arXiv:1508.07119v1 [] 28 Aug 2015

BI-COHEN-MACAULAY GRAPHS
JÜRGEN HERZOG AND AHAD RAHIMI
Abstract. In this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General bi-Cohen-Macaulay graphs are classified up to separation. The inseparable
bi-Cohen-Macaulay graphs are determined. We establish a bijection between the
set of all trees and the set of inseparable bi-Cohen-Macaulay graphs.

Introduction
A simplicial complex ∆ is called bi-Cohen-Macaulay (bi-CM), if ∆ and its Alexander dual ∆∨ are Cohen-Macaulay. This concept was introduced by Fløystad and
Vatne in [8]. In that paper the authors associated to each simplicial complex ∆ in
a natural way a complex of coherent sheaves and showed that this complex reduces
to a coherent sheaf if and only if ∆ is bi-CM.
The present paper is an attempt to classify all bi-CM graphs. Given a field K and
a simple graph on the vertex set [n] = {1, 2, . . . , n}, one associates with G the edge
ideal IG of G, whose generators are the monomials xi xj with {i, j} an edge of G. We
say that G is bi-CM if the simplicial complex whose Stanley-Reisner ideal coincides
with IG is bi-CM. Actually, this simplicial complex is the so-called independence
complex of G. Its faces are the independent sets of G, that is, subsets D of [n] with
{i, j} 6⊂ D for all edges {i, j} of G.
By its very definition, any bi-CM graph is also a Cohen-Macaulay graph (CM
graph). A complete classification of all CM graphs is hopeless if not impossible.
However, such a classification is given for bipartite graphs [10, Theorem 3.4] and for
chordal graphs [11]. We refer the reader to the books [9] and [14] for a good survey
on edge ideals and its algebraic and homological properties.
Based on the classification of bipartite and chordal CM graphs, we provide in
Section 2 a classification of bipartite and chordal bi-CM graphs, see Theorem 2.1
and Theorem 2.2. In Section 1 we first present various characterizations of bi-CM
graphs. By using the Eagon-Reiner theorem [5], one notices that the graph G is
bi-CM if and only if it is CM and IG has a linear resolution. Cohen-Macaulay ideals
generated in degree 2 with linear resolution are of very special nature. They all
arise as deformations of the square of the maximal ideal of a suitable polynomial
2010 Mathematics Subject Classification. 05E40, 13C14.
Key words and phrases. Bi-Cohen–Macaulay, Bipartite and chordal graphs, Generic graphs,
Inseparability.
This paper was written during the visit of the second author at Universität Duisburg-Essen,
Campus Essen. He is grateful for its hospitality.
1

ring. From this fact arise constraints on the number of edges of the graph and on
the Betti numbers of IG .
Though a complete classification of all bi-CM graphs seems to be again impossible,
a classification of all bi-CM graphs up to separation can be given, and this is the
subject of the remaining sections.
A separation of the graph G with respect to the vertex i is a graph G′ whose vertex
set is [n] ∪ {i′ } having the property that G is obtained from G′ by identifying i with
i′ and such that xi − xi′ is a non-zerodivisor modulo IG′ . The algebraic condition
on separation makes sure that the essential algebraic and homological invariants of
IG and IG′ are the same. In particular, G is bi-CM if and only if G′ is bi-CM. A
graph which does not allow any separation is called inseparable, and a inseparable
graph which is obtained by a finite number of separation steps from G is called
a separable model of G. Any graph admits separable models and the number of
separable models of a graph is finite. Separable and inseparable graphs from the
view point of deformation theory have been studied in [1].
In Section 4 we determine all inseparable bi-CM graphs on [n] vertices. Indeed,
in Theorem 4.4 it is shown that for any tree T on the vertex set [n] there exists a
unique inseparable bi-CM graph GT determined by T , and any inseparable bi-CM
graph is of this form. Furthermore, if G is an arbitrary bi-CM graph and T is the
relation graph of the Alexander dual of IG , then GT is a separable model of G.
For a bi-CM graph G, the Alexander dual J = (IG )∨ of IG is a Cohen-Macaulay
ideal of codimension 2 with linear resolution. As described in [3], one attaches to
any relation matrix of J a relation tree T . Replacing the entries in this matrix
by distinct variables with the same sign, one obtains the so-called generic relation
matrix whose ideals of 2-minors JT and its Alexander has been computed in [13].
This theory is described in Section 3. The Alexander dual of JT is the edge ideal
of graph, which actually is the graph GT mentioned before and which serves as a
separable model of G.
1. Preliminaries and various characterizations of
Bi-Cohen-Macaulay graphs
In this section we recall some of the standard notions of graph theory which are
relevant for this paper, introduce the bi-CM graphs and present various equivalent
conditions of a graph to be bi-CM.
The graphs considered here will all be finite, simple graphs, that is, they will
have no double edges and no loops. Furthermore we assume that G has no isolated
vertices. The vertex set of G will be denoted V (G) and will be the set [n] =
{1, 2, . . . , n}, unless otherwise stated. The set of edges of G we denote by E(G).
A subset F ⊂ [n] is called a clique of G, if {i, j} ∈ E(G) for all i, j ∈ F with
i 6= j. The set of all cliques of G is a simplicial complex, denoted ∆(G).
A subset C ⊂ [n] is called a vertex cover of G if C ∩ {i, j} =
6 ∅ for all edges {i, j}
of G. The graph G is called unmixed if all minimal vertex covers of G have the
same cardinality. This concept has an algebraic counterpart. We fix a field K and
consider the ideal IG ⊂ S = K[x1 , . . . , xn ] which is generated by all monomials xi xj
2

with {i, j} ∈ E(G). The ideal IG is called the edge ideal of G. Let C ⊂ [n]. Then the
monomial prime ideal PC = ({xi : i ∈ C}) is a minimal prime ideal of IG if and only
if C is a minimal vertex cover of G. Thus G is unmixed if and only if IG is unmixed in
the algebraic sense. A subset D ⊂ [n] is called an independent set of G if D contains
no set {i, j} which is an edge of G. Note that D is an independent set of G if and
only if [n] \ D is a vertex cover. Thus the minimal vertex covers of G correspond to
the maximal independent sets of G. The cardinality of a maximal independent is
called the independence number of G. It follows that the Krull dimension of S/IG
is equal to c, where c is the independence number of G.
The graph G is called bipartite if V (G) is the disjoint union of V1 and V2 such that
V1 and V2 are independent sets, and G is called disconnected if V (G) is the disjoint
union of W1 and W2 and there is no edge {i, j} of G with i ∈ W1 and j ∈ W2 . The
graph G is called connected if it is not disconnected.
A cycle C (of length r) in G is a sequence of edges {ik , jk } with k = 1, 2, . . . , r
such that jk = ik+1 for k = 1, . . . , r − 1 and jr = i1 . A cord of C is an edge {i, j}
of G with i, j ∈ {i1 , . . . , ir } and {i, j} is not an edge of C. The graph G is called
chordal if each cycle of G of length ≥ 4 has a chord. A graph which has no cycle
and which is connected is called a tree.
Now we recall the main concept we are dealing with in this paper. Let I ⊂ S be
T
a squarefree monomial ideal. Then I = m
j=1 Pj where each of the Pj is a monomial
∨
prime ideal of I. The ideal I which is minimally generated by the monomials
Q
uj = xi ∈Pj xi is called the Alexander dual of I. One has (I ∨ )∨ = I. In the case that
I = IG , each Pj is generated by the variables corresponding to a minimal vertex
cover of G. Therefore, (IG )∨ is also called the vertex cover ideal of G.
According to [8] a squarefree monomial ideal I ⊂ S is called bi-Cohen-Macaulay
(or simply bi-CM) if I as well as the Alexander dual I ∨ of I is a Cohen-Macaulay
ideal. A graph G is called Cohen-Macaulay or bi-Cohen-Macaulay (over K) (CM or
bi-CM for short), if IG is CM or bi-CM. One important result regarding the Alexander dual that will be used frequently in this paper is the Eagon-Reiner theorem
which says that I is a Cohen-Macaulay ideal if and only if I ∨ has a linear resolution. Thus the Eagon-Reiner theorem implies that I is bi-CM if and only if I is a
Cohen-Macaulay ideal with linear resolution. From this description it follows that
a bi-CM graph is connected. Indeed, if this is not the case, then there are induced
subgraphs G1 , G2 ⊂ G such that V (G) is the disjoint union of V (G1 ) and V (G2 ). It
follows that IG = IG1 + IG2 , and the ideals IG1 and IG2 are ideals in a different set
of variables. Therefore, the free resolution of S/IG is obtained as the tensor product
of the resolutions of S/IG1 and S/IG2 . This implies that IG has relations of degree
4, so that IG does not have a linear resolution.
From now on we will always assume that G is connected, without further mentioning it.
Proposition 1.1. Let K be an infinite field and G a graph on the vertex set [n]
with independence number c. The following conditions are equivalent:
(a) G is a bi-CM graph over K;
3

(b) G is a CM graph over K, and S/IG modulo a maximal regular sequence of
linear forms is isomorphic to T /m2T where T is the polynomial ring over K
in n − c variables and mT is the graded maximal ideal of T .
Proof. We only need to show that IG has a linear resolution if and only if condition
(b) holds. Since K is infinite and since S/IG is Cohen-Macaulay of dimension c, there
exists a regular sequence x of linear forms on S/IG of length c. Let T = S/(x). Then
T is isomorphic to a polynomial ring in n − c variables. Let J be the image of IG
in T . Then J is generated in degree 2 and has a linear resolution if and only if IG
has linear resolution. Moreover, J is mT -primary. The only mT -primary ideals with
linear resolution are the powers of mT . Thus, IG has a linear resolution if and only
if J = m2T .

Corollary 1.2. Let G be a graph on the vertex set [n] with independence number c.
The following conditions are equivalent:
(a) G is a bi-CM graph over K;


(b) G is a CM graph over K and |E(G)| = n−c+1
;
2
(c) G is a CM graph over K and the number of minimal vertex covers of G is
equal to n − c +1;

(d) βi (IG ) = (i + 1) n−c+1
for i = 0, . . . , n − c − 1.
i+2

Proof. For the proof of the equivalent conditions we may assume that K is infinite
and hence we may use Proposition 1.1.
2
(a) ⇐⇒ (b): With the notation of Proposition

1.1 we have J = mT if and only
n−c+1
if the number of generators of J is equal to
. Since IG and J have the same
2
number of generators and since the number of generators of IG is equal to |E(G)|,
the assertion follows.
(b) ⇐⇒ (c): Since S/IG is Cohen-Macaulay, the multiplicity of S/IG is equal to
the length ℓ(T /J) of T /J. On the other hand, the multiplicity is also the number
of minimal prime ideals of IG which coincides with the number of minimal vertex
covers of G. Thus the length of T /J is equal to the number of minimal vertex covers
of G. Since J = m2T if and only if ℓ(T /J) = n − c + 1, the assertion follows.
(a) ⇒ (d): Note that βi (IG ) = βi (J) for all i. Since J is isomorphic to the ideal
of 2-minors of the matrix
!
y1 y2 . . . yn−c
0
0 y1 . . . yn−c−1 yn−c
in the variables y1 , . . . , yn−c , the Eagon-Northcott complex ([4], [6]) provides a free
resolution of J and the desired result follows.
(d) ⇒ (a): It follows from the description of the Betti numbers of IG that
proj dim S/IG = n − c. Thus, depth S/IG = c. Since dimS/IG = c, it follows
that IG is a Cohen-Macaulay ideal. Since |E(G)| = β0 (IG ) = n−c+1
, condition (b)
2
is satisfied, and hence G is bi-CM, as desired.

Finally we note that G is a bi-CM graph over K if and only if the vertex cover ideal
of G is a codimension 2 Cohen-Macaulay ideal with linear relations. Indeed, let JG
4

be the vertex cover ideal of G. Since JG = (IG )∨ , it follows from the Eagon-Reiner
theorem JG is bi-CM if and only if IG is bi-CM.
2. The classification of bipartite and chordal bi-CM graphs
In this section we give a full classification of the bipartite and chordal bi-CM
graphs.
Theorem 2.1. Let G be a bipartite graph on the vertex set V with bipartition
V = V1 ∪ V2 where V1 = {v1 , . . . , vn } and V2 = {w1 , . . . , wm }. Then the following conditions are equivalent:
(a) G is a bi-CM graph;
(b) n = m and E(G) = {{vi , wj } : 1 ≤ i ≤ j ≤ n}.
Proof. (a) ⇒ (b): Since G is a bi-CM graph, it is in particular a CM-graph, and
so n = m, and by [9, Theorem 9.1.13] there exists a poset P = {p1 , . . . , pn } such
that G = G(P ). Here G(P ) is the bipartite graph on V = {v1 , . . . , vn , w1, . . . , wn }
whose edges are those 2-element subset {vi , wj } of V such that pi ≤ pj . Thus
IG = IG(P ) = HP∨ , where
\
HP =
(xi , yj )
pi ≤pj

is an ideal of S = K[{xi , yi}pi ∈P ], the polynomial ring in 2n variables over K. Since
G is bi-CM, it follows that HP is Cohen–Macaulay, and hence
proj dim S/HP = 2n − depth S/HP = 2n − dim S/HP = height HP = 2.
Thus proj dim HP = 1, and hence, by [10, Corollary 2.2], the Sperner number of P ,
i.e., the maximum of the cardinalities of antichains of P equals 1. This implies that
P is a chain, and this yields (b).
(b) ⇒ (a): The graph G described in (b) is of the form G = G(P ) where P is a
chain. By what is said in (a) ⇒ (b), it follows that G is bi-CM.

The following picture shows a bi-CM bipartite graph for n = 4.

x1
•

x2
•

x3
•

x4
•

•
y1

•
y2

•
y3

•
y4

Figure 1. A bi-CM bipartite graph.
Theorem 2.2. Let G be a chordal graph on the vertex set [n]. The following conditions are equivalent:
(a) G is a bi-CM graph;
5

(b) Let F1 , . . . , Fm be the facets of the clique complex of G. Then m = 1,
or m > 1 and
(i) V (G) = V (F1 ) ∪ V (F2 ) ∪ . . . ∪ V (Fm ), and this union is disjoint;
(ii) each Fi has exactly one free vertex ji ;
(iii) the restriction of G to [n] \ {j1 , . . . , jm } is a clique.
Proof. Let In,d be the ideal generated by all squarefree monomials of degree d in
∨
S = K[x1 , . . . , xn ]. It is known (and easy to prove) that In,d
= In,n−d+1 , and that
all these ideals are Cohen-Macaulay, and hence all bi-CM. If m = 1, then IG = In,2
and the result follows.
Now let m > 1. A bi-CM graph is a CM graph. The CM chordal graphs have
been classified in [11]: they are the chordal graphs satisfying (b)(i). Thus for the
proof of the theorem we may assume that (b)(i) holds and simply have to show that
(b)(ii) and (b)(iii) are satisfied if and only if IG has a linear resolution.
Let Pi be the monomial prime ideal generated by the variables xk with k ∈
V (Fi ) \ {ji }, and let G′ be subgraph of G whose edges do not belong to any Fi . It
is shown in the proof of [11, Corollary 2.1] that there exists a regular sequence on
S/IG such that after reduction modulo this sequence one obtains the ideal J ⊂ T
where T is the polynomial ring on the variables xk with k 6= ji for i = 1, . . . , m and
where
(1)

J = (P12 , . . . , Pm2 , IG′ ).

By Proposition 1.1, it follows that IG has a linear resolution if and only if J = m2T ,
where mT denotes the graded maximal ideal of T .
So, now suppose first that IG has a linear resolution, and hence J = m2T . Suppose
that some Fi has more than one free vertex, say Fi has the vertex k with k 6= ji .
Choose any Ft different from Fi and let l ∈ Fj with l 6= jt . Then xk and xl belong
to T but xk xl 6∈ J as can be seen from (1). This is a contradiction. Thus (b)(ii)
follows.
Suppose next that the graph G′′ which is the restriction of G to [n] \ {j1 , . . . , jm }
is not a clique. Then there exist i, j ∈ V (G′′ ) such that {i, j} 6∈ E(G′′ ). However,
since all xk with k ∈ V (G′′ ) belong to T and since J = m2T , it follows xi xj ∈ J.
Thus, by (1), xi xj ∈ Pk2 for some k or xi xj ∈ IG′ . Since (b)(ii) holds, this implies in
both cases that {i, j} ∈ E(G′′ ), a contradiction. Thus (b)(iii) follows.
Conversely, suppose (b)(ii) and (b)(iii) hold. We want to show that J = m2T . Let
xi , xj ∈ T . We have to show that xi xj ∈ J. It follows from the description of J
that x2k ∈ J for all xk ∈ T . Thus we may assume that i 6= j. If {i, j} is not an
edge of any Fk , then by definition it is an edge of G′ , and hence xi xj ∈ IG′ ⊂ J.
On the other hand, if {i, j} is an edge of Fk for some k, then i, j 6= ik , and hence
xi xj ∈ Pk2 ⊂ J. Thus the desired conclusion follows.

Let G be a chordal bi-CM graph as in Theorem 2.2(b) with m > 1. We call the
complete graph G′′ which is the restriction of G to [n] \ {j1 , . . . , jm } the center of G.
The following picture shows, up to isomorphism, all bi-CM chordal graphs whose
center is the complete graph K4 on 4 vertices:
6

•

•

•

•

•

•

•

•

•

•

•

•

•
•

•

•
•

•

•

•

•

•
•

•

•

•

•
Figure 2.

3. Generic Bi-CM graphs
As we have already seen in the first section, the Alexander dual J = IG∨ of the edge
ideal of a bi-CM graph G is a Cohen–Macaulay ideal of codimension 2 with linear
resolution. The ideal J may have several distinct relation matrices with respect to
the unique minimal monomial set of generators of J. As shown in [3], one may
attach to each of the relation matrices of J a tree as follows: let u1 , . . . , um be the
unique minimal set of generators of J. Let A be one of the relation matrices of J.
Because J has a linear resolution, the generating relations of J may be chosen all of
the form xk ui − xl uj = 0. This implies that in each row of the (m − 1) × m-relation
matrix A there are exactly two non-zero entries (which are variables with different
signs). We call such relations, relations of binomial type.
Example 3.1. Consider the bi-CM graph G on the vertex set [5] and edges {1, 2}
{2, 3}, {3, 1}, {2, 4}, {3, 4}, {4, 5} as displayed in Figure 3.
x2
•
x1 •

x4
•

x5
•

•
x3
Figure 3.
The ideal J = IG∨ is generated by u1 = x2 x3 x4 , u2 = x1 x3 x4 , u3 = x2 x3 x5 and
u4 = x1 x2 x4 . The relation matrices with respect to u1 , u2, u3 and u4 are the matrices

and









x1 −x2
0
0

0 −x4
0 
A1 = x5
,
x1
0
0 −x3
x1 −x2
0
0

0 −x4
0 
A2 = x5
.
0 x2
0 −x3
7

Coming back to the general case, one assigns to the relation matrix A the following
graph Γ: the vertex set of Γ is the set V (Γ) = {1, 2, . . . , m}, and {i, j} is said to
be an edge of Γ if and only if some row of A has non-zero entries for the ith- and
jth-component. It is remarked in [3] and easy to see that Γ is a tree. This tree is in
general not uniquely determined by G.
In our Example 3.1 the relation tree of A1 is
x3
•
•
x4

•
x1

•
x2

Figure 4.
while the relation tree of A2 is

•
x3

•
x1

•
x2

•
x4

Figure 5.
Now let J be any codimension 2 Cohen-Macaulay monomial ideal with linear
resolution. Then, as observed in Section 1, J ∨ = IG where G is a bi-CM graph.
Now we follow Naeem [13] and define for any given tree T on the vertex set [m] =
{1, . . . , m} with edges e1 , . . . , em−1 the (m − 1) × m-matrix AT whose entries akl are
defined as follows: we assign to the kth edge ek = {i, j} of T with i < j the kth row
of AT by setting
(2)





xij ,
akl = −xji ,

 0,

if l = i,
if l = j,
otherwise.

The matrix AT is called the generic matrix attached to the tree T .
By the Hilbert-Burch theorem [2], the matrix AT is the relation matrix of the ideal
JT of maximal minors of AT , and JT is a Cohen-Macaulay ideal of codimension 2
with linear resolution.
We let GT be the graph such that IGT = J ∨ , and call GT the generic bi-CM graph
attached to T .
Our discussion so far yields
Proposition 3.2. For any tree T , the graph GT is bi-CM.
8

In order to describe the vertices and edges of GT , let i and j be any two vertices
of the tree T . There exists a unique path P : i = i0 , i1 , . . . , ir = j from i to j. We
set b(i, j) = i1 and call b(i, j) the begin of P , and set e(i, j) = ir−1 and call e(i, j)
the end of P .
It follows from [13, Proposition 1.4] that IGT is generated by the monomials
xib(i,j) xje(i,j). Thus the vertex set of the graph GT is given as
V (GT ) = {(i, j), (j, i) : {i, j} is an edge of T }.
In particular, {(i, k), (j, l)} is an edge of GT if and only if there exists a path P from
i to j such that k = b(i, j) and l = e(i, j).
In Example 3.1, let T1 and T2 be the relation trees of A1 and A2 , respectively.
Then the generic matrices corresponding to these trees are








x12 −x21
0
0


x
0
−x
0
B1 =  13
,
31
x14
0
0
−x41

and

x12 −x21
0
0

0
−x31
0 
B2 = x13
.
0
x24
0
−x42
The generic graphs corresponding to the trees T1 and T2 are displayed in Figure 6.
• x12

x21
•

• x21
x14 •

• x31
•
x13

•
x41

x31
•

x12 •

• x13

•
x42
GT2

GT1
Figure 6.

It follows from this description
 that
 GT has 2(m − 1) vertices. Since GT is bi-CM,
n−c+1
the number of edges of GT is
, see Corollary 1.2. Here n − c is the degree of
2
 

the generators of IG∨ which is m − 1. Hence GT has m2 edges. Among the edges of
GT are in particular the m − 1 edges {(i, j), (j, i)} where {i, j} is an edge of T .

Proposition 3.3. Let A be the relation matrix of a codimension 2 Cohen-Macaulay
monomial ideal J with linear resolution, and assume that all the variables appearing
in A are pairwise distinct. Let T be the relation tree of A. Then J is isomorphic to
JT and J admits the unique relation tree, namely T .
9

Proof. Since all variables appearing in A are pairwise distinct, we may rename the
variables appearing in a binomial type relation and call them as in the generic matrix
xij and xji . Then A becomes AT and this shows that J ∼
= JT .
To prove the uniqueness of the relation tree, we first notice that the shifts in the
multigraded free resolution of J are uniquely determined and independent of the
particular choice of the relation matrix A. A possibly different relation matrix A′
can arise from A only be row operations with rows of the same multidegree. Let
r1 , . . . , rl by rows of A with the same multidegree corresponding to binomial type
relations, and fix a column j. Then the non-zero jth columns of each of the ri must
be the same, up to a sign. Since we assume that the variables appearing in A are
pairwise distinct, it follows that l = 1. In particular, there is, up to the order of the
rows, only one relation matrix with rows corresponding to binomial type relations.
This shows that T is uniquely determined.

4. Inseparable models of Bi-CM graphs
In order to state the main result of this paper we recall the concept of inseparability introduced by Fløystad et al in [7], see also [12].
Let S = K[x1 , . . . , xn ] be the polynomial ring over the field K and I ⊂ S a
squarefree monomial ideal minimally generated by the monomials u1 , . . . , um . Let y
be an indeterminate over S. A monomial ideal J ⊂ S[y] is called a separation of I
for the variable xi if the following holds:
(i) the ideal I is the image of J under the K-algebra homomorphism S[y] → S
with y 7→ xi and xj 7→ xj for all j;
(ii) xi as well as y divide some minimal generator of J;
(iii) y − xi is a non-zero divisor of S[y]/J.
The ideal I is called separable if it admits a separation, otherwise inseparable. If J is
an ideal which is obtained from I by a finite number of separation steps, then we say
that J specializes to I. If moreover, J is inseparable, then J is called an inseparable
model of I. Each monomial ideal admits an inseparable model, but in general not
only one. For example, the separable models of the powers of the graded maximal
ideal of S have been considered by Lohne [12].
Forming the Alexander dual behaves well with respect to specialization and separation.
Proposition 4.1. Let I ⊂ S be a squarefree monomial ideal. Then the following
holds:
(a) If J specializes to I, then J ∨ specializes to I ∨ .
(b) The ideal I is separable if and only I ∨ is separable.
Proof. (a) It follows from [7, Proposition 7.2] that if L ⊂ S[y] is a monomial ideal
such that y−xi is a regular element on S[y]/L with (S[y]/L)/(y−xi)(S[y]/L) ∼
= S/I,
∨
∨
∨ ∼
then y −xi is a regular element on S[y]/L and (S[y]/L )/(y −xi )(S[y]/L ) = S/I ∨ .
Repeated applications of this fact yields the desired result.
(b) We may assume that the ideal L as in (a) is a separation of I with respect to
xi . Since (a) holds, it remains to show that y as well as xi divides some generator
10

of L∨ . By assumption this is the case for L. Suppose that y does not divide any
generator of L∨ . Then it follows from the definition of the Alexander dual that y also
does not divide any generator of (L∨ )∨ . This is a contradiction, since L = (L∨ )∨ .
Similarly it follows that xi divides some generator of L∨ .

We now apply these concepts to edge ideals. Let G be a graph on the vertex
set [n]. We call G separable if IG is separable, and otherwise inseparable. Let J
be a separation of IG for the variable xi . Then by the definition of separation,
J is again an edge ideal, say J = IG′ where G′ is a graph with one more vertex
than G. The graph G is obtained from G′ by identifying this new vertex with
the vertex i of G. Algebraically, this identification amounts to say that S/IG ∼
=
(S ′ /IG′ )/(y−xi )(S ′ /IG′ ), where S ′ = S[y] and y−xi is a non-zerodivisor of S ′ /IG′ . In
particular, it follows that IG and IG′ have the same graded Betti-numbers. In other
words, all important homological invariants of IG and IG′ are the same. It is therefore
of interest to classify all inseparable graphs. An attempt for this classification is
given in [1].
Example 4.2. Let G be the triangle and G′ be the line graph displayed in Figure 7.
x3
•
•
x1

x3 •
•
x2

•
x1

• x4
•
x2

Figure 7. A triangle and its inseparable model
Then IG′ = (x1 x2 , x1 x3 , x2 x4 ). Since Ass(IG′ ) = {(x1 , x2 ), (x1 , x4 ), (x2 , x3 )}, it
follows that x3 − x4 is a non-zero divisor on S ′ /IG′ where S ′ = K[x1 , x2 , x3 , x4 ].
Moreover, (S ′ /IG′ )/(x3 − x4 )(S ′ /IG′ ) ∼
= S/IG . Therefore, the triangle in Figure 7
is obtained as a specialization from the line graph in Figure 7 by identifying the
vertices x3 and x4 .
We denote by G(i) the complementary graph of the restriction GN (i) of G to
N(i) where N(i) = {j : {j, i} ∈ E(G)} is the neighborhood of i. In other words,
V (G(i) ) = N(i) and E(G(i) ) = {{j, k} : j, k ∈ N(i) and {j, k} 6∈ E(G)}. Note that
G(i) is disconnected if and only if N(i) = A ∪ B, where A, B 6= ∅, A ∩ B = ∅ and all
vertices of A are adjacent to those of B.
Here we will need the following result of [1, Theorem 3.1].
Theorem 4.3. The following conditions are equivalent:
(a) The graph G is inseparable;
(b) G(i) is connected for all i.
Now we are ready to state our main result.
11

Theorem 4.4. (a) Let T be a tree. Then GT is an inseparable bi-CM graph.
(b) For any inseparable bi-CM graph G, there exists a unique tree T such that
G∼
= GT .
(c) Let G be any bi-CM graph. Then there exists a tree T such that GT is an
inseparable model of G.
Proof. (a) By Corollary 3.2, GT is a bi-CM graph. In order to see that GT is
inseparable we apply the criterion given in Theorem 4.3, and thus we have to prove
that for each vertex (i, j) of GT and for each disjoint union N((i, j)) = A ∪ B of the
neighborhood of (i, j) for which A 6= ∅ =
6 B, not all vertices of A are adjacent to
those of B.
As follows from the discussion in Section 3,
N((i, j)) = {(k, l) : there exists a path from i to l, and j = b(i, l) and k = e(i, l)}.
In particular, (j, i) ∈ N((i, j)). Let N((i, j)) = A ∪ B, as above. We may assume
that (j, i) ∈ A. Since T is a tree, then there is no path from j to any l with
(k, l) ∈ N((i, j)), because otherwise we would have a loop in T . This shows that
(j, i) is connected to no vertex in B, as desired.
(b) Let A be a relation matrix of J = IG∨ and T the relation tree of A. The nonzero entries of A are variables with sign ±1. Say the kth row of A has the non-zero
entries akik and akjk with ik < jk . We may assume that the variable representing
akik has a positive sign while that akjk has a negative sign, and that this is so for
each row. We claim that the variables appearing in the non-zero entries of A are
pairwise distinct. By Proposition 3.3 this then implies that T is the only relation
tree of J and that G ∼
= GT .
In order to prove the claim, we consider the generic matrix AT corresponding to T .
Let S ′ be the polynomial ring over S in the variables xij and xji with {i, j} ∈ E(T ).
For each k we consider the linear forms ℓk1 = xik jk − akik and ℓk2 = xjk ik − akjk .
For example, for the matrix A2 in Example 3.1 the linear forms are ℓ11 = x12 − x1 ,
ℓ12 = x21 − x2 , ℓ21 = x13 − x5 , ℓ22 = x31 − x4 , ℓ31 = x24 − x2 and ℓ32 = x42 − x3 .
We let ℓ be the sequence of linear form ℓ11 , ℓ12 , . . . , ℓm−1,1 , ℓm−1,2 in S ′ . Then
(S ′ /JT S ′ )/(ℓ)(S ′/JT S ′ ) ∼
= S/J. Since both ideals, J as well as JT , are CohenMacaulay ideals of codimension 2, it follows that ℓ is a regular sequence on S ′ /JT S ′ .
Thus, assuming the variables appearing in the non-zero entries of A are not all
pairwise distinct, we see that J is separable. Indeed, suppose that the variable xk
appears at least twice in the matrix. Then we replace only one of the xk by the
corresponding generic variable xij to obtain the matrix A′ . Let J ′ be the ideal of
maximal minors of A′ . It follows from the above discussions that xij −xk is a regular
element of S[xij ]/J ′ . In order to see that J ′ is a separation of J it remains to be
shown that xij as well as xk appear as factors of generators of J ′ . Note that J ′ is
a specialization of JT . The minors of AT which are the generators of JT are the
Q
monomials m+1
i=1 xib(i,j) for j = 1, . . . , m + 1, see [13, Proposition 1.2]. From this
i6=j

description of the generators of JT it follows that all entries of AT appear as factors
of generators of JT . Since J ′ is a specialization of JT , the same holds true for J ′ ,
and since xij as well as xk are entries of A′ , the desired conclusion follows.
12

Now since we know that J is separable, Proposition 4.1(b) implies that G is
separable as well. This is a contradiction.
(c) Let A be a relation matrix of J = IG∨ and T the corresponding relation tree.
As shown in the proof of part (b), JT specializes to J, and hence IGT specializes
to IG , by Proposition 4.1(a). By part (a), the graph GT is inseparable. Thus we
conclude that GT is an inseparable model of G, as desired.

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(1995), 245–257.
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coherent sheaves, Comm. Algebra 33 (2005), 3121–3136.
[9] J. Herzog and T. Hibi, Monomial Ideals. GTM 260. Springer 2010.
[10] J. Herzog and T. Hibi, Distributive lattices, Bipartite graphs and Alexander duality, J. Algebraic Combin., 22(2005), 289–302.
[11] J. Herzog and T. Hibi, X. Zheng, Cohen–Macaulay chordal graphs, J. Combin. Theory Ser.
A, 113(2006), 911–916.
[12] H. Lohne, The many polarizations of powers of maximal ideals, arXiv:1303.5780 (2013).
[13] M. Naeem, Cohen–Macaulay monomial ideals of codimension 2, Manuscripta math,
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[14] R. H. Villarreal, Monomial Algebras, Second Edition, Monographs and Research Notes in
Mathematics, CRC Press, 2015.
Jürgen Herzog, Fachbereich Mathematik, Universität Duisburg-Essen, Fakultät
für Mathematik, 45117 Essen, Germany
E-mail address: juergen.herzog@uni-essen.de
Ahad Rahimi, Department of Mathematics, Razi University, Kermanshah, Iran
E-mail address: ahad.rahimi@razi.ac.ir

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