arXiv:1310.2903v1 [] 10 Oct 2013

EXTREMAL BETTI NUMBERS OF SOME CLASSES OF BINOMIAL EDGE
IDEALS
AHMET DOKUYUCU
A BSTRACT. Let G be a cycle or a complete bipartite graph. We show that the binomial
edge ideal JG and its initial ideal with respect to the lexicographic order have the same
extremal Betti number. This is a partial positive answer to a conjecture proposed in [2].

I NTRODUCTION
Let G be a simple graph on the vertex set [n] with edge set E(G) and let S be the
polynomial ring K[x1 , . . . , xn , y1 , . . . , yn ] in 2n variables endowed with the lexicographic
order induced by x1 > · · · > xn > y1 > · · · > yn . The binomial edge ideal JG ⊂ S associated
with G is generated by all the binomials fi j = xi y j − x j yi with {i, j} ∈ E(G). The binomial
edge ideals were introduced in [5] and, independently, in [8]. Meanwhile, many algebraic
and homological properties of these ideals have been investigated; see, for instance, [1],
[2], [3], [5], [7], [9], [10], [11], [12], [13], [14].
In [2], the authors conjectured that the extremal Betti numbers of JG and in< (JG ) coincide for any graph G. Here, < denotes the lexicographic order in S induced by the natural
order of the variables. In this article, we give a positive answer to this conjecture when the
graph G is a complete bipartite graph or a cycle. To this aim, we use some results proved in
[12] and [14] which completely characterize the resolution of the binomial edge ideal JG
when G is a cycle or a complete bipartite graph. In particular, in this case, it follows that
JG has a unique extremal Betti number. In the first section we recall all the known facts
on the resolutions of binomial edge ideals of the complete bipartite graphs and cycles. In
Section 2, we study the initial ideal of JG when G is a bipartite graph or a cycle. We show
that proj dim in< (JG ) = proj dim JG and reg in< (JG ) = reg JG , and, therefore, in< (JG ) has
a unique extremal Betti number as well. Finally, we show that the extremal Betti number
of in< (JG ) is equal to that of JG .
To our knowledge, this is the first attempt to prove the conjecture stated in [2] for
extremal Betti numbers. In our study, we take advantage of the known results on the
resolutions of binomial edge ideals of cycles and complete bipartite graphs and of the
fact that their initial ideals have nice properties. For instance, as we show in Section 2,
the initial ideal of JG for a complete bipartite graph has linear quotients and is generated
in degrees 2 and 3. Therefore, it is componentwise linear and its Betti numbers may
2010 Mathematics Subject Classification. 13D02,05E40.
Key words and phrases. Binomial edge ideals, regularity, projective dimension.
1

be computed easily (Theorem 2.2). The initial ideal of JG when G is a cycle does not
have linear quotients, but by ordering its generators in a suitable way, we may easily
compute its extremal Betti number (Theorem 2.9). It is interesting to remark that even if
the admissible paths of the cycle (in the sense of [7, Section 3]) determine the minimal
set of monomial generators of in< (JG ), the Lyubeznik resolution [6] does not provide a
minimal resolution of in< (G).
1. P RELIMINARIES
1.1. Binomial edge ideals of complete bipartite graphs. Let G = Km,n be the complete
bipartite graph on the vertex set {1, . . ., m} ∪ {m + 1, . . ., m + n} with m ≥ n ≥ 1 and let
JG be its binomial edge ideal. JG is generated by all the binomials fi j = xi y j − x j yi where
1 ≤ i ≤ m and m + 1 ≤ j ≤ m + n. In [12, Theorem 5.3] it is shown that the Betti diagram
of S/JG has the form
0 1
2
0 1 0
0
1 0 mn 0
2 0 0 β24

···
p
···
0
···
0
· · · β p,p+2



m,
if n = 1,
2m + n − 2, if n > 1.
In particular, from the above Betti diagram we may read that S/JG has a unique extremal Betti number, namely β p,p+2 .
Moreover, in [12, Theorem 5.4] all the Betti numbers of S/JG are computed. Since we
are interested only in the extremal Bettinumber, we recall here its value as it was given
m − 1, if p = m,
in [12, Theorem 5.4], namely, β p,p+2 =
n − 1, if p = 2m + n − 2.
Since we will study the initial ideal of JG with respect to the lexicographic order induced by the natural order of the variables, we need to recall the following definition and
result of [5].
where p = proj dim S/JG =

Definition 1.1. Let i < j be two vertices of an arbitrary graph G. A path i = i0, i1 , . . . , ir−1, ir =
j from i to j is called admissible if the following conditions are fulfilled:
(i) ik 6= iℓ for k 6= ℓ;
(ii) for each k = 1, . . . , r − 1, one has either ik < i or ik > j;
(iii) for any proper subset { j1 , . . . , js} of {i1 , . . . , ir−1}, the sequence i, j1, . . . , js, j is
not a path in G.
Given an admissible path π of G from i to j, we set uπ = (∏ik > j xik )(∏iℓ <i yiℓ ).
Obviously, any edge of G is an admissible path. In this case, the associated monomial
is just 1.
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Theorem 1.2 (HHHKR). Let G be an arbitrary graph. The set of binomials
Γ=

[

{uπ fi j : π is an admissible path from i to j}

i< j

is the reduced Gröbner basis of JG with respect to lexicographic order on S induced by
the natural order of indeterminates, x1 > · · · > xn > y1 > · · · > yn .
One may easily see that the only admissible paths of the complete graph G = Km,n
are the edges of G, the paths of the form i, m + k, j with 1 ≤ i < j ≤ m, 1 ≤ k ≤ m, and
m +i, k, m + j with 1 ≤ i < j ≤ n, 1 ≤ k ≤ m. Therefore, we get the following consequence
of the above theorem.
Corollary 1.3. Let G = Km,n be the complete bipartite graph on the vertex set V (G) =
{1, . . ., m} ∪ {m + 1, . . ., m + n}. Then
in< (JG ) = ({xi y j }

1≤i≤m

m+1≤ j≤m+n

, {xi xm+k y j }1≤i< j≤m , {xm+1 yk ym+ j } 1≤i< j≤n ).
1≤k≤n

1≤k≤m

1.2. Binomial edge ideals of cycles. In this subsection, G denotes the n–cycle on the
vertex set [n] with edges {1, 2}, {2, 3}, . . ., {n − 1, n}, {1, n}.
In [14] it was shown that the Betti diagram of S/JG has the form
1 2
3
···
n
0
0 0
0
···
0
n 0
0
···
0
1
0
···
0
2
0 β24
3
0 0
β36 · · ·
0
..
..
..
..
..
..
.
.
.
.
.
.
n − 2 0 0 β2,n β3,n+1 · · · βn,2n−2
and all the Betti numbers were computed. One
sees that we have a unique extremal Betti
n−1
number and, by [14], we have βn,2n−2 = 2 − 1.
We now look at the initial ideal of JG . It is obvious by Definition 1.1 and by the labeling
of the vertices of G that the admissible paths are the edges of G and the paths of the form
i, i − 1, . . ., 1, n, n − 1, . . ., j + 1 with 2 ≤ j − i ≤ n − 2. Consequently, we get the following
system of generators for the initial ideal of JG .
0
1
0
0
0
..
.

Corollary 1.4. Let G be the n–cycle with the natural labeling of its vertices. Then
in< (JG ) = (x1 y2 , . . . , xn−1 yn , x1 yn , {xi x j+1 · · · xn y1 · · · yi−1 y j }2≤ j−i≤n−2 ).
2. E XTREMAL B ETTI

NUMBERS

2.1. Complete bipartite graphs. Let G = Km,n be the complete bipartite graph on the
vertex set {1, . . ., m} ∪ {m + 1, . . . , m + n} with m ≥ n ≥ 1 and let JG be its binomial
edge ideal. The initial ideal in< (JG ) has a nice property which is stated in the following
proposition.
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Proposition 2.1. Let G = Km,n be the complete graph. Then in< (JG ) has linear quotients.
Proof. Let u1 , . . . , ur be the minimal generators of in< (JG ) given in Corollary 1.3 where,
for i < j, either deg ui < deg u j or deg ui = deg u j and ui > u j . We show that we respect
to this order of its minimal monomial generators, in< (JG ) has linear quotients, that is, for
any ℓ > 1, the ideal quotient (u1 , . . ., uℓ−1 ) : (uℓ ) is generated by variables.
Let uℓ = xi y j for some 1 ≤ i ≤ m and m + 1 ≤ j ≤ m + n. In this case, one may easily
check that
(1)

(u1 , . . ., uℓ−1 ) : (uℓ ) = (x1 , . . . , xi−1 , ym+1 , . . ., y j−1 ).

Let uℓ = xi xm+k y j for some 1 ≤ i < j ≤ m and 1 ≤ k ≤ n. Then we get
(2)
(u1 , . . . , uℓ−1 ) : (uℓ ) = (ym+1 , . . . , ym+n , x1 , . . ., xi−1 , xm+1 , . . . , xm+k−1 , yi+1 , . . . , y j−1 ).
Finally, if uℓ = xm+i yk ym+ j for some 1 ≤ i < j ≤ n and 1 ≤ k ≤ m, we have
(3)
(u1 , . . . , uℓ−1 ) : (uℓ) = (x1 , . . . , xm , xm+1 , . . . , xm+i−1 , y1 , . . . , yk−1 , ym+i+1 , . . . , ym+ j−1 ).

Theorem 2.2. Let G = Km,n be the complete graph. Then


i+ j−m−2
,
βt,t+2 (in< (JG )) =
∑
t
1≤i≤m
m+1≤ j≤m+n

βt,t+3 (in< (JG )) =

n+k+ j−3
,
t
1≤k≤n

j−3
n+k+ j−3
,
+ ∑1≤i< j≤n m+k+
∑1≤i< j≤m
t
t
1≤k≤m
1≤k≤n


 ∑1≤i< j≤m


if n = 1,
if n > 1.

Proof. Since in< (JG ) has linear quotients, we may apply [4, Exercise 8.8] and get
 
m
qℓ
βt,t+d (in< (JG )) = ∑
t
ℓ=1
deg uℓ =d

where qℓ is the number of variables which generate (u1 , . . . , uℓ−1 ) : (uℓ ). Hence, by using
equations (1)-(3) for counting the number of variables which generate (u1 , . . . , uℓ−1 ) : (uℓ ),
we get all the graded Betti numbers of in< (JG ).

In particular, by the above theorem, it follows the following corollary which shows that
for G = Km,n the extremal Betti numbers of S/JG and S/ in< (JG ) coincide.
Corollary 2.3. Let G = Km,n be the complete graph. Then:

m,
(a) proj dim(S/ in< (JG )) = proj dim(in< (JG )) + 1 =
2m + n − 2,
4

if n = 1,
if n > 1.

(b) S/ in< (JG ) has a unique extremal Betti number, namely

m − 1,
β p,p+2 (S/ in< (JG )) = β p−1,p+2 (in< (JG )) =
n − 1,

if n = 1,
if n > 1.

Proof. (a) follows immediately from Betti number formulas of Theorem 2.2.
Let us prove (b). By using again Theorem 2.2, we get
(
m−1
m−1
if n = 1,
∑1≤i<m p−1 = ∑1≤i<m m−1 = m − 1,
β p−1,p+2 (in< (JG )) =
2m+n−3
2m+n−3
∑1≤i<n p−1 = ∑1≤i<n 2m+n−3 = n − 1, if n > 1.


2.2. Cycles. In this subsection, the graph G is an n–cycle. If n = 3, then G is a complete
graph, therefore the ideals JG and in< (JG ) have the same graded Betti numbers. Thus, in
the sequel, we may consider n ≥ 4.
As we have already seen in Corollary 1.4, in< (JG ) is minimally generated by the initial
monomials of the binomials corresponding to the edges of G and by m = n(n − 3)/2
monomials of degree ≥ 3 which we denote by v1 , . . ., vm where we assume that if i < j,
then either deg vi < deg v j or deg vi = deg v j and vi > v j . Let us observe that if vk =
xi x j+1 · · · xn y1 · · · yi−1 y j , we have deg vk = n − j + i + 1. Hence, there are two monomials
of degree 3, namely, v1 = x1 xn yn−1 and v2 = x2 y1 yn , three monomials of degree 4, namely,
v3 = x1 xn−1 xn yn−2 , v4 = x1 xn y1 yn−1 , v5 = x1 y1 y2 yn , etc.
We introduce the following notation. We set J = (x1 y2 , x2 y3 , . . ., xn−1 yn ), I = J +(x1 yn ),
and, for 1 ≤ k ≤ m, Ik = Ik−1 + (vk ), with I0 = I. Therefore, Im = in< (JG ).
Lemma 2.4. The ideals quotient J : (x1 yn ) and Ik−1 : (vk ), for k ≥ 1, are minimally generated by regular sequences of monomials of length n − 1.
Proof. The statement is obvious for J : (x1 yn ) since J is minimally generated by a regular
sequence. Now let k ≥ 1 and let vk = xi x j+1 · · · xn y1 · · · yi−1 y j for some i, j with 2 ≤
j − i ≤ n − 2. Then Ik−1 : (vk ) = I : (vk ) + (v1 , . . ., vk−1 ) : (vk ). One easily observes that
(v1 , . . ., vk−1 ) : (vk ) = (x1 , . . . , xi−1 , y j+1 , . . . , yn ). Hence,
I : (vk ) = (x1 , . . ., xi−1 , xi yi+1 , xi+1 yi+2 , . . ., x j−2 y j−1 , x j−1 y j , y j+1 , . . . , yn ) : (vk ) =
(x1 , . . . , xi−1 , yi+1 , xi+1 yi+2 , . . . , x j−2 y j−1 , x j−1 , y j+1 , . . . , yn )

Remark 2.5. From the above proof we also note that if vk = xi x j+1 · · · xn y1 · · · yi−1 y j ,
then the regular sequence of monomials which generates Ik−1 : (vk ) contains j − i − 2
monomials of degree 2 and n − j + i + 1 variables.
In the following lemma we compute the projective dimension and the regularity of S/I.
This will be useful for the inductive study of the invariants of S/Ik .
Lemma 2.6. We have proj dim S/I = n − 1 and reg S/I = n − 2.
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Proof. In the exact sequence
(4)

0→

S
S
x1 yn S
(−2) −→ −→ → 0,
J : (x1 yn )
J
I

we have proj dimS/J : (x1 yn ) = proj dim S/J = n − 1 since both ideals are generated by
regular sequences of length n − 1. Moreover, since J is generated by a regular sequence
of monomials of degree 2, by using the Koszul complex, we get

0,
j = 2i,
βi j (S/J) =
n−1
i , j 6= 2i.
In particular, it follows that Torn−1 (S/J, K) ∼
= K(−2n + 2).
Analogously, since J : (x1 yn ) is generated by a regular sequence of n − 3 monomials of
degree 2 and two variables, it follows that Torn−1 (S/J : (x1 yn ), K) ∼
= K(−2n + 4), which
∼
implies that Torn−1 (S/J : (x1 yn )(−2), K) = K(−2n + 2). By the long exact sequence of
Tor’s derived from sequence (4), we get Torn (S/I, K) = 0, hence proj dimS/I ≤ n − 1.
On the other hand, we have

j = 2n − 5,
 2,
βn−2, j (S/J : (x1 yn )) =
n − 3, j = 2n − 6,

0,
otherwise.

From the exact sequence of Tor’s applied to (4), as

Torn−1 (S/J, K)2n−3 = Torn−2 (S/J, K)2n−3 = 0,
we get the following exact sequence
0 → Torn−1 (S/I, K)2n−3 → Torn−2 (S/J : (x1 yn ), K)2n−5 → 0.
Thus, Torn−1 (S/I, K) 6= 0 which implies that proj dim S/I = n − 1.
For the regularity, we first observe that since the Koszul complex of the minimal generators gives the minimal graded free resolution of S/J and, respectively, S/J : (x1 yn ), we
have reg S/J = reg(S/J : (x1 yn )(−2)) = n − 1. Then sequence (4) implies that reg S/I ≤
n − 1. We have observed above that Torn−1 (S/I, K)2n−3 6= 0, thus reg S/I ≥ n − 2. In order
to derive the equality reg S/I = n − 2 we have to show that βi,i+n−1 (S/I) = 0 for all i.
For i = n − 1, as Torn−2 (S/J : (x1 yn ), K)2n−4 = 0, we get
0 → Torn−1 (S/J : (x1 yn ), K)2n−4 → Torn−1 (S/J, K)2n−2 → Torn−1 (S/I, K)2n−2 → 0.
But dimK Torn−1 (S/J : (x1 yn ), K)2n−4 = dimK Torn−1 (S/J, K)2n−2 = 1, which implies that
Torn−1 (S/I, K)2n−2 = 0. For i < n − 1, Tori (S/I, K)i+n−1 = 0 since Tori (S/J, K)i+n−1 = 0
and Tori−1 (S/J : (x1 yn ), K)i+n−3 = 0. The latter equality holds since, as we have already
observed, reg S/J : (x1 yn ) = n − 3.

Lemma 2.7. For 1 ≤ k ≤ m, we have proj dim S/Ik ≤ n and reg S/Ik ≤ n − 2.
6

Proof. We proceed by induction on k, by using the following exact sequence and Lemma 2.6
for the initial step,
S
S
S
(5)
0→
(− deg vk ) →
→ → 0.
Ik−1 : (vk )
Ik−1
Ik
Indeed, as Ik−1 : (vk ) is generated by a regular sequence of length n − 1, it follows
that proj dimS/Ik−1 : (vk ) = n − 1. Thus, if proj dim S/Ik−1 ≤ n, by (5), it follows that
proj dim S/Ik ≤ n.
In addition, we have reg(S/Ik−1 : (vk ))(− deg vk ) = reg(S/Ik−1 : (vk )) + deg vk . If vk =
xi x j+1 · · · xn y1 · · · yi−1 y j , by using Remark 2.5, we obtain reg(S/Ik−1 : (vk )) = j − i − 2.
As deg vk = n − j + i + 1, we get reg(S/Ik−1 : (vk ))(− deg vk ) = n − 1. Let us assume, by
induction, that reg S/Ik−1 ≤ n − 2. Then, by using the sequence (5), we obtain
reg S/Ik ≤ max{reg(S/Ik−1 : (vk ))(− deg vk ) − 1, reg S/Ik−1 } = n − 2.

Proposition 2.8. We have proj dimS/ in< (JG ) = n and reg S/ in< (JG ) = n − 2.
Proof. As in< (JG ) = Im , by Lemma 2.7, we get proj dim S/ in< (JG ) ≤ n and reg S/ in< (JG ) ≤
n − 2. For the other inequalities, we use [4, Theorem 3.3.4] which gives the inequalities
n = proj dimS/JG ≤ proj dim S/ in< (JG ) and n − 2 = reg S/JG ≤ reg S/ in< (JG ).

Theorem 2.9. Let G be a cycle. Then S/ in< (JG ) and S/JG have
the same extremal Betti
n−1
number, namely βn,2n−2 (S/JG ) = βn,2n−2 (S/ in< (JG )) = 2 − 1.

Proof. We only need to show that βn,2n−2 (S/ in< (JG )) = m since m = (n2 − 3n)/2 =
n−1
2 − 1.
We use again the sequence (5). By considering its long exact sequence of Tor’s and
using the equality Torn−1 (S/Ik−1 , K)2n−2 = 0, we get
0 → Torn (S/Ik−1, K)2n−2 → Torn (S/Ik , K)2n−2 → Torn−1 (S/Ik−1 : (vk ), K)2n−2−deg vk → 0,

for 1 ≤ k ≤ m. By Remark 2.5, if vk = xi x j+1 · · · xn y1 · · · yi−1 y j , then Ik−1 : (vk ) is generated
by a regular sequence of monomials which contains j − i − 2 elements of degree 2 and
n − j + i + 1 variables. This implies that dimK Torn−1 (S/Ik−1 : (vk ), K)2n−2−deg vk = 1.
Therefore, we get
dimK Torn (S/Ik , K)2n−2 = dimK Torn (S/Ik−1 , K)2n−2 + 1
for 1 ≤ k ≤ m. By summing up all these equalities, it follows that

βn,2n−2 (S/ in< (JG )) = dimK Torn (S/Im , K)2n−2 = dimK Torn (S/I, K)2n−2 + m = m.


The last equality is due to Lemma 2.6.

Remark 2.10. There are examples of graphs whose edge ideal have several extremal Betti
numbers. For instance, the graph G displayed below has two extremal Betti numbers
which are equal to the extremal Betti numbers of in< (JG ).
7

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FACULTY OF M ATHEMATICS AND C OMPUTER S CIENCE , OVIDIUS U NIVERSITY , B D . M AMAIA 124,
900527 C ONSTANTA , AND U NIVERSITY OF S OUTH -E AST E UROPE L UMINA , S OS . C OLENTINA NR .
64 B , B UCHAREST , ROMANIA
E-mail address: ahmet.dokuyucu@lumina.org

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