The relevance of Freiman's theorem for combinatorial commutative algebra
aa r X i v : . [ m a t h . A C ] J a n THE RELEVANCE OF FREIMAN’S THEOREM FORCOMBINATORIAL COMMUTATIVE ALGEBRA
J ¨URGEN HERZOG, TAKAYUKI HIBI AND GUANGJUN ZHU ∗ Abstract.
Freiman’s theorem gives a lower bound for the cardinality of thedoubling of a finite set in R n . In this paper we give an interpretation of his theoremfor monomial ideals and their fiber cones. We call a quasi-equigenerated monomialideal a Freiman ideal, if the set of its exponent vectors achieves Freiman’s lowerbound for its doubling. Algebraic characterizations of Freiman ideals are given,and finite simple graphs are classified whose edge ideals or matroidal ideals of itscycle matroids are Freiman ideals. Introduction
Let X be a finite subset of Z n , and let A ( X ) be the affine hull of the set X , thatis, the smallest affine subspace of Q n containing X . The doubling of X is the set2 X = { a + b : a, b ∈ X } . The starting point of this paper is the following celebratedtheorem of Freiman [7]: | X | ≥ ( d + 1) | X | − (cid:18) d + 12 (cid:19) , (1)where d is the dimension of the affine space A ( X ).Now let K be a field and let I be a graded ideal in the polynomial ring S = K [ x , . . . , x n ]. We denote by µ ( I ) the minimal number of generators of I and by ℓ ( I ) the analytic spread of I , that is, the Krull dimension of the fiber cone F ( I ) = ⊕ k ≥ I k / m I k . Here m = ( x , . . . , x n ) is the graded maximal ideal of S . It has beennoticed in [10, Theorem 1.9] that Freiman’s theorem has an interesting consequenceregarding the minimal number of generators of the square of a monomial ideal.Namely, it was shown that if I ⊂ S is a monomial ideal with the property that allgenerators of I have the same degree, then µ ( I ) ≥ ℓ ( I ) µ ( I ) − (cid:0) ℓ ( I )2 (cid:1) .For the application of Freiman’s theorem to a monomial ideal I ⊂ S and its fibercone F ( I ), it is sufficient to require that I is quasi-equigenerated, by which we meanthat the exponent vectors of all generators of I lie in a hyperplane of Z n . Thisguarantees that the set of exponent vectors of the monomials u ∈ G ( I ) is obtainedfrom the set of exponent vectors of the monomials v ∈ G ( I ) by doubling. Here, G ( I ) denotes the unique minimal set of monomial generators of a monomial ideal I . Mathematics Subject Classification.
Primary 13C99; Secondary 13A15, 13E15, 13H05,13H10.
Key words and phrases.
Monomial ideal, Freiman ideal, Freiman graph, Freiman matroid, fibercone.* Corresponding author. e call a quasi-equigenerated monomial ideal I ⊂ S a Freiman ideal, if µ ( I ) = ℓ ( I ) µ ( I ) − (cid:0) ℓ ( I )2 (cid:1) . Of course, I is Freiman if and only if equality holds in (1) for theset X of exponent vectors of the monomials u ∈ G ( I ). The sets X ⊂ Z n for which | X | = ( d +1) | X |− (cid:0) d +12 (cid:1) are characterized by Stanescu in [19]. Here, in Theorem 1.3,we give several equivalent conditions for a quasi-equigenerated monomial ideal I ⊂ S to be a Freiman ideal. For example, it is shown that I is Freiman if and only if thefiber cone F ( I ) of I is Cohen–Macaulay and the defining ideal of F ( I ) has a 2-linearresolution. This homological characterization of Freiman ideals can be deduced fromthe fact that for a Freiman ideal not only one has µ ( I ) = ℓ ( I ) µ ( I ) − (cid:0) ℓ ( I )2 (cid:1) , but also µ ( I k ) = (cid:0) ℓ + k − k − (cid:1) µ ( I ) − ( k − (cid:0) ℓ + k − k (cid:1) for all k ≥
1. Indeed, this fact is a consequenceof a result of B¨or¨oczky, Santos and Serra [4, Corollary 7]. Now this formula for theminimal number of monomial generators of I k of a Freiman ideal I easily yields that h i = 0 for all i ≥
2, where (1 , h , h , . . . ) is the h -vector of the fiber cone F ( I ) of I . Then we apply a result of Eisenbud and Goto [5, Corollary 4.5] and obtain that F ( I ) is normal. Since F ( I ) is a toric ring, Hochster’s theorem [14] finally yields that F ( I ) is Cohen–Macaulay.Even more surprising than the inequality µ ( I ) ≥ ℓ ( I ) µ ( I ) − (cid:0) ℓ ( I )2 (cid:1) , and seeminglynot noticed before in the study of toric rings, are the following consequences of theabove mentioned results about the h -vector of the fiber cone F ( I ) and of Freiman’stheorem, which may phrased as follows: let A be a standard graded toric K -algebrawith h -vector (1 , h , h , . . . ). Then h ≥
0, and if h = 0, then h i = 0 for all i ≥ h ≥ G . We call G a Freiman graph, if its edgeideal I ( G ) is a Freiman ideal. This classification can be reduced to the case that G isconnected, see Corollary 2.2. In Theorem 2.3 it is shown that if G is connected and H is the subgraph of G whose edges are the edges of all the 4-cycles of G , then G isa Freiman graph if and only if there exist no primitive even walks in G , or otherwise H is bipartite of type (2 , s ) for some integer s , and there exist no primitive evenwalks in G of length >
4. The proof is inspired by the classification of edge idealswith 2-linear resolution, due to Ohsugi and Hibi [16]. In the special case that G is bipartite this classification is even more explicit. Indeed, in Corollary 2.4 it isshown, that if G is a connected bipartite graph, then G is Freiman if and only if G is a tree, or otherwise there exists a bipartite subgraph H of G of type (2 , s ) forsome integer s , a subset S of V ( H ) and for each w ∈ S an induced tree T w of G with V ( H ) ∩ V ( T w ) = { w } .The other case studied here is that of the matroidal ideal I M of the cycle matroid M of G . We call a matroid M whose matroidal ideal I M is a Freiman ideal, aFreiman matroid. In Theorem 2.5 we show that the cycle matroid of G is Freimanif and only if G contains at most one cycle. It is a challenging open problem toclassify all Freiman matroids. A few results in this direction, even for polymatroids,can be found in [11]. s an interesting consequence of the classification of the cycle matroids which areFreiman, it is shown in Theorem 2.13 that if r denotes the regularity of the basering of the cycle matroid of a graph G , then, unless the base ring is a polynomialring, one has 3 ≤ r ≤ e , where e is the number of the edges of G .1. Freiman ideals and their fiber cones
Let K be a field and S = K [ x , . . . , x n ] the polynomial ring in n indeterminatesover K , and let I ⊂ S be a monomial ideal. The unique minimal set of monomialgenerators of I will be denoted by G ( I ). The number µ ( I ) = | G ( I ) | is the minimalnumber of generators of I . We write x c for the monomial x c · · · x c n n where c =( c , . . . , c n ).Let a = ( a , . . . , a n ) ∈ Z n be an integer vector with all a i >
0. We say that I is equigenerated (of degree d ) with respect to a or simply say that I is quasi-equigenerated , if there exists an integer d such that d = h a , c i for all x c ∈ G ( I ) . Here h a , c i = P ni =1 a i c i denotes the standard scalar product of a and c . The ideal I is called equigenerated if a = (1 , , . . . , F ( I ) = L k ≥ I k / m I k is called the analyticspread of I , and denoted ℓ ( I ). One always hasheight I ≤ ℓ ( I ) ≤ min { µ ( I ) , n } . Let G ( I ) = { u , . . . , u m } , and let T = K [ y , . . . , y m ] be the polynomial ring over K in the indeterminates y , . . . , y m . Consider the K -algebra homomorphism ϕ : T → F ( I ) with y j u j + m I for j = 1 , . . . , m . Then F ( I ) ∼ = T /J , where J = Ker ϕ .Since I is quasi-equigenerated, it follows that F ( I ) ∼ = K [ u t, . . . u m t ] ⊂ T [ t ], where t is a new indeterminate over T . Thus we see that J is a toric prime ideal generatedby binomials in T which are homogeneous with respect to the standard grading of T .The following theorem is a consequence of a famous theorem of Freiman [7] andits generalizations by B¨or¨oczky et al [4]. Theorem 1.1.
Let I ⊂ S be a quasi-equigenerated monomial ideal whose analyticspread is ℓ ( I ) . Then µ ( I k ) ≥ (cid:18) ℓ ( I ) + k − k − (cid:19) µ ( I ) − ( k − (cid:18) ℓ ( I ) + k − k (cid:19) for all k ≥ .Proof. Let S ( I ) ⊂ Z n be the set of exponent vectors of the elements of G ( I ). Since I is a quasi-equigenerated monomial ideal, I k are quasi-equigenerated monomial idealsfor all k ≥
1. It follows that µ ( I k ) = | S ( I k ) | = | kS ( I ) | for all k ≥
1. Let d be the reiman dimension of S ( I ), then from [4, Corollary 2], | kS ( I ) | ≥ (cid:18) d + k − k − (cid:19) | S ( I ) | − ( k − (cid:18) d + k − k (cid:19) = (cid:18) d + k − k − (cid:19) µ ( I ) − ( k − (cid:18) d + k − k (cid:19) . On the other hand, by [10, Theorem 1.9] one has ℓ ( I ) = d + 1. Thus the assertionfollows. (cid:3) The above theorem implies in particular that µ ( I ) ≥ l ( I ) µ ( I ) − (cid:0) l ( I )2 (cid:1) . In [11], thefirst and third author of this paper called an equigenerated monomial ideal a Freimanideal, if equality holds. Here we extend this definition and call a quasi-equigeneratedideal I a Freiman ideal , if µ ( I ) = l ( I ) µ ( I ) − (cid:18) l ( I )2 (cid:19) . In the next theorem we will give some characterizations of Freiman ideals in termsof the Hilbert series of their fiber cones. To this end we recall some concepts fromcommutative algebra.Let R be a standard graded K -algebra. We denote its Krull dimension bydim R and its embedding dimension by emb dim R . The Hilbert series Hilb R ( t ) = P k ≥ dim K R k t k of R is of the formHilb R ( t ) = h ( R ; t ) / (1 − t ) dim R , where h ( R ; t ) = 1 + h t + h t + · · · is a polynomial with h = emb dim R − dim R .The polynomial h ( R ; t ) is called the h -polynomial of R , and the finite coefficientvector of h ( R ; t ) is called the h -vector of R .The multiplicity of R is defined to be e ( R ) = P i ≥ h i . By Abhyankar [1] it isknown that emb dim R ≤ e ( R ) + dim R − , if R is a domain. The same inequality holds if R is Cohen–Macaulay [17]. The K -algebra R is said to have minimal multiplicity if emb dim( R ) = e ( R ) + dim R − . We first observe
Corollary 1.2.
Let I ⊂ S be a quasi-equigenerated monomial ideal with analyticspread ℓ , and let (1 , h , h , . . . ) be the h -vector of the fiber cone F ( I ) of I . Then k X i =2 (cid:18) ℓ + k − i − k − i (cid:19) h i ≥ for all k ≥ . In particular, h ≥ .Proof. The Hilbert series of the fiber cone F ( I ) of I isHilb F ( I ) ( t ) = X k ≥ µ ( I k ) t k = 1 + h t + h t + h t + h t + · · · (1 − t ) ℓ (1 + h t + h t + · · · )(1 + ℓt + (cid:18) ℓ + 12 (cid:19) t + (cid:18) ℓ + 23 (cid:19) t + · · · )= 1 + ( h + ℓ ) t + ( h ℓ + (cid:18) ℓ + 12 (cid:19) + h ) t + · · · + [ (cid:18) ℓ + k − k (cid:19) + k X i =1 (cid:18) ℓ + k − i − k − i (cid:19) h i ] t k + · · · . It follows that µ ( I ) = h + ℓ, and µ ( I ) = h ℓ + (cid:18) ℓ + 12 (cid:19) + h = ℓµ ( I ) − (cid:18) ℓ (cid:19) + h . Moreover, for all k ≥ µ ( I k ) = (cid:18) ℓ + k − k (cid:19) + (cid:18) ℓ + k − k − (cid:19) h + (cid:18) ℓ + k − k − (cid:19) h + · · · + (cid:18) ℓ (cid:19) h k − + h k = (cid:18) ℓ + k − k (cid:19) + (cid:18) ℓ + k − k − (cid:19) ( µ ( I ) − ℓ ) + k X i =2 (cid:18) ℓ + k − i − k − i (cid:19) h i = (cid:18) ℓ + k − k − (cid:19) µ ( I ) − ( k − (cid:18) ℓ + k − k (cid:19) + k X i =2 (cid:18) ℓ + k − i − k − i (cid:19) h i . Therefore, Theorem 1.1 implies that h ≥ k P i =2 (cid:0) ℓ + k − i − k − i (cid:1) h i ≥ k ≥ (cid:3) Let I be a graded ideal. An ideal J ⊆ I is called a reduction of I if I k +1 = J I k forsome nonnegative integer k . The reduction number of I with respect to J is definedto be r J ( I ) = min { k | I k +1 = J I k } . A reduction J of I is called a minimal reduction if it does not properly contain anyother reduction of I . If I is equigenerated and | K | = ∞ , then a graded minimalreduction of I exists. In this case the reduction number of I is defined to be thenumber r ( I ) = min { r J ( I ) | J is a minimal reduction of I } . Now we are ready to present the main result of this section.
Theorem 1.3.
Let I ⊂ S be a quasi-equigenerated monomial ideal with analyticspread ℓ , and let (1 , h , h , . . . ) be the h -vector of the fiber cone F ( I ) of I . Write F ( I ) = T /J , where T is a polynomial ring over K , and J is contained in the squareof the graded maximal ideal of T . Then the following conditions are equivalent: (a) I is a Freiman ideal. (b) µ ( I k ) = (cid:0) ℓ + k − k − (cid:1) µ ( I ) − ( k − (cid:0) ℓ + k − k (cid:1) for all k ≥ . (c) µ ( I k ) = (cid:0) ℓ + k − k − (cid:1) µ ( I ) − ( k − (cid:0) ℓ + k − k (cid:1) for some k ≥ . (d) h = 0 . e) h i = 0 for all i ≥ . (f) F ( I ) has minimal multiplicity. (g) F ( I ) is Cohen–Macaulay and the defining ideal of F ( I ) has a -linear free T -resolution.Moreover, if | K | = ∞ , then the above conditions are equivalent to (h) F ( I ) is Cohen–Macaulay and r ( I ) = 1 .Proof. (a) ⇒ (b) follows from [4, Corollary 7], (b) ⇒ (c) is trivial, and (c) ⇒ (b)also follows from [4, Corollary 7].In the proof of Corollary 1.2, we have seen that µ ( I ) = h + ℓ, µ ( I ) = ℓµ ( I ) − (cid:0) ℓ (cid:1) + h and µ ( I k ) = (cid:18) ℓ + k − k − (cid:19) µ ( I ) − ( k − (cid:18) ℓ + k − k (cid:19) + k X i =2 (cid:18) ℓ + k − i − k − i (cid:19) h i . By definition, I is a Freiman ideal, if and only if µ ( I ) = ℓµ ( I ) − (cid:0) ℓ (cid:1) . This isequivalent to saying that h = 0. This proves (a) ⇐⇒ (d).(b) ⇐⇒ (e) follows from [11, Proposition 1.8 (b)].(e) ⇒ (f): Notice that e ( R ) = P i ≥ h i . If h i = 0 for i ≥
2, then e ( R ) = h + h = 1 + h = 1 + µ ( I ) − ℓ. This means that F ( I ) has minimal multiplicity.(f) ⇒ (g): After a base field extension, we may assume that K is algebraicallyclosed. Since F ( I ) has minimal multiplicity, it follows that F ( I ) is normal by [5,Corollary 4.5]. Since F ( I ) is a toric ring, Hochster’s theorem [14] implies that F ( I )is Cohen–Macaulay and h i = 0 for all i ≥
2. Thus from [11, Corollary 1.5] we obtainthat the defining ideal of F ( I ) has a 2-linear free T -resolution.(g) ⇒ (d): We may assume that K is infinite. Since F ( I ) is Cohen–Macaulay wecan choose a regular sequence t , . . . , t ℓ of linear forms of T which is also regularon F ( I ). We let ¯ T = T / ( t , . . . , t ℓ ) and denote by ¯ J the image of J in ¯ T . Since t , . . . , t ℓ is a regular sequence on T /J and J has a 2-linear resolution, it followsthat dim ¯ T / ¯ J = 0 and ¯ J has a 2-linear resolution. It follows that ¯ J = m T , where m T is the graded maximal ideal of ¯ T , since the only m ¯ T -primary ideals with linearresolution are powers of the maximal ideal. Thus we see that F ( I ) i = 0 for i ≥ F ( I ) = F ( I ) / ( y , . . . , y ℓ ), and y i = t i + J for i = 1 , . . . , ℓ . Since y , . . . , y ℓ isa regular sequence of linear form, the h -vector of F ( I ) and that of F ( I ) coincide.Hence the conclusion follows.(g) ⇒ (h): We assume that K is infinite, and let y , . . . , y ℓ be as in the proof (g) ⇒ (d). We write y i = f i + m I with f i ∈ I for i = 1 , . . . , ℓ , and let J = ( f , . . . , f ℓ ). Now F ( I ) i = 0 for i ≥ I = J I + m I . Thus, Nakayama’slemma yields I = J I , as desired.(h) ⇒ (d): Since I is quasi-equigenerated, K is infinite and F ( I ) is Cohen–Macaulay, a reduction ideal J of I with r J ( I ) = 1 can be chosen of the form J =( f , . . . , f ℓ ) such that the sequence y i = f i + m I with i = 1 , . . . , ℓ is a regularsequence on F ( I ). Since I = J I it follows that F ( I ) i = 0 for i ≥
2, where ( I ) = F ( I ) / ( y , . . . , y ℓ ). Since h -vector of F ( I ) and that of F ( I ) coincide, theassertion follows. (cid:3) Classes of Freiman ideals
Freiman graphs.
Let K be a field and G be a finite simple graph on [ n ]. Theideal I ( G ) ⊂ S = K [ x , . . . , x n ] generated by the monomials x i x j with { i, j } ∈ E ( G )is called the edge ideal of G . We say that G is a Freiman graph , if I ( G ) is aFreiman ideal. In this subsection we want to classify all Freiman graphs. Noticethat F ( I ( G )) is isomorphic to the edge ring K [ G ] = K [ x i x j : { i, j } ∈ E ( G )] of G .Let T = K [ z e : e ∈ E ( G )] be the polynomial ring in the indeterminates z e , and let I G be the kernel of the K -algebra homomorphism T → K [ G ] with z e x i x j for e = { i, j } . Then K [ G ] ∼ = T /I G .Certainly G is Freiman, if K [ G ] is the polynomial ring. The monomials x i x j with { i, j } ∈ E ( G ) are algebraic independent if and only if K [ G ] is a polynomial ring,which is the case if and only if I G = 0.In order to describe the generators of I G one has to consider walks in G . A walk W of length r of G is a sequence of vertices i , i , . . . , i r such that { i j , i j +1 } is anedge of G for j = 0 , . . . , r −
1. If i = i r , then W is called a closed walk . The closedwalk is called even (odd) , if r is even (odd). It is called a cycle , if the vertices i j with 0 ≤ j ≤ r − Proposition 2.1.
The edge ring K [ G ] of G is a polynomial ring, if and only if eachconnected component of G contains at most one cycle, and this cycle is odd.Proof. Let G , . . . , G s be the connected components of G . Then K [ G ] = N si =1 K [ G i ].Thus K [ G ] is a polynomial ring if and only if each K [ G i ] is a polynomial ring. Hencewe may assume that G is connected and have to show that K [ G ] is a polynomialring if and only if G contains at most one cycle, and this cycle is odd.Let W : i , i , . . . , i s be an even closed walk, and set e j = { i j , i j +1 } for j =0 , . . . , s −
1. Then f W = s − Y k =0 z e k − s − Y k =0 z e k +1 ∈ I G , and all binomial generators of I G are of this form, see [16, Lemma 1.1]. Thus K [ G ]is a polynomial ring if and only if f W = 0 for each even closed walk in G .Suppose that K [ G ] is a polynomial ring and G contains an even cycle C . Then f C = 0, a contradiction. Therefore, G contains no even cycles. Suppose G containsat least two odd cycles, say C and C . Let i be a vertex of C and j be a vertex of C . Then, since G is connected, there exists a walk W connecting i with j . Now weconsider the following even closed walk W ′ : we first walk around C starting with i . Then we walk along W from i to j . Then we continue our walk around C tocome back to j . From there we walk back to i along W , but backwards. Obviously, f W ′ = 0. Thus there cannot be two odd cycles. This proves one direction of theproposition. onversely, suppose that G contains only one odd cycle, it is enough to prove that I G = 0. By [16, Lemma 3.1 and Lemma 3.2], the toric ideal I G is generated by thebinomials f W , where W is an even closed walk of G of one of the following types:(1) W is an even cycle of G ; (2) W = ( C , C ), where C and C are odd cycles of G having exactly one common vertex; (3) W = ( C , W , C , W ), where C and C are odd cycles of G having no common vertex and where W and W are walks of G both of which combine a vertex v of C and a vertex v of C . Walks of thesethree types are called primitive.By the hypothesis that G contains only one odd cycle, there cannot exist primitiveeven closed walks of G . Therefore, I G = 0. (cid:3) Corollary 2.2.
Let G be a finite simple graph with r connected components. Thenfollowing conditions are equivalent: (a) G is a Freiman graph. (b) All connected components of G are Freiman graphs and at least r − of itsconnected components contain at most one cycle, and this cycle is odd.Proof. Let G , . . . , G r be the connected components of G , and let Hilb K [ G i ] ( t ) = h ( K [ G i ]; t ) / (1 − t ) d i with d i = dim K [ G i ]. Then, since K [ G ] = N ri =1 K [ G i ], itfollows that Hilb K [ G ] ( t ) = r Y i =1 Hilb K [ G i ] ( t ) = h ( K [ G ]; t ) / (1 − t ) d , where h ( K [ G ]; t ) = Q ri =1 h ( K [ G i ]; t ) and d = P ri =1 d i = dim K [ G ].(a) ⇒ (b): Since G is Freiman, Theorem 1.3 implies that deg h ( K [ G ]; t ) ≤ h ( K [ G i ]; t ) ≤ i . Thus (b) followsfrom Proposition 2.1 together with Theorem 1.3.(b) ⇒ (a): The assumptions in (b) together with Theorem 1.3 and Proposition 2.1imply that deg h ( K [ G i ]; t ) ≤ h ( K [ G i ]; t ) = 0 for all h ( K [ G i ]; t ) but possi-bly one of them. Thus deg h ( K [ G ]; t ) ≤
1, which by Theorem 1.3 implies that G isFreiman. (cid:3) Due to Corollary 2.2 it is enough to consider only connected graphs for the clas-sification of Freiman graphs.
Theorem 2.3.
Let G be a finite simple connected graph, and let H be the subgraphof G whose edges are the edges of all the -cycles of G . Then G is Freiman, if andonly if there exist no primitive even walks in G , or otherwise H is bipartite of type (2 , s ) for some integer s , and there exist no primitive even walks in G of length > .Proof. If G has no primitive even walks, then K [ G ] is the polynomial ring, and hence G is Freiman. Suppose now that H is bipartite of type (2 , s ) for some s , and thereexist no primitive even walks in G of length >
4. Since G admits no primitive evenwalks of length >
4, the ideal I G is generated by the binomials f C with C a 4-cycleof G . In particular, I G = I H , and K [ G ] is a polynomial extension of K [ H ]. Since H is bipartite of type (2 , s ), the ideal I H may be viewed as the ideal of 2-minors of a × s -matrix of indeterminates. Hence the Eagon-Northcott complex [6] provides a2-linear resolution of I H (= I G ). By Theorem 1.3, this implies that G is Freiman.Conversely, suppose that G is Freiman. Then Theorem 1.3 implies that I G has nogenerators of degree >
2, and hence G admits no primitive even walks of length > H is bipartite of type ( r, s ) with r ≤ s . Since the edge set of H is the union of the edges of the 4-cycles of G , we must have r ≥
2. Also it followsfrom this fact that H is a complete bipartite graph if r = 2. Indeed, suppose thatone edge is missing, say the edge { , r } . Then { , r } ∈ E ( H ) and { , r } belongs toa 4-cycle. Any 4-cycle containing the edge { , r } is of the form r, , i, , r for some i = r . Hence { , r } ∈ E ( H ), a contradiction.It remains to be show that r = 2. Suppose r >
2, and let { v , . . . , v r } ∪{ w , . . . , w s } be the bipartition of the graph H . Since H is the union of theedges of the 4-cycles of G , we may assume that, after a relabeling of the vertices, C : v , w , v , w , v is a 4-cycle of H . Since r >
2, there must exist another 4-cycle C of H which contains the vertex v . Then C and C have only one vertex or oneedge in common, or the cycles C and C are disjoint. But in the last case there is awalk in G connecting a vertex of C with a vertex of C since G is connected. In anycase we have in this situation a primitive even walk of length >
4, a contradiction. (cid:3)
Corollary 2.4.
Let G be a finite connected bipartite graph. Then G is Freiman ifand only if G is a tree, or there exists a bipartite subgraph H of G of type (2 , s ) , asubset S of V ( H ) and for each w ∈ S an induced tree T w of G with V ( H ) ∩ V ( T w ) = { w } .Proof. If G is a tree or if there exists a bipartite subgraph H of G satisfying theconditions of the corollary, then the subgraph H is the union of all 4-cycles of G ,and moreover there exist no primitive even walks in G of length >
4. Therefore, G is Freiman, by Theorem 2.3.Conversely, assume that G is Freiman and let H be as defined in Theorem 2.3.Then Theorem 2.3 implies that H = ∅ or H is bipartite of type (2 , s ) for some s . If H = ∅ , then Theorem 2.3 implies that G has no even cycles of any length. Since G is bipartite and connected, this implies that G is a tree.Suppose now that H is bipartite of type (2 , s ). If G = H , then there is nothing toprove. Otherwise there exists v ∈ V ( G ) \ V ( H ). Since G is connected there exists awalk P outside of H connecting v with w ∈ V ( H ). Assume there is another walk P outside H connecting v with w ∈ V ( H ) with w = w . Then there exists w belonging to P and P such that the subwalk of P connecting w with w and thesubwalk of P connecting w with w have only the vertex w in common. Then weget a cycle walking from w to w to w , and from w back to w (inside H ). Since G is bipartite, this cycle must be even, a contradiction because by Theorem 2.3, G has even cycles of length at most 4. But all even cycles of length 4 belong to H . Itfollows that each vertex v of G not belonging to H is connected with at most onevertex w ∈ H .Now let w ∈ H , and let V be the set of vertices v ∈ V ( G ) \ V ( H ) which areconnected to w via a walk. We denote by T w the restriction of G to V . As shown bove, V ( T w ) ∩ V ( T w ) = ∅ . Since G has no even closed walks of length >
4, itfollows that there is for each v ∈ V ( T w ) a unique walk, whose edges are all distinctfrom one another, and which connects v with w . In fact, if there exist two suchwalks connecting v with w , say P , P , then their lengths must be both odd or botheven, since G is bipartite. By Theorem 1.3, the lengths of P and P are both 2.In fact, since G is Freiman, there exist no primitive even walks in G of length > P and P are both 1 or 2. If their lengths are both 1, then G have a multiple edge connecting v with w . But the graph G is simple withoutmultiple edge, thus the union of P and P is a 4-cycle of G . This implies v ∈ V ( H ),a contradiction. Now the property that for each v ∈ V ( T w ) there exists a uniquewalk connecting v with w , implies that T w is a tree. (cid:3) A typical example of a bipartite Freiman graph is shown in the next figure. bb b b b bb bbb b bb b b
Figure 12.2.
Cycle matroids.
Given a finite simple graph G on [ n ]. The cycle matroid of G is the matroid whose ground set is E ( G ), and whose bases are the sets E ( F ) with F a spanning forest of G . A spanning forest of G is a maximal acyclic subgraph of G .A matroid M is called graphic , if for some simple graph G , M is isomorphic tothe cycle graph of G .Let K be a field. Let M be a matroid on the ground set [ m ], and B the set of itsbases. To M we attach the ideal I M ⊂ S = K [ x , . . . , x m ], whose generators are themonomials u B with u B = Q i ∈ B x i and B ∈ B . The ideal I M is called the matroidalideal of M . The fiber cone of I M is the base ring of M . We say that M is a Freimanmatroid , if I M is a Freiman ideal.Let E ( G ) = { e , . . . , e m } , and M the cycle matroid of G . Then I M ⊂ K [ x , . . . , x m ]is the ideal generated by the monomials u F = Q e i ∈ F x i with F a spanning forestof G .As the main result of this subsection we have heorem 2.5. Let M be the cycle matroid of G . Then the following conditions areequivalent: (a) M is a Freiman matroid. (b) G contains at most one cycle. (c) F ( I M ) is a polynomial ring. We will need the following lemmas before proving Theorem 2.5. First, we have
Lemma 2.6.
Let G be a finite simple graph whose cycle matroid is M . Let E be asubset of the edges of G , Γ the graph with E (Γ) = E and N be the cycle matroid of Γ . Then N is a Freiman matroid.Proof. We observe that F ( I N ) is a combinatorial pure subring of F ( I M ) in thesense of [15]. We can write F ( I M ) = B/J and F ( I N ) = A/I , where A and B are polynomial rings and J ⊂ B and I ⊂ A are the binomial relation ideals of F ( I M ) and F ( I N ), respectively. By [15, Corollary 2.5] we have β Aij ( I ) ≤ β Bij ( J ) forall i, j . Since M is a Freiman matroid, it follows from Theorem 1.3 that J has a2-linear B -resolution. Therefore the above inequality of Betti numbers implies that I has a 2-linear A -resolution. Furthermore, since F ( I N ) is normal, F ( I N ) is Cohen–Macaulay, by Hochster’s theorem, see [14]. We now apply again Theorem 1.3 andconclude that N is a Freiman matroid. (cid:3) Lemma 2.7.
Let G be a finite simple graph consisting of two cycles C and C which have at most one vertex in common. Then the cycle matroid of G is not aFreiman matroid.Proof. Let M i be the cycle matroid of C i , and let r i = | E ( C i ) | . Then I M i can be iden-tified with the squarefree Veronese ideal I r i ,r i − of degree r i − r i indeterminates.Hence the fiber cone of I M i is isomorphic to K [ I r i ,r i − ] and this ring is isomorphic toa polynomial ring S i over K in r i indeterminates. Since C and C have at most onevertex in common, each spanning forest of G is obtained from G by removing oneedge from C and another edge from C . This shows that I M = I M I M . Hence thefiber cone of I M is isomorphic to the Segre product T = S S of the polynomialrings S and S .By [8, Theorem 4.2.3] we know that dim T = r + r −
1. Thus Hilb T ( t ) = h ( T ; t ) / (1 − t ) r + r − , and so the a -invariant a ( T ) of T is equal to deg h ( T ; t ) − ( r + r −
1) (see [3, Definition 4.4.4]). Hence we getdeg h ( T ; t )( t ) = a ( T ) + ( r + r − . (2)On the other hand, if ω T is the canonical module of T , then a ( T ) = − min { i : ( ω T ) i = 0 } , (3)see [8, Definition 3.1.4]. We now use another result of Goto and Watanabe [8,Theorem 4.2.3] which says that ω T ∼ = ω S ω S . Since ω S i = S i ( − r i ), we have ω T ∼ = S ( − r ) S ( − r ). The i th graded component ( ω T ) i of ω T is isomorphic to( S ( − r ) S ( − r )) i = ( S ) i − r ⊗ ( S ) i − r . herefore, min { i : ( ω T ) i = 0 } = max { r , r } . Combining (2) with (3), we getdeg h ( T ; t ) = r + r − − max { r , r } . Since each cycle has at least 3 edges, we see that r i ≥
3. It follows that deg h ( T ; t ) ≥
2. Thus Theorem 1.3 implies that M is not a Freiman matroid.In the case that C and C are connected by a walk P , we have I M = uI M I M ,where u is the monomial whose factors correspond to the edge of P . Then K [ I M ] = K [ uI M I M ] ∼ = K [ I M I M ], and the rest of the proof is as before. (cid:3) Lemma 2.8.
Let G be a finite simple graph consisting of a cycle C of length r anda walk W of length s which connects two non-consecutive vertices of C . Then thecycle matroid M of G is not a Freiman matroid.Proof. By hypothesis, the walk W divides cycle C into two walks W and W , where W , W and W have common starting point and endpoint. Thus C = W ∪ W . Forsimplicity we set W = W . Let M i be the cycle matroid of W i , r i = | E ( W i ) | , andlet M be the cycle matroid of C and r = | E ( C ) | . Then r = s , r = r + r = r .By the proof of Lemma 2.7, the fiber cone F ( I M i ) of I M i is isomorphic to K [ I r i ,r i − ]and this ring is isomorphic to a polynomial ring S i over K in r i indeterminates.Therefore, we get I M = I r ,r − I r + r ,r + r − + J I r ,r , where J = I r ,r − I r ,r − . Wehave I r ,r ⊂ I r ,r − and I r + r ,r + r − ⊂ J , Let S = S ⊗ S and Hilb F ( I M ) ( t ) = h ( S ; t ) / (1 − t ) ℓ , where ℓ = dim( F ( I M )). By [13, Theorem 1.6], we haveHilb S/I M ( t ) = Hilb S/ ( I r ,r − I r r ,r r − + JI r ,r ) ( t )= Hilb S /I r ,r − ( t ) Hilb I r r ,r r − ( t ) + Hilb S /J ( t ) Hilb I r ,r ( t )+ Hilb S /I r ,r ( t ) Hilb S /I r r ,r r − ( t )= Hilb I r r ,r r − ( t ) + Hilb S /J ( t ) Hilb I r ,r ( t ) + Hilb S /I r ,r ( t ) . HenceHilb F ( I M ) ( t ) = 1(1 − t ) r + r − Hilb
S/I M ( t )= 1(1 − t ) r + r − [ ∞ X m =0 (cid:18) m + r − m (cid:19) t m + ( 1(1 − t ) r − h ( S ; t )(1 − t ) r − ) ∞ X m =0 t m + 1(1 − t ) r − ]= 1(1 − t ) r + r − [ 1(1 − t ) r + [ 1(1 − t ) r − h ( S ; t )(1 − t ) r − ] 11 − t + 1(1 − t ) r − ]= 1(1 − t ) r + r − (1 − t ) r + (1 − t ) r − [1 − (1 − t ) h ( S ; t )] + (1 − t ) r +1 (1 − t ) r + r = 1 − (1 − t ) r − (1 − t ) r − [1 − (1 − t ) h ( S ; t )] − (1 − t ) r +1 (1 − t ) r + r = 1 − (1 − t ) s − (1 − t ) s − [1 − (1 − t ) h ( S ; t )] − (1 − t ) r +1 (1 − t ) r + s , here deg h ( S ; t ) = min { r , r } −
1, as seen in the proof of Lemma 2.7.We consider the following two cases:(1) If s = 1, then Hilb F ( I M ) ( t ) = h ( S ; t ) − − (1 − t ) r (1 − t ) r and h ( S ; t ) = h ( S ; t ) − − (1 − t ) r ,Since deg h ( S ; t ) = min { r , r } − r = r + r , deg h ( S ; t ) ≥ max { r , r } + 1.Notice that r i ≥
2, we have that deg h ( S ; t ) ≥ s ≥
2, then h ( S ; t ) = 1 − (1 − t ) s − (1 − t ) s − [1 − (1 − t ) h ( S ; t )] − (1 − t ) r +1 such that h ( S ; 1) = 1 and deg h ( S ; t ) ≥ M of G is not a Freiman matroid. (cid:3) Now we are ready to prove Theorem 2.5.
Proof. (a) ⇒ (b): Suppose there is a cycle C for which there exists a walk W whichconnects two non-consecutive vertices of C . Then let Γ be the graph with edges E ( W ) ∪ E ( C ). By Lemma 2.8 the cycle matroid of Γ is not a Freiman matroid, andso M is not a Freiman matroid by Lemma 2.6, a contradiction. Thus there is nocycle C of G such that there exists a walk W which connects two non-consecutivevertices of C .Let us now assume that G contains at least two cycles C and C . We discuss twocases: (i) V ( C ) ∩ V ( C ) = ∅ , and (ii) V ( C ) ∩ V ( C ) = ∅ .Let Γ be the graph with E (Γ) = E ( C ) ∪ E ( C ). In case (i), the cycle matroidof Γ is not a Freiman matroid, see Lemma 2.7. Therefore, according to Lemma 2.6,also M is not a Freiman matroid.In case (ii), by using Lemma 2.6, we may assume that Γ = G . Let S = V ( C ) ∩ V ( C ), and Σ the graph which is obtained from G by restriction to S . Then Σ isa disjoint union of walks and some isolated vertices. If Σ is just a single vertex, M is not a Freiman matroid by Lemma 2.7, and if Σ is just a single walk, then theedges E ( C ) ∪ E ( C ) \ E (Σ) form a cycle C and Σ is a walk which connects twonon-consecutive vertices of C . Thus Lemma 2.8 implies that M is not a Freimanmatroid. Finally assume that Σ has more than one connected component. Let V ( C ) = { v , . . . , v r } and V ( C ) = { w , . . . , w s } with a counter-clockwise labelingof the vertices. We may assume without loss of generality that v = w and v = w ,since Σ is not connected. There exist integers a, b with the property that v a = w b and v i = w j for all i, j with 1 < i < a and 1 < j < b . It then follows that the walk W ′ with V ( W ′ ) = { w , w , . . . , w b } is a walk connecting the vertices v and v a of C . Thus M is not a Freiman matroid.(b) ⇒ (c): Let G , . . . , G s be the connected components of G and let M i be thecycle matroid of G i . Since G contains at most one cycle, we may assume that thecomponents G i are trees for i ≥
2. Then, as in the proof of Lemma 2.8, we seethat I M = uI r,r − if G contains a cycle C with | E ( C ) | = r , or otherwise I M is aprincipal monomial ideal. Moreover, all I M i for i ≥ I M = I M · · · I M s , we have K [ I M ] ∼ = K [ I M ], and hence K [ I M ] is a polynomialring.(c) ⇒ (a) is obvious. (cid:3) e close this paper with discussing the regularity of the base ring of the cyclematroid of a simple graph. Let, as before, G be a finite simple graph on [ n ] and E ( G ) = { e , . . . , e m } its edge set. Let e , . . . , e m denote the canonical coordinatevectors of R m . Given a subset F ⊂ E ( G ), one defines ρ ( F ) = P e i ∈ F e i ∈ R m .For example, ρ ( { e , e , e } ) = e + e + e = (0 , , , , , , . . . , ∈ R m . The basepolytope P ( M ) of the cycle matroid M of G is the convex polytope which is theconvex hull of the finite set { ρ ( F ) : ρ ( F ) ∈ B} in R m . One has dim F ( I M ) =dim P ( M ) + 1.Recall that, when G is connected, a vertex i ∈ [ n ] of G is said to be a cut vertex ifthe induced subgraph G | [ n ] \{ i } of G on [ n ] \ { i } is disconnected. A finite connectedsimple graph G is called 2 -connected if no vertex of G is a cut vertex of G . It thenfollows from [20, Theorem 5] that Lemma 2.9.
Let G be a finite -connected simple graph on [ n ] with e edges. Let M be the cycle matroid of G and P ( M ) the base polytope of M . Then ℓ ( I M ) = dim P ( M ) + 1 = e. Corollary 2.10.
Let G be a finite connected simple graph on [ n ] with e edges andwrite c for the number of cut vertices of G . Let M be the cycle matroid of G and P ( M ) the base polytope of M . Then ℓ ( I M ) = dim P ( M ) + 1 = e − c. Proof.
Let, say, n ∈ [ n ] be a cut vertex of G . Let G ′ and G ′′ be the connectedcomponents of the induced subgraph G | [ n − . Write U for the vertex set of G ′ and V that of G ′′ . Let M ′ be the cycle matroid of G | U ∪{ n } and M ′′ that of G | V ∪{ n } .Let e ′ be the number of edges of G | U ∪{ n } and e ′′ that of G | V ∪{ n } . Since the basering F ( I M ) is the Segre product of F ( I M ′ ) and F ( I M ′′ ), it follows that dim F ( I M ) =dim F ( I M ′ ) + dim F ( I M ′′ ) −
1. Let c ′ be the number of cut vertices of G ′ and c ′′ thatof G ′′ . Then using induction yields dim F ( I M ′ ) = e ′ − c ′ and dim F ( I M ′′ ) = e ′′ − c ′′ .Hence dim F ( I M ) = ( e ′ − c ′ ) + ( e ′′ − c ′′ ) −
1. Since e = e ′ + e ′′ and c = c ′ + c ′′ + 1,the desired formula dim F ( I M ) = dim P ( M ) + 1 = e − c follows. (cid:3) Corollary 2.11.
Let G be a finite disconnected simple graph on [ n ] with e edges and G , . . . , G s its connected component. Let c i denote the number of cut vertices of G i for ≤ i ≤ s . Let M be the cycle matroid of G and P ( M ) the base polytope of M .Then ℓ ( I M ) = dim P ( M ) + 1 = e − c − s + 1 . Proof.
Let M i be the cycle matroid of G i for 1 ≤ i ≤ s . Since the base ring F ( I M ) isthe Segre product of the base rings F ( I M ) , . . . , F ( I M s ), the required formula followsimmediately from Corollary 2.10. (cid:3) Let G be a finite 2-connected simple graph on [ n ] with e edges and P ( M ) thebase polytope of the cycle matroid M of G . Thus dim P ( M ) = e −
1. We introducethe sequence of integers ( δ , δ , δ , . . . ) by the formula(1 − λ ) e ∞ X i =0 δ i t i = ∞ X i =0 | i P ( M ) ∩ Z m | t i , here i P ( M ) is the i th dilated polytope { iα : α ∈ P ( M ) } of P ( M ). It then followsthat δ i = 0 for i > dim P ( M ) = e −
1. We say that δ ( P ( M )) = ( δ , δ , . . . , δ e − )is the δ -vector of P ( M ) and δ ( P ( M ); t ) = P e − i =0 δ i t i is the δ -polynomial of P ( M ).Furthermore, introducing the integer r ≥ r = min { r ∈ Z : r ≥ r ( P ( M ) \ ∂ P ( M )) ∩ Z m = ∅} , where P ( M ) \ ∂ P ( M ) is the relative interior of P ( M ), one has the formuladeg δ ( P ( M ); t ) = e − r . (4)We refer the reader to [12] for the detailed information about δ -vectors.Since the base polytope P ( M ) possesses the integer decomposition property ([9,p. 250]), it follows that Lemma 2.12.
The δ -polynomial δ ( P ( M ); t ) of P ( M ) coincides with the h -polynomialof F ( I M ) . Since the base ring F ( I M ) is Cohen–Macaulay, it is well-known that the regularityreg( F ( I M )) of F ( I M ) is one more than the degree of the h -polynomial h ( F ( I M ); t ) of F ( I M ). In other words, the regularity reg( F ( I M )) is one more than the degree of the δ -polynomial δ ( P ( M ); t ) of P ( M ). Since the hyperplanes x i = 0 and x i = 1 of R m are supporting hyperplanes of P ( M ) for 1 ≤ i ≤ n , it follows that no lattice point of R m belongs to the relative interior of P ( M ). Hence, by using the formula (4), one hasdeg δ ( P ( M ); t ) = deg h ( F ( I M )); t ) ≤ e −
2. Thus, in particular reg( F ( I M )) ≤ e − F ( I M ) is the polynomial ring,one has deg h ( F ( I M ) , t ) ≥ I M ) ≥
3. As a result,
Theorem 2.13.
Let G be a finite -connected simple graph on [ n ] with e edges. Let M be the cycle matroid of G . Unless F ( I M ) is the polynomial ring, one has ≤ reg( F ( I M )) ≤ e − . It is natural to ask, given integers 3 ≤ r < e , if there exists a 2-connected simplegraph G with e edges for which reg( F ( I M )) = r , where M is the cycle matroid of G and F ( I M ) is the base ring of M . Corollary 2.14.
Let G be a finite disconnected simple graph on [ n ] with e edges and G , . . . , G s its connected components. Let c i denote the number of cut vertices of G i for ≤ i ≤ s . Let M be the cycle matroid of G . Unless F ( I M ) is the polynomialring, one has ≤ reg( F ( I M )) ≤ e − c − s. Finally, since the toric ideal of the base ring of a cycle matroid of a finite simplegraph is generated by quadratic binomials [2], it follows that no base ring of a cyclematroid has a linear resolution.
Acknowledgement.
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E-mail address : [email protected] Takayuki Hibi, Department of Pure and Applied Mathematics, Graduate Schoolof Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
E-mail address : [email protected] Guangjun Zhu, School of Mathematical Sciences, Soochow University, Suzhou215006, P. R. China
E-mail address : [email protected]@suda.edu.cn