Monomial ideals and toric rings of Hibi type arising from a finite poset
aa r X i v : . [ m a t h . A C ] J u l MONOMIAL IDEALS AND TORIC RINGS OF HIBI TYPEARISING FROM A FINITE POSET
VIVIANA ENE, J ¨URGEN HERZOG AND FATEMEH MOHAMMADI
Abstract.
In this paper we study monomial ideals attached to posets, introducegeneralized Hibi rings and investigate their algebraic and homological properties.The main tools to study these objects are Gr¨obner basis theory, the concept ofsortability due to Sturmfels and the theory of weakly polymatroidal ideals.
Introduction
In 1985 Hibi [12] introduced a class of algebras which nowadays are called Hibirings. They are toric rings attached to finite posets, and may be viewed as naturalgeneralizations of polynomial rings. Indeed, a polynomial ring in n variables over afield K is just the Hibi ring of the poset [ n ] = { , , . . . , n } .Hibi rings appear naturally in various combinatorial and algebraic contexts, forexample in invariant theory. Hodge algebras may be viewed as flat deformationsof Hibi rings. In this sense the coordinate ring of the flag variety for GL n is thedeformation of the Hibi ring for the so-called Gelfand-Tsetlin poset.Given a finite poset P = { p , . . . , p n } , let I ( P ) be the ideal lattice of P . ByBirkhoff’s theorem any finite distributive lattice arises in this way. Let K be field.Then the Hibi ring over K attached to P is the toric ring K [ I ( P )] generated bythe set of monomials { x I t : I ∈ I ( P ) } where x I = Q p i ∈ I x i . Let T = K [ { y I : y I ∈I ( P ) } ] be the polynomial ring in the variables y I over K , and ϕ : T → K [ I ( P )] the K -algebra homomorphism with y I x I t . One fundamental result concerning Hibirings is that the toric ideal L P = Ker ϕ has a reduced Gr¨obner basis consisting ofthe so-called Hibi relations: y I y I − y I ∩ J y I ∪ J with I J and J I. Hibi showed [12, Theorem 2.c] that any Hibi ring is a normal Cohen–Macaulaydomain, and that it is Gorenstein if and only if the attached poset P is pure [12,Corollary 3.d], that is, all maximal chains of P have the same cardinality.More generally, for any lattice L , not necessarily distributive, one may considerthe K algebra K [ L ] with generators y α , α ∈ L , and relations y α y β = y α ∧ β y α ∨ β where ∧ and ∨ denote meet and join in L . Hibi showed that K [ L ] is a domain if and onlyif L is distributive, in other words, if L is an ideal lattice of a poset.The starting point of this paper are the so-called Hibi ideals which were firstintroduced in [6]. Attached to each finite poset P = { p , . . . , p n } , one defines theHibi ideal H P as the monomial ideal in the polynomial ring K [ x , . . . , x n , y , . . . , y n ] Mathematics Subject Classification. enerated by the monomials x I y P \ I with I ∈ I ( P ). Note that the toric ring gener-ated over K by these monomials is isomorphic to the Hibi ring attached to P . Thesignificance of Hibi ideals is that their Alexander dual can be interpreted as the edgeideal of a bipartite graph. To be precise, if we define the bipartite graph G on thevertex set { x , . . . , x n , y , . . . , y n } by saying that { x i , y j } is an edge of G if and only p i ≤ p j , then H ∨ P is the edge ideal of G in the sense of Villarreal [19]. It turned outthat bipartite graphs obtained in this way are exactly the Cohen–Macaulay bipartitegraphs.Motivated by the dual relationship between Hibi ideals and edge ideals of bipartitegraphs we introduce in this paper the following ideals attached to a finite poset P = { p , . . . , p n } : fix integers r ≥ s ≥
1, a field K and let S be the polynomial ringover K in the variables x ij with i = 1 , . . . , r and j = 1 , . . . , n .We denote by C r ( P ) the set of multichains of length r . In other words, the elementsof C r ( P ) are subsets { p j , . . . , p j r } of P with p j ≤ p j ≤ · · · ≤ p j r .For C ∈ C r ( P ), C = { p j , . . . , p j r } and ∅ 6 = S ⊆ [ r ] we set u C,S = Y i ∈ S x ij i , and u C = u C, [ r ] . Then we define the monomial ideals I r,s ( P ) = ( u C,S : C ∈ C r ( P ) and S ⊂ [ r ] with | S | = s ) , and H r,s ( P ) = I r,s ( P ) ∨ .We call I r,s ( P ) the multichain ideal of type ( r, s ), and H r,s ( P ) the generalizedHibi ideal of type ( r, s ) of the poset P , since H , ( P ) is just the classical Hibi ideal H P . For simplicity the ideals H r,r ( P ) will be denoted by H r ( P ). It is worthwhileto notice that the ideals I r,s ( P ) may be interpreted as facet ideals of completelybalanced simplicial complexes, as introduced by Stanley [17].In Theorem 1.1 we compute explicitly the minimal monomial set of generators of H r ( P ) and show that H r,s ( P ) = H r ( P ) h r − s +1 i , where I h k i denotes the k th squarefreepower of a squarefree monomial ideal. It turns out that the generators of H r ( P )correspond bijectively to chains of length r I : I ⊆ I ⊆ · · · ⊆ I r = P of poset ideals of P .Based on this explicit description we show in Theorem 2.2 that all powers H r ( P ) k of the ideal H r ( P ) are weakly polymatroidal. The concept of weakly polymatroidalideals has been introduced by Hibi and Kokubo in [15] where they showed that theyshare with polymatroidal ideals the nice property of having linear quotients. Inparticular, we conclude from this that the ideals H r ( P ) k have a linear resolution forall k ≥
1. Now we use the fact shown in Theorem 1.1 that H r,s ( P ) is a suitablesquarefree power of the ideal H r ( P ), and observe that the squarefree part of aweakly polymatroidal ideal is again weakly polymatroidal (see Lemma 2.3) to finallydeduce in Theorem 2.4 that all Hibi ideals H r,s ( P ) are weakly polymatroidal. Bythe Eagon–Reiner Theorem [2] this implies that for any finite poset and all integers1 ≤ s ≤ r the chain ideals I r,s ( P ) are Cohen–Macaulay. Thus we obtain a rich familyof completely balanced simplical complexes whose facet ideals are Cohen–Macaulay. n the case that s = 2 this yields a class of r -partite graphs with Cohen–Macaulayedge ideal.In Corollary 2.5 we show that the ideals I r ( P ) are Gorenstein, if and only if theyare generated by a regular sequence which is the case if and only all elements of P are pairwise incomparable.Section 3 is devoted to the study of the resolution of the ideal H r ( P ). As H r ( P )has linear quotients, the resolution can in principle be obtained as an iterated map-ping cone. To get an explicit description of the maps in the resolution one has toknow all the linear quotients. This is described in Lemma 3.1. With this infor-mation at hand we can describe the projective dimension of H r ( P ) as the maximallength of antichains in P , see Corollary 3.3. Applying then a result of Terai whichrelates the projective dimension of an ideal to the regularity of its Alexander dualwe obtain a nice formula for the regularity for the chain ideals I r ( P ). Next we showthat the ideals H r ( P ) have regular decomposition functions in the sense of [11], andthen apply a result of the same paper to finally obtain in Theorem 3.6 the explicitresolution of the ideals H r ( P ).The remaining sections of the paper are devoted to the study of the toric rings R r,s ( P ) which naturally generalize the classical Hibi rings. The toric ring R r,s ( P ),respectively R r ( P ), is defined to be the standard graded K -algebra generated over K by the unique minimal minimal set G r,s ( P ) of monomial generators of H r,s ( P ),respectively of H r ( P ). In Theorem 4.1 we first show that R r ( P ) has a quadraticreduced Gr¨obner basis consisting of Hibi type relations. This result is used to showin Corollary 6.3 that all the toric rings R r ( P ) are normal Cohen–Macaulay domainsand to identify in Theorem 4.3 the toric ring R r ( P ) as a classical Hibi ring attachedto the direct product P × Q r − of the poset P and the poset Q r − = [ r − R r ( P ) is Gorenstein if and only if P ispure.The situation for the toric rings R r,s ( P ) is more complicated. Among their rela-tions are also relations which are not of Hibi type and so these algebras cannot beidentified with classical Hibi rings for suitable posets. Indeed, if we choose for P the poset consisting just of one element, then H r,s ( P ) is nothing than the squarefreeVeronese K -algebra which is generated over K by all squarefree monomials of degree s in r variables. If we choose the same monomial order to compute the Gr¨obner basisof the corresponding toric ideal of this algebra as we did in the proof for the algebras R r ( P ), then in this particular case this term order is just the reverse lexicographicorder induced by the order of the variables which is given by the lexicographic orderof the generators of the algebra. For this monomial order the algebra R r,s ( P ) has areduced Gr¨obner basis consisting of binomials of degree ≤ r = 6 and s = 3.The question arises whether there is a monomial order for which the algebras R r,s ( P ) has a quadratic Gr¨obner basis. The answer is yes, and the method toprove this is due to Sturmfels who used a sorting order to show that all algebras ofVeronese type have a quadratic Gr¨obner basis. What we need to show is that the et of monomial G r,s ( P ) is sortable. This then implies by a theorem of Sturmfels[18, Theorem 14.2] that the quadratic sorting relations with the unsorted pairs asleading terms form a Gr¨obner basis with respect to the sorting order induced by thesorting of the monomials. We prove in Theorem 5.3 that the set G r,s ( P ) is indeedsortable. As a consequence we obtain that the algebras R r,s ( P ) are again all normalCohen–Macaulay domains.In the last section we study the Rees algebra of the ideals H r,s ( P ). In [9] theauthors introduce the so-called ℓ -exchange property which guarantees that the Reesalgebra of a monomial ideal which is generated in one degree has a Gr¨obner basiscomposed of the Gr¨obner basis of the fibre of the Rees algebra and binomial relationswhich are linear in the variables of the base ring. It is shown in Proposition 6.1that the ℓ -exchange property is satisfied for sortable, weakly polymatroidal ideals.Thus we may apply the result of [9] and find that the Rees algebra R ( H r,s ( P ))has a quadratic Gr¨obner basis. As applications we find that all powers of H r,s ( P )are normal and have a linear resolution, and that R ( H r,s ( P )) is a normal Cohen–Macaulay Koszul algebra.While we can characterize the Gorenstein ideals I r ( P ) and the Gorenstein rings R r ( P ), we do not have such a characterization for the ideals I r,s ( P ) and the rings R r,s ( P ).1. Multichain ideals of a poset and their Alexander dual
In this section we determine the Alexander dual H r,s ( P ) of the multichain ideal I r,s ( P ). Recall from the introduction that I r,s ( P ) = ( u C,S : C ∈ C r ( P ) and S ⊂ [ r ] with | S | = s ) , where C r ( P ) is the set of multichains of length r in P , and where u C,S = Y i ∈ S x ij i . In order to formulate the main result of this section we introduce some notation.Given a multichain I : I ⊆ I ⊆ . . . ⊆ I r = P of poset ideals in P , we attach to it the following monomial in S : u I = x J x J · · · x rJ r , where x jJ j = Y p k ∈ J j x jk and J j = I j \ I j − for j = 1 , . . . , r .We denote by I r ( P ) the set of multichains of poset ideals of length r in P , and forany squarefree monomial ideal L we denote by L h k i the k th squarefree power of L ,that is, the ideal generated by all squarefree elements in L k . Theorem 1.1.
Let P be a finite poset. Then (a) The Alexander dual H r ( P ) of I r ( P ) is the ideal ( u I : I ∈ I r ( P )) ; (b) The Alexander dual H r,s ( P ) of I r,s ( P ) is the ideal H r ( P ) h r − s +1 i . roof. First we show that for any multichain of poset elements p ℓ ≤ · · · ≤ p ℓ r , theideal Q = ( x ℓ , . . . , x rℓ r )is a minimal prime ideal of H r ( P ).In order to see that Q contains H r ( P ), we show that for each u I ∈ H r ( P ) thereexists some j ∈ [ r ] such that x j,ℓ j divides u I .By contrary assume that no x j,ℓ j divides u I . Then p ℓ r I r \ I r − , and so p ℓ r ∈ I r − . Since p ℓ r − ≤ p ℓ r it follows that p ℓ r − ∈ I r − . On the other hand, since p ℓ r − I r − \ I r − we have p ℓ r − ∈ I r − which implies that p ℓ r − ∈ I r − . Continuingin this way we get p ℓ ∈ I , a contradiction.Suppose Q is not a minimal prime ideal, then there exists an integer i suchthat Q i = ( x ℓ , . . . , x i − ℓ i − , x i +1 ℓ i +1 , . . . , x rℓ r ) contains H r ( P ). However x iP is agenerator of I r ( P ) which does not belong to Q i , a contradiction.Next we show that for any multichain of poset elements p ℓ ≤ · · · ≤ p ℓ r and eachsubset { ℓ t , . . . , ℓ t r − s } of { ℓ , . . . , ℓ r } , the ideal Q = ( { x ℓ , . . . , x rℓ r } \ { x t ℓ t , . . . , x t r − s ℓ tr − s } )(1)is a prime ideal containing H r ( P ) h r − s +1 i .Let u = u I · · · u I r − s +1 be an arbitrary element in H r ( P ) h r − s +1 i . We show thatthere exists some j ∈ [ r ] \ { t , . . . , t r − s } such that x j,ℓ j divides u .We know already that ( x ℓ , . . . , x rℓ r ) is a minimal prime ideal of H r ( P ). So foreach 1 ≤ k ≤ r − s + 1, there exists an index j k such that x j k ,ℓ jk | u I k . Since u isa squarefree monomial, the elements x j ,ℓ j , . . . , x j r − s +1 ,ℓ jr − s +1 are pairwise distinct,and hence at least one of them must belong to { x ℓ , . . . , x rℓ r }\{ x t ℓ t , . . . , x t r − s ℓ tr − s } ,as we wanted to show.In order to show that Q is a minimal prime ideal of H r ( P ) h r − s +1 i we first observethe following fact: for each minimal prime ideal Q ′ of H r ( P ) h r − s +1 i , there exist s indices j < j < · · · < j s in [ r ] such that Q ′ ∩ { x j i , . . . , x j i n } 6 = ∅ for i = 1 , . . . , s. By contrary, there exist t < t < · · · < t r − s +1 with t i ∈ [ r ] such that Q ′ ∩ { x t i , . . . , x t i n } = ∅ for i = 1 , . . . , r − s + 1 . Then the monomials u = Q r − s +1 i =1 x t i P ∈ H r ( P ) h r − s +1 i , but u Q ′ , a contradiction.It follows from these considerations that each minimal prime ideal of H r ( P ) h r − s +1 i has at least s variables as generators. Since Q has precisely s variables as generators,the prime ideal Q must be a minimal prime ideal of H r ( P ) h r − s +1 i .It remains to be shown that each minimal prime of H r ( P ) h r − s +1 i is of the form(1). So let Q be an arbitrary minimal prime ideal of H r ( P ) h r − s +1 i . We know fromthe proof before that Q has to have exactly s variables as generators. Assume that Q = ( x i ℓ , x i ℓ , . . . , x i s ℓ s ) for some i < i < · · · < i s . We are going to show that p l ≤ p l ≤ · · · ≤ p l s . By contrary, assume that for some j , p ℓ j (cid:2) p ℓ j +1 . Then onsider the multichain I : I ⊆ I ⊆ · · · ⊆ I r of poset ideals of P with I = · · · = I i j − = ∅ , I i j = · · · = I i j +1 − = h p ℓ j +1 i , and I i j +1 = · · · = I r = P, where for p ∈ P we set h p i = { q ≤ p : q ∈ P } . Therefore, u I = x i j h p ℓj +1 i x i j +1 P \h p ℓj +1 i .Let { t , t , . . . , t r − s } = [ r ] \ { i , i , . . . , i s } , and let u = u I Q r − si =1 x t i P . Then u ∈ H r ( P ) h r − s +1 i , but u Q , a contradiction. (cid:3) Generalized Hibi ideals and their powers
Kokubo and Hibi in [15] introduced weakly polymatroidal ideals as a generaliza-tion of polymatroidal ideals. They show [15, Theorem 1.4] that weakly polyma-troidal ideals have linear quotients. In particular, this implies that weakly polyma-troidal ideals have a linear resolution.Let R = K [ x , . . . , x n ] be a polynomial ring over the field K . Recall that amonomial ideal I ⊂ R which is generated in one degree is called weakly polymatroidal if for any two monomials u = x a · · · x a n n and v = x b · · · x b n n in G ( I ) for which thereexists an integer t with a = b , . . . , a t − = b t − and a t > b t , there exists ℓ > t suchthat x t ( v/x ℓ ) ∈ I . Here we denote as usual the unique minimal set of monomialgenerators of I . Note that the concept weakly polymatroidal depends on the orderof the variables of R .Observe that a squarefree monomial ideal I which is generated in one degree isweakly polymatroidal if for any two monomials u = x i · · · x i d with i < i < · · · < i d ,and v = x j · · · x j d with j < j < · · · < j d in G ( I ) such that i = j , . . . , i t − = j t − and i t < j t , there exists ℓ ≥ t such that x i t ( v/x j ℓ ) ∈ I .We define a partial order on the set I r ( P ) by setting I ≤ I ′ if I i ⊆ I ′ i for i = 1 , . . . , r . Observe that the partially ordered set I r ( P ) is a distributive lattice,if we define the meet of I : I ⊆ · · · ⊆ I r and I ′ : I ′ ⊆ · · · ⊆ I ′ r as I ∩ I ′ where( I ∩ I ′ ) i = I i ∩ I ′ i for i = 1 , . . . , r , and the join as I ∪ I ′ where ( I ∪ I ′ ) i = I i ∪ I ′ i for i = 1 , . . . , r ,The following lemma was proved in [12, p.99] for r = 2 and k = 1. Lemma 2.1.
Any element in the minimal generating set of H r ( P ) k can be writtenas u I · · · u I k , where I i ∈ I r ( P ) and I ≤ · · · ≤ I k .Proof. We first claim that u I u I ′ = u I∩I ′ u I∪I ′ . (2)Indeed, the equation (2) is valid if and only if x tI t x tI t − · x tI ′ t x tI ′ t − = x tI t ∩ I ′ t x tI t − ∩ I ′ t − · x tI t ∪ I ′ t x tI t − ∪ I ′ t − for t = 1 , . . . r .In order to see that this identity holds, just observe that x tI t ∩ I ′ t = gcd { x tI t , x tI ′ t } , x tI t − ∩ I ′ t − = gcd { x tI t − , x tI ′ t − } , nd x tI t ∪ I ′ t = x tI t · x tI ′ t gcd { x tI t , x tI ′ t } x tI t − ∪ I ′ t − = x tI t − · x tI ′ t − gcd { x tI t − , x tI ′ t − } . Now let u = u J · · · u J k ∈ H r ( P ) h k i . By induction we show that u can be writtenas u J ′ · · · u J ′ k such that J ′ i ≤ J ′ k for i = 1 , . . . , k −
1. Indeed, applying the inductionhypothesis we may assume u J · · · u J k − = u J ′ · · · u J ′ k − with J ′ i ≤ J ′ k − for i =1 , . . . , k −
2. Then, by using (2) we obtain u J · · · u J k = ( u J ′ · · · u J ′ k − )( u J ′ k − u J k )= ( u J ′ · · · u J ′ k − )( u J ′ k − ∩J k u J ′ k − ∪J k ) . We have J ′ k − ∩ J k ≤ J ′ k − ∪ J k and J ′ i ≤ J ′ k − ∪ J k for i = 1 , . . . , k −
2. Hencewe may indeed assume from the very beginning that in u = u J · · · u J k , we have J i ≤ J k for i = 1 , . . . , k − u J · · · u J k = u I · · · u I k with I ≤ · · · ≤ I k .By induction assume that u J · · · u J k − = u I · · · u I k − with I ≤ · · · ≤ I k − . Since S k − i =1 I i = S k − i =1 J i ≤ J k , setting I k = J k we have I ≤ · · · ≤ I k and u J · · · u J k = u I · · · u I k . (cid:3) Theorem 2.2.
For any positive integer k , the ideal H r ( P ) k is weakly polymatroidal.Proof. Let P = { p , . . . , p n } . We may assume that if p i < p j , then i < j . We aregoing to show that H r ( P ) k is weakly polymatroidal with respect to the followingorder x > x > · · · > x n > x > · · · > x n > · · · > x r > · · · > x rn of the variables.Let u = u I · · · u I k with I k ≤ · · · ≤ I and v = u J · · · u J k with J k ≤ · · · ≤ J be two monomials in the minimal generating set of H r ( P ) k such that deg x m ′ ℓ ′ u =deg x m ′ ℓ ′ v for all x m ′ ℓ ′ > x mℓ and deg x mℓ u > deg x mℓ v .We claim: I sm ′ = J sm ′ for all m ′ < m and all s. (3)We prove the claim by induction on m ′ . Let m ′ = 1. We have to show that I s = J s for all s . Let p ℓ ∈ P . Then p ℓ ∈ I s if and only if deg x ℓ u ≥ s . Similarly p ℓ ∈ J s ifand only if deg x ℓ v ≥ s . Since deg x ℓ u = deg x ℓ v , the desired conclusion follows.Now let m ′ < m and assume that I sm ′′ = J sm ′′ for all m ′′ < m ′ and all s . Againlet p ℓ ∈ P . Then p ℓ ∈ I sm ′ \ I s,m ′ − if and only if deg x m ′ ℓ u ≥ s , and similarly p ℓ ∈ J sm ′ \ J s,m ′ − if and only if deg x m ′ ℓ v ≥ s . Since deg x m ′ ℓ u = deg x m ′ ℓ v , itfollows that I sm ′ \ I s,m ′ − = J sm ′ \ J s,m ′ − . Our induction hypothesis implies that I s,m ′ − = J s,m ′ − . Thus the desired conclusion follows.Next we claim:for all ℓ ′ < ℓ and all s, p ℓ ′ ∈ I sm ⇔ p ℓ ′ ∈ J sm . (4)As in the proof of claim (3) we see that p ℓ ′ ∈ I sm \ I s,m − if deg x mℓ ′ u ≥ s , and p ℓ ′ ∈ J sm \ J s,m − if deg x mℓ ′ v ≥ s . Hence since deg x mℓ ′ u = deg x mℓ ′ v for ℓ ′ < ℓ , we onclude that p ℓ ′ ∈ I sm \ I s,m − if and only if p ℓ ′ ∈ J sm \ J s,m − . However by claim(3) we have that I s,m − = J s,m − . Thus the result follows.Since deg x m,ℓ u > deg x m,ℓ v , there exists some t such that p ℓ ∈ I t,m \ I t,m − and p ℓ J t,m \ J t,m − . By (3), we have I t,m − = J t,m − . Therefore p ℓ ∈ J t,m ′ \ J t,m ′ − forsome m ′ > m .Now, we consider the multichain L : L ⊆ L ⊆ · · · ⊆ L r = P of subsets of P ,where L = J t, , . . . , L m − = J t,m − L m = J t,m ∪ { p ℓ } , . . . , L m ′ − = J t,m ′ − ∪ { p ℓ } ,L m ′ = J t,m ′ , . . . , L r = J t,r Observe that L is indeed a multichain of poset ideals of P . We have already L r = J t,r = P . Therefore, it is enough to show that L j is a poset ideal for each j . If j < m or j ≥ m ′ , then L j = J t,j is a poset ideal. Assume that m ≤ j < m ′ , then L j = J t,j ∪ { p ℓ } . Since I t,m is a poset ideal containing p ℓ , for any element p ℓ ′ < p ℓ we have p ℓ ′ ∈ I t,m , and so by (4) we see that p ℓ ′ ∈ J t,m . Hence the monomial u J · · · u J t − u L u J t +1 · · · u J k = x m,ℓ ( v/x m ′ ,ℓ ) is a monomial in H r ( P ) k which fulfillsthe condition of weakly polymatroidal ideals. (cid:3) We shall use Theorem 2.2 to show that the ideals H r,s ( P ) are weakly polyma-troidal. For the proof of this fact we need the following simple result. Let I bea monomial ideal generated in one degree. The squarefree part of I is the idealgenerated by all squarefree generators of I . Lemma 2.3.
The squarefree part of every weakly polymatroidal ideal is again weaklypolymatroidal.Proof.
Let I be a weakly polymatroidal ideal in K [ x , . . . , x n ]. Let u = x i x i · · · x i d and v = x j x j · · · x j d be two monomials in the minimal generating set of the square-free part of I with i = j , . . . , i t − = j t − and i t < j t . Since I is weakly polyma-troidal, there exists some ℓ ≥ t such that w = x i t ( v/x j ℓ ) is in I . Since w is again asquarefree monomial, it follows that the squarefree part of I fulfills the condition ofweakly polymatroidal ideals. (cid:3) Now we are ready to show
Theorem 2.4.
The ideal H r,s ( P ) is weakly polymatroidal. In particular, I r,s ( P ) isa Cohen–Macaulay ideal.Proof. In Theorem 1.1 we observed that H r,s ( P ) = H r ( P ) h r − s +1 i . Therefore Theo-rem 2.2 and Lemma 2.3 imply that H r,s ( P ) is weakly polymatroidal.By [15, Theorem 1.4] weakly polymatroidal ideals have linear quotients. Thus,since H r,s ( P ) is weakly polymatroidal, it follows from [8, Proposition 8.2.5] that I r,s ( P ) is Cohen–Macaulay. (cid:3) Corollary 2.5.
The ring
S/I r ( P ) is Cohen–Macaulay of dimension n ( r − , andthe following conditions are equivalent: (a) S/I r ( P ) is Gorenstein; b) S/I r ( P ) is a complete intersection; (c) all elements of P are incomparable.Proof. The degrees of the minimal generators of H r ( P ) correspond to the heights ofthe minimal prime ideals of I r ( P ). Since all generators of H r ( P ) are of degree n itfollows that height I r ( P ) = n . Thus dim S/I r ( P ) = nr − n = n ( r − s of elements x j − x ij with i = 2 , · · · , r and j = 1 , . . . , n .Then S/I r ( P ) / ( s ) S/I r ( P ) ∼ = K [ x , . . . , x n ] /J where J is generated by the monomi-als x i · · · x i r with p i ≤ p i ≤ · · · ≤ p i r . Thus J contains the elements x r , . . . , x r n .In particular, dim S/I r ( P ) / ( s ) S/I r ( P ) = 0, which implies that s is a regular se-quence, since the length of s is n ( r − S/I r ( P ) is Gorenstein ifand only if K [ x , . . . , x n ] /J is Gorenstein. Since J is a monomial ideal and sincedim K [ x , . . . , x n ] /J = 0, this is the case if and only if J is generated by pure pow-ers of the variables. This happens if and only if I r ( P ) is generated by the monomials x i · · · x ri with i = 1 , . . . , n . This yields the desired conclusions. (cid:3) Each of the ideals I r ( P ) may be considered as the edge ideal of an uniform Cohen-Macaulay admissible clutter (see [5], [16]). However, there exist uniform Cohen-Macaulay admissible clutters which do not arise from a poset. Such an exampleis given in [16, Example 3.4]. Namely, the ideal I = ( x y z , x y z , x y z ) ⊂ K [ x , y , z , x , y , z ] is the edge ideal of an uniform Cohen-Macaulay admissibleclutter. If I came from a poset P , then P would have 2 elements since height( I ) = 2 , thus I = I ( P ) . But in this case I cannot have three minimal monomial generators.3. The resolution of H r ( P )We recall that, for a monomial ideal I ⊂ R = K [ x , . . . , x n ] which has linearquotients with respect to the order u , . . . , u m of its minimal generators, we denoteby set( u i ), for i ≥
1, the set of all the variables which generate the quotient ideal( u , . . . , u i − ) : u i . By [11, Lemma 1.5], the symbols f ( σ ; u ) , σ ⊂ set( u ) , | σ | = i − ,u ∈ G ( I ) , form a homogeneous basis of the i th module in the minimal resolutionof R/I.
Therefore, the computation of the sets of I allows the computation of theBetti numbers of R/I .We now compute the sets associated with the minimal generators of H r ( P ) . Lemma 3.1.
Let
I ∈ I r ( P ) be a multichain of ideals in P, I : I ⊆ . . . ⊆ I r − ⊆ I r = P , and u I the corresponding generator of H r ( P ) . Then set( u I ) = r − [ m =1 { x mj : p j ∈ Min( P \ I m ) } , where, for a poset ideal I ⊂ P, Min( P \ I ) is the set of all the minimal elements in P \ I. In particular, we have reg
S/I r ( P ) = proj dim H r ( P ) = max r − X m =1 | Min( P \ I m ) | , where the maximum is taken over all the multichains of poset ideals I : I ⊆ . . . ⊆ I r − ⊆ I r = P . roof. Let x mj with p j ∈ Min( P \ I m ). It follows that there exists ℓ > j such that p j ∈ I ℓ . Let t = min { ℓ : p j ∈ I ℓ } , that is, p j ∈ I t \ I t − . Then I ′ : I ′ = I ⊆ . . . ⊆ I ′ m − = I m − ⊆ I ′ m = I m ∪ { p j } ⊆ · · · ⊆ I ′ t − = I t − ∪ { p j } ⊆⊆ I ′ t = I t ⊆ . . . ⊆ I ′ r = I r = P. is a multichain of poset ideals, u I ′ = x mj ( u I /x tj ) , and u I ′ > lex u I . Therefore, x mj = u I ′ / gcd( u I ′ , u I ) ∈ set( u I ) . For the other inclusion, let u L > lex u I . Thus there exist m and j such thatdeg x m ′ ′ u L = deg x m ′ ′ u I for all x m ′ ′ > x mj and deg x mj ( u L ) > deg x mj ( u I ) . Thus p j I m . As in the first part of the proof we define the monomial u I ′ and we getthat u I ′ = x mj u I /x tj > lex u I . Obviously x mj divides u L / gcd( u L , u I ), which endsthe proof.The formula for the projective dimension is an immediate consequence [11, Lemma1.5], while the equality with the regularity is implied by a result of Terai (see [8,Prop. 8.1.10]). (cid:3) In a similar way one can prove the following slightly more general
Lemma 3.2.
Let I , . . . , I k ∈ I r ( P ) be multichains of ideals in P with I l : I l ⊆ . . . ⊆ I lr − ⊆ I lr = P , and u I · · · u I k the corresponding generator of H r ( P ) k . Then set( u I · · · u I k ) = k [ l =1 r − [ m =1 { x mj : p j ∈ Min( P \ I lm ) } . Recall that an antichain of P is a subset A of P which any two of whose elementsare incomparable in P . By using this concept we get the following interpretation ofthe regularity of I r ( P ). Corollary 3.3.
We have reg
S/I r ( P ) = proj dim H r ( P ) = ( r − s, where s is the maximal cardinality of an antichain of P .Proof. Let A ⊆ P be an antichain with | A | = s , and let B be the following ideal of P : B = { q ∈ P : q < p for some p ∈ A } . Now, consider the following multichain of poset ideals I : I = B = · · · = I r − = B ⊂ I r = P. Then r − X m =1 | Min( P \ I m ) | = ( r − s, and obviously the maximum of the numbers P r − m =1 | Min( P \ I ′ m ) | taken over allmultichains I ′ of I ( P ) cannot be bigger than that for I . The desired conclusionfollows from Lemma 3.1. (cid:3) n order to determine the resolution of H r ( P ) we apply the method developed in[11]. We first recall the needed tools. For a monomial ideal I ⊂ R = K [ x , . . . , x n ]which has linear quotients with respect to the order u , . . . , u m of its minimalmonomial generators, one defines its decomposition function g : M ( I ) → G ( I )by g ( u ) = u j if j is the smallest number such that u ∈ ( u , . . . , u j − ) : u j . Here M ( I ) denotes the set of all monomials of the ideal I. The function g is regular ifset( g ( x s u )) ⊂ set( g ( u )) for all s ∈ set( u ) and u ∈ G ( I ) . In order to show that the decomposition function associated with H r ( P ) is regular,we fix a notation. For u I ∈ G ( H r ( P )) and x mj ∈ set( u I ) we denote by u I ′ thegenerator of H r ( P ) corresponding to the following multichain of poset ideals I ′ : I ′ = I ⊆ . . . ⊆ I ′ m − = I m − ⊆ I ′ m = I m ∪ { p j } ⊆ · · · ⊆ I ′ t − = I t − ∪ { p j } ⊆⊆ I ′ t = I t ⊆ . . . ⊆ I ′ r = I r = P. where, as before, t = min { ℓ : p j ∈ I ℓ } . We have seen in the proof of Lemma 3.1 that u I ′ = x mj ( u I /x tj ) and u I ′ > lex u I . Lemma 3.4.
Let g be the decomposition function of H r ( P ) , u I a minimal generatorof H r ( P ) and x mj ∈ set( u I ) . Then g ( x mj u I ) = u I ′ . Proof.
Let g ( x mj u I ) = u L for some multichain of poset ideals L : L ⊆ . . . ⊆ L r − ⊆ L r = P. This implies that x mj u I = u L x νq for some variable x νq . Let us suppose that x νq = x tj . Since x mj | u L it follows that p j ∈ L m \ L m − . On the other hand, since x tj | u I and x νq = x tj , it follows that x tj | u L , thus p j ∈ L t \ L t − . Therefore, we musthave t = m, which is impossible. (cid:3) Corollary 3.5. set( x mj u I ) ⊂ set( u I ) . Proof.
By the definition of the multichain I ′ , we have that I ′ µ = I µ for all µ < m or µ ≥ t. Let m ≤ µ < t. Then Min( P \ I ′ µ ) = Min( P \ I µ ) \ { p j } ⊂ Min( P \ I µ ). Byapplying Lemma 3.1, we get the inclusion. (cid:3) [11, Theorem 1.12] gives the minimal resolution of monomial ideals with linearquotients which admit a regular decomposition function. Applying this theorem toour situation we obtain Theorem 3.6.
Let F • be the graded minimal free resolution of S/H r ( P ) . Then thesymbols f ( σ ; u I ) , σ ⊂ set( u I ) , | σ | = i − , u I ∈ G ( H r ( P )) , form a homogeneousbasis of F i for i ≥ . The chain map of F • is given by ∂ ( f ( σ ; u I )) = X x mj ∈ σ ( − α ( σ,x mj ) ( x tj f ( σ \ { x mj } ; u I ′ ) − x mj f ( σ \ { x mj } ; u I )) if σ = ∅ , and ∂ ( f ( ∅ ; u I )) = u I otherwise. Here α ( σ, x mj ) = |{ x m ′ ℓ ′ ∈ σ : x m ′ ℓ ′ > x mj }| . . Generalized Hibi rings
In this section we introduce a class of K -algebras which can be identified with clas-sical Hibi rings. Let R r ( P ) be the toric ring generated over K by all the monomials u I with I ∈ I r ( P ), and let T be the polynomial ring over K in the indeterminates y I with I ∈ I r ( P ). Furthermore let ϕ : T → R be the surjective K -algebra ho-momorphism with ϕ ( y I ) = u I for all I ∈ I r ( P ). We choose a total order on thevariables y I with the property that I < I ′ implies that y I > y I ′ . Theorem 4.1.
The set G of elements y I y I ′ − y I∪I ′ y I∩I ′ ∈ T with I , I ′ ∈ I r ( P ) incomparable,is a reduced Gr¨obner basis of the ideal L r = Ker ϕ with respect to the reverse lexico-graphic order induced by the given order of the variables y I .Proof. The equation (2) shows that y I y I ′ − y I∪I ′ y I∩I ′ is indeed an element of L r .Now let Q ts =1 y I s − Q ts =1 y I ′ s be a primitive binomial in L r with initial monomial Q ts =1 y I s . We are going to show that there are two indices k and ℓ such that I k and I ℓ are incomparable multichains of ideal, and that y I k y I ℓ is the leading monomialof y I k y I ℓ − y I k ∪I ℓ y I k ∩I ℓ . This will then show that G is Gr¨obner basis of L r . It isobvious that G is actually reduced.Suppose to the contrary that I ≤ I ≤ . . . ≤ I t . We will show that I ′ s < I t forall s . Indeed, since Q ts =1 y I s − Q ts =1 y I ′ s ∈ L r we see that Q ts =1 u I s = Q ts =1 u I ′ s . Itfollows that t Y s =1 ( ℓ Y k =1 x kI sk \ I sk − ) = t Y s =1 ( ℓ Y k =1 x kI ′ sk \ I ′ sk − ) for all ℓ = 1 , · · · , r. Here I s is the multichain of ideals I s ⊆ I s ⊆ · · · ⊆ I sr = P , and I ′ s the multichainof ideals I ′ s ⊆ I ′ s ⊆ · · · ⊆ I ′ sr = P .Now for all j and k we apply the substitution x kj x j , and obtain t Y s =1 x I sℓ = t Y s =1 x I ′ sℓ , ℓ = 1 , . . . , r, where x J = Q j ∈ J x j for J ⊂ [ n ].Since I ≤ I ≤ . . . ≤ I t , it follows that supp( Q ts =1 x I sℓ ) = I tℓ . Thus the equation Q ts =1 x I sℓ = Q ts =1 x I ′ sℓ implies that x I ′ sℓ | x I tℓ for all ℓ and all s . It follows that I ′ s ≤ I t .We cannot have equality, since Q ts =1 y I s − Q ts =1 y I ′ s is a primitive binomial. Thiscontradicts the fact that Q ts =1 y I s is the initial monomial of Q ts =1 y I s − Q ts =1 y I ′ s .Finally, y I k y I ℓ is the leading monomial of y I k y I ℓ − y I k ∪I ℓ y I k ∩I ℓ thanks to themonomial order on T . (cid:3) Corollary 4.2.
For any poset P and all integers r ≥ , the toric ring R r ( P ) is anormal Cohen–Macaulay domain.Proof. Since the defining ideal of R r ( P ) has a squarefree initial ideal, it followsfrom a result of Sturmfels [18, Corollary 8.8] that R r ( P ) is normal, and a result ofHochster [13, Theorem 1] that R r ( P ) is Cohen–Macaulay. (cid:3) ur next goal is to find out which of the toric rings R r ( P ) is Gorenstein. Theanswer to this will be a consequence of the next theorem where it will be shownthat R r ( P ) can be interpreted as Hibi ring of a suitable poset. To be precise, let Q r = [ r ] with usual order. Recall that the direct product P × Q of two poset P and Q is poset on product of the underlying sets of P , Q with partial order given by( p, q ) ≤ ( p ′ , q ′ ) ⇔ p ≤ p ′ and q ≤ q ′ . Theorem 4.3.
Let P be any finite poset. Then R r ( P ) ∼ = R ( P × Q r − ) for all r ≥ .Proof. We first notice that the partially ordered set I r ( P ) is a distributive latticewith meet and join defined as intersection and union of multichains of ideals. InTheorem 4.1 we have seen that the defining relations of R r ( P ) are just the Hibirelations of the distributive lattice I r ( P ). In particular, it follows that R r ( P ) is theHibi ring of I r ( P ). Let P ′ be the subposet of join irreducible elements in I r ( P ). Thenwe obtain that R r ( P ) ∼ = R ( P ′ ). Thus it remains to be shown that P ′ ∼ = P × Q r − .For this purpose we have to identify the join irreducible elements in I r ( P ).Let I : ∅ ⊆ · · · ⊆ ∅ ⊂ I k ⊆ · · · ⊆ I r = P be an element of I r ( P ). We claim that I is join irreducible if and only if I k is a join irreducible element of the ideal latticeof P and I k = I k +1 = · · · = I r − . Indeed, suppose that I k = J ∪ J ′ where J and J ′ are ideals in P properly contained in I k . Then I = J ∪ J ′ where J : ∅ ⊆ · · · ⊆ ∅ ⊂ J ⊆ I k +1 ⊆ · · · ⊆ I r and J ′ : ∅ ⊆ · · · ⊆ ∅ ⊂ J ′ ⊆ I k +1 ⊆ · · · ⊆ I r , a contradiction.Suppose now that there exists an integer s with k ≤ s < r − I j = I k for j = k, . . . , s and I k ⊂ I s +1 . Then I = J ∪ J ′ where J : ∅ ⊆ · · · ⊆ ∅ ⊂ I k ⊆ I k ⊆ · · · ⊆ I k ⊆ I r and J ′ : ∅ ⊆ · · · ⊆ ∅ ⊂ I s ⊆ I s +1 ⊆ · · · ⊆ I r , a contradiction.Thus we have shown the ”only if” part of the claim. The ”if” part is obvious.Let I : ∅ ⊆ · · · ⊆ ∅ ⊂ I = I = · · · = I ⊂ P with k copies of I where I is a joinirreducible element of I ( P ) . Then I is a principal ideal in I ( P ) , hence there existsa unique element p ∈ I such that I = { a ∈ P : a ≤ p } . Finally we define the poset isomorphism between the poset of join irreducibleelements of I r ( P ) and P × Q r − as follows. To I : ∅ ⊆ · · · ⊆ ∅ ⊂ I = I = · · · = I ⊂ P with k copies of I we assign ( p, k ) ∈ P × Q r − . (cid:3) Corollary 4.4.
Let P be poset of cardinality n . Then dim R r ( P ) = n ( r −
1) + 1 .Proof.
It is known [12] and easy to see that the classical Hibi ring on a poset ofcardinality m has Krull dimension m + 1. Since R r ( P ) ∼ = R ( P × Q r − ) and since | P × Q r − | = n ( r −
1) the assertion follows. (cid:3)
A poset P is called pure , if all maximal chains have the same length. orollary 4.5. Let P be a finite poset. Then the following conditions are equivalent: (a) R r ( P ) is Gorenstein; (b) R ( P ) is Gorenstein; (c) P is pure.Proof. A well-known theorem of Hibi [12, Corollary 3.d] says that R ( P ′ ) is Goren-stein if and only if P ′ is pure. Since P is pure if and only if P × Q r − is pure, itfollows that all the statements are equivalent. (cid:3) The algebra R r,s ( P )Now we want to study the algebra R r,s ( P ) and show that it has a quadraticGr¨obner basis with squarefree initial ideal. Let G r,s be the minimal set of monomialgenerators of H r,s ( P ). Then the elements of G r,s generate R r,s ( P ). Let k = r − s + 1,then G r,s consist of all squarefree monomials of the form u I u I · · · u I k with I > I > · · · > I k and I j ∈ I r ( P ). Corresponding to each such monomial we intro-duce the variable y I , I ,..., I k , and let T be the polynomial ring over K in this set ofvariables. Let L r,s = Ker ϕ where ϕ : T → R r,s ( P ) is the K -algebra homomorphismwith y I , I ,..., I k u I u I · · · u I k .Defining in a similar way the order of the variables y I , I ,..., I k as we did it in thecase of R r ( P ), we choose a total order on the variables with the property that y I , I ,..., I k > y J , J ,..., J k if ( I , I , . . . , I k ) < ( J , I , . . . , J k ) , where by definition ( I , I , . . . , I k ) ≤ ( J , J , . . . , J k ), if I l ≤ J l for l = 1 , . . . , k .In analogy to Theorem 4.1 one would expect that L r,s has a quadratic Gr¨obnerbasis with respect to the reverse lexicographic order induced by the above orderof the variables. This is however not the case. To see this we choose for P theposet consisting of only one element p . Then H r,s ( P ) = I r,s where I r,s denotes thesquarefree Veronese ideal of degree s in r variables, that is, the ideal generated byall squarefree monomials of degree s in T = K [ x , . . . , x r ]. In this particular casethe above defined order of the variables is the y u > y v if u > lex v . Theorem 5.1.
Let G be the reduced Gr¨obner basis of L r,s = Ker ϕ with respect to thereverse lexicographical order on T induced by the above order of the variables, where ϕ : T → R r,s ( P ) is the K algebra homomorphism with y u u . Then the binomialsof G have squarefree initial monomials and are generated in degree at most . Proof.
Let g = y u · · · y u q − y v · · · y v q ∈ G with u ≥ lex . . . ≥ lex u q and v ≥ lex . . . ≥ lex v q . Let in < ( g ) = y u · · · y u q . Then, since g is a primitive binomial, we have y u q > y v q , that is, u q > lex v q . Obviously, there are at least two different variables in the support of in < ( g ) , thatis, q ≥ u > lex u q . In the first place we assume that there exist 1 ≤ a < b ≤ q with u a = u b and i ∈ supp( u a ) \ supp( u b ) , j ∈ supp( u b ) \ supp( u a ) , such that i > j. Let u ′ a = x j u a /x i and u ′ b = x i u b /x j . Then we have u a u b = u ′ a u ′ b , that is h = y u a y u b − y u ′ a y u ′ b ∈ L r,s ( P ),and u b > lex u ′ b , whence in < ( h ) = y u a y u b . Since G is a reduced Gr¨obner basis, we musthave in < ( g ) = in < ( h ) , thus in < ( g ) is a squarefree monomial of degree 2. ow we assume that for all 1 ≤ a < b ≤ q with u a = u b we have(5) max(supp( u a ) \ supp( u b )) < min(supp( u b ) \ supp( u a )) . Since u q > lex v q we also have(6) ℓ = min(supp( u q ) \ supp( v q )) < h = min(supp( v q ) \ supp( u q )) . In particular, we get x ℓ | u q , x ℓ v q , x h | v q , and x h u q . We then obtain that there exists 1 ≤ a, b ≤ q − x ℓ u a and x h | u b . Notethat u a = u b . Indeed, if u a = u b we get h ∈ supp( u a ) \ supp( u q ) and ℓ ∈ supp( u q ) \ supp( u a ) . Since a < q, by using (5), we obtain h < ℓ which is in contradiction to (6).Therefore, the monomials u a , u b , and u q are distinct. Let j ∈ supp( u a ) \ supp( u b )and consider the monomials u ′ a = x ℓ u a /x j , u ′ b = x j u b /x h , and u ′ q = x h u q /x ℓ . It follows that u a u b u q = u ′ a u ′ b u ′ q , thus h = y u a y u b y u q − y u ′ a y u ′ b y u ′ q ∈ G , and u q > lex u ′ q ,hence in < ( h ) = y u a y u b y u q . Since G is reduced, it follows that in < ( g ) is a squarefreemonomial of degree at most 3 . (cid:3) The following example shows that the reduced Gr¨obner basis G of L r,s does ingeneral indeed contain monomials of degree 3. With CoCoA we compute G for thering R , ( P ). The binomials of degree 3 in G are the following: kps − lmt, ejs − f gt, bjp − cdt, drs − gmt, cqs − f lt, ajp − cds, bqr − ekt, aqr − eks, ano − bkp, aio − bdr, ahi − bej, ahn − bcq .Here, we denoted for simplicity the variables of T by a, b, c, · · · , t and defined T → R , ( P ) by mapping the variables in their natural (lexicographical order) to thecorresponding monomial of H , ( P ) in the lexicographic order. As can been seen,there are 12 binomials of degree 3 in the reduced Gr¨obner basis of L , .If R , ( P ) would be isomorphic to Hibi ring, then it would have to have a quadraticGr¨obner basis with respect to the reverse lexicographic order induced by some orderof the variables. This is not the case, at least for the natural order of the variables u , as we have seen above, and very likely for any other order of the variables.Unfortunately this is not so easy to check because there exist quite a lot of differentorders, even if one takes into account all the symmetries.The following simple argument shows that the even smaller ring R , ( P ) with P = { p } can not be a Hibi ring. It is generated over K by the monomials x x , x x , x x , x x , x x , x x . Suppose R , ( P ) is the Hibi ring of a poset P . Then its ideal lattice I ( P ) shouldhave cardinality 6. The only posets with this property are P = { p , p , p , p , p } , p < p < p < p < p ; P = { p , p , p , p } , p < p and p < p , p ; P = { p , p , p , p } , p , p < p and p < p ; P = { p , p , p } , p < p . or the posets P , P , P the toric ideal of the associated Hibi ring is generated by atmost one binomial, and for P by three binomials, while the defining ideal of R , ( P )has two binomial generators.In order to obtain a squarefree Gr¨obner basis of the defining ideal of R r,s ( P ) weuse a result of Sturmfels [18, Theorem 14.2] and show that the set G r,s is sortable.Recall that a set B of monomials which are of same degree d in the polynomialring S = K [ x , . . . , x n ] is called sortable , if image of the mapsort : B × B → S d × S d is contained in B × B . The map sort is defined as follows: let u, v ∈ B where uv = x i x i · · · x i d with i ≤ i ≤ · · · ≤ i d . Then sort( u, v ) = ( u ′ , v ′ ) where u ′ = x i x i · · · x i d − and v ′ = x i x i · · · x i d . We call the pair of monomial unsorted ,if ( u, v ) = ( u ′ , v ′ ). Theorem 5.2 (Sturmfels) . Let R be the toric ring generated over K by a sortableset B of monomials. Then R = K [ y u : u ∈ B ] /I , and the binomials y u y v − y u ′ y v ′ where ( u, v ) is unsorted and ( u ′ , v ′ ) = sort( u, v ) ,form a Gr¨obner basis of I . Theorem 5.2 will be used to prove
Theorem 5.3.
For all integers ≤ s ≤ r the set of monomials G r,s ( P ) is sortable.In particular, R r,s ( P ) has a quadratic Gr¨obner basis and is a normal Cohen–Macaulaydomain. Before giving the proof of the theorem we need the following two lemmata.
Lemma 5.4.
Let u I = u I · · · u I k be an element in G r,s with I > · · · > I k where I j : I j ⊂ I j ⊆ . . . ⊆ I jr for j = 1 , . . . , k, and let Q c ∈ C x ct | u I for some C ⊆ [ r ] and some t ∈ [ n ] . Then for any element p m < p t of P , there exists a set C ′ and a bijection C → C ′ with c c ′ ≤ c such that Y c ′ ∈ C ′ x c ′ m divides u I . Proof.
First observe that x ct | u I if and only if p t belongs to I jc \ I j,c − for some j .Note that j is uniquely determined by c and t , since u I is a squarefree monomial.Then for pairwise disjoint indices j , . . . , j | C | we have p t ∈ I j i ,c i \ I j i ,c i − for all i . Since I j i is a poset ideal, there exists some c ′ i ≤ c i with p m ∈ I j i ,c ′ i \ I j i ,c ′ i − .Then our first observation shows that x c ′ i ,m | u I . Since u I is a squarefree monomial, c ′ , c ′ , . . . , c ′| C | are again pairwise disjoint indices and so Q c ′ ∈ C ′ x c ′ m divides u I , where C ′ = { c ′ , c ′ , . . . , c ′| C | } . (cid:3) Lemma 5.5.
Let u = x ,A x ,A · · · x r,A r ∈ S kn be a squarefree monomial satisfyingthe following condition ( ∗ ) : for each j ∈ [ n ] there exist exactly k of the sets A i , say A i , . . . , A i k , such that j ∈ A i l for l = 1 , . . . , k .Then u = u u · · · u k where u i = x ,A i x ,A i · · · x r,A ir such that for each i = 1 , . . . , k and for each j ∈ [ n ] there exists a unique l ∈ [ r ] suchthat j ∈ A il ; (2) A i +1 ,j ⊆ S j − l =1 A il for all i and j .Proof. Let A i = A i \ S i − l =1 A l . Since [ n ] is the disjoint union of the sets A , . . . , A r ,condition (1) is satisfied for i = 1. Let v = u/u = x ,B · · · x r,B r . Then v is asquarefree monomial of degree ( k − n , and since condition (1) is satisfied for i = 1it follows from ( ∗ ) that for each j ∈ [ n ] there exist exactly k − B i ,say B i , . . . , B i k − , such that j ∈ B i l for l = 1 , . . . , k −
1. By using induction on k we may assume that v = u u · · · u k with u i = x ,A i x ,A i · · · x r,A ir such that theconditions (1) and (2) are satisfied for i ≥
2. Thus it remains to be shown that A j ⊆ S j − l =1 A l for all j . We actually show that B j ⊂ S j − l =1 A l for all j . Indeed, B j = A j \ A ,j − = A j ∩ j − [ l =1 A l ⊆ j − [ l =1 A l = j − [ l =1 A l . (cid:3) First observe that with the similar notation as in the above lemma one has t ∈ A ls for some l ⇔ x st divides u. (7) Proof of Theorem 5.3.
By Theorem 1.1 we have H r,s ( P ) = H r ( P ) h k i where k = r − s + 1. Consider the following order x > x > · · · > x r > x > · · · > x r > · · · > x n > · · · > x rn of the variables of S .Let u I = u I · · · u I k and u J = u J · · · u J k be two elements in G r,s with I > · · · > I k and J > · · · > J k , where I j : I j ⊆ I j ⊆ . . . ⊆ I jr and J j : J j ⊆ J j ⊆ . . . ⊆ J jr for j = 1 , . . . , k. Let sort( u I , u J ) = ( u, v ) . We first notice that both monomials u and v are again squarefree. Indeed, since u I u J = uv it follows that each variable x ij appears at most to the power 2 in uv . Ifthis happens, then the sorting operator moves one x ij to u and the other x ij to v .Now we may decompose u and v as in Lemma 5.5. Say, u = u u · · · u k with u i = x ,A i x ,A i · · · x r,A ir , and v = v v · · · v k with v i = x ,B i x ,B i · · · x r,B ir . The proof of the theorem is completed once we have shown that for all i and j thesets j [ l =1 A il and j [ l =1 B il are poset ideals in P . et t ∈ A ij and suppose that p m < p t . We want to show that m ∈ S jl =1 A il . Thisthen proves that S jl =1 A il is poset ideal.Since t ∈ A ij and since A ij ⊂ S j − l =1 A i − ,l , it follows that there exists s i − < s i = j such that t ∈ A i − ,s i − Proceeding in this way we find a sequence s < s < · · · < s i such that t ∈ T il =1 A l,s l . By the definition of the sorting operator, there must exista sequence s ′ < s ′ < · · · < s ′ i − with s l ≤ s ′ l ≤ s l +1 for l = 1 , . . . , i − t ∈ T i − l =1 B l,s ′ l .Since t ∈ A ls for some l , if and only if x st | u , and similarly t ∈ B ls for some l , ifand only if x st | v (see (7)), it follows that i Y l =1 x s l t i − Y l =1 x s ′ l t divides uv. Since uv = u I u J there exists C and D such that Y c ∈ C x ct | u I , Y d ∈ D x dt | u J and Y c ∈ C x ct Y d ∈ D x dt = i Y l =1 x s l t i − Y l =1 x s ′ l t with c, d ≤ j for all c ∈ C and d ∈ D .Applying Lemma 5.4 we conclude that there exist sets C ′ and D ′ , and bijections C → C ′ with c c ′ ≤ c and D → D ′ with d d ′ ≤ d such that Y c ′ ∈ C ′ x c ′ m | u I , Y d ′ ∈ D ′ x d ′ m | u J . It is clear that c ′ , d ′ ≤ j for c ′ ∈ C ′ and d ′ ∈ D ′ .It follows from the definition of the sorting operator that i of the factors of Q c ′ ∈ C ′ x c ′ m Q d ′ ∈ D ′ x d ′ m appear in u and i − v . Therefore, by (7) andstatement (1) of Lemma 5.5 there exist pairwise different integers l , . . . , l i and in-tegers c , . . . , c i ≤ j such that m ∈ A l a ,c a for a = 1 , . . . , i . Therefore there is at leastone a such that l a ≥ i . By using part (2) of Lemma 5.5 we see that m ∈ A ic forsome c < c a ≤ j , as desired.In the same way one shows that S jl =1 B il is a poset ideal for all i and j . (cid:3) The Rees algebra of H r,s ( P )In this section we study the Rees algebra of H r,s ( P ). Before stating our mainresult we recall some results of [9], since we shall use them to show that the toricideal of each power of H r,s ( P ) has again a quadratic Gr¨obner basis. First we recallsome definitions and results on Rees algebra.Let I = ( u , . . . , u m ) be a monomial ideal in K [ x , . . . , x n ] which is generated inone degree. Let R = K [ y , . . . , y m ] and L be the toric ideal of K [ u , . . . , u m ] whichis the kernel of the surjective homomorphism ϕ : R → K [ u , . . . , u m ]defined by ϕ ( y i ) = u i for all i .Let T be the polynomial ring over K [ x , . . . , x n ] in the variables y , . . . , y m . Wemay regard T a bigraded K -algebra by setting deg( x i ) = (1 ,
0) for i = 1 , . . . , n nd deg( y j ) = (0 ,
1) for j = 1 , . . . , m . For any two vectors a = ( a , . . . , a n ) and b = ( b , . . . , b m ) with all 0 ≤ a i , b j in Z we write x a for the monomial x a · · · x a n n and y b for the monomial y b · · · y b m m .Let ≺ be a monomial order on R . A monomial y a in R is called a standardmonomial of L with respect to ≺ , if it does not belong to the initial ideal of L . Werecall the ℓ - exchange property which was introduced in [9]:The ideal I satisfies the ℓ -exchange property with respect to the monomial order ≺ on R , if for any two standard monomials y a and y b in L of same degree satisfying(i) deg x t ϕ ( y a ) = deg x t ϕ ( y b ) for t = 1 , . . . , q − q ≤ n − x q ϕ ( y a ) < deg x q ϕ ( y b ),there exists a factor u δ of ϕ ( y a ) and q < j ≤ n such that x q u δ /x j ∈ I .The following result is a slight generalization of [9, Theorem 4.3]. Proposition 6.1.
Let I ⊂ K [ x , . . . , x n ] be a weakly polymatroidal ideal which issortable. Then I satisfies the ℓ -exchange property with respect to the sorting order.Proof. Let y a and y b be two standard monomials in L satisfying (i) and (ii). Sup-pose that ϕ ( y a ) = u i · · · u i d and ϕ ( y b ) = u j · · · u j d , and that all pairs ( u i l , u i l ′ )and ( u j l , u j l ′ ) are sorted. It follows from (i) that deg x t ( u i l ) = deg x t ( u j l ) for l =1 , . . . , d and for t = 1 , . . . , q −
1, and (ii) implies that there exists 1 ≤ l ≤ d with deg x q ( u i l ) < deg x q ( u j l ). Since deg x t ( u i l ) = deg x t ( u j l ) for t = 1 , . . . , q − x q ( u i l ) < deg x q ( u j l ), and since I weakly polymatroidal there exists j > q with x q u i l /x j ∈ I , as desired. (cid:3) Let t be a variable over K [ x , . . . , x n ]. Then the Rees ring R ( I ) = ∞ M j =0 I j t j ⊂ K [ x , . . . , x n , t ]is a bigraded algebra with deg( x i ) = (1 ,
0) for i = 1 , . . . , n and deg( u j t ) = (0 ,
1) for j = 1 , . . . , m . We recall that the toric ideal of R ( I ) is the ideal P R ( I ) ⊂ T which isthe kernel of the surjective homomorphism ϕ : T → R ( I ) with x i x i for all i and y j u j t for all j .Let < ♯ be an arbitrary monomial order on R and < lex the lexicographic order on K [ x , . . . , x n ] with respect to x > · · · > x n . The new monomial order < ♯ lex is definedon T as follows: For two monomials x a y b and x a ′ y b ′ in T , we have x a y b < ♯ lex x a ′ y b ′ if and only if (i) x a < lex x a ′ or (ii) x a = x a ′ and y b < ♯ y b ′ .Let H r,s ( P ) ⊂ S be generated by the monomials in G r,s ( P ) = { u , · · · , u m } . Wehave shown in Theorem 2.4 that H r,s ( P ) is weakly polymatroidal for the followingorder of the monomials x > x > · · · > x r > x > · · · > x r > · · · > x n > · · · > x rn . With respect to the same order of the variables the set of monomials is sortable,as shown in Theorem 5.3. Thus if we let < ♯ be the monomial order given by theproperty that G r,s ( P ) is sortable, we may apply [9, Theorem 5.1] to obtain heorem 6.2. The reduced Gr¨obner basis of the toric ideal P R ( H r,s ( P )) with respectto < ♯ consists of all binomials belonging to G < ♯ ( L r,s ) together with the binomials x ir y k − x js y l , where x ir > x js with x ir u k = x js u l and x js is the largest monomial for which x ir u k /x js belongs to H r,s ( P ) . Corollary 6.3.
The Rees ring R ( H r,s ( P )) is a normal Cohen–Macaulay domain.In particular all powers of H r,s ( P ) are normal.Proof. We see from the description of G < ♯ lex ( P R ( H r,s ( P )) ) that the initial ideal of P R ( H r,s ( P )) is squarefree. Therefore the result follows from [18, Corollary 8.8] to-gether with [13, Theorem 1]. (cid:3) Corollary 6.4.
The Rees ring R ( H r,s ( P )) is Koszul.Proof. Since the initial ideal of P R ( H r,s ( P )) is generated in degree 2, the assertionfollows from a theorem of Backelin and Fr¨oberg [1, Theorem 4(b)]. (cid:3) Corollary 6.5.
All powers of H r,s ( P ) have a linear resolution.Proof. Since for all monomials u in the minimal set of generators of the initial idealof P R ( H r,s ( P )) we have deg x ij u ≤ i, j , the so-called x -condition for H r,s ( P )is satisfied. Thus [10, Corollary 1.2] yields the desired conclusion. (cid:3) As an extension [7, Corollary 3.8] we have
Corollary 6.6.
Let P be a poset of cardinality n . Then lim k →∞ depth( S/I r ( P ) k ) = n − . Proof.
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Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia124, 900527 Constanta, Romania
E-mail address : [email protected] J¨urgen Herzog, Fachbereich Mathematik, Universit¨at Duisburg-Essen, CampusEssen, 45117 Essen, Germany
E-mail address : [email protected] Fatemeh Mohammadi, Department of Pure Mathematics, Ferdowsi University ofMashhad P.O. Box 1159-91775, Mashhad, Iran
E-mail address : [email protected]@gmail.com