arXiv:1301.0784v1 [] 4 Jan 2013

ON APPROXIMATELY COHEN-MACAULAY BINOMIAL EDGE
IDEAL
SOHAIL ZAFAR
Abstract. Binomial edge ideals IG of a graph G were introduced by [4]. They found
some classes of graphs G with the property that IG is a Cohen-Macaulay ideal. This
might happen only for few classes of graphs. A certain generalization of being CohenMacaulay, named approximately Cohen-Macaulay, was introduced by S. Goto in [3]. We
study classes of graphs whose binomial edge ideal are approximately Cohen-Macaulay.
Moreover we use some homological methods in order to compute their Hilbert series.

INTRODUCTION
Let K denote a field. Let G denote a connected undirected graph over the vertices
labeled by [n] = {1, 2, . . . , n}. Let R = K[x1 , . . . , xn ] denote the polynomial ring in n
variables over the field K. The edge ideal I of G is generated by all xi xj , i < j ,
such that {i, j} forms an edge of G. This notion was studied by Villarreal [7] where it is
also discussed under which circumstances R/I is a Cohen-Macaulay ring. It seems to be
hopeless to characterize all of the graphs G such that R/I is a Cohen-Macaulay ring.
In a similar way one might define the binomial edge ideal IG ⊆ S = K[x1 , . . . , xn , y1, . . . , yn ].
It is generated by all binomials xi yj − xj yi , i < j , such that {i, j} is an edge of G. In the
paper of V. Ene, J. Herzog, T. Hibi [2], the authors start with the systematic investigation
of IG . There is a primary decomposition of IG . Moreover there are examples such that
S/IG is a Cohen-Macaulay ring. Furthermore, the authors believe that it is hopeless to
characterize those G such that the binomial edge ideal is Cohen-Macaulay in general.
A generalization of the notion of a Cohen-Macaulay ring was introduced by S. Goto
[3] under the name approximately Cohen-Macaulay (see also the Definition 1.4). Not so
many examples of approximately Cohen-Macaulay rings are known. Note that they are
not a domain. In the present paper we collect a few graphs G such that the associated
ring S/IG is an approximately Cohen-Macaulay ring, where IG denotes the binomial edge
ideal.
In fact, we give a complete characterization of all connected trees whose binomial edge
ideal are approximately Cohen-Macaulay. Such trees are described as 3-star like. We also
prove that the cycle of length n is approximately Cohen-Macaulay.
It is well known fact that the canonical module ω(S/I) of a Cohen-Macaulay ring is a
Cohen-Macaulay module. The converse is not true. Examples of non Cohen-Macaulay
rings such that ω(S/I) is Cohen-Macaulay are approximately Cohen-Macaulay rings (as
follows by the definition). So we investigate the canonical modules and the modules
2000 Mathematics Subject Classification. 05C05, 05C38, 05E40, 13H10.
Key words and phrases. Binomail edge ideal, Approximately Cohen-Macaulay, Trees, Cycle.
This research was partially supported by Higher Education Commission, Pakistan.
1

2

SOHAIL ZAFAR

of deficiency. As applications of our investigations we study the Hilbert series of our
examples.
The paper is organized as follows: In the first section there is a summary of preliminary results. In section 2 we give the characterization of approximately Cohen-Macaulay
binomial edge ideals for trees. In section 3 we prove that the binomial edge ideal of any
cycle is approximately Cohen-Macaulay. In addition we compute the Hilbert series of the
corresponding ideals. In order to do that for the n-cycle, we include some investigations
on the canonical module of the binomial edge ideal of a complete graph.
1. PRELIMINARY RESULTS
In this section we will fix the notations we use in the sequel. Moreover we summarize
auxiliary results that we need in our paper. We have tools from combinatorial algebra,
canonical modules and commutative algebra. In general we fix an arbitrary field K and
the polynomial ring S = K[x1 , . . . , xn , y1 , . . . , yn ] in 2n variables x1 , . . . , xn and y1 , . . . , yn .

(a) Combinatorial Algebra.
By G a graph on the vertex set [n] we always understand a connected simple graph.
The binomial edge ideal JG ⊂ S is defined as the ideal generated by all binomials fij =
xi yj − xj yi , 1 ≤ i < j ≤ n, where {i, j} is an edge of G. This construction has been
found in [4]. For a subset T ⊂ [n] let be G[n]\T the graph obtained from G by deleting all
vertices that belongs to T . Let c = c(T ) denote the number of connected components
of G[n]\T . Let G1 , ...Gc denote these components of G[n]\T . Let G̃ denotes the complete
graph on the vertex set of G. Define
PT (G) = (∪i∈T {xi , yi }, JG̃1 , . . . , JG̃C(T ) )
where G̃i , i = 1, ..., c = c(T ), denotes the complete graph on the vertex set of the connected
component of Gi . Then PT (G) ⊂ S is a prime ideal of height n+ |T | −c, where |T | denotes
the number of elements of T . Moreover
JG = ∩T ⊆[n] PT (G).
For these and related facts we refer to [4]. For us the following lemma is important.
Lemma 1.1. With the previous notations it follows that JG ⊂ PT (G) is a minimal prime
if and only if either T = ∅ or T 6= ∅ and for each i ∈ T we have that c(T \{i}) < c(T ).
For the proof see corollary 3.9 of [4]. It is noteworthy to say that JG is the intersection
of prime ideals. That is, S/JG is reduced. Finally we observe that S/JG is a graded ring
with natural grading induced by the grading of S.

(b) Canonical Modules.
Let S as above denote the polynomial ring. Let M denote the finitely generated graded
S−module. For some of our arguments we use a few basic facts about duality. In the
following we introduced the modules of deficiency related to M.

ON APPROXIMATELY COHEN-MACAULAY BINOMIAL EDGE IDEAL

3

Definition 1.2. For an integer i define
ω i (M) := ExtS2n−i (M, S)
the i − th module deficiency to M. Note that ω i (M) = 0 for i < depth(M) and i >
dim(M) = d. For i = d we call
ω(M) = ω d (M)
the canonical module of M.
These modules have been introduced in [6]. See also [6] for some properties of ω i (M)
and related facts. In particular we have
dim(ω(M)) = dim(M)
and dim(ω i (M)) ≤ i for all 0 ≤ i < d.
Lemma 1.3. With the previous notation let I ⊂ S be the homogenous ideal then :
(1) There is a natural homomorphism
S/I → ω(ω(S/I)) ∼
= Hom(ω(S/I), ω(S/I))
which is an isomorphism if S/I is a Cohen-Macaulay ring.
(2) Suppose that S/I is Cohen-Macaulay ring with dim(S/I) > 0. Suppose that ω(S/I)
is an ideal of S/I then (S/I)/ω(S/I) is a Gorenstein ring with dim(S/I) − 1.
For the proof we refer to [6] and [1].

(c) Approximately Cohen-Macaulay rings.
For our purpose here we need a certain generalization of Cohen-Macaulay rings that
was originally introduced by S.Goto (see [3]) in the case of local rings.
Definition 1.4. Let R denote a commutative ring of finite dimension d and I ⊂ R an
ideal. Then
(1) AsshR (R/I) = {p ∈ AssR (R/I) : dim R/p = dim R/I}.
(2) UR (I) = ∩p∈AsshR (R/I) I(p) where I = ∩p∈AssR (R/I) I(p) denotes a minimal primary
decomposition of the ideal I.
That is UR (I) describes the equidimensional part of the primary decomposition of the
ideal I.
As an analogue to Goto’s notion of approximately Cohen-Macaulay rings we define a
graded version of it.
Definition 1.5. Let R = ⊕ Ri denote a standard K−algebra, where K = R0 . It is called
i≥0

approximately Cohen-Macaulay if R+ = ⊕ Ri contains a homogeneous element x such
i>0

that R/xn R is a Cohen-Macaulay ring of dim(R) − 1 for all n ≥ 1.
A characterization of approximately Cohen-Macaulay rings can be done by the following
theorem.
Theorem 1.6. Let I ⊂ S denote a homogenous ideal and d = dim(S/I). Then the
following conditions are equivalent.

4

SOHAIL ZAFAR

(i): S/I is approximately Cohen-Macaulay.
(ii): S/US (I) is a d−dimensional Cohen-Macaulay ring and depth(S/I) ≥ d − 1.
(iii): ω d (S/I) is Cohen-Macaulay module of dimension d and ω d−1(S/I) is either
zero or a (d − 1)−dimensional Cohen-Macaulay module.
Proof. In the case of local ring the above theorem was proved by Goto in the paper of
[3]. The proof in the graded case follows by the same arguments. We have to adopt the
graded situation from the local one. We omit the details.

Remark 1.7. A necessary condition for S/I to be an approximately Cohen-Macaulay ring
is that dim S/p ≥ dim S/I − 1 for all p ∈ Ass(S/I)
2. TREES
In this section we will characterize all trees which the property that the associated
binomial edge ideal defines an approximately Cohen-Macaulay ring. A graph G is called
tree if it is connected and has no cycles. The simplest tree is the line. The binomial edge
ideal of a line is a complete intersection. So it defines a Cohen-Macaulay ring (see [4]).
In our consideration we do not consider the line in detail. To classify all the trees that
are approximately Cohen-Macaulay, we have to introduce a new terminology called 3-star
like trees.
Definition 2.1. A tree G is called 3-star like if there are no vertices of degree ≥ 4 and if
either there is at most one vertex of degree 3 or there is at most one edge with the property
that both of its vertices are of degree 3. Then the line is of course 3-star like. Other types
of 3-star like trees (see the figures 1 and 2).

1

3

4

2

Figure 1.
Example 2.2. Consider the simplest example of the 1st type of 3-star like tree as shown
in figure 1. Its binomial edge ideal is approximately Cohen-Macaulay of dimension 6,
depth 5 and the Hilbert Series is
H(S/JG , t) =

1 + 2t − 2t3
.
(1 − t)6

ON APPROXIMATELY COHEN-MACAULAY BINOMIAL EDGE IDEAL

5

5

1

4

2
3

6

Figure 2.
Example 2.3. The simplest example in case of 2nd type of 3-star like tree is shown above.
Its binomial edge ideal is approximately Cohen-Macaulay of dimension 8, depth 7 and the
Hilbert Series is
1 + 4t + 5t2 − 3t4
.
H(S/JG , t) =
(1 − t)8
In order to prove the approximately Cohen-Macaulay property of binomial edge ideals
we need the following construction principle. It will be useful also in different circumstances.
Lemma 2.4. Let G be any connected graph with vertices set [n] having at least one vertex
of degree 1, choose one of them and label it by n. Let G′ be a graph on vertices set [n + 1]
by attaching one edge {n, n+1} to the graph G. Now G is approximately Cohen-Macaulay
if and only if G′ is approximately Cohen-Macaulay.
Proof. Suppose that JG and JG′ denotes the binomial edge ideal of the corresponding
graphs. Let dim(S/JG ) = d and depth(S/JG ) = d − 1. Now JG′ = (JG , f ) where f =
xn yn+1 −xn+1 yn and S ′ = S[xn+1 , yn+1 ], therefore dim(S ′ /JG ) = d+2 and depth(S ′ /JG ) =
d + 1.
Now n ∈
/ T for all T ⊆ [n] such that c(T \{i}) < c(T ). Which implies xn , yn ∈
/ PT (G) for
all PT (G) ∈ Ass(S/JG ), hence f is not a zero divisor in S/JG and is regular. Therefore
dim(S ′ /JG′ ) = d + 1 and depth(S ′ /JG′ ) = d.
Consider the exact sequence
f

0 → S ′ /JG (−2) → S ′ /JG → S ′ /JG′ → 0.
Apply Hom(S ′ , .) to above sequence we get the long exact sequence
f

0 → ω d+2 (S ′ /JG ) → ω d+2 (S ′ /JG )(2) → ω d+1 (S ′ /JG′ )
f

→ ω d+1 (S ′ /JG ) → ω d+1 (S ′ /JG )(2) → ω d(S ′ /JG′ ) → 0.
Clearly f is ω d+2(S ′ /JG )−regular. Moreover f is ω d+1 (S ′ /JG )−regular since f is S/JG regular. So multiplication by f is injective and we get the following two isomorphisms:
ω d+1 (S ′ /JG′ ) ∼
= ω d+2 (S ′ /JG )/f ω d+2 (S ′ /JG )(2)
and
ω d (S ′ /JG′ ) ∼
= ω d+1 (S ′ /JG ))/f ω d+1(S ′ /JG )(2).

6

SOHAIL ZAFAR

Hence ω d+2 (S ′ /JG ) and ω d+1 (S ′ /JG ) is of dimension d + 2 and d + 1 resp. Therefore
both are Cohen-Macaulay if and only if ω d+1 (S ′ /JG′ ) and ω d (S ′ /JG′ ) are Cohen-Macaulay
of dimension d+1 and d resp. Therefore from theorem 1.6 S ′ /JG′ is approximately CohenMacaulay if and only if S/JG is approximately Cohen-Macaulay.

Corollary 2.5. Let G be a tree with vertices set [n], then S/JG is approximately CohenMacaulay if and only if G is 3-star like.
Proof. Suppose the contrary that G is not 3-star like then we have two cases:
Case 1: If G contains at least one vertex of degree d ≥ 4 say i. Then T = {i} and
c(T ) = d so we have dim(S/PT (G)) = n + d − 1 ≥ n + 3.
Case 2: If G has two vertices of degree 3 which are not adjacent say i and j. Then
T = {i, j} and c(T ) = 5. Therefore we have dim(S/PT (G)) = n + 3.
Therefore in both cases dim(S/JG ) ≥ n + 3.
On the other hand JG̃ ∈ Ass(S/JG ) and dim(S/JG̃ ) = n + 1 so depth(S/JG ) ≤ n + 1
and hence S/JG is not approximately Cohen-Macaulay.
Conversely, in order to prove that any 3-star like tree is approximately Cohen-Macaulay
we will use induction on n, the number of vertices. For the case of a line there is nothing
to prove. For n = 4 and n = 6 it is true see example 2.2. and 2.3 above. The general
case follows by the construction principle in Lemma 2.4.

Now we will discuss some properties of the trees which are approximately CohenMacaulay. In Corollary 2.5. we have shown that 3-star like trees with vertices set [n]
have dimension n + 2 and depth n + 1. Now the Hilbert series of of 3-star like trees can
be easily computed.
Lemma 2.6. With the notations of Lemma 2.4, We have
H(S ′/JG′ , t) = (1 − t2 )H(S ′/JG , t).
Proof. Consider the exact sequence
f

0 → S ′ /JG (−2) → S ′ /JG → S ′ /JG′ → 0.
Hence we have a required result.



Corollary 2.7. Let G be a 3-star like tree with vertex set [n]. Then the Hilbert Series of
S/JG is
(1 + 2t − 2t3 )(1 + t)n−4
for n > 3
(1 − t)n+2
(1 + 4t + 5t2 − 3t4 )(1 + t)n−6
and
for n > 5 respectively
(1 − t)n+2
for the first resp. the second type of 3-starlike trees.
H(S/JG , t) =

Proof. We will prove this by induction on n. Consider the first case of 3-starlike trees.
For n = 4 it is true, see example 2.2 resp. 2.3. Suppose the claim is true for n. That is,
(1 + 2t − 2t3 )(1 + t)n−4
.
H(S ′/JG , t) =
(1 − t)n+4

ON APPROXIMATELY COHEN-MACAULAY BINOMIAL EDGE IDEAL

7

Now by Lemma 2.6. we have
H(S ′ /JG′ , t) =

(1 + 2t − 2t3 )(1 + t)n−3
(1 − t)n+3

as required.
Similar arguments might be used in order to calculate the Hilbert Series in the second
case of 3-starlike trees.

3. CYCLE
In this section we study the algebraic properties of the binomial edge ideal associated
to a cycle. A graph G is called cycle if it is a closed directed path, with no repeated
vertices other than the starting and ending vertices. We denote the cycle of length n by
C and its binomial edge ideal by IC . To study further properties of IC or S/IC we shall
need few basic properties from [4]. If
IC = ∩T ⊆[n] PT (C)
then
dimS/PT (C) = n + 1 if T = ∅ and dimS/PT (C) ≤ n if T 6= ∅.
Hence
dim(S/IC ) = n + 1 and US (IC ) = P∅ (C) = JG̃ .
Moreover PT (C) is minimal prime of S/IC if either (i) T = ∅ or (ii) if T 6= ∅ and |T | > 1
and no two elements i, j ∈ T belongs to the same edge of C.
Theorem 3.1. x1 , y1 − x2 , . . . , yn−1 − xn , yn is the system of parameters for S/IC .
Proof. Let x = x1 , y1 − x2 , . . . , yn−1 − xn , yn , M = S/IC and I = Ann(M/xM). Now
M/xM = S/(IC , x) . If we replace y1 = x2 , y2 = x3 , . . . , yn−1 = xn in IC , we get
I = (x1 , x1 x3 − x22 , x2 x4 − x23 , . . . , xn−2 xn − x2n−1 , x2n , xn x2 ).
Clearly x1 , xn ∈ Rad(I), we need to prove that x2 , . . . , xn−1 ∈ Rad(I). If x1 , x1 x3 − x22 ∈ I
then x22 ∈ I. It follows that x2 ∈ Rad(I), hence we have a basis of induction. If xk ∈
Rad(I) for 2 ≤ k ≤ n−2 then xk xk+2 −x2k+1 ∈ I as k+1 ≤ n−1. Therefore x2k+1 ∈ Rad(I)
it follows that xk+1 ∈ Rad(I). This then implies that Rad(I) = (x1 , x2 , . . . , xn ). Hence I
is m − primary in the ring K[x1 , . . . , xn ] so x is the system of parameters of S/IC .

Lemma 3.2. Let IL be the binomial edge ideal of a line L of length n , g = x1 yn − xn y1
and JG̃ be binomial edge ideal of a complete graph on [n] then
(a): If IL = ∩T ⊆[n] PT (L) then g ∈
/ PT (L) for T 6= ∅.
(b): IL = (IL : g) ∩ JG̃ .
(c): IL : (IL : g) = JG̃ .
(d): IL : g = IL : JG̃ .
Proof.

(a): Because of IL = ∩T ⊆[n] PT (L), it is known from [4] that PT (L) is minimal
prime of IL if either (i) T = ∅ or (ii) if T 6= ∅ and 1, n ∈
/ T and if |T | > 1 then
there are no two elements i, j ∈ T such that {i, j} is an edge of L. If T = ∅, then
PT (L) = JG̃ is the ideal of complete graph. Now let T 6= ∅ and 1, n ∈
/ T . Suppose

8

SOHAIL ZAFAR

that |T | > 1 then g ∈
/ PT (L) because x1 , y1 , xn , yn does not belongs to ∪i∈T {xi , yi }
and g does not belongs to any JG̃ for any connected component of [n]\T
(b): IL : g = ∩g∈P
/ T (L) PT (L), Using (a) we have
IL : g = ∩T 6=∅ PT (L)
and IL = ∩T 6=∅ PT (L) ∩ P∅ (L) therefore,
IL = (IL : g) ∩ JG̃ .
(c): Using (b) we have
IL : (IL : g) = ((IL : g) ∩ JG̃ ) : (IL : g) = JG̃ : (IL : g).
In order to finish we have to prove that JG̃ = JG̃ : (IL : g). From the definition
JG̃ ⊆ JG̃ : (IL : g). Now for the other inclusion let IL : g = (h1 , h2 , ..., hr ), then
JG̃ : (IL : g) = ∩ri=1 JG̃ : hi . Now JG̃ : hi = JG̃ for at least one i, since JG̃ is a prime
ideal and hi ∈
/ JG̃ for at least one i so JG̃ : (IL : g) ⊆ JG̃ and we are done.
(d): IL : g ⊆ IL : JG̃ is trivial. For another inclusion let f ∈ IL : JG̃ then f JG̃ ⊆
IL ⊆ PT (L), now JG̃ * PT (L) for T 6= ∅, so f ∈ PT (L) for T 6= ∅ which implies
f ∈ IL : g.

Definition 3.3. [5] Two ideals I and J of height g in S are said to be linked if there is
a regular sequence α of height g in their intersection such that I = α : J and J = α : I.
It is also known from [5] that I and J are two linked ideals of S then S/I is CohenMacaulay if and only if S/J is Cohen-Macaulay.
Lemma 3.4. Let IL be the binomial edge ideal of a line of length n then S/IL : g is
Cohen-Macaulay of dimension n + 1.
Proof. IL is a complete intersection and using (b), (c) and (d) of Lemma 3.2 we have
IL : g and JG̃ are linked ideals. Now it follows from above theorem that S/IL : g is
Cohen-Macaulay because S/JG̃ is Cohen-Macaulay.

Theorem 3.5. Any cycle of length n ≥ 3 is approximately Cohen-Macaulay.
Proof. First we will compute the depth of S/IC for n ≥ 3. From above notations IC =
(IL , g). Consider the exact sequence
0 → S/IL : g(−2) → S/IL → S/IC → 0.
Now it follows from the Depth’s Lemma that
depth(S/IC ) ≥ min{depth(S/IL : g) − 1, depth(S/IL )} = n.
Hence depth(S/IC ) ≥ n.
Now S/US (IC ) ∼
= S/JG̃ , which is n + 1−dimensional Cohen-Macaulay ring, so from
Theorem 1.6. S/IC is approximately Cohen-Macaulay.

Furthermore we will find the Hilbert series of S/IC . For this we have to introduce a
monomial ideal M = (x2 x3 · · · xn−1 , x2 x3 · · · xn−2 yn−1, . . . , x2 y3 · · · yn−1 , y2 y3 · · · yn−1 ). We
need also the expression for the Hilbert series of S/JG̃ . Recall the result in [1], if R is

ON APPROXIMATELY COHEN-MACAULAY BINOMIAL EDGE IDEAL

9

Cohen-Macaulay ring with dimension d and x = x1 , . . . , xd be the homogenous system
of parameters of degree 1 then
H(R/xR, t)
H(R, t) =
.
(1 − t)d
Now in our case S/JG̃ is Cohen-Macaulay ring with dimension n + 1, x = x1 , y1 −
x2 , . . . , yn−1 − xn , yn is system of parameter of degree 1 of S/JG̃ and
S/(xS, J ) ∼
= K[y1 , . . . , yn−1]/(y1 , . . . , yn−1)2
G̃

therefore H(S/(xS, JG̃ ), t) = 1 + (n − 1)t and we have
H(S/JG̃ , t) =

1 + (n − 1)t
.
(1 − t)n+1

Lemma 3.6. With the notations above we have
ω(S/JG̃ ) ∼
= (JG̃ , M)/JG̃ .
Proof. JG̃ is the ideal of all 2-minors of a generic 2 × n−matrix which implies all 2-minors
of a generic 2 × n−matrix are zero in S/JG̃ hence both rows of this matrix are linearly
dependent, therefore S/JG̃ ∼
= K[x1 , . . . , xn , x1 t, . . . , xn t]. It is known that [1]
ω(S/J ) ∼
= (x1 , y1)n−2 S/J
G̃

G̃

therefore

ω(S/JG̃ ) ∼
= (x1 , x1 t)n−2 K[x1 , . . . , xn , x1 t, . . . , xn t].
Next we consider the monomial ideal M in S/JG̃
MS/JG̃ ∼
= (x2 x3 · · · xn−1 , x2 x3 · · · xn−1 t, . . . , x2 x3 · · · xn−1 tn−2 )K[x1 , . . . , xn , x1 t, . . . , xn t].
Now after multiplying ω(S/JG̃ ) by x2 x3 · · · xn−1 and MS/JG̃ by x1 respectively, we see
that both are isomorphic that is
ω(S/J ) ∼
= (J , M)/J .
= MS/J ∼
G̃

G̃

G̃

G̃


Theorem 3.7. S/(JG̃ , M) is Gorenstein of dimension n.
Proof. From Lemma 1.3. and 3.6. We have
(S/JG̃ )/MS/JG̃ ∼
= S/(JG̃ , M).
which is Gorenstein of dimension n.



Lemma 3.8. With the notations above we have
IL : g = (IL , M) and hence (IL : g, JG̃ ) = (M, JG̃ ).
Proof. First we will show that M ⊆ IL : g. Because IL ⊆ IL : g it will be enough to prove
that
(IL , M)/IL ⊆ IL : g/IL
in S/IL . That is, we have always
xi yi+1 ≡ xi+1 yi mod IL

10

SOHAIL ZAFAR

for i = 1, . . . , n − 1. Now let x2 x3 · · · xn−1 ∈ M. With g = x1 yn − xn y1 , we get
x2 x3 · · · xn−1 g ≡ x1 · · · xn−1 yn − x2 · · · xn−1 xn y1 mod IL .
Now put xn−1 yn ≡ xn yn−1 mod IL and xn−2 yn−1 ≡ xn−1 yn−2 mod IL and so on. After
(n − 1) steps it follows that
x2 x3 · · · xn−1 g ≡ 0 mod IL .
This proves that x2 x3 · · · xn−1 ∈ IL : g. Similarly y2 y3 · · · yn−1 ∈ IL : g. Now take any
arbitrary element x2 · · · xi yi+1 · · · yn−1 ∈ M then
x2 · · · xi yi+1 · · · yn−1 g ≡ y1 x2 · · · xi yi+1 · · · yn−1 xn − x1 · · · xi yi+1 · · · yn mod IL .
We have xi yi+1 ≡ xi+1 yi mod IL . Therefore
x2 · · · xi yi+1 · · · yn−1 g ≡ y1 x2 · · · xi yi+1 · · · yn−1 xn −x1 · · · xi−2 (xi−1 yi )(xi+1 yi+2 )yi+3 · · · yn mod IL .
Now replace xi−1 yi = xi yi−1 and xi+1 yi+2 = xi+2 yi+1 in S/IL , we have
x2 · · · xi yi+1 · · · yn−1 g ≡ y1 x2 · · · xi yi+1 · · · yn−1 xn −x1 · · · (xi−2 yi−1 )xi yi+1 (xi+2 yi+3 ) · · · yn mod IL .
If we continue such replacements, we get that
x2 · · · xi yi+1 · · · yn−1 g ≡ 0 mod IL .
So M ⊆ IL : g hence
(M, JG̃ ) ⊆ (IL : g, JG̃ ).
S/IL is a Gorenstein ring and S/IL ։ S/JG̃ , therefore from Lemma 1.3
ω(S/J ) ∼
= (IL : J )/IL .
= HomS/I (S/J , S/IL ) ∼
G̃

L

G̃

G̃

Using Lemma 3.2.(d) we get
ω(S/JG̃ ) ∼
= (IL : g)/IL .
Now from Lemma 3.6 we have that
ω(S/JG̃ ) ∼
= (JG̃ , M)/JG̃ .
So there are two expressions for ω(S/JG̃ ). As the canonical module is unique up to isomorphism. We want to describe an isomorphism. To this end define a map
φ : (JG̃ , M)/JG̃ → (IL : g)/IL
which sends

n−1
P
i=1

ri mi + JG̃ to

n−1
P

ri mi + IL where m1 , . . . , mn−1 are generators of M. If

i=1

n−1
X

ri mi −

i=1

n−1
X

ri′ mi ∈ JG̃ .

i=1

Then this implies
n−1
X

(ri − ri′ )mi ∈ JG̃ ∩ M ⊆ JG̃ ∩ (IL , M) = IL .

i=1

This follows because of the inclusion
IL ⊆ JG̃ ∩ (IL , M) ⊆ JG̃ ∩ (IL : g) = IL

ON APPROXIMATELY COHEN-MACAULAY BINOMIAL EDGE IDEAL

11

as follows by Lemma 3.2 (b). Hence it’s a well define map and clearly a homomorphism.
φ ∈ HomS/J (ω(S/J ), ω(S/J )) ∼
= S/J (see Lemma 1.3.).
G̃

G̃

G̃

G̃

So any homomorphism ω(S/JG̃ ) → ω(S/JG̃ ) is given by multiplication by an element of
S/JG̃ . Because φ is a non-zero homomorphism of degree zero it is in fact an isomorphism.
That is
Im(φ) = (IL , M)/IL .
So finally we get
IL : g = (IL , M) and hence (IL : g, JG̃ ) = (M, JG̃ )

Lemma 3.9. The Hilbert series of S/(IL : g, JG̃ ) is
H(S/(IL : g, JG̃ ), t) =

1 + (n − 1)t − (n − 1)tn−2 − tn−1
(1 − t)n+1

Proof. It is known from [1] that
1
H(ω(S/JG̃ ), t) = (−1)n+1 H(S/JG̃ , ).
t
Therefore,
H(ω(S/JG̃ ), t) =

(n − 1)tn + tn+1
.
(1 − t)n+1

As we know

ω(S/JG̃ ) ∼
= (M, JG̃ )/JG̃ .
From the above formula the initial degree of ω(S/JG̃ ) is n while the initial degree of
(M, JG̃ )/JG̃ is n − 2. Therefore by dividing the non-zero divisor t2 we get the Hilbert
series of (M, JG̃ )/JG̃
H((M, JG̃ )/JG̃ , t) =

(n − 1)tn−2 + tn−1
.
(1 − t)n+1

Consider the exact sequence
0 → (M, JG̃ )/JG̃ → S/JG̃ → S/(M, JG̃ ) → 0.
From Lemma 3.8.
(IL : g, JG̃ ) = (M, JG̃ ).
Therefore
H(S/(IL : g, JG̃ ), t) = H(S/JG̃ , t) − H((M, JG̃ )/JG̃ , t).
Hence we get the required result.
Theorem 3.10. Hilbert series of S/IC is
(1 + t)n−1 − t2 (1 + t)n−1 + (n − 1)tn + tn+1
H(S/IC , t) =
.
(1 − t)n+1
In particular, the multiplicity of S/IC is e(S/IC ) = n.



12

SOHAIL ZAFAR

Proof. Hilbert series of binomial edge ideal of IL is easy to compute because IL is a
complete intersection generated by n − 1 forms of degree 2. Namely we have
(1 + t)n−1
(1 − t2 )n−1
=
.
H(S/IL , t) =
(1 − t)2n
(1 − t)n+1
Consider the exact sequence
0 → S/IL → S/IL : g ⊕ S/JG̃ → S/(IL : g, JG̃ ) → 0.
Therefore
H(S/IL : g, t) = H(S/(IL : g, JG̃ ), t) + H(S/IL , t) − H(S/JG̃ , t).
So by using Lemma 3.9. We get
(1 + t)n−1 − (n − 1)tn−2 − tn−1
.
(1 − t)n+1
Consider the another exact sequence and replace (IL , g) = IC
H(S/IL : g, t) =

0 → S/IL : g(−2) → S/IL → S/IC → 0.
We have
H(S/IC , t) = H(S/IL , t) − t2 H(S/IL : g, t).
and after putting values we get the desired formula.



References
[1] W. Burns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1993.
[2] V. Ene, J. Herzog, T. Hibi, Cohen Macaulay Binomial edge ideals, Nagoya Math. J. 204 (2011),
57-68.
[3] S Goto, Approximately Cohen-Macaulay Rings. J. Algebra 76 (1982), 214-225.
[4] J. Herzog, T. Hibi, F. Hreinsdotir, T.Kahle, J, Rauh, Binomial edge ideals and conditional independence statements, Adv. Appl. Math. 45 (2010), 317-333.
[5] C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Inv. math, 26 (1974), 271-302.
[6] P. Schenzel, On the use of local cohomology in algebra and geometry. Elias, J. (ed.) et al., Six
lectures on commutative algebra. Lectures presented at the summer school, Bellaterra, Spain, July
16-26, 1996. Basel: Birkhuser. Prog. Math. 166 (1998), 241-292.
[7] R. H. Villarreal, Monomial Algebras, New York: Marcel Dekker Inc. (2001).
Abdus Salam School of Mathematical Sciences, GCU, Lahore Pakistan
E-mail address: sohailahmad04@gmail.com

