumption 2 a ≤ n − m + 2 implies that there exists at least onemonomial generator of I a,m,n , namely x m x m +2 . . . x m +2( a − . For example,we have I , , = ( x x , x x , x x ).2.1. Cyclic polytopes.
Recall from [1, Section 5.2] the definition of cyclicpolytopes. We fix two integers m, d , with 2 ≤ d < m , and define the cyclicpolytope C d ( m ) ⊂ R d as follows: Fix, for 1 ≤ i ≤ m , t i ∈ R with t Assume W ⊂ { , . . . , m } is a nonempty subset with W ≤ d . W is a face of ∆( d, m ) if and only if the number of odd contiguous subsetsof W is at most d − W . In particular, if W ≤ [ d/ then W is a face of ∆( d, m ) . Kustin–Miller unprojection. We recall the definition of Kustin–Miller unprojection from [10]. Assume R is a local (or graded) Goren-stein ring, and J ⊂ R a codimension 1 ideal with R/J Gorenstein. Fix φ ∈ Hom R ( J, R ) such that Hom R ( J, R ) is generated as an R -module by thesubset { i, φ } , where i denotes the inclusion morphism. The Kustin–Millerunprojection ring S of the pair J ⊂ R is the quotient ring S = R [ T ]( T u − φ ( u ) (cid:12)(cid:12) u ∈ J ) , where T is a new variable. The ring S is, up to isomorphism, independentof the choice of φ . The original definition of Kustin and Miller [8] was usingprojective resolutions, compare Subsection 2.3 below.2.3. The Kustin–Miller complex construction. The following construc-tion, which is due to Kustin and Miller [8], will be important in Section 6,where we identify the minimal graded free resolution of k [∆( d, m )]. TANLEY-REISNER RINGS ASSOCIATED TO CYCLIC POLYTOPES 5 Assume R is a polynomial ring over a field with the degrees of all variablespositive, and I ⊂ J ⊂ R are two homogeneous ideals of R such that bothquotient rings R/I and R/J are Gorenstein and dim R/J = dim R/I − k , k ∈ Z such that ω R/I = R/I ( k ) and ω R/J = R/J ( k ), com-pare [1, Proposition 3.6.11], and assume that k > k . We fix a graded homo-morphism φ ∈ Hom R/I ( J, R/I ) of degree k − k such that Hom R/I ( J, R/I )is generated as an R/I -module by the subset { i, φ } , where i denotes the in-clusion morphism, compare Subsection 2.2. We denote by S = R [ T ] /Q theKustin–Miller unprojection ring of the pair J ⊂ R/I defined by φ , where T is a new variable of degree k − k . We have that Q = ( I, T u − φ ( u ) (cid:12)(cid:12) u ∈ J )and that S is a graded algebra.We denote by g = dim R − dim R/J the codimension of the ideal J of R .Let C J : 0 → R = A g → A g − → · · · → A → R = A and C I : 0 → R = B g − → · · · → B → R = B be the minimal graded free resolutions of R/J and R/I respectively as R -modules. Due to the Gorensteiness of R/J and R/I they are both self-dual.We denote by a i : A i → A i − and b j : B j → B j − the differential maps.In the following, for an R -module M we denoted by M ′ the R [ T ]-module M ⊗ R R [ T ].Kustin and Miller constructed in [8] a graded free resolution C S of S as R [ T ]-module of the form C S : 0 → F g → F g − → · · · → F → F → S → , where, when g ≥ F = B ′ , F = B ′ ⊕ A ′ ( k − k ) ,F i = B ′ i ⊕ A ′ i ( k − k ) ⊕ B ′ i − ( k − k ) , for 2 ≤ i ≤ g − ,F g − = A ′ g − ( k − k ) ⊕ B ′ g − ( k − k ) , F g = B ′ g − ( k − k ) , cf. [8, p. 307, Equation (3)]. When g = 2 we have F = B ′ , F = A ′ ( k − k ) , F = B ′ ( k − k ) . We will now describe the differentials of the complex C S . We denote therank of the free R -module A by t , since C J is self-dual t is also the rankof the free R -module A g − . We fix R -module bases e , . . . , e t of A andˆ e , . . . , ˆ e t of A g − . We define, for 1 ≤ i ≤ t , c i , ˆ c i ∈ R by a ( e i ) = c i R and a g (1 R ) = P t i =1 ˆ c i ˆ e i . By Gorensteiness we have that c i , ˆ c i ∈ J for all1 ≤ i ≤ t . For 1 ≤ i ≤ t , let l i ∈ R be a lift in R of φ ( c i ) and let ˆ l i ∈ R bea lift in R of φ (ˆ c i ). For an R-module A we set A ∗ = Hom R ( A, R ). For an R -basis f , . . . f t of A we denote by f ∗ , . . . , f ∗ t the basis of A ∗ dual to it. JANKO B ¨OHM AND STAVROS ARGYRIOS PAPADAKIS Denote by ˜ α dg − : A ∗ g − → R = B ∗ g − the R -homomorphism with ˜ α dg − (ˆ e ∗ i ) =ˆ l i R for 1 ≤ i ≤ t . Taking into account the self-duality of C I , C J , we havethat ˜ α dg − extends to a chain map ˜ α d : C ∗ J → C ∗ I . We denote by ˜ α : C I → C J the chain map dual to ˜ α d . The map ˜ α : B = R → R = A is multiplicationby an invertible element, say w , of R , cf. [9], and we set α = ˜ α/w .We will now define a chain map β : C J → C I [ − β : A → R = B by β ( e i ) = − l i R . We obtain a chain map β : C J → C I [ − 1] byextending β . Moreover, β g : A g = R → R = B g − is multiplication bya nonzero constant u ∈ R . By [8, p. 308] there exists a homotopy map h : C I → C I with h : B → B and h g − : B g − → B g − being the zeromaps and β i α i = h i − b i + b i h i , for 1 ≤ i ≤ g .Finally, following [8, p. 307], we have that the differential maps f i : F i → F i − of the complex C S are given in block format by the following formulas f = (cid:2) b β + T a (cid:3) , f = (cid:20) b β h + T I − a − α (cid:21) ,f i = b i β i h i − + ( − i T I i − − a i − α i − b i − for 3 ≤ i ≤ g − ,f g − = β g − h g − + ( − g − T I g − − a g − − α g − b g − ,f g = (cid:20) − α g − + ( − g u − T a g b g − (cid:21) , where I t denotes the identity rank B t × rank B t matrix.The resolution C S is, in general, not minimal [3, Example 5.2]. How-ever, in the cases of stacked and cyclic polytopes it is minimal, see [3] andTheorem 6.1. In the following we will call C S the Kustin–Miller complexconstruction . We refer the reader to Subsection 6.2 for explicit examples ofthis construction. 3. The main theorem for d even We fix a field k , and assume that d, m are integers with d even and2 ≤ d < m − 1. (The case m = d + 1 is discussed in Subsection 3.2.) Weset a = ( d + 2) / 2, and denote by k [∆( d, m )] the Stanley-Reisner ring of thesimplicial complex ∆( d, m ).The following lemma is an almost immediate consequence of Theorem 2.1. TANLEY-REISNER RINGS ASSOCIATED TO CYCLIC POLYTOPES 7 Lemma 3.1. We have k [∆( d, m )] ∼ = k [ x , . . . , x m ] / ( I a, ,m − , I a, ,m ) . Proof. Denote by A the set of minimal monomial generators of the ideal( I a, ,m − , I a, ,m ). We first show that if x V ∈ A , then V is not a face of∆( d, m ). Assume x V is a monomial generator of I a, ,m − , the case x V is amonomial generator of I a, ,m follows by the same arguments. Since V = a ,we have that the number of odd contiguous subsets of V is at least a − a − d/ > d/ − d − a , by Theorem 2.1 V is not a face of∆( d, m ).Assume now W ⊂ { , . . . , m } is a subset with W ≤ d . We will showthat if W is not a face of ∆( d, m ) then there exists a monomial generator x V ∈ A with V ⊂ W . By Theorem 2.1 W ≥ a . We will argue by inductionon the cardinality of W .Denote by p the number of the odd contiguous subsets of W consideredas a subset of { , . . . , m } , and, for w ∈ W , by p w the number of the oddcontiguous subsets of W \ { w } also considered as a subset of { , . . . , m } . ByTheorem 2.1 p > d − W . If W = a , then p > d − W implies that W has at least d − a + 1 = a − W − V = W .Assume for the rest of the proof that W > a . By the inductive hypoth-esis it is enough to show that there exists w ∈ W such that W \ { w } is nota face of ∆( d, m ). Hence, by Theorem 2.1 it is enough to show that thereexists w ∈ W with p w > d − W + 1.We call a nonempty X ⊂ W a gc-subset if there exist i ≤ j with i − / ∈ W , j + 1 / ∈ W such that X = { i, i + 1 , . . . , j } . It is obvious that a contiguoussubset of W is a gc-subset, and that a gc-subset of W is contiguous if andonly if contains neither 1 nor m .If W contains a gc-subset of even cardinality, say { i, i + 1 , . . . , j } we set w = m if j = m , while if j = m we set w = i . In the first case, since i = 1contradicts W ≤ d , we have that p w = p + 1, so p w > d − W + 1 follows.Similarly, for the second case again p w = p + 1 and p w > d − W + 1 follows.Assume for the rest of proof that all gc-subsets of W are of odd cardinality.First assume that W contains a gc-subset { i, i + 1 , . . . , j } of odd cardinalityat least 3, and set w = i + 1. Since ( i, j ) = (1 , m ) is impossible by W ≤ d ,it is clear that p w = p + 1, so again p w > d − W + 1.So we can assume for the rest of the proof that all gc-subsets of W are ofcardinality 1. We either set w = m if m ∈ W , or if m / ∈ W we set w to bethe smallest element of W . If m ∈ W and 1 ∈ W we have p w = p = W − p > d − W implies 2 W − > d , so since d is even 2 W > d + 3,hence p w > d − W + 1. If m ∈ W and 1 / ∈ W , we have p w = p = W − JANKO B ¨OHM AND STAVROS ARGYRIOS PAPADAKIS and p w > d − W + 1 is equivalent to 2 W > d + 2, which is true by theassumption W > a = ( d + 2) / 2. If m / ∈ W and 1 ∈ W the argument isexactly symmetric to the case m ∈ W and 1 / ∈ W . If m / ∈ W and 1 / ∈ W ,we have p w = p − W − p w > d − W + 1 is equivalent to2 W > d + 2, which is true by the assumption W > a = ( d + 2) / 2. Thisfinishes the proof of Lemma 3.1. (cid:3) We now further assume that d is an even integer with d ≥ 4, the case d = 2 is discussed in Subsection 3.1. We set R = k [ x , . . . , x m , z ], where weput degree 1 for all variables. We consider the ideals I = ( I a, ,m − , I a, ,m )and J = ( I a − , ,m − , zI a − , ,m − ) of R . (When we need to be more precisewe will also use the notations I d,m for I and J d,m for J .) It is clear that I ⊂ ( I a − , ,m − ), hence I ⊂ J . Moreover, using Lemma 3.1, R/I ∼ = k [∆( d, m )][ z ]and R/J ∼ = k [∆( d − , m − x , x m ]. Consequently, both rings R/I and R/J are Gorenstein by [1, Corollary 5.6.5], and dim R/J = dim R/I − Lemma 3.2. There exists unique φ ∈ Hom R/I ( J, R/I ) such that φ ( v ) = 0 for all v ∈ I a − , ,m − and φ ( zw ) = wx x m for all w ∈ I a − , ,m − . Moreover,the R/I -module Hom R/I ( J, R/I ) is generated by the set { i, φ } , where i : J → R/I denotes the inclusion homomorphism. Taking into account Lemma 3.2, the Kustin–Miller unprojection ring S of the pair J ⊂ R/I is equal to S = ( R/I )[ T ]( T u − φ ( u ) (cid:12)(cid:12) u ∈ J ) . We extend the grading of R to a grading of S by putting the degree of thenew variable T equal to 1. By Lemma 3.2 S is a graded k -algebra. Ourmain result for the case d even is the following theorem. Theorem 3.3. The element z ∈ S is S -regular, and there is an isomorphismof graded k -algebras S/ ( z ) ∼ = k [∆( d, m + 1)] . Proof. Denote by Q ⊂ R [ T ] the ideal Q = ( I, z ) + ( T u − φ ( u ) (cid:12)(cid:12) u ∈ J ) ⊂ R [ T ] . By the definition of S we have S/ ( z ) ∼ = R [ T ] /Q . By the definition of φ wehave Q = ( I a, ,m , T I a − , ,m − , z ). Hence, Lemma 3.1 implies that S/ ( z ) ∼ = k [∆( d, m + 1)]. As a consequence, dim S/ ( z ) = dim S − 1, and since by[10, Theorem 1.5] S is Gorenstein, hence Cohen–Macaulay, we get that z is S -regular. (cid:3) TANLEY-REISNER RINGS ASSOCIATED TO CYCLIC POLYTOPES 9 Example 3.4. Assume d = 4 and m = 6. We have I = ( x x x , x x x ) , J = ( x x , x x , x x , zx , zx )and S = k [ x , . . . , x , T, z ] / ( I, T x x , T x x , T x x , x ( zT − x x ) , x ( zT − x x )) . The case d = 2 and d + 1 < m . Assume d = 2 and d + 1 < m . Itis clear that ∆( d, m ) is just the (unique) triangulation of the 1-sphere S having m vertices. Hence ∆( d, m + 1) is a stellar subdivision of ∆( d, m ),and the results of [3] apply.In more detail, set R = k [ x , . . . , x m , z ], with the degree of all variablesequal to 1. Consider the ideals I = ( I , ,m − , I , ,m ) and J = ( I , ,m − , z )of R . (When we need to be more precise we will also use the notations I ,m for I and J ,m for J .) Clearly k [∆( d, m )][ z ] ∼ = R/I . Moreover, we have that I ⊂ J , that J ⊂ R/I is a codimension 1 ideal of R/I with R/J Gorenstein,and that if we denote by S the Kustin–Miller unprojection ring of the pair J ⊂ R/I we have S/ ( z ) ∼ = k [∆( d, m + 1)]. Moreover, arguing as in the proofof Theorem 3.3 we get that z is an S -regular element.3.2. The case d is even and m = d + 1 . Assume d ≥ m = d + 1. We have that k [∆( d, m )] ∼ = k [ x , . . . , x m ] / ( d +1 Y i =1 x i )and k [∆( d, m + 1)] ∼ = k [ x , . . . , x m +1 ] / ( d/ Y i =0 x i +1 , ( d/ Y i =1 x i ) . We set R = k [ x , . . . , x m , z ], with the degree of all variables equal to 1.Consider the ideals I = ( Q d +1 i =1 x i ) and J = ( Q d/ i =1 x i , z Q ( d/ − i =1 x i +1 ) of R . (When we need to be more precise we will also use the notations I d,m for I and J d,m for J .) We have I ⊂ J , that J ⊂ R/I is a codimension 1 idealof R/I with R/J Gorenstein, and that if we denote by S the Kustin–Millerunprojection ring of the pair J ⊂ R/I we have S/ ( z ) ∼ = k [∆( d, m + 1)].Moreover, arguing as in the proof of Theorem 3.3 we get that z is an S -regular element.3.3. Proof of Lemma 3.2. We start the proof of Lemma 3.2. Recall that I = ( I a, ,m − , I a, ,m ) and J = ( I a − , ,m − , zI a − , ,m − ). Since J is a codi-mension 1 ideal of R/I and R/I is Gorenstein, hence Cohen–Macaulay, thereexists b ∈ J which is R/I -regular. Write b = b + zb , with b ∈ I ea − , ,m − 10 JANKO B ¨OHM AND STAVROS ARGYRIOS PAPADAKIS and b ∈ I ea − , ,m − , where I e ∗ denotes the ideal of R/I generated by I ∗ .Consider the element s = b x x m b ∈ K ( R/I ) , where K ( R/I ) denotes the total quotient ring of R/I , that is the localizationof R/I with respect to the multiplicatively closed subset of regular elementsof R/I , cf. [6, p. 60]. We need the following lemma. Lemma 3.5. (a) We have that x x m vw = 0 (equality in R/I ) for all v ∈ I a − , ,m − and w ∈ I a − , ,m − .(b) We have s zw = wx x m (equality in K ( R/I ) ) for all w ∈ I a − , ,m − .Proof. Proof of (a). It is enough to show that x x m x V x W = 0 in k [∆( d, m )],whenever x V is a generating monomial of I a − , ,m − and x W is a generatingmonomial of I a − , ,m − , with V ⊂ { , . . . , m − } and W ⊂ { , . . . , m − } .Consider the set A = { , m } ∪ V ∪ W . If 2 / ∈ V it is clear that x x V = 0and, similarly, if m − / ∈ V we have x m x V = 0.Hence for the rest of the proof we can assume that 2 ∈ V and m − ∈ V .Denote by A = { , . . . , p } the initial segment of A , and by A the finalsegment of A . Since 2 , m − / ∈ W , we necessarily have that all odd elementsof A \{ } are in W \ V , and all even elements of A are in V \ W . If the largestelement p of A is not in V , the monomial with support ( V \ A ) ∪{ , , . . . , p } is in I , hence x x V x W = 0. By a similar argument, if the smallest elementof A is not in V we get x m x V x W = 0. So we can assume that both thelargest element of A and the smallest element of A are in V . By the abovediscussion, this implies that A ∩ V ) = A ∩ W ) + 1 and A ∩ V ) = A ∩ W ) + 1, hence W a = V a + 1, where we set V a = V \ ( A ∪ A ) and W a = W \ ( A ∪ A ). Hence there exists a contiguous subset of V a ∪ W a , say A = { i, i +1 , . . . , j } , which starts with an element of W \ V then either stopsor continuous with an element of V \ W and finally finishes with an elementof W \ V . The monomial with support in ( V \ A ) ∪ { i, i + 2 , . . . , j } is in I ,hence we get x V x W = 0 which finishes the proof of part (a) of Lemma 3.5.We now prove part (b) of the lemma. It is enough to show that ( b + zb ) wx x m = zw ( b x x m ), for all w ∈ W . For that it is enough to show x x m b w = 0, which follows from part (a). (cid:3) Using Lemma 3.5, multiplication by s , which a priori is only an R/I -homomorphism R/I → K ( R/I ), maps J inside R/I , so defines an R/I -homomorphism φ : J → R/I . By the same Lemma 3.5, we have that φ ( v ) =0, for all v ∈ I a − , ,m − , and φ ( zw ) = wx x m , for all w ∈ I a − , ,m − . Sincean R/I -homomorphism is uniquely determined by its values on a generatingset, the uniqueness of φ stated in Lemma 3.2 follows. TANLEY-REISNER RINGS ASSOCIATED TO CYCLIC POLYTOPES 11 We will now prove the part of Lemma 3.2 stating that the R/I -moduleHom R/I ( J, R/I ) is generated by the set { i, φ } . By the arguments containedin the proof of [1, Theorem 5.6.2], we have isomorphisms ω k [∆( d,m )] ∼ = k [∆( d, m )](0) , ω k [∆( d − ,m − ∼ = k [∆( d − , m − , of graded k -algebras, where ω R denotes the canonical R -module. Conse-quently, since R/I ∼ = k [∆( d, m )][ z ], R/J ∼ = k [∆( d − , m − x , x m ] weget(3.1) ω R/I ∼ = ( R/I )( − 1) and ω R/J ∼ = ( R/J )( − . Combining (3.1) with the short exact sequence ([10, p. 563])0 → ω R/I → Hom R/I ( J, ω R/I ) → ω R/J → , we get the short exact sequence0 → R/I → Hom R/I ( J, R/I ) → ( R/J )( − → . As a consequence, Hom R/I ( J, R/I ) is generated as an R/I -module by thesubset { i, ψ } , whenever ψ ∈ Hom R/I ( J, R/I ) has homogeneous degree 1and is not contained in the R/I -submodule of Hom R/I ( J, R/I ) generatedby the inclusion homomorphism i . Hence, to prove Hom R/I ( J, R/I ) = ( i, φ )is enough to show that there is no c ∈ R/I with φ = ci . Assume such c exists. Let w ∈ I a − , ,m − be a fixed monomial generator. We then have czw = φ ( zw ) = wx x m (equality in R/I ), and since R/I is a polynomialring with respect to z we get wx x m = 0, which is impossible, since I =( I a, ,m − , I a, ,m ). Hence Hom R/I ( J, R/I ) = ( i, φ ), which finishes the proofof Lemma 3.2. 4. The main theorem for d odd Assume k is a fixed field, and d, m two integers with d odd and 5 ≤ d 1, the cases d = 3 and m = d + 1 are discussed in Subsections 4.1 and4.2 respectively. We set a = ( d + 1) / 2. Combining Proposition 3.1 with [1,Exerc. 5.2.18] we get the following proposition. Proposition 4.1. We have k [∆( d, m )] ∼ = k [ x , . . . , x m ] / ( I a, ,m − , x x m I a − , ,m − ) . Remark . By Proposition 4.1 and [1, Exerc. 5.2.18], for d ≥ k [∆( d, m )] is related to the ideal defining k [∆( d − , m − d odd to the easiercase d even. A similar remark also applies when d = 3. We set R = k [ x , . . . , x m , z , z ], where we put degree 1 for all variables.Consider the ideals I = ( I a, ,m − , x x m I a − , ,m − ) and J = ( I a − , ,m − ,z z I a − , ,m − ) of R . It is clear that I ⊂ ( I a − , ,m − ), hence I ⊂ J .By Proposition 4.1 we have that R/I ∼ = k [∆( d, m )][ z , z ] and R/J ∼ = k [∆( d − , m − x , x m − , x m ]. Consequently, both rings R/I and R/J are Gorenstein by [1, Corollary 5.6.5], and dim R/J = dim R/I − 1. Thefollowing lemma is the analogue of Lemma 3.2 for the case d odd. Lemma 4.3. There exists unique φ ∈ Hom R/I ( J, R/I ) such that φ ( v ) = 0 for all v ∈ I a − , ,m − and φ ( z z w ) = x x m − x m w for all w ∈ I a − , ,m − .Moreover, the R/I -module Hom R/I ( J, R/I ) is generated by the set { i, φ } ,where i : J → R/I denotes the inclusion homomorphism.Proof. Taking into account Proposition 4.1 and Remark 4.2, Lemma 4.3follows by the same arguments as Lemma 3.2. (cid:3) Taking into account Lemma 4.3, the Kustin–Miller unprojection ring S of the pair J ⊂ R/I is equal to S = ( R/I )[ T ]( T u − φ ( u ) (cid:12)(cid:12) u ∈ J ) . We extend the grading of R to a grading of S by putting the degree of thenew variable T equal to 1. Lemma 4.3 tells us that S is a graded k -algebra.Our main result for the case d odd is the following theorem. Theorem 4.4. The sequence z , z ∈ S is S -regular, and there is an iso-morphism of graded k -algebras S/ ( z , z ) ∼ = k [∆( d, m + 1)] . Proof. Denote by Q ⊂ R [ T ] the ideal Q = ( I, z , z ) + ( T u − φ ( u ) (cid:12)(cid:12) u ∈ J ) ⊂ R [ T ] . By the definition of S we have S/ ( z , z ) ∼ = R [ T ] /Q .Denote by g : R [ T ] → R [ x m +1 ] the k -algebra isomorphism which is uniquelyspecified by g ( z i ) = z i for i = 1 , g ( x i ) = x i for 1 ≤ i ≤ m − g ( x m ) = x m +1 and g ( T ) = x m . It is easy to see that g ( Q ) = ( I d,m +1 , z , z ).Since g is an isomorphism, we have using Proposition 4.1 that R [ T ] /Q ∼ = R [ x m +1 ] / ( I d,m +1 , z , z ) ∼ = k [∆( d, m + 1)] , hence S/ ( z , z ) ∼ = k [∆( d, m + 1)]. As a consequence, dim S/ ( z , z ) =dim S − 2, and since by [10, Theorem 1.5] S is Gorenstein, hence Cohen–Macaulay, we get that z , z is an S -regular sequence. (cid:3) TANLEY-REISNER RINGS ASSOCIATED TO CYCLIC POLYTOPES 13 The case d = 3 and d + 1 < m . Assume d = 3 and d + 1 < m .Combining [1, p. 229, Exerc. 5.2.18] with the discussion of Subsection 3.1 wehave the following picture. Set R = k [ x , . . . , x m , z , z ], where we put degree1 for all variables. Consider the ideals I = ( I , ,m − , x x m I , ,m − ) and J = ( I , ,m − , z z ) of R . Then k [∆( d, m )][ z , z ] ∼ = R/I . Moreover, we have I ⊂ J , that J ⊂ R/I is a codimension 1 ideal of R/I with R/J Gorenstein,and that if we denote by S the Kustin–Miller unprojection ring of the pair J ⊂ R/I then z , z is an S -regular sequence and S/ ( z , z ) ∼ = k [∆( d, m +1)] . The case d is odd and m = d +1 . Assume d ≥ m = d +1.We have k [∆( d, m )] ∼ = k [ x , . . . , x m ] / ( d +1 Y i =1 x i )and k [∆( d, m + 1)] ∼ = k [ x , . . . , x m +1 ] / ( ( d +1) / Y i =0 x i +1 , ( d +1) / Y i =1 x i ) . Set R = k [ x , . . . , x m , z , z ], where we put degree 1 for all variables. Con-sider the ideals I = ( Q d +1 i =1 x i ) and J = ( Q ( d +1) / i =1 x i , z z Q ( d − / i =1 x i +1 )of R . We have I ⊂ J , that J ⊂ R/I is a codimension 1 ideal of R/I with R/J Gorenstein, and that if we denote by S the Kustin–Miller unpro-jection ring of the pair J ⊂ R/I then z , z is an S -regular sequence and S/ ( z , z ) ∼ = k [∆( d, m + 1)].5. Combinatorial interpretation of our construction We fix d ≥ m ≥ d + 1, and we will give a combinatorialinterpretation of the constructions of Section 3. We introduce the notation R ( m ) = k [ x , . . . , x m , z ]. Consider the ideals I d,m and J d,m of R ( m ) as definedin Section 3 if d ≥ m ≥ d + 2, as defined in Subsection 3.1 if d = 2and m ≥ d + 2, and as defined in Subsection 3.2 if d ≥ m = d + 1.Note that I d,m is the Stanley–Reisner ideal of ∆( d, m ). We will inductivelyidentify J d,m . We set P d,m = I d,m : ( x x m ), then P d,m = I star ∆( d,m ) ( { ,m } ) + ( x i (cid:12)(cid:12) i is not a vertex of star ∆( d,m ) ( { , m } )) , It is clear that the ideal P d,m of R ( m ) is monomial, and that no minimalmonomial generator of it involves the variables x , x m and z . We denote byˆ P d,m the ideal of k [ x , . . . x m − , z ] which has the same minimal monomialgenerating set.If d = 2 we have J d,m = ( P d,m , z ). Assume now d ≥ 4. It is easyto see that the ideal ˆ P d,m is equal to the image of the ideal I d − ,m − of R ( m − under the k -algebra isomorphism R ( m − → k [ x , . . . x m − , z ] that sends z to z and x i to x i +1 for 1 ≤ i ≤ m − 2, hence ˆ P d,m is the Stanley–Reisner ideal of a simplicial complex isomorphic to ∆( d − , m − I d − ,m − of R ( m − to the ideal I d − ,m − of R ( m − , which is the Stanley–Reisner ideal of ∆( d − , m − Q d,m ⊂ k [ x , . . . , x m , z ] the image of the ideal I d − ,m − under the k -algebra isomorphism R ( m − → k [ x , . . . , x m , z ] that sends z to x m , x i to x i +1 for 1 ≤ i ≤ m − 2, and x m − to z . It is then easy to see that J d,m is the ideal of R ( m ) generated by the image of Q d,m under the inclusionof k -algebras k [ x , . . . , x m , z ] → R ( m ) . In particular, R ( m ) / ( J d,m , x , x m ) ∼ = k [∆( d − , m − d ≥ m ≥ d + 1. Consider the ideal J as definedin Section 4. Using Remark 4.2, a similar combinatorial interpretation existsfor J in terms of the ∆( d − , m − 2) related to the star of the face { , m } of ∆( d, m ) when d ≥ 5, and an analogous statement when d = 3. We leavethe precise formulations to the reader.6. The minimal resolution of cyclic polytopes Combining the results of Sections 3 and 4, we have that for d ≥ d + 1 < m , the Stanley-Reisner ring k [∆( d, m + 1)] can be constructed fromthe Stanley–Reisner rings k [∆( d, m )] and k [∆( d − , m − d = 2 , m = d + 1. Using the Kustin–Miller complexconstruction discussed in Subsection 2.3, we can inductively build a gradedfree resolution of S , hence using Proposition 6.3 below of k [∆( d, m + 1)],starting from the minimal graded free resolutions of k [∆( d, m )] and k [∆( d − , m − Theorem 6.1. For d ≥ and d + 1 < m , the graded free resolution of k [∆( d, m +1)] obtained from the minimal graded free resolutions of k [∆( d, m )] and k [∆( d − , m − using the Kustin–Miller complex construction isminimal. For d = 2 or and d + 1 < m , the graded free resolution of k [∆( d, m +1)] obtained from the minimal graded free resolution of k [∆( d, m )] and the appropriate Koszul complex (see Subsections 3.1 and 4.1) using theKustin–Miller complex construction is also minimal. We remark that in the proof of Theorem 6.1 we do not use the calcula-tion of the graded Betti numbers of k [∆( d, m )] obtained by Schenzel [12]for even d , and by Terai and Hibi [13] for odd d . Not only that, but in TANLEY-REISNER RINGS ASSOCIATED TO CYCLIC POLYTOPES 15 Proposition 6.6 we recover their results, without using Hochster’s formulaor Alexander duality.6.1. Proof of Theorem 6.1. For the proof of Theorem 6.1 we will needthe following combinatorial discussion.Assume d ≥ d + 1 < m and 1 ≤ i ≤ m − d − 1. We set η ( d, m, i ) = (cid:18) m − [ d/ − d/ 2] + i (cid:19)(cid:18) [ d/ 2] + i − d/ (cid:19) , compare [13, p. 291]. We also set η ( d, m, 0) = η ( d, m, m − d ) = 0. Proposition 6.2. We have, for ≤ i ≤ m − d , (6.1) η ( d, m + 1 , i ) = η ( d, m, i ) + η ( d, m, i − 1) + η ( d − , m − , i ) . (By our conventions, for i = 1 the equality becomes η ( d, m +1 , 1) = η ( d, m, η ( d − , m − , , while for i = m − d it becomes η ( d, m + 1 , m − d ) = η ( d − , m − , m − d ) + η ( d, m, m − d − .)Proof. Assume first 2 ≤ i ≤ m − d − 1. We will use twice the Pascal triangleidentity (cid:0) kd (cid:1) = (cid:0) k − d (cid:1) + (cid:0) k − d − (cid:1) . We have η ( d, m + 1 , i ) = m + 1 − [ d/ − d/ 2] + i ! [ d/ 2] + i − d/ ! = m − [ d/ − d/ 2] + i ! + m − [ d/ − d/ 2] + i − !! [ d/ 2] + i − d/ ! = m − [ d/ − d/ 2] + i ! [ d/ 2] + i − d/ ! + m − [ d/ − d/ 2] + i − ! [ d/ 2] + i − d/ ! = η ( d, m, i ) + m − [ d/ − d/ 2] + i − ! [ d/ 2] + i − d/ ! + [ d/ 2] + i − d/ − !! = η ( d, m, i ) + η ( d, m, i − 1) + η ( d − , m − , i ) . The special cases i = 1 and i = m − d are proven by the same argument. (cid:3) For the proof of Theorem 6.1 we will also need the following generalpropositions, the first of which is well-known. Proposition 6.3. ([1, Proposition 1.1.5]). Assume R = k [ x , . . . , x n ] is apolynomial ring over a field k with the degrees of all variables positive, and I ⊂ R a homogeneous ideal. Moreover, assume that x n is R/I -regular. De-note by cF the minimal graded free resolution of R/I as R -module. We thenhave that cF ⊗ R R/ ( x n ) is the minimal graded free resolution of R/ ( I, x n ) as k [ x , . . . , x n − ] -module, where we used the natural isomorphisms R ⊗ R R/ ( x n ) ∼ = R/ ( x n ) ∼ = k [ x , . . . , x n − ] . The following proposition is an immediate consequence of Equation (2.1). Proposition 6.4. Assume k is a field and R = k [ x , . . . , x n ] , R = k [ y , . . . , y n ] are two polynomial rings with the degrees of all variables positive. Assume I ⊂ R is a monomial ideal, and denote by I the ideal of R generated bythe image of I under the k -algebra homomorphism R → R , x i y i , for ≤ i ≤ n . Obviously I is a homogeneous ideal of R . We claim that forall i ≥ we have b i ( R /I ) = b i ( R /I ) (of course the graded Betti numbers b ij of R /I and R /I may differ). Proposition 6.5. Assume k is a field, R = k [ x , . . . , x n , T ] and R = k [ y , . . . , y n , T , T ] are two polynomial rings with the degrees of all variablespositive, deg x i = deg y i , for ≤ i ≤ n , and deg T = deg T +deg T . Assume I ⊂ R is a homogeneous ideal, and denote by I ⊂ R the ideal generatedby the image of I under the graded k -algebra homomorphism φ : R → R specified by φ ( x i ) = y i , for ≤ i ≤ t , and φ ( T ) = T T . Denote by cF the minimal graded free resolution of R /I as R -module. Then I is ahomogeneous ideal R , and the complex cF ⊗ R R is a minimal graded freeresolution of R /I as R -module. In particular, the corresponding gradedBetti numbers b ij of R /I and R /I are equal.Proof. It is clear that I is a homogeneous ideal of R . By [6, Theorem 18.16] φ is flat. As a consequence, [6, Proposition 6.1] implies that the natural map I ⊗ R R → I is an isomorphism of graded R -modules. By flatness, ten-soring the minimal graded free resolution of I as R -module with R we getthe minimal graded free resolution of I as R -module, and Proposition 6.5follows. (cid:3) Theorem 6.1 will follow from the following more precise statement. Noticethat, as we already mentioned before, the statements about the graded Bettinumbers have been proven before by different arguments in [12, 13], but wedo not need to use their results. Proposition 6.6. Assume d ≥ and d + 1 < m . Set b ij = b ij ( k [∆( d, m )]) .Then the statement of Theorem 6.1 is true for ( d, m ) . Moreover, we havethat if d is even then b ij = 1 for ( i, j ) ∈ { (0 , , ( m − d, m ) } , b i,d/ i = η ( d + 1 , m + 1 , i ) + η ( d + 1 , m + 1 , m − d − i ) , for ≤ i ≤ m − d − , and b ij = 0 otherwise. If d is odd, then b ij = 1 for ( i, j ) ∈ { (0 , , ( m − d, m ) } , b i, [ d/ i = η ( d, m, i ) , b i, [ d/ i +1 = η ( d, m, m − d − i ) , for ≤ i ≤ m − d − , and b ij = 0 otherwise.Proof. We use induction on d and m . If d ≥ m = d +2 then k [∆( d, m )]is a codimension 2 complete intersection and everything is clear. TANLEY-REISNER RINGS ASSOCIATED TO CYCLIC POLYTOPES 17 The next step, is to notice that, for d = 2 and m ≥ 3, Proposition 6.6follows from [3, Proposition 5.7], since ∆(2 , m ) is equal to ∆ P ( m ) definedin [3, Section 5].Now assume that d is even with d ≥ d + 3 ≤ m , and, by theinductive hypothesis, Proposition 6.6 holds for the values ( d − , m − 1) and( d, m ). An easy computation, taking into account Proposition 6.2, showsthat the Kustin–Miller complex construction resolving k [∆( d, m + 1)] hasthe conjectured graded Betti numbers. Since no degree 0 morphisms appearit is necessarily minimal. This finishes the proof for d even.Assume now d ≥ ≤ i ≤ m − d ,(6.2) b i ( k [∆( d, m )]) = b i ( k [∆( d − , m − . (Of course the graded Betti numbers b ij can, and in fact are, different for k [∆( d, m )] and k [∆( d − , m − d odd tothe case d − 1, by doing an almost identical induction on ( d, m ) as in thecase ( d − , m − k [∆( d, m + 1)] has to be minimal, since we proved that the one for k [∆( d − , m )] is minimal and the corresponding numbers b i = P j b ij areequal by Equation (6.2). This finishes the proof of Proposition 6.6. (cid:3) Examples and implementation. In this subsection we demonstratethe construction of the cyclic polytope resolution with a sequence of two ex-amples. First we carry out the Kustin-Miller complex construction describedin Subsection 2.3 for the step passing from the codimension 4 complete in-tersection J , and the Pfaffian I , to the codimension 4 ideal I , . In thesecond step we pass from J , and the Pfaffian I , to I , , using that J , is equal to I , after a change of variables. At the end of the subsection wegive a link to computer algebra code where we implement our constructions.Using the notation of Subsection 2.3, we will explicitly compute for eachstep the auxiliary data α i , β i , h i , u and hence the differentials f i from theinput data a i and b i . The ideals I , and I , are Gorenstein codimension3, hence Pfaffian, and we will fix below a certain resolution for each ofthem. In addition, we will also fix below a certain Koszul complex resolving J , = ( z, x , . . . , x ).Assume q ≥ M is a skew-symmetric q × q matrixwith entries in a commutative ring. For 1 ≤ i ≤ q , we denote by pf i M thePfaffian ([1, Section 3.4]) of the submatrix of M obtained by deleting the i -th row and column of M . The main property of pf i M is that its square isthe determinant of the corresponding submatrix. We will use the notation R ( m ) = k [ x , . . . , x m , z ] introduced in Section 5.For d ≥ M d the ( d + 3) × ( d + 3) skew-symmetricmatrix with entries in R ( d +3) whose ( i, j ) entry for i ≤ j is zero except thatfor 1 ≤ i ≤ d + 2 we have ( M d ) i,i +1 = x i and that ( M d ) ,d +3 = − x d +3 . It isan easy calculation that I d,d +3 = (pf i ( M d ) (cid:12)(cid:12) ≤ i ≤ d + 3) . In addition, according to the Buchsbaum-Eisenbud theorem [2], the minimalgraded free resolution of R ( d +3) /I d,d +3 is given by(6.3) 0 → R ( d +3) v td −→ R d +3( d +3) M d −−→ R d +3( d +3) v d −→ R ( d +3) where v d denote the 1 × ( d +3) matrix with (1 , i ) entry equal to ( − i pf i ( M d )and v td denotes the transpose of v d .We set R = R (5) and fix the following Koszul complex resolution of R/J , (6.4) 0 → R a −→ R a −→ R a −→ R a −→ R where a = (cid:0) z x x x (cid:1) , a = x x x − z x − x − z − x x − z x − x ,a = − x x z x − x z − x x z x x x t , a = x x x − z . We now discuss the Kustin–Miller complex construction for the steppassing from ( I , , J , ) to I , , which corresponds to the unprojection of J , ⊂ R/I , . We will use as input for the Kustin–Miller complex construc-tion the resolution (6.4) of R/J , and the case d = 2 of (6.3), which is aresolution of R/I , . Performing the computations we obtain, in the nota-tion of Subsection 2.3, the complex C S specified by h = h = 0, u = − α : R → R , P i =1 c i e i x ( c e + c e ) + x c e + x ( c e + c e ) α : R → R , P i =1 c i e i x ( c e + c e ) + x c e α : R → R , e x x e and β : R → R , P i =1 c i e i 7→ − x x c e β : R → R , P i =1 c i e i 7→ − x ( c e + c e ) − x c e β : R → R , P i =1 c i e i 7→ − x ( c e + c e ) − x c e − x ( c e + c e ) , where ( e i ) ≤ i ≤ q denotes the canonical basis of R q as R -module. Substituting x for T and 0 for z in the differential maps of C S we get the minimalgraded free resolution of R (6) /I , . Moreover, substituting z for x in thedifferential maps of the resolution of R (6) /I , just constructed we get theminimal graded free resolution of R (7) /J , .We now set R = R (7) and discuss the Kustin–Miller complex construc-tion for the step passing from ( I , , J , ) to I , , which corresponds to theunprojection of J , ⊂ R/I , . We will use as input for the Kustin–Millercomplex construction the resolution of R/J , constructed above and thecase d = 4 of (6.3), which is a resolution of R/I , . Performing the com-putations we obtain, in the notation of Subsection 2.3, the complex C S specified by h = h = 0, u = − α : R → R , P i =1 c i e i x ( c e + c e + c e ) + x c e + x ( c e + c e + c e ) α : R → R , P i =1 c i e i x ( c e + c e ) − x ( c e + c e − c e + c e ) α : R → R , e x x ( x e − x e − x e ) and β : R → R , P i =1 c i e i x x ( − c x − c x + c x ) β : R → R , P i =1 c i e i 7→ − x ( c e + c e + c e − c e ) − x ( c e + c e ) β : R → R , P i =1 c i e i 7→ − x c e − x ( c e + c e + c e ) + x ( c e + c e − c e ) . Substituting x for T and 0 for z in the differential maps of C S we get theminimal graded free resolution of R (8) /I , .Under the link [4], a related package for the computer algebra systemMacaulay2 [5] is available. Applying the ideas of the present paper, it con-structs the resolution of the ideal I d,m for d even and m ≥ d + 1 startingfrom Koszul complexes and the skew-symmetric Buchsbaum-Eisenbud reso-lution (6.3) of I d,d +3 . The functions in the package provide the user with theoption to output all the intermediate data a i , b i , α i , β i , h i , u , f i in additionto the final resolution. Acknowledgements . The authors wish to thank Christos Athanasiadis foruseful discussions and suggestions, and an anonymous referee for suggestionsthat improved the presentation of the material. References [1] W. Bruns and J. Herzog, Cohen-Macaulay Rings, revised edition, Cambridge Studiesin Advanced Mathematics , Cambridge University Press, Cambridge, 1998.[2] D. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, andsome structure theorems for ideals of codimension (1977),447–485.[3] J. B¨ohm and S. Papadakis, Stellar subdivisions and Stanley–Reisner rings of Goren-stein complexes , preprint, 2009, 15 pp, arXiv:0912.2151v1 [math.AC].[4] J. B¨ohm and S. Papadakis, CyclicPolytopeRes – The unprojec-tion structure of the Stanley-Reisner ring of the boundary com-plex of a cyclic polytope , Springer-Verlag, 1995.[7] D. Eisenbud, The geometry of syzygies. A second course in commutative algebra andalgebraic geometry, Graduate Texts in Mathematics, , Springer-Verlag, 2005.[8] A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones , J.Algebra (1983), 303–322.[9] S. Papadakis, Kustin-Miller unprojection with complexes , J. Algebraic Geom. (2004), 249–268.[10] S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes , J. Alge-braic Geom. (2004), 563–577.[11] M. Reid, Graded rings and birational geometry , in Proc. of Algebraic Geome-try Symposium ∼ masda/3folds.[12] P. Schenzel, ¨Uber die freien Aufl¨osungen extremaler Cohen-Macaulay-Ringe , J.Algebra (1980), 93–101.[13] N. Terai and T. Hibi, Computation of Betti numbers of monomial ideals associatedwith cyclic polytopes , Discrete Comput. Geom. (1996), 287–295. Department of Mathematics, University of California, Berkeley, CA 94720,USA, and Department of Mathematics, Universit¨at des Saarlandes, Campus E24, D-66123, Saarbr¨ucken, Germany E-mail address : [email protected] Centro de An´alise Matem´atica, Geometria e Sistemas Dinˆamicos, Departa-mento de Matem´atica, Instituto Superior T´ecnico, Universidade T´ecnica deLisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal E-mail address ::