Cohen-Macaulay binomial edge ideals

Abstract

We study the depth of classes of binomial edge ideals and classify all closed
graphs whose binomial edge ideal is Cohen--Macaulay.

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arXiv:1004.0143v1 [math.AC] 1 Apr 2010
COHEN–MACAULAY BINOMIAL EDGE IDEALS
VIVIANA ENE, J¨URGEN HERZOG AND TAKAYUKI HIBI
ABSTRACT. We study the depth of classes of binomial edge ideals and classify all closed graphs
whose binomial edge ideal is Cohen–Macaulay
INTRODUCTION
Binomial edge ideals were introduced in [5]. They appear independently, and at about the same
time, also in the paper [6]. In simple terms, a binomial edge ideal is just an ideal generated by an
arbitrary collection of 2-minors of a 2×n-matrix whose entries are all indeterminates. Thus the
generators of such an ideal are of the form fij=xiyj−xjyiwith i< j. It is then natural to associate
with such an ideal the graph G on the vertex set [n] for which {i, j} is an edge if and only if fij
belongs to our ideal. This explains the naming for this type of ideals. The binomial edge ideal
of graph G is denoted by JG. In [5] the relevance of this class of ideals for algebraic statistics is
explained.
The goal of this paper is to characterize Cohen–Macaulay binomial edge ideals for simple
graphs with vertex set [n]. Similar to ordinary edge ideals which were introduced by Villarreal
[7], a general classification of Cohen–Macaulay binomial edge ideals seems to be hopeless. Thus
we have to restrict our attention to special classes of graphs. In Section 1 we first consider the
class of chordal graphs with the property that any two maximal cliques of it intersect in at most
one vertex. These graphs include of course all forests. We show in Theorem 1.1 that for these
graphs we have depthS/JG= n+c, where n is the number of vertices of G and c is the number of
connected components of G. As an application we show that the binomial edge ideal of a forest is
Cohen–Macaulay if and only if each of its connected components is a path graph, and this is the
case if and only if S/JGis a complete intersection.
In Section 3 we use the results of Section 2 to give in Theorem 3.1 a complete characterization
of all closed graphs whose binomial edge ideal is Cohen–Macaulay. Surprisingly this is the case
if and only if its initial ideal is Cohen–Macaulay. Even more is true: if for a closed graph G, the
ideal JGis Cohen–Macaulay, then the graded Betti numbers of JGand its initial ideal coincide.
For a closed graph whose binomial edge ideal is Cohen–Macaulay, the Hilbert function and the
multiplicity of S/JGcan be easily computed, see Proposition 3.5 and Proposition 3.6. Then by
using the associativity formula of multiplicities in combination with the information given in [5]
concerning the minimal prime ideals of binomial edge ideals we deduce in Corollary 3.7 certain
numerical identities.
The term “closed graph” is not standard terminology in graph theory. It was introduced in [5] to
characterize those graphs, which, for certain labeling of their edges, do have a quadratic Gr¨ obner
basis with respect to the lexicographic order induced by x1> ··· > xn> y1··· > yn. It is easy to
1991 Mathematics Subject Classification. 13C05, 13C14, 13C15, 05E40, 05C25.
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see, as shown in [5], that any closed graph must be chordal. But by far not all chordal graphs are
closed. In Theorem 2.2 we give a description of the closed graphs which is then used in the proof
of Theorem 3.1.
1. CLASSES OF CHORDAL GRAPHS WITH COHEN–MACAULAY BINOMIAL EDGE IDEAL
In this section wewill compute the depth of S/JGfor a very special class of chordal graphs. This
class includes all forests. As a consequence it will be shown that a forest has a Cohen–Macaulay
binomial edge ideal if and only if all its components are path graphs.
We shall need a few results from [5]. There in Corollary 3.9 and Corollary 3.3 the following
fact is shown: Suppose that G is connected. Let S ⊂ [n], and let G1,...,GcG(S)be the connected
components of G[n]\S. For each Giwe denote by˜Githe complete graph on the vertex setV(Gi). If
there is no confusion possible we simply write c(S) for cG(S), and set
PS(G) = (
?
i∈S
{xi,yi},J˜ G1,...,J˜Gc(S)).
Then JG=?
dimS/JG= max{(n−|S|)+c(S) : S ⊂ [n]}.
S⊂[n]PS(G), and PS(G) is a minimal prime ideal of JGif and only if S = / 0, or S ?= / 0
and for each i ∈ S one has c(S\{i}) < c(S). Moreover, heightPS(G) = n+|S|−c(S) and hence
Theorem 1.1. Let G be a chordal graph on [n] with the property that any two distinct maximal
cliques intersect in at most one vertex. Then depthS/JG= n+c, where c is the number of con-
nected components of G.
Moreover, the following conditions are equivalent:
(a) JGis unmixed.
(b) JGis Cohen–Macaulay.
(c) Each vertex of G is the intersection of at most two maximal cliques.
Proof. LetG1,...,Gcbetheconnected components ofGandset Si=K[{xj,yj}j∈Gi]. ThenS/JG∼=
S1/JG1⊗···⊗Sc/JGc, so that depthS/JG= depthS1/JG1+···+depthSc/JGc. Thus in order to
prove the desired result, we may assume that G is connected.
Let ∆(G) be the clique complex of G and let F1,...,Frbe a leaf order on the facets of ∆(G). We
make induction on r. If r = 1, then G is a simplex and the statement is true. Let r > 1; since Fris a
leaf, there exists a unique vertex, say i ∈ Fr, such that Fr∩Fj= {i} for some j. Let Ft1,...,Ftqbe
the facets of ∆(G) which intersect the leaf Frin the vertex i.
Let M(G) denote the set of all sets S ⊂ [n] such that PS(G) is a minimal prime ideal of JG. We
have JG= Q1∩Q2where Q1=?
(1)0 → S/JG→ S/Q1⊕S/Q2→ S/(Q1+Q2) → 0.
S∈M(G), i?∈SPS(G) and Q2=?
S∈M(G), i∈SPS(G).
Consider the exact sequence
The ideal Q1is the binomial edge ideal associated to the graph G′which is obtained from G by
replacing the facets Ft1,...,Ftq, and Frby the clique on the vertex set Fr∪(
q?
j=1Ftj). Note that G′
is a connected chordal graph which has again the property that any two cliques intersect in at
most one vertex, and it has a smaller number of cliques than G. Therefore, by induction, we have
depth(S/Q1) = depth(S/JG′) = n+1.
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In order to determine Q2we first observe that for all S ⊂ [n] with i ∈ S we have that PS(G) =
(xi,yi)+PS\{i}(G′′), where G′′is the restriction of G to the vertex set [n]\{i}. From this we
conclude that Q2= (xi,yi)+JG′′. Let Sibe the polynomial ring S/(xi,yi). Then S/Q2∼= Si/JG′′.
Hence, since G′′is a graph on n−1 vertices and with q+1 components satisfying the conditions
of the theorem, our induction hypothesis implies that depthS/Q2= (n−1)+q+1 = n+q.
Next we observe that Q1+Q2=JG′ +((xi,yi)+JG′′)=(xi,yi)+JG′. Thus S/(Q1+Q2)∼=Si/JH
where H is obtained form G′by replacing the clique on the vertex set Fr∪(
q?
j=1Ftj) by the clique on
the vertex set Fr∪(
q?
j=1Ftj)\{i}. Thus our induction hypothesis implies that depthS/(Q1+Q2)=n.
Hence the depth lemma applied to the exact sequence (1) yields the desired conclusion concerning
the depth of S/JG.
For the proof of the equivalence of statements (a), (b) and (c), we may again assume that G
is connected. Let JGbe unmixed. Then dim(S/JG) = n+1 since JGhas a minimal prime of
dimension n+1, namely P/ 0(S). Since depth(S/JG) = n+1, it follows that JGis Cohen-Macaulay,
whence (a) ⇒ (b). The converse, (b) ⇒ (a), is well known.
(a)⇒(c): Let us assume that there is a vertex i of G where at least three cliques intersect. Then,
for S = {i}, we get a minimal prime PS(G) of JGof height strictly smaller than n−1, which is in
contradiction with the hypothesis on JG.
(c) ⇒ (a): Let {i1,...,ir−1} be the intersection vertices of the maximal cliques of G, and PS(G)
a minimal prime of JG. Let H1,...,Htbe the connected components of G[n]\S. Suppose that there
exists i ∈ S\{i1,...,ir−1}. We have c(S \{i}) < c(S). This implies that there exists Ha,Hb,
two connected components of G[n]\S, such that i is connected to Haand Hb. Let u ∈ V(Ha) and
v ∈V(Hb) such that {i,u} and {i,v} are edges of G. Since i ∈ S\{i1,...,ir−1}, it follows that u,v
and i belong to the same clique of G, which implies that {u,v} is an edge of G. Therefore, Haand
Hbare connected, a contradiction. By induction on the cardinality of S we see that c(S) = |S|+1.
Therefore, all the minimal primes of JGhave the same height.
?
As a consequence of Theorem 1.1 we obtain the following
Corollary 1.2. Let G be a forest on the vertex set [n]. Then depth(S/JG) = n+c, where c is the
number of the connected components of G. Moreover, the following conditions are equivalent:
(a) JGis unmixed;
(b) JGis Cohen-Macaulay;
(c) JGis a complete intersection;
(d) Each component of G is a path graph.
Proof. The implications (c) ⇒ (b) ⇒ (a) are obvious, while (a ) ⇒ (d) follows from Theorem 1.1.
For the proof of (d) ⇒ (c) we may assume that G is a path, and the vertices are labeled in such
a way such that E(G) = {{i,i+1}: i = 1,...,n−1}. Then in<(JG) = (x1y2,x2y3,...,xn−1yn),
where < is the lexicographic order induced by x1> x2··· > xn> y1> y2> ··· > yn. Since the
initial ideal of JGis a complete intersection, JGitself is complete intersection.
?
The depth formula that we proved in Theorem 1.1 is not valid for arbitrary chordal graphs.
For example for the graph G displayed in Figure 1 we have depthS/JG= 5 (and not 6 as one
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would expect by Theorem 1.1). It is also an example of a graph for which JGis unmixed but not
Cohen–Macaulay.
•
•
•
•
•
FIGURE 1.
2. CLOSED GRAPHS
In [5] the concept of closed graphs was introduced. In that paper a simple graph G on the vertex
set [n] is called closed with respect to the given labeling, if the following condition is satisfied:
• For all {i, j},{k,l} ∈ E(G) with i < j and k < l one has {j,l} ∈ E(G) if i = k but j ?= ℓ,
and {i,k} ∈ E(G) if j = l but i ?= k.
The definition was motivated by the following result [5, Theorem 1.1]: G is closed with re-
spect to the given labeling, if and only if JGhas a quadratic Gr¨ obner basis with respect to the
lexicographic order induced by x1> x2> ··· > xn> y1> ··· > yn.
Itisshown in[5, Proposition 1.4] that the graph G on[n] isclosed withrespect to thegiven label-
ing, if and only if for any two integers 1≤i< j ≤n the shortest walk {i1,i2},{i2,i3},...,{ik−1,ik}
between i and j has the property that i = i1< i2< ··· < ik= j. In particular, for each i < n one
has that {i,i+1} ∈ E(G).
Definition 2.1. We say a graph is closed if there exists a labeling for which it is closed.
It arises the question to characterize the closed graphs. It is known from [5, Proposition 1.2]
that if G is closed, then G is chordal.
Recall that by a result of Dirac [2] (see also [4]) that a graph G is chordal if and only if it admits
a perfect elimination order, that is, its vertices can be labeled 1,...,n such that for all j ∈ [n], the
set Cj= {i: i ≤ j} is a clique of G. A clique is simply a complete subgraph of G.
There is an equivalent characterization of chordal graphs in terms of its maximal cliques. To
describe it we introduce some terminology. Let ∆ be a simplicial complex. A facet F of ∆ is
called a leaf, if either F is the only facet, or else there exists a facet G, called a branch of F, which
intersects F maximally. In other words, for each facet H of ∆ with H ?= F one has H ∩F ⊂ G∩F.
Each leaf F has at least one free vertex, that is, a vertex which belongs only to F. On the other
hand, if a facet admits a free vertex it needs not to be a leaf.
The simplicial complex ∆ is a called a quasi-forest if its facets can be ordered F1,...,Frsuch
that for all i > 1 the facet Fiis a leaf of the simplicial complex with facets F1,...,Fi−1. Such an
order of the facets is called a leaf order. A connected quasi-forest is called a quasi-tree.
Now let G be a graph. The collection of cliques of G forms a simplicial complex, called the
clique complex of G. It is denoted ∆(G). The equivalent statement to Dirac’s theorem now says
that G is chordal if and only if ∆(G) is a quasi-forest.
Theorem 2.2. Let G be a graphs on [n]. The following conditions are equivalent:
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(a) G is closed;
(b) there exists a labeling of G such that all facets of ∆(G) are intervals [a,b] ⊂ [n].
Moreover, if the equivalent conditions hold and the facets F1,...,Frof ∆(G) are labeled such that
min(F1) < min(F2) < ··· < min(Fr), then F1,...,Fris a leaf order of ∆(G).
Proof. (a) ⇒ (b): Let G be a closed graph on [n] and F = {j: {j,n} ∈ E(G)}, and let k =
min{j: j ∈ F}. Then F = [k,n]. Indeed, if j ∈ F with j < n, then as observed above, it follows
that {j, j+1} ∈ E(G), and then because G is closed we see that since {j,n} ∈ E(G), then also
{j+1,n} ∈ E(G). Thus j+1 ∈ F.
Next observe that F is a maximal clique of G, that is, a facet of ∆(G). First of all it is a clique,
because i, j ∈ F with i < j < n, then, since {i,n} and {j,n} are edges of G, it follows that {i, j} is
an edge as well, since G is closed. Secondly, it is maximal, since {j,n} ?∈ E(G), if j ?∈ F.
Let H ?= F be a facet of ∆(G) with H ∩F ?= / 0, and let ℓ = max{j: j ∈ H ∩F}. We claim that
H ∩F = [k,ℓ]. There is nothing to prove if k = ℓ. So now suppose that k < ℓ and let k ≤t < ℓ and
s ∈ H \F. Then s,t < ℓ and {s,ℓ} and {t,ℓ} are edges of G. Hence since G is closed it follows
that {s,t} ∈ E(G). This implies that s ∈ H, as desired.
It follows from the claim that the facet H for which max{j: j ∈ H∩F} is maximal, is a branch
of F. In particular, F is a leaf. Let H ∩F = [k,ℓ], where H is a branch of F, and denote by Gℓ
the restriction of G to [ℓ]. Since Gℓis again closed and since ℓ < n, we may assume, by applying
induction on the cardinality of the vertex set of G, that all facets of ∆(Gℓ) are intervals. Now let F′
be any facet of ∆(G). If F = F′, then F is an interval, and if F ?= F′, then, as we have seen above,
it follows that F′∈ ∆(G′). This yields the desired conclusion.
(b) ⇒ (a): Let {i, j} and {k,ℓ} be edges of G with i < j and k < ℓ. If i = k, then {i,k} and {i,ℓ}
belong to the same maximal clique, that is, facet of ∆(G) which by assumption is an interval. Thus
if j ?= ℓ, then {j,ℓ} ∈ E(G). Similarly one shows that if j = ℓ, but i ?= k, then {i,k} ∈ E(G). Thus
G is closed.
Finally it is obvious that the facets of ∆(G) ordered according to their minimal elements is a
leaf order, because for this order Fi−1has maximal intersection with Fifor all i.
?
3. CLOSED GRAPHS WITH COHEN–MACAULAY BINOMIAL EDGE IDEAL
With the description of closed graphs given in Theorem 2.2 it is not hard to classify all closed
graphs with Cohen–Macaulay binomial edge ideal.
Theorem 3.1. LetG be a connected graph on [n] which is closed with respect to the given labeling.
Then the following conditions are equivalent:
(a) JGis unmixed;
(b) JGis Cohen-Macaulay;
(c) in<(JG) is Cohen-Macaulay;
(d) G satisfies the condition that whenever {i, j+1} with i < j and {j,k+1} with j < k are
edges of G, then {i,k+1} is an edge of G;
(e) there exist integers 1 = a1< a2< ... < ar< ar+1= n and a leaf order of the facets
F1,...,Frof ∆(G) such that Fi= [ai,ai+1] for all i = 1,...,r.
Proof. We begin by proving (a) ⇒ (e). By Theorem 2.2, ∆(G) has facets F1,...,Frwhere each
facet is an interval. We may order the intervals Fi=[ai,bi] such that 1= a1< a2< ··· <ar≤ br=
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