Betti numbers of binomial ideals

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Journal of Symbolic Computation www.elsevier.com/locate/jsc

Betti numbers of binomial ideals Hernán de Alba a , Marcel Morales b,c a

CONACYT, Universidad Autónoma de Zacatecas, Unidad Académica de Matemáticas, Calzada Solidaridad entronque Paseo de la Bufa, Zacatecas, Zac. 98000, Mexico b Université de Grenoble I, Institut Fourier, UMR 5582, B.P. 74, 38402 Saint-Martin D’Hères Cedex, France c ESPE de Lyon, Université de Lyon 1, France

a r t i c l e

i n f o

Article history: Received 8 July 2015 Accepted 13 May 2016 Available online xxxx Keywords: Betti numbers Binomial ideal Lattice ideal Clique complex Square-free quadratic monomial ideal

a b s t r a c t Let us consider the family of binomial ideals B = I + J , where J is lattice ideal and I is a square-free quadratic monomial ideal. We give a formula for calculating the Betti numbers of B. Moreover we bound the Green–Lazarsfeld invariant of a family of quadratic binomial ideals B using this formula. This result extends a previous result of Eisenbud et al. for square-free quadratic monomial ideals and extends completely Fröberg’s theorem. We describe also a subfamily where we can calculate the Green–Lazarsfeld invariant of any ideal B and we also compute its first non-linear Betti number. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Let k[x] be the polynomial ring with variables x = (x1 , . . . , xn ) over a field k. A classical problem in algebraic geometry and commutative algebra consists of studying minimal free resolutions of an ideal of k[x]. In 1977 Hochster computed the multigraded Betti numbers of a squarefree monomial ideal using simplicial homology. Since this time a line of research is to calculate the Betti numbers of monomial and binomial ideals using simplicial complexes and simplicial homology. Campillo and Marijuan (1991, Theorem 1.2) obtained a similar formula for the ideal of a toric affine curve. This result was independently extended to any dimension by Campillo and Pison (1993) (see also Sturmfels, 1996, Theorem 12.12) and Herzog and Aramova (2000, Lemma 4.1). Later on, Bruns and Herzog (1997, Proposition 1.1) gave a formula for the Betti numbers of a binomial ideal B = I + J , where I is a

E-mail addresses: [email protected] (H. de Alba), [email protected] (M. Morales). http://dx.doi.org/10.1016/j.jsc.2016.06.001 0747-7171/© 2016 Elsevier Ltd. All rights reserved.

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square-free monomial ideal and J a positive toric ideal, by using relative simplicial homology. Bruns et al. (2008) gave a similar formula for the Betti numbers of face toric ideals. Since graded Betti numbers are the main subject of this paper, we give the precise definition in a more general frame described in lemma 4.1 of Bruns et al. (2008) and section 1.5 of de Alba Casillas (2012). Let  be an abelian positive semigroup (there are not invertible elements except 0) and cancellative with respect to 0, i.e. if a + b = a implies b = 0 for a, b ∈  . Let k[x] be the polynomial ring in n-variables with a  grading (i.e., there exists a decomposition k[x] = ⊕σ ∈ R σ of k[x] as abelian groups such that R σ · R σ  ⊂ R σ +σ  for all σ , σ  ∈  ). Let M = ⊕σ ∈ M σ be a  -graded module over k[x] of finite type. We define for all σ ∈ 



M (−σ )a =

if a ∈ / σ + , ⊕b∈| a=σ +b M b if a ∈ σ + .

0

M has a graded minimal free resolution of the form:

0→

nρ 

j=1 a j ,ρ ∈ 

βρ ,a j,ρ

k[x](−a j ,ρ )

→ ··· →

n0 

β0,a j,0

k[x](−a j ,0 )

→ M → 0,

j=1 a j ,0 ∈ 

where βi ,a j,i ∈ N∗ and ρ is the projective dimension of M. This resolution is unique up to isomorphism. The ranks βi ,a j,i ( M ) = βi ,a j,i are called  -graded Betti numbers of M. If  = N the graduation is the standard graduation. When M is N standard-graded, for p , q ∈ N, we say that the resolution is q-linear up to the step p, if a j ,0 = q, a j ,1 = q + 1, . . . , a j , p = q + p for all j, but a j , p +1 = q + p + 1 for some j. In this case the number p + 1 is called the Green–Lazarsfeld index and it will be denoted by p q ( M ). If a j ,l = q + l for all j , l we say that the resolution is q-linear and that p q ( M ) = ∞. The goal of Section 2 is to prove our first result (Theorem 2.7). We obtain a formula involving relative simplicial homology to compute the multigraded Betti numbers of ideals B = I + J , where I is a squarefree monomial ideal and J is a lattice ideal. Actually this formula is an extension of the formula obtained by Bruns and Herzog (1997, Proposition 1.1). In Section 3 we study the Green–Lazarsfeld index of homogeneous quadratic binomial ideals. By using Gröbner bases we are able to bound the Green–Lazarsfeld index in terms of a monomial ideal. In Lemma 3.3 we give a procedure to calculate the Green–Lazarsfeld index for some quadratic binomial ideals and in Section 4 we introduce a family of binomial ideals, for which we can use this lemma. We want to remember some important results about the Green–Lazarsfeld index of quadratic monomial ideals before introducing Section 4, since in this section we obtain similar results for a special family of binomial ideals. Let us recall that any squarefree quadratic monomial ideal arises from a graph. Namely let G be a graph and (G ) its clique complex, the Stanley–Reisner ideal I (G ) is a squarefree quadratic monomial ideal. For a squarefree quadratic monomial ideal the Green–Lazarsfeld index was studied by Eisenbud et al. (2005, Theorem 2.1), where they proved that p 2 ( I (G ) ) + 3 is the shortest length of a minimal cycle in G. This result is a generalization of Fröberg’s theorem (Fröberg, 1990), which says that I (G ) is 2-linear if and only if G is chordal. The goal of Section 4 is to give lower and upper bounds for the Green–Lazarsfeld index of a binomial ideal B(G ) = I (G ) + J(G ) , where (G ) is some extension of (G ), I (G ) is the Stanley–Reisner monomial ideal of (G ) and J(G ) is a sum of scroll ideals following some conditions. This family of ideals, which geometrically consist in the union of rational normal scrolls, appears in the classification of varieties of minimal degree by Del Pezzo, Bertini and Xambó (see Eisenbud and Goto, 1984 for literature). Equations of varieties of minimal degree were described in Barile and Morales (1998, 2000, 2004); and they correspond to ideals B (G ) , such that G is a chordal graph. In Ha and Morales (2009) extended this class of ideals to any graph and proved that some properties of I (G ) can be extended to B(G ) . In order to give lower and upper bounds for the Green–Lazarsfeld index of a binomial ideal B(G ) , first we show that the set of generators B(G ) is a Gröbner basis of B(G ) for an order defined in section 4. It turns out that the initial ideal of B(G ) is squarefree and it is completely described in

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terms of (G ) and a system of generators of J(G ) . This allows us to bound p 2 (B(G ) ) by introducing a family of cycles of G that we will call the set of virtual minimal cycles of G (Theorem 4.15). As a consequence we obtain an extension of Fröberg’s theorem (Corollary 4.18), which says B(G ) is 2-linear if and only if G is chordal. In Theorem 4.19 we are able to compute p 2 (B(G ) ) and also the first non-linear Betti number in terms of minimal cycles of G for a subfamily of ideals B(G ) , which extends the Theorem 2.1 of Eisenbud et al. (2005) and also the main result of Fernández-Ramos and Gimenez (2009). In order to compute the Betti numbers by our methods, we need to be able to compute some relative homology modules. Note that the projective dimension is bounded by the number of variables in k[x] and the degrees of Betti numbers can be bounded in terms of the Castelnuovo–Mumford regularity. For the family of ideals considered in Section 4 the Castelnuovo–Mumford regularity is bounded by the dimension of k[x]/B(G ) + 1. 2. Betti numbers of binomial ideals α

αi

α

Given α = (α1 , . . . , αn ) ∈ Nn . We will denote the monomial x1 1 . . . xn n by xα , Supp(α ) = {i ∈ N :   = 0} and |α | = ni=1 αi . Given F ⊂ {1, . . . , n}, we will denote by x F the monomial i ∈ F xi .

Definition 2.1. An ideal B ⊂ k[x] is said to be binomial if there exists a family of binomials (and monomials) that generates B, i.e., B is generated by a finite family of binomials {xα − axβ : α , β ∈ Nn , a ∈ k}. We will say that B is pure binomial if there are not monomials in B. We will say that B is a monomial ideal if there exists a family of monomials which generates B. By Eisenbud and Sturmfels (1996, Proposition 1.11) if B is a pure binomial ideal, then the algebra

k[x] can be graded by a semigroup ( B ) = Nn / ∼ B where

∀α , β ∈ Nn , α ∼ B β, if there exists a binomial xα − axβ ∈ B with a ∈ k∗ . We set the morphism between semigroups deg( B ) : Nn → ( B ) defined by deg( B ) (α ) = [α ]∼ = B , and by extension deg( B ) (xα ) = deg( B ) (α ). So k[x] = ⊕σ ∈( B ) S σ , where S σ = ⊕deg( B ) (α )=σ k · xα , is a ( B )-graduation of k[x]. Let R = k[x]/ B, also by Eisenbud and Sturmfels (1996, Proposition 1.11), each homogeneous component R σ of R is 1-dimensional over k. Let  be an abelian semigroup. We say that s is cancellative when s + x = s + y, for x, y ∈  , implies x = y. If all elements of  are cancellative, we say  is cancellative. We get immediately from the definitions the following lemma: Lemma 2.2. Let B a pure binomial ideal of k[x] and R = k[x]/ B. Then, σ ∈ ( B ) is cancellative if and only if for all a ∈ ( B ), dimk R [−σ ]a ≤ 1. Definition 2.3. A lattice is a finitely generated free abelian group. A partial character ( L , ρ ) on Zn is a morphism ρ from a sublattice L of Zn to the multiplicative group k∗ = k − {0}. Given a partial character ( L , ρ ) on Zn , we define the ideal: +

−

I L ,ρ = (xα − ρ (α )xα |α = α + − α − ∈ L ) ⊂ k[x] and we call it a lattice ideal. Here α + , α − ∈ Nn denote the positive and negative part of α respectively. A prime lattice ideal is called a toric ideal, while the set of zeroes in kn is an affine toric variety. In this paper we consider only lattice ideals I L ,ρ , such that L is a positive lattice, i.e., L ∩ Nn = {0}. The fact that L is a positive lattice ideal is equivalent with the fact that the semigroup ( I L ,ρ ) is positive, i.e., there are not invertible elements except 0. Let I ⊂ k[x] be an ideal. We denote by ( I : (x1 · · · xn )∞ ) the set of all the polynomials p ∈ k[x] such that xα p ∈ I for some α ∈ Nn . Actually ( I : (x1 · · · xn )∞ ) is an ideal and we have the following well

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known equivalences (see de Alba Casillas, 2012; Eisenbud and Sturmfels, 1996; Miller and Sturmfels, 2005): Proposition 2.4. Let J ⊂ k[x] a pure binomial ideal. The following statements are equivalent (1) (2) (3) (4)

J is a lattice ideal.

( J ) is a cancellative semigroup. ( J : (x1 · · · xn )∞ ) = J . The image of all variable xi over k[x]/ J is a non-zero divisor over k[x]/ J .

Definition 2.5. Let J ⊂ k[x] be a lattice ideal. We define for all b ∈ ( J )

b = { F ⊂ {1, . . . , n} : b − deg( J ) (x F ) ∈ ( J )}. Lemma 2.6. Let K be the Koszul complex of the sequence x = (x1 , . . . , xn ). Let J ⊂ k[x] be a pure binomial ideal. Then J is a lattice ideal if and only if (K ⊗ k[x]/ J )b is the chain complex C (b ; k)(−1), for all b ∈ ( J ). Proof.

⇒) K ⊗ k[x]/ J is the complex:

K ⊗ k[x]/ J

:= 0 →

n 

∂n

(k[x]/ J )n −→

n −1

(k[x]/ J )n → · · · →

1 

∂1

(k[x]/ J )n −→ k[x]/ J → 0,

j

where ∂ j (e i 1 ∧ · · · ∧ e i j ) = s=1 (−1) j +1 xi s e i 0 ∧ · · · ∧ e i s−1 ∧ e i s+1 ∧ . . . e i j . We set R = k[x]/ J . As R is ( J )-graded, the complex (K ⊗ k[x]/ J )b is also ( J )-graded, for all b ∈ ( J ), is:

0 → R (−a1 − · · · − an )b → ⊕i 1 <···
→ R → 0, where ai = deg( J ) (xi ). As J is a lattice ideal, ( J ) is cancellative, by Proposition 2.4. Then, by Lemma 2.2, for all σ ∈ ( J ), dim(k[x]/ J (−σ )b ) ≤ 1. Moreover, if F ⊂ {1, . . . , n}

R (−





ai )b =

i∈ F



R b−

 =  =



if b ∈ /(

0 i∈ F

ai

i ∈ F ai ) + ( J ),  if b ∈ ( i ∈ F ai ) + ( J )

0

if c ∈ ( J ), c +

Rc

if ∃c ∈ ( J ), c +

0

 

i∈ F

ai = b,

i∈ F

ai = b

if F ∈ / b ,

k if F ∈ b .

This last equality is due to the definition of b . Thus (K ⊗ k[x]/ J )b is the chain complex C (b ; k)(−1). ⇐) We suppose J is not a lattice ideal. Thus there exists a variable xi such that deg( J ) (xi ) is not cancellative in ( J ) and by Lemma 2.2 there exists a homogeneous component of k[x][−deg( J ) (xi )] which is a vector space over k of dimension ≥ 2. As in the proof of the necessity we can show that (K ⊗ k[x]/ J )b is not a chain complex of any simplicial complex. 2 Theorem 2.7. Let J be a lattice ideal and I be a squarefree monomial ideal of k[x]. Let B = I + J ; R = k[x]/ B. Set

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b = { F ∈ b : b − deg( J ) x F = deg( J ) (xγ ), for some xγ ∈ I } Then for all b ∈ ( J ) βi ,b ( R ) = dimk Hi −1 (b , b ; k). Proof. As Ker(k[x]/ J → R ) = (xα + J : xα ∈ I ) := I  , we have the following exact sequence:

0 → I  → k[x]/ J → R → 0, which induces the following exact sequence:

0 → K ⊗ I  → K ⊗ k[x]/ J → K ⊗ R → 0. K⊗k[x]/ J

Thus K ⊗ R ∼ = K⊗ I  . Similarly as in the proof of the necessity of Lemma 2.6 we can show that (K ⊗ I  )b = C ( b ; k)(−1). Moreover, from Lemma 2.6 we know that

(K ⊗ k[x]/ J )b = C (b ; k)(−1). By consequence

K⊗R ∼ = (C (b ; k)/C ( b ; k))(−1).

(2.7.1)

We know that the Koszul complex K is a minimal free resolution of k[x]/m and it is well known that x] βi ,b ( R ) = dimk (Tork[ ( R , k))b ; so i x] βi ,b ( R ) = dimk (Tork[ (k, R ))b i = dimk H i ((K ⊗ R )b )

(by commutativity of functor Tor)

(by definition of functor Tor) = dimk H i −1 (b , b : k) (by (2.7.1)).

2

Remark 2.8. When B = I + J is homogeneous, with I , J as in Theorem 2.7, we have that the graduation by ( J ) is finer than the standard graduation. Hence

βi , j ( B ) =



βi ,b ( B ).

(2.8.1)

b∈( J )&|b|= j

Remark 2.9. For our purpose we need to express b and b in terms of a set of generators of I + J , that is possible because ( J ) is defined by an equivalence relation in terms of the binomials of J . As follows: (1) First, we see that if b ∈ ( J ), then there exists

α ∈ Nn such that b = deg( J ) (xα . Now, if b − σ ∈ Nn and a ∈ k∗ such that xα −

deg( J ) (x F ) ∈ ( J ), for some F ⊂ {1, . . . , n}, then there exist

axσ x F ∈ J . (2) Since deg( J ) (α ) = deg( J ) (β) implies xα − a1 xβ ∈ J for some a1 ∈ k∗ , if xβ − a2 xγ ∈ J for some a2 ∈ k∗ , then xα − a1 a2 xγ ∈ J . So, if b := deg( J ) (α ), we can write b and b as

b = { F ⊂ {1, . . . , n} : ∃γ ∈ Nn , xα − axγ x F ∈ J for some a ∈ k∗ }, and

b = { F ∈ b : ∃γ ∈ Nn , xα − axγ x F ∈ J for some a ∈ k∗ , and xγ ∈ I }. (3) In the proof of Theorem 2.7 we need that the Kernel of k[x]/ J → k[x]/( I + J ) is generated by a system of monomial classes such that any class of this monomials is represented by a squarefree monomial. Thus, if there is an ideal generated by monomials I 1 such that the Kernel of k[x]/ J → k[x]/( I 1 + J ) is generated by those kind of classes, so we can affirm that there exists an squarefree monomial ideal I such that I 1 + J = I + J . (4) If we fix a lattice ideal J ⊂ k[x] and there is a monomial ideal I 1 such that I + J = I 1 + J , then the sets b and b are equal for I and I 1 and for all b ∈ ( J ). We can see an application of this remark in Example 2.10.

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Example 2.10. Before giving the example we note that if b ∈ ( J ), there exists an α ∈ Nn such that deg( J ) (xα ) = b, and in this example we will use xα instead of b. We consider the ideal B = ( w 2 − x2 , w z, xy ) ⊂ k[ w , x, y , z] = k[x]. The binomial ideal J = ( w 2 − x2 ) corresponds to the lattice Z(2, −2, 0, 0), hence J is a lattice ideal; and J is not a prime ideal, so J is not a toric ideal. We can use Theorem 2.7 to calculate the Betti-numbers of k[x]/ B. For illustrating our methods we calculate the Betti numbers of degree 4. For βi , w 3 z (k[x]/ B ) we have that  w 3 z = { w , x, z} (the simplicial complex whose all faces are subsets of { w , x, z}) and w 3 z = { w }, {x} (the simplicial complex whose all faces are the subsets of { w } or {x}). So x3 z is homotopically equivalent to the 2-dimensional disc D 2 and w 3 z is homotopically equivalent to the 0-dimensional sphere S 0 . Thus, using Theorem 2.7 and the long exact sequence of relative homology:

· · · → H n ( A ) → H n ( X ) → H n ( X , A ) → H n−1 ( A ) → · · · , where A ⊂ X simplicial complexes, we obtain that

βi , w 3 z (k[x]/ B ) = dimk ( H i −1 ( w 3 z , w 3 z )) = dimk ( H i −1 ( D 2 , S 0 )) = dimk ( H i −2 ( S 0 ))  1 if i = 2; = 0

otherwise.

In a similar way we obtain

βi ,x3 y (k[x]/ B ) = dimk H i −1 ({ w , x, y }, { w }, {x}) = dimk ( H i −1 ( D 2 , S 0 )) = dimk ( H i −2 ( S 0 ))  1 if i = 2; = 0

otherwise,

βi ,x2 yz (k[x]/ B ) = dimk H i −1 ({x, y , z}, { w , y , z}, {x, z}, { w , y }) = dimk ( H i −1 ( D 2 ∗ D 2 , P 1 ∪ P 1 )) = dimk ( H i −2 ( S 0 ))  1 if i = 2; = 0

otherwise,

βi , wxyz (k[x]/ B ) = dimk ({ w , x, y , z}, {x, y }, { w , z}) = dimk ( H i −1 ( D 3 , P 1 ∪ P 1 )) H i −2 ( S 0 )) = dimk (  1 if i = 2; = 0

otherwise,

where P 1 denotes the path-graph of two vertices and ∗ is the wedge union. We can also see that βi ,m (k[x]/ B ) = 0 for all m ∈ ( J ) of degree 4 distinct to w 3 z, x3 y, x2 yz and wxyz. Thus, by Remark 2.8,



βi ,4 (k[x]/ B ) =

4 0

if i = 2; otherwise.

Actually, using Macaulay2 (Grayson et al., 1992) we obtain that the homogeneous minimal free resolution of k[x]/ B is:

0 → k[x](−5)2 → k[x](−4)4 → k[x](−2)3 → k[x]. Note that Theorem 2.7 cannot be applied if the binomial ideal J is not a lattice ideal:

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Example 2.11. We consider the ideal B = (xz − u w , y w − v z, xv − u y , uv w ). The binomial ideal J = (xz − u w , y w − v z, xv − u y ) is not a lattice ideal by Proposition 2.4, since using Macaulay 2 we can see that

( J : (u · · · z)∞ ) = ( w y − v z, wx − uz, vx − u y , w 2 − z2 , v w − yz, u w − xz, v 2 − y 2 , uv − xy , u 2 − x2 ) = J . Actually the Betti numbers of B were calculated in Bruns et al. (2008, Example 4.6). Remark 2.12. The formula in Theorem 2.7 is an extension of the formula obtained in Bruns and Herzog (1997, Proposition 1.1) when J is toric and the complexes which appear in their formula are the same than our complexes b and b , respectively. When J is equal to zero in Theorem 2.7 we have that the formula given in this proposition is reduced to the Hochster’s formula (Hochster, 1977). Corollary 2.13. Let J be a lattice ideal, I be a squarefree monomial ideal of k[x]. Let deg( J ) (α ).

α ∈ Nn and b =

(1) If for all binomial xα − axβ ∈ J , Supp(β) ⊂ Supp(α ), then

βi ,b ( I + J ) = dimk H|Supp(α )|−i −2 (( b ) A ; k), where ( b ) A is the Alexander dual of b . Moreover, if we suppose that xα is squarefree and I satisfies the property that for any binomial xβ − axγ ∈ J we have that xβ ∈ I if and only if xγ ∈ I , then βi ,b ( I + J ) = β i ,α ( I ) . / I + J then βi ,b ( I + J ) = βi ,α ( J ). (2) If xα ∈ Proof. (1) By Proposition 2.7

βi ,b ( I + J ) = βi +1,b (k[x]/( I + J )) = dimk Hi (b , b ; k), where

b = { F ⊂ {1, . . . , n} : ∃γ ∈ Nn , xα − axγ x F ∈ J , for some a ∈ k} and

b = { F ∈ b : ∃γ ∈ Nn , xα − xγ x F ∈ J , for some a ∈ k, xγ ∈ I ∗ }. As for every binomial xα − axβ ∈ J , Supp(β) ⊂ Supp(α ), then b = Supp(α ) (the simplicial complex whose all faces are all the subsets of Supp(α )). Now, using the Alexander duality and the long exact sequence of relative homology we have that

βi ,b ( I + J ) = dimk Hi (b , b ; k) H|Supp(α )|−i −2 (( b ) A , (b ) A ; k) = dimk H|Supp(α )|−i −2 (( b ) A ; k). = dimk Now, we assume that xα is squarefree and I satisfies the property that for any binomial xβ − axγ ∈ J we have that xβ ∈ I if and only if xγ ∈ I . With this assumption we claim that

b = { F ⊂ Supp(α ) : (Supp(α ) \ F ) ∈ / ( I )}, where ( I ) is the Stanley–Reisner complex associated to I . In fact: / ( I )}. Since xα − xα ∈ J , xα − xSupp(α )\ F x F ∈ J and Let F ∈ { F ⊂ Supp(α ) : (Supp(α ) \ F ) ∈ Supp(α )\ F x ∈ I , so F ∈ b . On the other hand, let F ∈ b , then there exists γ ∈ Nn and a ∈ k∗ such that xα − axγ x F ∈ J with xγ ∈ I . But

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xα − axγ x F = xSupp(α )\ F x F − axγ x F = x F (xSupp(α )\ F − axγ ); as J is a lattice ideal, hence xSupp(α )\ F − xγ ∈ J . Moreover by the assumption over I and the fact that xγ ∈ I , we conclude that xSupp(α )\ F ∈ I . Thus we have proved our claim. By definition of Alexander dual of a simplicial complex

b = { F ⊂ Supp(α ) : (Supp(α ) \ F ) ∈ / ( I )} = (( I )Supp(α ) ) A . Thus

βi ,b ( I + J ) = dimk H|Supp(α )|−i −2 (( b ) A ; k) H|Supp(α )|−i −2 (( I )Supp(α ) ; k) = dimk = βi ,α ( I ). The last equality follows from Hochster’s Formula. (2) As xα ∈ / I + J , there does not exist γ ∈ Nn neither a1 ∈ k∗ such that xα − axγ ∈ J and xγ ∈ I . Thus

b = ∅ and by Proposition 2.7

βi ,b ( I + J ) = dimk Hi −1 (b , ∅; k) = dimk Hi −1 (b ; k) = βi ,α ( J ). The last equality follows from Remark 2.12.

2

3. The Green–Lazarsfeld index of quadratic binomial ideals We recall that a graph G is a pair of sets ( V (G ), E (G )), where E (G ) is a family of subsets of V (G ) of cardinality 2 called edge-set of G. The edge ideal of G is I (G ) = (xi xk : {xi , xk } ∈ E (G )). A clique of G is a subset T of vertices of G such that for all v , w ∈ T { v , w } ∈ E (G ), i.e. the restriction of G on the vertex subset T is a complete graph. The clique complex of G is the simplicial complex (G ) whose faces are the cliques of G. The Stanley–Reisner ideal associated to (G ) is I (G ) = I (G c ) where G c is the graph with vertex-set V (G ) and edge-set E (G c ) = {{ v , w } ⊂ V (G ) : { v , w } ∈ / E (G )}. So I (G ) is a square-free quadratic monomial ideal and reciprocally for any square free quadratic monomial ideal I there exists a graph G ( I ) such that I = I (G ( I )) , for short we set ( I ) = (G ( I )). Definition 3.1. A cycle C of a graph G is a subgraph of G with vertex-set V (C ) = { v 1 , . . . , v q } and edge-set is E (C ) = {{ v 1 , v 2 }, { v 2 , v 3 }, . . . , { v q−1 , v q }, { v q , v 1 }} ⊂ E (G ). We call the cardinality of E (C ) the length of C and we denote it as |C |. We say that the cycle C of length > 3 of G has a chord if there is an edge { v i , v j } ∈ ( E (G ) \ E (C )) for some v i , v j ∈ V (C ). We say that the cycle C is minimal if |C | > 3 and it does not have any chord. The graph G is called chordal graph if all cycles of length > 3 have a chord. Proposition 3.2. Let  = (G ) be the clique complex of the graph G and I = I  . Then: (1) The minimal length of a minimal cycle of G is p 2 ( I ) + 3. (2) Besides, if p 2 ( I ) = ∞, β p 2 ( I ), p 2 ( I )+3 ( I ) is equal to the number of minimal cycles of length p 2 ( I ) + 3 of G. The first statement of Proposition 3.2 is Theorem 2.1 of Eisenbud et al. (2005). And the second statement of this proposition was first proved by Fernández-Ramos and Gimenez (2009) and by de Alba Casillas (2012) who gave another proof. From now on we assume that B = I + J is a binomial ideal, where I is a monomial ideal and J is a toric ideal and B has a square-free quadratic Gröbner basis for some order. Then, as a consequence of upper-semicontinuity (Miller and Sturmfels, 2005, Theorem 8.29) and Proposition 3.2,

min (|C |) − 3 = p 2 (in≥ ( B )) ≤ p 2 ( B ),

C ∈Cin≥ B

where Cin≥ B is the family of all minimal cycles of the graph G (in≥ B ).

(3.2.1)

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Table 1 Betti-table of B.

β j , j +i

0

1

2

3

4

5

6

7

8

9

2 3

48 0

266 0

729 0

1224 0

1344 0

972 0

441 4

108 10

10 6

0 1

Lemma 3.3. Let I ⊂ k[x] be a monomial ideal and J ⊂ k[x] a toric ideal, such that for any binomial xα − xγ ∈ J , we have that xα ∈ I if and only if xγ ∈ I . Let B = I + J . We suppose that B and in≥ ( B ) for some order ≥ are quadratic. Let C be a minimal cycle of G (in≥ ( B )) such that for all α ∈ Nn with x V (C ) − xα ∈ J then supp(xα ) ⊂ V (C ). If we set b = deg( J ) (x V (C ) ), then β|C |−3,b ( B ) = H1 (( I )C ; k). Moreover if C is of minimal length and H1 (( I )C ; k) = 0, then p 2 ( B ) = p 2 (in≥ ( B )).

Proof. By Corollary 2.13 and Hochster’s formula

β|C |−3,b ( B ) = β|C |−3,b ( I ) = H1 (( I )C ; k). Moreover if C is of minimal length and H1 (( I )C ; k) = 0, then β|C |−3,b ( B ) = 0. By Proposition 3.2, p 2 (in( B )) = |C | − 3, so using the equation (2.8.1) we have



β p 2 (in≥ ( B ))−3, p 2 (in≥ ( B )) ( B ) =

β p 2 (in≥ ( B ))−3,α ( B ) ≥ β p 2 (in≥ ( B ))−3,b ( B ) = 0,

α ∈(J )

|α | = p 2 (in≥ ( B ). Thus p 2 ( B ) ≤ p 2 (in≥ ( B )), and by (3.2.1) we have the equality.

2

In the following example we show that it is not always possible to apply Lemma 3.3. However in Section 4, we are going to study a family of ideals in which we can apply Lemma 3.3. Example 3.4. Let us consider I the monomial ideal associated to the clique complex generated by the graph G of Fig. 1 and J the binomial ideal generated by the minors of the matrices

M1 =

a x

x y

y z

z c

v w

w b



, M2 =

d u

u s

s e

 .

It can be proved that J is a toric ideal. We set B = I + J . We remark that

B = (ad, ce , be , bd, xd, xe , xs, xu , yd, ye , ys, yu , zd, ze , zs, zu , vd, ve , vs, vu , wd, we , ws, wu , ua, ub, uc , sa, sb, sc , ay − x2 , az − xy , ac − xz, aw − xv , ab − xw , xz − y 2 , zc − yz, xw − yv , xb − y w , yc − z2 , y w − zv , yb − c w , zw − cv , zb − c w , vb − w 2 , ds − u 2 , de − us, ue − s2 ) ⊂ k[a, x, y , z, v , w , d, u , s, c , b, e ]. and in( B ) is the monomial ideal associated to the clique complex generated by the graph G  of Fig. 1 with respect the lexicographic order. We can see that the minimal cycle C of G with edges E (C ) = {{a, v }, { v , c }, {c , d}, {d, u }, {u , s}, {s, e }, {e , a}} has length 7 and actually is a minimal cycle of G  of minimal length. Then p 2 (in( B )) = 7 − 3 = 4, by Proposition 3.2. We can remark that avcduse − azwduse ∈ J , but supp(azwduse )  V (C ) = {a, v , c , d, u , s, e }. So, we can not apply Lemma 3.3 to calculate β4,C ( B ). Now, using Macaulay2 we get the table of graded Betti numbers of B (see Table 1). Thus from the table we have that p 2 ( B ) = 6. A modification of our methods yields an extension of our result that allows us to cover this kind of cases. This is work in progress.

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Fig. 1. Bad bound of p 2 .

4. A family of quadratic binomial ideals In this section we are going to build a family of quadratic binomial ideals which satisfies the hypothesis of Lemma 3.3. Definition 4.1. Let G be a graph and  = (G ) its clique complex and T ⊂ G a forest in which each (e ) (e ) edge of T is a maximal clique of G. To each edge e = {x0 , x1 } of T we associate a new finite set of (e )

(e )

vertices Y (e) = { y 1 , . . . , yne } = ∅ and an ideal Ie , which is the ideal generated by the 2 × 2 minors of the scroll matrix:



Me =

(e)

(e)

x0

y1

y1

y2

(e)

(e)

(e)

. . . y n −1 y n e (e) . . . yn x1

.

We set e = e ∪ Y (e) and G as the graph generated by G and the new cliques e for all e ∈ T . We define the binomial extension of I  as the ideal

B = I  + J , where J =



Ie and  = (G ).

e∈T

Remark 4.2. (1) Using the notation of Definition 4.1 and the definition of clique complex, it is clear that all the facets of  are e, where e ∈ T ; and the facets F of  distinct to any e ∈ T . Hence, we will denote a facet of  by F and the facet of  associated to F by F . (2) I  is a squarefree quadratic monomial ideal generated by all product zi z j such that zi , z j does not belong to the same facet  (or to the same maximal clique of G) and we will denote this set of generators of I  by NF . Moreover, if we denote the set of 2 × 2 minors of the matrix M e by Je and B  = NF (∪e∈ E ( T ) Je ), then B  is a generator system of B . Example 4.3. Consider the graph G of the Fig. 2. In order to get a binomial extension of the clique complex (G ) we have to choose a forest of G whose edges are maximal cliques. For example, the tree T given by the edges {c , d}, {d, e }, {a, e } satisfies this condition. But the tree T 1 given by the edges {a, c }, {c , d}, {d, e } does not satisfy the condition, because {a, c } is not a maximal clique of G. We take the tree T to produce a binomial extension and the matrices

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Fig. 2. A graph.

M1 =

c x

x y

y d



, M2 =

d u

u e



, M3 =

a z

z e

 .

Thus a binomial extension of the ideal I (G ) is

B = ( ad, ce , be , bd, ax, ay , au , bx, by , bu , bz, cu , cz, xu , xe , xz, yu , ye , yz, dz, uz, c y − x2 , cd − xy , xd − y 2 , de − u 2 , ae − z2 ). Proposition 4.4. Let  be a clique complex and  be a binomial extension of  . Then J is a toric ideal. Proof. The proof follows by induction using the properties of the tree T . For a complete proof see de Alba Casillas and Morales (2012, Theorem 2.9). 2 Lemma 4.5. Let e ∈ T . For every 2 × 2 minor of M e , zα − zβ , and all γ ∈ Nm we have that zγ +α ∈ I  if and only if zγ +β ∈ I  . In particular if zγ +α ∈ I  , then zγ (zα − zβ ) ∈ I  . Proof. It is enough to show that if zγ +α ∈ I  then zγ +β ∈ I  , since zβ − zα = −(zα − zβ ) is also a 2 × 2 minor of M e . Note that γ has to be different from 0, otherwise zα ∈ I  , but by the definition of M e , this is not possible. Furthermore, as zγ +α ∈ I  , there exist z1 , z2 ∈ V () such that z1 z2 ∈ I  and z1 z2 | zγ +α . We have three cases: (1) If z1 |zα and z2 |zα , then zα ∈ I  , we have already seen that this is not possible. (2) If z1 |zα or z2 |zα but not both at the same time, we can suppose z1 |zα and z2  zα ; so z2 |zγ . As z1 |zα , we have that z1 ∈ e. Moreover z1 z2 ∈ I  , thus z2 ∈ / e and for every F  facet of  such α β   that z2 ∈ F we have z1 ∈ / F . As z − z is a minor 2 × 2 of M e , there exist z1 , z2 ∈ e such that   β z = z1 z2 and we have two cases: (a) z1 = z2 , which implies z1 ∈ Y (e) and z1 is only in the facet e, hence z1 z2 ∈ I  and zγ +β ∈ I  . (b) z1 = z2 . Let us show that either z1 z2 ∈ I  or z2 z2 ∈ I  , which would imply that zγ +β ∈ I  . Suppose that neither z1 z2 ∈ I  nor z2 z2 ∈ I  , then there exists a facet F  of  such that  z2 , z1 , z2 ∈ F , thus { z1 , z2 } ∈ e ∩ F  = e ∩ F  and e = { z1 , z2 } ⊂ F  which implies that z1 ∈ F  .

This is a contradiction since z2 ∈ F  and z1 z2 ∈ I  . (3) If z1  zα and z2  zα , then z1 z2 |zγ and z1 z2 |zγ +β , hence zγ +β ∈ I  .

2

Proposition 4.6. If zα − zβ ∈ J and zα ∈ I  , then zβ ∈ I  . m Proof. As zα − zβ ∈ J , there exist a1 , . . . , an ∈ k, γ1 , . . . , γn ∈ N , e 1 , . . . , en edges of T and b1 , . . . , bn n 2 × 2 minors of M e1 , . . . , M en respectively, such that zα − zβ = i =1 ai zγi b i . Let

K := {i ∈ {1, . . . , n} : one of the monomials of zγi b i ∈ I  }.

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K = ∅, since there must exist i ∈ {1, . . . , n} such that zα is a monomial of zγi bi . Let i ∈ K, by Lemma 4.5, if one of the monomials of zγi b i ∈ I  , then the two monomials of zγi b i belong to I  . So zα − zβ =



ai zγi b i +

i ∈K



ai zγi b i .

i∈ /K



• If zβ appears in a term of i ∈K ai zγi b i , then zβ ∈ I  .  • If zβ does not appear as a term of i ∈K ai zγi bi , then zβ must to appear as a term of i ∈/ K ai zγi bi . Since every monomial of the first addition does not appear as a term of the second addition, we have:

zα −



ai zγi b i = zβ −

i ∈K



ai zγi b i = 0.

i∈ /K

Thus zα ∈ J , but this is a contradiction to the fact that J is a toric ideal. So this case is not possible. 2 If T is a forest we can order the vertices of T by a total order > T , so we can write all the (e ) (e ) (e ) (e ) edges of T as e = {x0 , x1 }, where x0 > x1 . With this total order we order the edges of T by the (e ) (e )

lexicographic order over the words x0 x1 . Hence, let G be a graph and G its binomial extension, we order the vertices of G. Without lost of generality we can suppose that



Me =

(e)

y1

(e)

y2

x0

y1

(e)

. . . y n e −1

(e)

y ne

(e)

(e)

. . . yn(ee)

x1

(e)

, (e )

(e )

(e )

and we order the vertices of the first row of M e as x0 > y 1 > · · · > yne . Moreover if z is in the first row of M e , z is in the first row of M e and z = z , z > z if e > e  . The remaining vertices of G are ordered by an arbitrary order and we denote this order by >. Definition 4.7. We define the graph G obtained from G deleting all the edges which correspond to all the diagonals of M e from top to bottom and from left to right, where e is an edge of T . We will  = ( denote by G ) the clique complex generated by G. Proposition 4.8. We order the vertices of G by >. Then (1) B  is a quadratic Gröbner basis of B for the lexicographic order on the monomials of k[z]. (2) The ideal in>lex (B ) is a squarefree quadratic monomial ideal and the associated simplicial complex of . in>lex (B ) is Proof. (1) Follows immediately from the Buchberger’s criterium. (2) As for every matrix M e ,

in([( M e )]1,i [( M e )]2,k − [( M e )]1,k [( M e )]2,i ) = [( M e )]1,i [( M e )]2,k with i < k and [( M e )]1,i = [( M e )]2,k , we have in>lex ( B  ) is a square-free monomial set. And by (1) B  is a Gröbner basis of B , so I  = in>lex (B ) is a square-free monomial ideal and by definition of (in>lex (B )) we get (in>lex (B )) = . 2 Remark 4.9. From item a) of last Proposition we have that k[x]/B is a Koszul algebra. Definition 4.10. Let e ∈ E (1 ), we will say e is virtual if e ∈ / E ( G ). Let C be a cycle of 1 , it will be a virtual minimal cycle if and only if either C is a minimal cycle of G, or if C is not a minimal cycle in G, then C satisfies the following properties:

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(1) Any chord of C is virtual. (2) If e , e  ∈ E (C ) are distinct, then e , e  are not in a same facet of  . We will denote by C v the family of all virtual minimal cycles of G. Lemma 4.11. For any C ∈ C v there exists V ⊂ V (C ) such that (G ) V is a minimal cycle of length ≥ 4. Proof. As C is a cycle there exists a cycle D in G such that V ( D ) ⊂ V (C ) and either its length is 3 or it is minimal. Assume that D has length 3 then V ( D ) = {a, b, c } is a clique of  , so there exists a facet F in  which contains D. This contradicts (2) of Definition 4.10. Thus D is minimal. 2 (e )

(e )

Definition 4.12. Let C be a virtual minimal cycle of 1 and e = {x0 , x1 } be a virtual edge of C , (e )

(e )

then e ∈ E ( T ) and e = e ∪ { y 1 , . . . , yne }. We define a cycle (e )

(e )

(e )

ϕ (C ) in G by replacing any virtual edge

(e )

(e )

(e )

(e )

(e )

e = {x0 , x1 } ∈ E (C ) by the path P e with edges {x0 , y 1 }, { y 1 , y 2 }, . . . { yne , x1 }. Remark 4.13.

 ϕ (C ) is a minimal cycle of  . Actually ϕ defines an application from (1) By definition of the minimal  virtual cycles of G to the minimal cycles of G. Also we can remark that |ϕ (C )| = |C | + e∈ R (C ) ne − 3, where R(C) is the set of virtual edges of C . . (2) If F = F , then G F = G F by the definition of . (3) If F = F , then F = e ∈ E ( T ) and ( G F )1 = P e , by the definition of

Proposition 4.14 (Going down for minimal cycles). ϕ is surjective. C be a minimal cycle of G , so its length is ≥ 4. Let f 1 , . . . , f q be its consecutive edges. Let Proof. Let us remark that (1) For any i = 1, . . . , q, the edge f i is either contained in a facet F of  without edges in T (in this case F is a clique of G), or f i is an edge of a path P e for some e ∈ T . (2) Two distinct edges f i , f j cannot be contained in the same facet F of  without edges in T , otherwise C would have a chord belonging to F . We form a path by putting together consecutive edges of C , which are contained in the same path P e for some e ∈ T . C as the join of consecutive paths Q 1 , . . . , Q s , which are either a As a consequence we can write simple edge contained in a facet F of  without edges in T , or a part of a path P e for some e ∈ T . Note also that Q 1 ∩ Q s ⊂ V (G ) and for any j = 1, . . . , s − 1 we have Q j ∩ Q j +1 ⊂ V (), hence Q j is either a simple edge e j contained in a facet F of  without edges in T , or coincides with P e j for some e j ∈ T . Let C be the cycle in G with consecutive edges e 1 , . . . , e s . If C has a chord this chord is necessarily virtual, otherwise it would appear in G, and it would be a chord for C . In conclusion C is a virtual minimal cycle and ϕ (C ) = C. 2 Theorem 4.15. If C v is not empty

minC ∈C v (|C | +



ne ) = minC ∈C v (|ϕ (C )|) ≤ p 2 (B ) + 3 ≤ minC ∈CG |ϕ (C )|

e ∈ R (C )

= minC ∈CG (|C | +



ne ),

e ∈ R (C )

where CG is the set of minimal cycles of G; otherwise p 2 (B ) = ∞.

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Fig. 3. Homotopy between  V (ϕ (C )) and C .

Proof. Let I  be the Stanley–Reisner ideal associated to  . By upper-semicontinuity p 2 ( I  ) ≤ p 2 (B ). Moreover, by Theorem 4.14 the set of minimal cycles of G is equal to ϕ (C v ), so if C v = ∅, then by v Proposition 3.2 p 2 ( I  ) = ∞ and p 2 (B ) = ∞. If C = ∅, then minC ∈C v (|C | +



ne ) − 3 = minC ∈C v (|ϕ (C )|) − 3 = p 2 ( I  ) ≤ p 2 (B ).

e ∈ R (C )

Let C be a minimal cycle of 1 and b = deg( J ) (x V (C ) ). We note from Propositions 4.4, 4.6, 4.8 and if x V (C ) − xα ∈ J , then supp(xα ) ⊂ V (C ), that

ϕ (C ) satisfies the hypothesis of Lemma 3.3; so

β|ϕ (C )|−3,b (B ) = H1 ( V (ϕ (C )) ; k). We note that the facets of  V (ϕ (C )) are the subset F e defined by:



Fe =

e e ∪ Y 1e

if e ∈ E (ϕ (C )), if e ∈ / E (ϕ (C )),

for every e ∈ E (C ). So  V (ϕ (C )) =< F e : e ∈ C > and is homotopically equivalent to  V (C ) = C , as we can see in the Fig. 3, where we set E (C ) = {{x1 , x2 }, {x2 , x3 }, . . . , {xn , x1 }} and

if {xi , xi +1 } ∈ / E ( C ), F {xi ,xi+1 } = {xi , xi +1 , y 1i , . . . , yni i }, where xn+1 = x1 . So H 1 ( V (C ) ; k) = 1 and by Equation (2.8.1)

β|ϕ (C )|−3,|ϕ (C )| (B ) =



β|ϕ (C )|−3),α (B ) ≥ β|ϕ (C )|−3,b (B ) = 1,

α ∈( J )&|α |=|ϕ (C )|

so

p 2 (B ) + 3 ≤ minC ∈CG |ϕ (C )| = minC ∈CG (|C | +



ne ).

2

e ∈ R (C )

In this way we obtain an algorithm to bound the Green–Lazarsfeld invariant of B : Algorithm A. • Input: A graph G, a forest T (which satisfies the property of Definition 4.1) and for every e ∈ T a matrix of scroll M e of length ne . • Output: Lower bound and upper bound of p 2 (B ), respectively. • Step 1: Construct G by Definition 4.7.

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• Step 2: Look for minimal cycles of G (CG ) and virtual minimal cycles of G (C v ): – Step 2.1 If G is chordal, then p 2 (B ) = ∞. Terminate.  – Step 2.2 If G is not chordal, then the lower bound  of p 2 (B ) is minC ∈C v (|C | + e∈ R (C ) ne ) and the upper bound of p 2 (B ) is minC ∈CG (|C | + e∈ R (C ) ne ). Terminate. Remark 4.16. If there is up to ten variables Algorithm A can be done by hand. Note that there is an algorithm by Adam Van Tuyl to calculate the minimal cycles of a graph, implemented in Macaulay2. Remark 4.17. Up to the computation of relative homology it is possible to compute the Betti numbers of the ideal B using (Bruns and Herzog, 1997, Proposition 1.1) or Theorem 2.7; since it is well known that the Castelnuovo–Mumford regularity of B is less than or equal to dim k[z]/in(B ) + 1, and the projective dimension of B is less than or equal to the number of variables of B . Let us recall that the regularity of Castelnuovo–Mumford regularity of an N-graded ideal I ⊂ k[x] is reg( I ) = max{ j ∈ N : βi ,i + j ( I ) = 0 for all i ∈ N}. Corollary 4.18. Let  be a clique complex and  be a binomial extension of  . Then, B is 2-linear if and only if G is chordal. Proof. By Theorem 4.15, B is 2-linear if and only if C v = ∅, if and only if G does not have minimal virtual cycles. By Lemma 4.11 the last statement is equivalent to G not having minimal cycles, which is the definition of a chordal graph. 2 Theorem 4.19. Let  be a clique complex and  a binomial extension of  such that any edge of the forest T (see Definition 4.1) is not a chord of any cycle. Then

p 2 (B ) + 3 = minC ∈CG (|C | +



ne ),

e ∈ R (C )

and β p 2 (B ), p 2 (B )+3 (B ) = #{C ∈ C1 () : |C | + cycles of G.



e ∈ R (C ) ne

= p 2 (B )}, where CG is the family of minimal

Proof. As any edge of the tree T (see Definition 4.1) is not a chord of any cycle, C v is equal to the set of all minimal cycles of G, so the first statement follows from Theorem 4.15. Moreover, by upper semicontinuity and Theorem 3.3.

β p 2 ( I (  )), p 2 ( I (  ))+3 ( I ( )) ≥ β p 2 (B ), p 2 (B )+3 (B )  = α ∈( J )&|α |= p2 (B )+3 β p 2 (B ),α (B )   β p 2 (B ),bC (B ) ≥ C ∈{C ∈CG :|C |+ e ∈ R (C ) ne = p 2 (B )}  ≥ #{C ∈ CG : |C | + e∈ R (C ) ne = p 2 (B )}, where b C = deg( J ) (x V (ϕ (C )) ), for any C ∈ CG . By (2) of Proposition 3.2 v β p 2 ( I (  )), p 2 ( I (  ))+3 ( I ( )) = #{C ∈ C : |C | +



ne = p 2 (B )}.

e ∈ R (C )

Thus

β p 2 (B ), p 2 (B )+3 (B ) = #{C ∈ C v : |C | +



ne = p 2 (B )}.

2

e ∈ R (C )

Example 4.20. We consider the binomial extension ideal B of Example 4.3. We remark that the unique minimal virtual cycle of G is given by the edges {a, c }, {c , d}, {d, e }, {e , a} (see Fig. 2). So by Theorem 4.15 p 2 (B ) + 3 = 4 + 2 + 1 + 1 = 8. Hence p 2 (B ) = 5 and β5,8 (B ) = 1.

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