HILBERT FUNCTION OF BINOMIAL EDGE IDEALS
FATEMEH MOHAMMADI AND LEILA SHARIFAN
Abstract. In this paper the numerical invariants of the binomial edge ideal of a graph are
studied.
1. Introduction
Let G be a simple graph on the vertex set [n] and R = K[x1 , . . . , xn , y1 , . . . , yn ] be the polynomial
ring over the field K. The binomial edge ideal of G is the ideal
JG = (fij : {i, j} ∈ E(G) and i < j) ⊂ R,
where fij = xi yj − xj yi . This notion was first introduced in [3] and independently in [6]. The
goal of this paper is to compute the numerical invariants of the binomial edge ideals of graphs like
depth and Hilbert function. The main idea is to sit the module R/JG∪{e} in a short exact sequence
0 → R/JG : fe (−2) → R/JG → R/JG∪{e} → 0
with the property that two other modules in that sequence are known.
For this purpose, we use the information given in [3] concerning the minimal prime ideals of
binomial edge ideals to find the minimal prime ideals of the JG : fe , where e %∈ E(G). In order to
get more information on the modules of the above short exact sequence, a crucial point is to give
a combinatorial description for the ideal JG : fe which is studied in Theorem 3.7. In the case that
e is a bridge, we will see in Theorem 3.4 that this ideal is just the binomial edge ideal of another
known graph.
The notion of closed graphs, which JG has a quadratic Gröbner basis with respect to the
lexicographic order induced by natural ordering on the variables, is introduced in [3], and CohenMacaulay closed graphs are completely classified in [4]. Applying the main result of the paper we
will use closed graphs with Cohen-Macaulay binomial edge ideals to compute the depth and the
Hilbert function of further graphs. In Section 4 we study the case that G \ {e} or G ∪ {e} is a closed
graph with Cohen-Macaulay binomial edge ideals (For the main idea see Corollaries 4.3, 4.5, 4.6).
As an example we compute the Hilbert function of a quasi cycle in Corollary 4.2.
In the last section we discuss the connection of the quotient ideal JG : fe and the conditional
independence ideals to check what can we say about the ideal JG : fe , where e %∈ E(G). In
some special cases it is indeed a CI-ideal and we can describe its associated minimal primes, see
Corollary 5.2.
2. preliminaries
In this section, we review some definitions and results from [3, 4]. Let G be a simple graph on
[n]. Corresponding to each subset S ⊂ [n] we have the prime ideal
!
PS (G) = ( {xi , yi }) + JG̃1 + . . . + JG̃c(S) ,
i∈S
2000 Mathematics Subject Classification. 13D40, 13C14, 13C15, 05E40, 13P10.
Key words and phrases. Binomial Edge Ideals, Cohen-Macaulay rings, Hilbert function.
The research of the second author was in part supported by a grant from IPM, No. 900130066.
1
where G1 , . . . , Gc(S) are the connected components of the induced subgraph on the vertices [n] \ S,
and G̃! is the complete graph on the vertices of G! for all !. Then
"
JG =
PS (G).
(1)
S⊂[n]
Moreover, dim R/JG = max{(n − |S|) + c(S) : S ⊂ [n]} and hence dim R/JG ≥ n + c(G), where
c(G) is the number of the connected components of G. Equation (1) also shows that JG is a radical
ideal. Note that if S is an arbitrary subset of [n] the prime ideal PS (G) is not necessary a minimal
prime ideal of JG . The next Lemma detects the minimal prime ideals of JG when G is a connected
graph. Note that for S ⊂ [n], by c(S) we mean c(G[n]\S ).
Lemma 2.1. Let G be a connected graph on the vertex set [n] and S ⊂ [n], then PS (G) is a minimal
prime ideal of JG if and only if S = ∅, or S %= ∅ and for each i ∈ S one has c(S\{i}) < c(S).
Let G be a graph with the
components G1 , . . . , Gc . Then the minimal prime ideals
#connected
c
of JG are exactly the ideals i=1 Pi where each Pi is a minimal prime ideal of JGi . In this case,
if we set Ri = K[{xj , yj }j∈V (Gi ) ], then
R/JG ∼
(2)
= R1 /JG1 ⊗ · · · ⊗ Rc /JGc .
So
depth(R/JG ) = depth(R1 /JG1 ) + · · · + depth(Rc /JGc ),
and the Hilbert series of R/JG is
c
$
HSR/JG (t) =
HSRi /JGi (t).
(3)
(4)
i=1
The graph G is called closed with respect to the given labeling if the following condition is satisfied:
For every two edges {i, j} and {k, !} in E(G) with i < j and k < !, one has {j, !} ∈ E(G) if
i = k, and {i, k} ∈ E(G) if j = !.
Let G be a graph, we recall that the clique complex of G, denoted by ∆(G) is the simplicial
complex on [n] whose faces are the cliques of G. It is shown in [4, Theorem 2.2] that the graph
G is closed if and only if there exists a labeling of G such that all facets of ∆(G) are intervals
[a, b] ⊂ [n]. In [4, Theorem 3.1] connected closed graphs with Cohen-Macaulay binomial edge
ideals are classified as those closed graphs whose binomial edge ideals have a Cohen-Macaulay
initial ideal, or equivalently there exist integers 1 = a1 < a2 < · · · < ar < ar+1 = n such that the
facets of ∆(G) are F1 , . . . , Fr , where Fi = [ai , ai+1 ] for i = 1, . . . , r. In this case the Hilbert series
of R/JG is
%r
[(|Fi | − 1)t + 1]
HSR/JG (t) = i=1
.
(5)
(1 − t)n+1
3. Combinatorial description of the quotient of binomial ideals
In this section we study the quotient ideal JG : fe , where JG is the binomial edge ideal of a
simple graph G. In order to get a combinatorial description for this ideal we need to introduce the
following graphs constructed from G.
Definition 3.1. Let G be a simple graph on the vertex set [n], e = {i, j} %∈ E(G) and e$ = {i$ , j $ } ∈
E(G). Then we define the graphs G \ {e$ }, G ∪ {e} and Ge on the vertex set [n] with the following
edge sets:
• E(G \ {e$ }) = E(G) \ {e$ };
• E(G ∪ {e}) = E(G) ∪ {e};
• E(Ge ) = E(G) ∪ {{k, !} : k, ! ∈ N (i) or k, ! ∈ N (j)}, where N (i) is the neighbor set of
the vertex i in G.
First we find a primary decomposition for the quotient ideal JG : fe in the following proposition,
and later we describe each component in the primary decomposition explicitly.
2
Proposition 3.2. Let G be a simple graph on [n] and e = {i, j} %∈ E(G). Then
"
JG : fe =
PS (G),
S∈Aij
where Aij is a set consisting of all subsets S of [n] \ {i, j} with the following properties:
(a) PS (G) is a minimal prime of JG ;
(b) For each path P : i, i1 , . . . , is , j there exists some t such that it ∈ S.
&
Proof. The ideal JG = S⊂[n] PS (G) is a radical ideal by [3, Theorem 3.2] and from [1, Lemma 4.4]
&
it follows that JG : fe = fe %∈PS (G) PS (G) which is again a radical ideal. Then it is enough to show
that fe %∈ PS (G) if and only if S ∈ Aij . Assume that fe %∈ PS (G) for some S ⊂ [n]. It is clear that
i, j %∈ S, since i ∈ S implies that xi , yi ∈ PS (G) and so fe ∈ PS (G). Moreover the construction of
PS (G) shows that i and j do not belong to the same connected component of G \ S. Thus at least
one of the vertices of each path P : i, i1 , . . . , is , j should be in S.
On the other hand, for S ∈ Aij , we have the variables xi , xj , yi , yj are not in PS (G), and i, j do
not belong to the same connected component of G \ S which implies fe %∈ PS (G).
!
Corollary 3.3. Let G be a simple graph on [n], e = {i, j} %∈ E(G). Then
• dim R/JG = max{(n − |S|) + c(S) : S ⊂ [n]},
• dim R/(JG : fe ) = max{(n − |S|) + c(S) : S ∈ Aij }.
In particular, dim R/JG : fe ≤ dim R/JG .
Let G be a simple graph on the vertex set [n] and let e = {i, j} %∈ E(G). If c(G) > c(G ∪ {e}),
then e is called a bridge of G ∪ {e}. In the first theorem we describe the relation between the
binomial edge ideals of G and Ge in the case that e is a bridge of G ∪ {e}.
Theorem 3.4. Let G be a simple graph and e = {i, j} %∈ E(G) be a bridge in G ∪ {e}. Then
J G : f e = J Ge .
&
Proof. First note that by Proposition 3.2, JG : fe = i,j%∈S PS (G). In order to prove the theorem
&
we show that i,j%∈S PS (G) is indeed a primary decomposition for JGe . It is enough to have that
each minimal prime of JGe is of the form PS (G) for some S with i, j %∈ S. By (1) each minimal
prime of Ge is of the form PS (Ge ) for some S. By contradiction assume that i ∈ S for some
minimal prime PS (Ge ). Therefore c(S \ {i}) < c(S) (see Lemma 2.1). So there exist two vertices s
and t in Ge \ (S \ {i}) such that i belongs to each path between s and t. Let P : s, . . . , s$ , i, t$ , . . . , t
be an arbitrary path between s and t. As we see in the construction of Ge two neighbors s$ and
t$ are adjacent which implies that s, . . . , s$ , t$ , . . . , t is a path in Ge \ S, a contradiction. Therefore
i, j %∈ S.
In the construction of Ge we just add some edges in G among the vertices in N (i) or the vertices
in N (j). So, if i, j ∈
/ S and G1 , · · · , Gc are the connected components of G[n]\S and G$1 , · · · , G$t
are the connected components of (Ge )[n]\S then t = c and V (G! ) = V (G$! ) for each !. Therefore
each minimal prime PS (Ge ) is of the form PS (G) which completes the proof.
!
Here we study the case that the Cohen-Macaulay property of the ideal JG transfers to the ideal
J Ge .
Corollary 3.5. Let G be a closed graph and e = {i, j} %∈ E(G) be a bridge in G ∪ {e}. If JG is
Cohen-Macaulay, then JGe is also a Cohen-Macaulay ideal. Moreover Ge is a closed graph.
Proof. Let G1 , . . . , Gc be the connected components of G and assume that JG is Cohen-Macaulay.
Then it is clear that each G! is a closed graph and by equations (2) and (3) the ideals JG! are
Cohen-Macaulay. Thus in order to prove the desired result, we may assume that G is connected.
Since G is a closed graph, by [4, Theorem 3.1] there exists a labeling on the vertices of G and the
sequence 1 = a1 < a2 < · · · < at < n such that the facets F1 = [a1 , a2 ], . . . , Fi = [ai , ai+1 ], . . . , Ft =
3
[at , n] are the facets of ∆(G). First note that i and j are in the different facets of ∆(G), since
e %∈ E(G).
Assume that i or j is equal to a! for some !. Then all the vertices of [a!−1 , a! ] and [a! , a!+1 ] are
adjacent in the graph Ge which implies that the clique [a!−1 , a!+1 ] belongs to ∆(Ge ). If i, j %= a!
for each !, then the corresponding facets of i and j will be in ∆(Ge ) too. Therefore corresponding
to each a! equal to one of the endpoints of e, the union of cliques containing a! will be in ∆(Ge ).
Thus Ge is closed and [4, Theorem 3.1] implies that JGe is Cohen-Macaulay.
!
Example 3.6. Let G be a simple graph on [2n] with E(G) = {{i, n}, {j, n + 1} : i = 1, . . . , n −
1, and j = n + 2, . . . , 2n}. For e = {n, n + 1} we have
J G : f e = JG e = JK 1 + JK 2 ,
where K1 is the complete graph on the vertex set [n] and K2 is the complete graph on the vertex
set {n+1, . . . , 2n}. As we see JGe is Cohen-Macaulay and Ge = K1 ∪K2 is a closed graph. Assume
that n > 3. Then G has a claw and so it is not closed. Moreover, by [4, Theorem 1.1], JG is not
Cohen-Macaulay. Therefore the converse of Corollary 3.5 does not hold.
In the following by ≺ we mean the lexicographic order induced by x1 > x2 > · · · > xn > y1 >
· · · > yn on the variables. Now we try to give a simple description for the quotient ideal JG : fe in
combinatorial terms of the graph G.
Theorem 3.7. Let G be a simple graph on [n] and e = {i, j} %∈ E(G). Then
JG : fe = JGe + (gP,t : P : i, i1 , . . . , is , j is a path between i, j and 0 ≤ t ≤ s),
where gP,0 = xi1 · · · xis and for each 1 ≤ t ≤ s, gP,t = yi1 · · · yit xit+1 · · · xis .
Before proving Theorem 3.7 we need the following lemma:
Lemma 3.8. Let G be a simple graph on [n], e = {i, j} %∈ E(G) and P : i, i1 , . . . , is , j be a path
in G. Assume that A, B are two arbitrary subsets of {i1 , . . . , is } with A ∪ B = {i1 , . . . , is } and
A ∩ B = ∅, then
$
$
hA,B =
xij
yij ∈ JGe + (gP,t : 0 ≤ t ≤ s).
(6)
ij ∈A
ij ∈B
Proof. For each given sets A, B with A ∪ B = {i1 , . . . , is } and A ∩ B = ∅, let MA,B = {j :
ij ∈ A and ij+1 ∈ B}. By induction on the number of elements of MA,B we show that hA,B ∈
JGe + (gP,t : 0 ≤ t ≤ s). If |MA,B | = 0 then there exists 1 ≤ t ≤ s such that B = {i1 , . . . , it } and
A = {it+1 , · · · , is }. So hA,B = gP,t and we are done.
Let k ≥ 0 and assume that for each partition A, B of {i1 , . . . , is } with |MA,B | ≤ k, hA,B ∈
JGe + (gP,t : 0 ≤ t ≤ s). Suppose that A$ , B $ ⊂ {i1 , . . . , is } is a partition of {i1 , . . . , is } and
|MA! ,B ! | = k + 1.
Let d = max MA! ,B ! then there exists 1 ≤ p ≤ d and d + 1 ≤ q ≤ s such that
{ip , ip+1 , . . . , id , iq+1 } ∩ {i1 , . . . , is } ⊂ A and {ip−1 , id+1 , id+2 , . . . , iq } ∩ {i1 , . . . , is } ⊂ B.
Now, note that hA! ,B ! = fid ,id+1 hA! ,B ! /(xid yid +1 ) + yid xid+1 hA! ,B ! /(xid yid +1 ) and fid ,id+1 ∈ JGe .
So hA! ,B ! ∈ JGe + (gP,t : 0 ≤ t ≤ s) if and only if
u1 = yid xid+1 hA! ,B ! /(xid yid +1 ) ∈ JGe + (gP,t : 0 ≤ t ≤ s).
But u1 = hA!1 ,B1! where A$1 = A$ ∪ {id+1 } \ {id } and B1$ = B $ ∪ {id } \ {id+1 }. If d + 1 < q we can
repeat the same process on hA!1 ,B1! using the binomial fid+1 ,id+2 ∈ JGe . After a finite number of
steps we can find the following representation for hA! ,B ! .
hA! ,B ! =
q−1
'
wz$ fiz ,iz+1 + hA!! ,B !! ,
z=d
where wi$ ∈ K[x1 , . . . , xn , y1 , . . . , yn ], A$$ = A$ ∪ {iq } \ {id } and B $$ = B $ ∪ {id } \ {iq }.
4
So in order to show the result it is enough to check that
hA!! ,B !! ∈ JGe + (gP,t : 0 ≤ t ≤ s).
If p = d then |MA!! ,B !! | = k and by induction hypothesis the proof is complete. So we assume that
p < d. In this case |MA!! ,B !! | = k + 1 and its maximum element is d − 1. We can repeat the above
procedure on hA!! ,B !! and after a finite number of steps find the following equation.
hA! ,B ! =
q−1
'
wz fiz ,iz+1 + hA,B ,
(7)
z=p
where A = A$ ∪ {ip+q−d , . . . , iq } \ {ip , . . . , ip+q−d−1 }, B = B $ ∪ {ip , . . . , ip+q−d−1 } \ {ip+q−d , . . . , iq }
and wp , wp+1 , . . . , wq−1 ∈ K[x1 , . . . , xn , y1 , . . . , yn ].
Note that in equation (7), fiz ,iz+1 ∈ JGe because {iz , iz+1 } ∈ E(Ge ). Also, |MA,B | ≤ k and by
induction hypothesis hA,B ∈ JGe +(gP,t : 0 ≤ t ≤ s). It follows that hA! ,B ! ∈ JGe +(gP,t : 0 ≤ t ≤ s)
which implies (6).
!
Proof of Theorem 3.7. By Proposition 3.2 we have that JG : fe = ∩S∈Aij PS (G). Let S be an
arbitrary subset in Aij . For each path P : i, i1 , . . . , is , j we have i! ∈ S for some !, and so
xi! , yi! ∈ PS (G) which implies that
gP,t = yi1 · · · yit xit+1 · · · xis ∈ PS (G)
for each 1 ≤ t ≤ s. Assume that k and ! are neighbors of i. If k ∈ S or ! ∈ S, then we have
fk,! ∈ PS (G). Otherwise they are in the same connected component of the graph induced on
V (G) \ S which implies that fk! ∈ PS (G), as desired. Therefore we have
JGe + (gP,t : where P : i, i1 , . . . , is , j is a path and 0 ≤ t ≤ s) ⊆ JG : fe .
To prove the other direction of inclusion we use an induction argument on the number of vertices
of G. The proof is clear for each graph with at most four vertices. So by induction assume that
the result holds for each graph with a fewer number of vertices than G. If there is no path between
i and j in G, then by Theorem 3.4 the result holds. So assume that h ∈ JG : fe , and Q, P1 , . . . , Pr
are all paths between i and j in G. We write h = h$ + h$$ , where
h$$ ∈ I = JGe + (gP! ,t : for all ! and t),
and no monomial of h$ lies in LT≺ (I). Then h$ ∈ JG : fe and so
h $ f e = h 1 f e1 + · · · + h r f er .
(8)
Let g = LT≺ (h$ ). Now we show that for each ic in the path
Q : i, i1 , . . . , is , j
we have xic | g or yic | g. By contradiction assume that xic ! g and yic ! g for some 1 ≤ c ≤ s. Set
xic = yic = 0 in equation (8). Let w = h$ |xic =yic =0 and h$! = h! |xic =yic =0 . Then LT≺ (w) is equal
to g again, since xic ! g and yic ! g. Also we get
'
wfe =
h$! fe! ,
ic %∈e!
Therefore w ∈ JG1 \{e} : fe , where G1 is the induced subgraph of G on the vertices V (G) \ {ic }.
Note that JG1 e ⊆ JGe and each path between i and j in G1 \{e} is one of the pathes P1 , . . . , Pr . By
our argument no monomial of w lies in LT≺ (JG1 e +(gP! ,t : for all ! and t)), which is a contradiction
by induction hypothesis for G1 . Therefore xic or yic divides g.
that there exists two subsets A and B with%A ∪ B =
% We conclude
%
% {i1 , . . . , is } such that
ij ∈A xij
ij ∈B yij divides the monomial g and since, by (6),
ij ∈A xij
ij ∈B yij ∈ JGe + (gQ,t :
t ≤ s) we have g ∈ JGe + (gQ,t : t ≤ s).
Now continuing the same argument for h$ − g shows that h$ ∈ JGe + (gQ,t : for all 0 ≤ t ≤ s)
which completes the proof.
!
5
Example 3.9. Consider the following graph on [10]. We have ∆(G) = F1 ∪ F2 ∪ . . . ∪ F6 , where
F1 = {1, 2}, F2 = {2, 3, 4}, F3 = {4, 5}, F4 = {5, 6, 7}, F5 = {7, 8}, F6 = {8, 9, 10}. Assume
that e = {2, 7}. Since the subgraph induced on the vertices {1, 2, 3, 7} is a claw, i.e. the complete
bipartite graph K1,3 , by [3, Proposition 1.2], the graph G ∪ {e} is not closed. But the both graphs
G and Ge are closed with respect to the given ordering. By Proposition 3.2, ∩S∈A27 PS (G) is the
minimal primary decomposition of JG : fe where A27 is the set of all subsets S of [10] \ {2, 7} that
satisfies the combinatorial criterion of Lemma 2.1 and for each path 2, i1 , · · · , is , 7 there exists t such
that it ∈ S. By checking these two properties we can easily see that A2,7 = {{5}, {5, 8}, {4}, {4, 8}}.
G ∪ {2, 7} :
G{2,7} :
9!
6!
!
!
!
!
!
!
!
!
!
!!
7
8
10
3!
4!
"
"
"
!
"!
1
2
5!
3!
4!
" !
"
!
! "
!!
"!
1
2
9!
5!
6!
!
" !
!
"
!
!
! "
!
!
!!
"!
7
8
10
4. Binomial edge ideals of quasi cycles
In this section, we study binomial edge ideals of the special class of graphs. Our main goal is to
apply Theorem 3.7 to compute the Hilbert function of the binomial edge ideals for graph G, where
G \ {e} or G ∪ {e} is a connected closed Cohen-Macaulay graph.
Let G be a connected closed graph with Cohen-Macaulay binomial edge ideal. Then there exist
integers 1 = i1 < i2 < · · · < is < is+1 = n such that F1 , . . . , Fs are the facets of ∆(G), where
F! = [i! , i!+1 ] for all !. Let e = {1, n} and H = G∪{e} be the graph on [n] with E(H) = E(G)∪{e}
then the graph H is called the quasi cycle graph associated to G.
We are going to compute Hilbert function of the binomial edge ideal of quasi cycle graphs and
for this purpose we need the following lemma.
Lemma 4.1. Let G be a connected closed graph with Cohen-Macaulay binomial edge ideal and
1 = i1 < i2 < · · · < is < is+1 = n be some integers such that F1 , . . . , Fs are the facets of ∆(G),
where F! = [i! , i!+1 ] for ! = 1, . . . , s. Assume that g1 = xi2 xi3 · · · xis and gt = yi2 · · · yit xit+1 · · · xis
for 2 ≤ t ≤ s. Let J = JG + (g1 , . . . , gs ) then we have
(1) LT≺ (JG ) : g1 = (xk y! : 1 ≤ k < ! ≤ i2 ) + (y! : ! > i2 ),
(2) LT≺ (JG ) + (g1 , . . . , gj−1 ) : gj = (xk y! : ij < k < ! ≤ ij+1 ) +
(xk : k ≤ ij ) + (y! : ! > ij+1 ) for j = 2, . . . s,
(3) The set B = {fk,! : ij ≤ k < ! ≤ ij+1 for some 1 ≤ j ≤ s} ∪ {g1 , . . . , gs } is a Gröbner
basis of J.
!s
(
)
[(|Fj |−1)t+1]
(4) HSR/J (t) = j=1(1−t)n+1
− ts−1 (n−s)t+s
.
n+1
(1−t)
6
Proof. The first and the second part follow from the fact that
C = {fk,! : ij ≤ k < ! ≤ ij+1 for some 1 ≤ j ≤ s}
is a Gröbner basis for JG .
In order to show that B is a Gröbner basis we apply Buchberger’s criterion, that is, we show
that all S-pairs S(f, g) for f, g ∈ B, reduce to zero. Since C is a Gröbner basis of JG it is clear
that S(f, g) reduce to zero when both f and g belong to C. Also S(f, gt ) reduce to zero when
f ∈ B, and the initial term of f and the monomial gt are coprime. So the only cases we should
check are the following ones:
a) S(fir ,j , gt ) for t + 1 ≤ r ≤ s and ir < j ≤ ir+1 . Then one can easily see that there exist
monomials wt+1 , · · · , wr−1 , w ∈ R such that
S(fir ,j , gt ) = −xj yir (yi2 · · · yit xit+1 · · · xis )/xir
=
r−1
'
wm fim ,im+1 + wgt+1
m=t+1
which implies that S(fir ,j , gt ) reduces to zero.
b) S(fj,ir , gt ) for 2 ≤ r ≤ t and ir−1 ≤ j < ir Again we can see that there exist monomials
wr , · · · , wt−1 , w ∈ R such that
S(fj,ir , gt ) = −xir yj (yi2 · · · yit xit+1 · · · xis )/yir .
=
t−1
'
wm fim ,im+1 + wgt−1 .
m=r
So it follows that S(fir ,j , gt ) reduces to zero.
In order to compute HSR/J (t) note that by part (3) the set B is a Gröbner basis for J. Thus
LT≺ (J) = LT≺ (JG ) + (g1 , . . . , gs ). Let J1 = LT≺ (JG ) : g1 and
for 2 ≤ j ≤ s.
Jj = LT≺ (JG ) + (g1 , . . . , gj−1 ) : gj
Using the corresponding short exact sequences and the parts (1) and (2),
HSR/LT≺ (J) (t)
= HSR/LT≺ (JG ) (t) − ts−1
where by equation (5),
HSR/LT≺ (JG ) (t) =
HSR/J1 (t)
%s
j=1
*
(1 − t)n+1
HSK[{x! ,y! }1≤!≤i2 ]/(xk y! :
×
HSK[{x! ,y! }i2 <!≤n ]/(y! :
!>i2 ) (t)
ij <k<!≤ij+1 ) (t)
!>ij+1 ) (t)
7
,
i1 ≤k<!≤i2 ) (t)
and for 2 ≤ j ≤ s,
× HSK[{x! ,y! }ij+1 <!≤n ]/(y! :
HSR/Jj (t),
j=1
+
(|Fj | − 1)t + 1
=
HSR/Jj (t) = HSK[{x! ,y! }ij <!≤ij+1 ]/(xk y! :
s
'
=
=
(|F1 | − 1)t + 1
,
(1 − t)(n+1)
× HSK[{x! ,y! }1≤!≤ij ]/(xk :
(|Fj | − 2)t + 1
.
(1 − t)n+1
k≤ij ) (t)
Therefore we have
HSR/LT≺ (J) (t) =
=
( %s
j=1 [(|Fj |
( %s
− 1)t + 1] )
(1 − t)n+1
j=1 [(|Fj |
− 1)t + 1] )
(1 − t)n+1
− ts−1
− ts−1
!
s
( (|F | − 1)t + 1 '
(|Fj | − 2)t + 1 )
1
+
(n+1)
(1 − t)n+1
(1 − t)
j=2
( (n − s)t + s )
(1 − t)n+1
.
Now we are going to show that how Lemma 4.1 can help to find the Hilbert function of the
binomial edge ideal of a new class of graphs. First we discuss the quasi cycle graph associated to
the closed graph G with Cohen-Macaulay binomial edge ideal.
Corollary 4.2. Let G be a connected closed graph on [n] with Cohen-Macaulay binomial edge ideal
and H be the quasi cycle graph associated to G. Then the Hilbert series of R/JH is
(
)( %
,
*
+)
s
s+1
1 − t2
(|F
|
−
1)t
+
1
+
t
(n
−
s)t
+
s
j
j=1
HSR/JH (t) =
,
(1 − t)n+1
where F1 , . . . , Fs are the facets of ∆(G). In particular, the multiplicity of R/JH is e(R/JH ) = n.
Proof. Using the same notation as Lemma 4.1 we let J = JG + (g1 , . . . , gs ). By Theorem 3.7, we
have JG : f1,n = J. So we can consider the following short exact sequence
0 → R/J(−2) → R/JG → R/JH → 0.
Thus HSR/JH (t) = HSR/JG (t)−t2 HSR/J (t) and the conclusion follows by equation (5) and Lemma
4.1 (Part 4).
!
From now on assume that G is a connected closed graph with a Cohen-Macaulay binomial edge
ideal and let 1 = j1 < j2 < · · · < js < js+1 = n be the integers such that F1 , . . . , Fs are the facets
of ∆(G), where F! = [j! , j!+1 ] for ! = 1, . . . , s.
Let H = G ∪ {e} for an arbitrary edge e. In order to study the numerical invariants of the
binomial edge ideal of H, as the proof of Corollary 4.2 we consider the following short exact
sequence
0 → R/J(−2) → R/JG → R/JH → 0,
where J = JG : fe . Indeed we have
HSR/JH (t) = HSR/JG (t) − t2 HSR/J (t).
In order to find HSR/J (t) we should consider several cases. Let us discuss one of them precisely:
Corollary 4.3. Let G be the closed graph described above, e = {jp , j! } for some p and ! with
1 < p < ! − 1 < s and H = G ∪ {e}. Then
HSR/JH (t) = HSR/JG (t) − t2 (1 − t)4 HSK[{xj ,yj }j∈V (H1 ) ]/J1 (t)HSK[{xj ,yj }j∈V (H2 ) ]/J2 (t),
where H1 is the closed graph with the facets F1 , . . . , Fp−2 , F!+1 , . . . , Fs , H2 is the closed graph with
$
$
the facets Fp−1
= [jp−1 , jp+1 ], Fp+1 , . . . , F!−2 , F!−1
= [j!−1 , j!+1 ], J1 = JH1 and J2 = JH2 + (gP,t :
where P : jp , jp+1 , . . . , j!−1 , jl and 1 ≤ t ≤ ! − p).
Proof. In order to compute the Hilbert series of R/JH we just need to compute HSR/J (t). Applying
Theorem 3.7 we have:
$
J = JH ! + (gP,t , where P : jp , jp+1 , . . . , j!−1 , jl and 1 ≤ t ≤ ! − p),
where H is a closed graph on the vertex set [n] with the cliques
$
$
F1 , . . . , Fp−2 , Fp−1
= [jp−1 , jp+1 ], Fp+1 , . . . , F!−2 , F!−1
= [j!−1 , j!+1 ], F!+1 , . . . , Fs .
Let T1 be the polynomial ring on the variables
{xj : j < jp−1 or j ≥ j!+1 } ∪ {yj : j ≤ jp−1 or j > j!+1 },
8
then LT≺ (fi,j ) ∈ T1 if {i, j} is an edge in one of the cliques F1 , . . . , Fp−2 or F!+1 , . . . , Fs .
Let T2 be the polynomial ring on the variables
{xj : jp−1 ≤ j < j!+1 } ∪ {yj : jp−1 < j ≤ j!+1 },
then LT≺ (fi,j ) ∈ T2 If {i, j} is an edge in one of the cliques
$
$
Fp−1
= [jp−1 , jp+1 ], Fp+1 , . . . , F!−2 , F!−1
= [j!−1 , j!+1 ].
Moreover for each 1 ≤ t ≤ ! − p, we have that gP,t ∈ T2 .
Thus LT≺ (J) = LT≺ (J1 ) + LT≺ (J2 ), J1 = JH1 for the closed graph H1 where facets of ∆(H1 )
are
F1 , . . . , Fp−2 , F!+1 , . . . , Fs ,
and J2 = JH2 + (gP,1 , . . . , gP,!−p ) where H2 is a closed graph and the facets of ∆(H2 ) are
$
$
Fp−1
= [jp−1 , jp+1 ], Fp+1 , . . . , F!−2 , F!−1
= [j!−1 , j!+1 ].
Moreover
HSR/J (t) =
=
HSR/LT≺ (J) (t) = HST1 /LT≺ (J1 ) (t)HST2 /LT≺ (J2 ) (t)
(1 − t)4 HSK[{xj ,yj }j∈V (H1 ) ]/J1 (t)HSK[{xj ,yj }j∈V (H2 ) ]/J2 (t).
!
Note that in Corollary 4.3 for the case that (p, !) = (2, s), H1 is the graph on two vertices 1, n,
and E(H1 ) = ∅.
Remark 4.4. For each graph H in Corollary 4.3 we can explicitly compute HSR/H (t) using
equations (4),(5) and Lemma 4.1 (part 4). Moreover a careful modification of Corollary 4.3 works
to find the Hilbert series of R/J in other cases, where e = {jp , z} with jp < j! < z < j!+1 or
e = {q, z} with jp < q < jp+1 ≤ j! < z < j!+1 .
In the following we are going to give another application of Theorem 3.7. Let e = {i, j} be an
arbitrary edge of G and H = G \ {e}. If we set J = JH : fe then we have the short exact sequence
0 → R/J(−2) → R/JH → R/JG → 0.
(9)
In order to study Hilbert function, depth and dimension of R/JH by using the sequence (9) it is
enough to characterize the same invariants for R/J because
HSR/JH (t) = HSR/JG (t) + t2 HSR/J (t).
(10).
Corollary 4.5. Let G be a graph as in Corollary 4.3, e = {j!., j!+1 } for some 1 ≤ ! ≤ s and
n + 2 if |F! | = 2,
H = G \ {e}. Then depth(R/JH ) = n − |F! | + 4, dim(R/JH ) =
n + 1 otherwise
and
HSR/JH (t) = HSR/JG (t) + t2 HSK[{xj ,yj }j∈V (G1 ) ]/JG1 (t)HSK[{xj ,yj }j∈V (G2 ) ]/JG2 (t),
where G1 is the closed graph with the facets F1 , . . . , F!−1 and G2 is the closed graph with the facets
F!+1 , . . . , Fs .
Proof. Let J = JH : fe by Theorem 3.7 we have:
J = JG1 + (xj , yj : j! < j < j!+1 ) + JG2 ,
where G1 and G2 are closed graphs and F1 , . . . , F!−1 are the facets of ∆(G1 ) and F!+1 , . . . , Fs are
the facets of ∆(G2 ). In this case
R/J ∼
= K[{xj , yj }j∈V (G ) ]/JG ⊗K K[{xj , yj }j∈V (G ) ]/JG .
1
1
2
2
Thus R/J is Cohen-Macaulay and using equations (4) and (5),
HSR/J (t) = HSK[{xj ,yj }j∈V (G1 ) ]/JG1 (t)HSK[{xj ,yj }j∈V (G2 ) ]/JG2 (t)
9
=
s (
%
r=1
r%=!
(|Fr | − 1)t + 1
(1 − t)n−|F! |+4
)
.
So dim(R/J) = depth(R/J) = n − |F! | + 4 and by equation (10) it is possible to compute the
Hilbert function of R/JH easily.
Moreover if |F! | < 4 it is clear that R/JH is Cohen-Macaulay and dim(R/JH ) = n − |F! | + 4
and if |F! | ≥ 4 then dim(R/JH ) = n + 1.
For computing the depth of R/JH we can apply depth formula to (9) and see that depth(R/JH ) =
n − |F! | + 4.
!
Corollary 4.6. Let G be a graph as in Corollary .
4.3, e = {j! , z} for some 1 ≤ ! ≤ s, j! <
n+1
if |F! | = 3,
z < j!+1 and H = G \ {e} then depth(R/JH ) =
, dim(R/JH ) =
n − |F! | + 5 otherwise
.
n + 2 if |F! | = 3,
and
n + 1 otherwise
HSR/JH (t) = HSR/JG (t) +
t2
HSK[{xj ,yj }j∈V (G1 ) ]/JG1 (t)HSK[{xj ,yj }j∈V (G2 ) ]/JG2 (t),
(1 − t)2
where G1 is the closed graph with the facets F1 , . . . , F!−1 and G2 is the closed graph with the facets
[j!+1 + 1, j!+2 ], F!+2 , . . . , Fs .
Proof. Let J = JH : fe by Theorem 3.7 we have:
J = JG1 + (xj , yj : j! < j ≤ j!+1 , j %= z) + JG2 ,
where G1 is a closed graph such that F1 , . . . , F!−1 are the facets of ∆(G1 ) and G2 is a closed graph
such that [j!+1 + 1, j!+2 ], F!+2 , . . . , Fs are the facets of ∆(G2 ). In this case
R/J ∼
= K[{xj , yj }j∈V (G1 ) ]/JG1 ⊗K K[xz , yz ] ⊗K K[{xj , yj }j∈V (G2 ) ]/JG2 .
Thus R/J is Cohen-Macaulay and using equation (5),
HSR/J (t) = HSK[{xj ,yj }j∈V (G1 ) ]/JG1 (t)HSK[xz ,yz ] (t)HSK[{xj ,yj }j∈V (G2 ) ](t)/JG2 (t)
=
(
(|F!+1 | − 2)t + 1
)
(1 −
s
%
(
(|Fr | − 1)t + 1
r=1
r%=!,!+1
t)n−|F! |+5
)
.
So dim(R/J) = depth(R/J) = n − |F! | + 5 and by equation (10) it is possible to compute the
Hilbert series of R/JH .
Moreover, if |F! | = 3 it is clear that dim(R/JH ) = n − |F! | + 5 = n + 2 and using depth formula
on the sequence (9), n + 1 ≤ depth(R/JH ) ≤ n + 2. On the other hand one can easily check
that h−vector of R/JH has negative entries so it can not be Cohen-Macaulay and it follows that
depth(R/JH ) = n + 1.
Finally if |F! | ≥ 4 then dim(R/JH ) = n + 1 and depth(R/JH ) = n − |F! | + 5.
!
Remark 4.7. With a similar argument as in Corollary 4.5 and Corollary 4.6 we can study JH
where H = G \ {e}, G is the closed graph described before and .
e = {p, z} for some j! < p < z <
n+1
if |F! | = 4,
j!+1 , 1 < ! < s. In this case we can see that depth(R/JH ) =
and
n
−
|F
|
+
6
otherwise
!
.
n + 2 if |F! | = 4,
dim(R/JH ) =
.
n + 1 otherwise
10
5. The quotient of binomial ideals and CI-ideals
One application of binomial edge ideals is to study the conditional independence ideals, see [3, 5].
We recall some definitions in CI-ideals from [2]. Consider a random vector X = (X0 , . . . , XN ),
where each Xi is a discrete random variable which takes values among the integers 1, . . . , di . Let
px0 ,...,xN be the probability of the event X0 = x0 , X1 = x1 , . . . , XN = xN . Then a joint probability
distribution
# of X is a non-negative real valued function p denoted by p = (px0 ,...,xN )x0 ∈[d0 ],...,xN ∈[dN ] ,
where x∈X p(x) = 1.
Let S be the polynomial ring S = C[px0 ,...,xN : x0 ∈ [d0 ], . . . , xN ∈ [dN ]], and for a subset
A ⊆ {0, 1, . . . , N } consider the subsets XA = {Xa : a ∈ A} and
{XA = xA } = {y ∈ [d0 ] × · · · × [dN ] : ya = xa for all a ∈ A}.
#
Then by p(xT ) we mean x∈{XA =xA } px .
Assume that A and B be two disjoint subsets of {0, . . . , N }, and C ⊂ {(x0 , . . . , xN ) : x0 ∈
[d0 ], . . . , xN ∈ [dN ]} and let p be a joint probability distribution.
The collection XA of random variables is said to be conditionally independent of XB given C
if and only if for all xA , yA ∈ ×a∈A [da ] and xB , yB ∈ ×b∈B [db ], the joint probability distribution p
admits the following equations:
p(xA , xB ; C)p(yA , yB ; C) − p(xA , yB ; C)p(yA , xB ; C) = 0.
Then we will denote it by XA ⊥ XB |C. The ideals generated by some equations of this form are
called CI-ideals.
A collection C of CI-statements is called a robustness specification if there exist subsets A1 , . . . , At
of {0, . . . , N } such that
C = {X0 ⊥ XAc1 |XA1 , . . . , X0 ⊥ XAct |XAt }.
Now assume that d0 = 2 and C be a robustness specification. Then there is a graph G associated
to C on the vertex set {(x1 , . . . , xN ) : x1 ∈ [d1 ], . . . , xN ∈ [dN ]} such that the vertices (x1 , . . . , xN )
and (y1 , . . . , yN ) are adjacent if and only if there exists some ! such that xi = yi for all i ∈ A! .
We quote the next result from [3, Corollary 4.5].
Corollary 5.1. Each minimal prime a CI-ideal of a robustness specification is determined by a subset S of structural zeros in the distribution of X[N ] which is common to all probability distributions
lying in the component corresponding to that prime ideal. The possible sets S are characterized by
[3, Corollary 3.9].
Let PS (G) be a minimal prime of G and G1 , . . . , GS be the connected components of G\S. Then
corresponding to each component Gi of G \ S we have the CI-statement X0 ⊥ X[N ] |(XN ∈ Gi ).
Therefore having such S, just by knowing the component of G \ S containing the random vector
X[N ] we will get all relevant information about X0 .
Given an arbitrary CI-ideal I of a robustness specification, and the binomial
fxy = p1,x1 ,...,x[N ] p2,y1 ,...,y[N ] − p1,y1 ,...,y[N ] p2,x1 ,...,x[N ] ,
what can be said about the ideal J = I : fxy ? Is it again a CI-ideal or can we find a combinatorial
criteria to describe the minimal primes of I : fxy ? In the case that J is a CI-ideal of a robustness
specification, how one can find a natural graph associated to J with binomial edge ideal equal to
J.
Assume that I be a CI-ideal of a robustness specification such that
fxy = p1,x1 ,...,x[N ] p2,y1 ,...,y[N ] − p1,y1 ,...,y[N ] p2,x1 ,...,x[N ]
does not belong to I. We observed that minimal prime ideals of the quotient ideal I : fxy are
among the minimal primes of I. Therefore as a consequence of Proposition 3.2 we have
Corollary 5.2. Let I be a CI-ideal of a robustness specification C such that
fxy = p1,x1 ,...,x[N ] p2,y1 ,...,y[N ] − p1,y1 ,...,y[N ] p2,x1 ,...,x[N ]
11
does not belong to I. Then the ideal I : fxy is a CI-ideal of a robustness specification if and only
if {x, y} is a bridge in the graph G ∪ {x, y} associated to I. In this case the graph associated to
I : fxy is Gx,y by Theorem 3.4.
Remark 5.3. Let I be a CI-ideal of a robustness specification C such that
fxy = p1,x1 ,...,x[N ] p2,y1 ,...,y[N ] − p1,y1 ,...,y[N ] p2,x1 ,...,x[N ]
does not belong to I, and let G be the associated graph to I. If {x, y} is not a bridge of G ∪ {x, y},
then the ideal I : fxy is not a CI-ideal of a robustness specification. But by Theorem 3.2 we know
its primary decomposition.
Acknowledgment: We wish to thank J. Herzog for useful discussions in connection with this
paper. This paper was written when the first author was a postdoc at the Mathematical Sciences
Research Institute (MSRI), and gratefully acknowledges support from the institute.
References
[1] M. F. Atiyah, I. G. Macdonald. Introduction to Commutative Algebra, Addison-Wesley (1969).
[2] M. Drton, B. Sturmfels, S. Sullivant, “Lectures on Algebraic Statistics,” Birkhäuser, (2009).
[3] J. Herzog, T. Hibi, F. Hreinsdotir, T. Kahle, J. Rauh. Binomial edge ideals and conditional independence
statements, Adv. Appl. Math 45, 317–333.
[4] V. Ene, J. Herzog, T. Hibi. Cohen-Macaulay Binomial Edge Ideals, Nagoya Math. J. 204 (2011) 57–68.
[5] A. Fink. The binomial ideal of the intersection axiom for conditional probabilities, J. Algebraic Combin. 33,
(2011) 455–463.
[6] M. Ohtani. Graphs and Ideals generated by some 2-minors, Comm. in Algebra 39, (2011) 905–917.
Fatemeh Mohammadi, Mathematical Sciences Research Institute, Berkeley, CA 94720-3840, U.S.A.
Current address: Philipps-Universitt Marburg Fachbereich Mathematik und Informatik Hans-MeerweinStrasse 35032 Marburg, Germany.
E-mail address: [email protected]
Department of Mathematics, Hakim Sabzevari University, Sabzevar, Iran, and School of Mathematics, Institute for research in Fundamental Sciences (IPM), P. O. Box: 19395-5746, Tehran.
E-mail address: [email protected]
12