Algebraic properties of the binomial edge ideal of complete bipartite graph

0.19 MB·English

by  Peter Schenzel

ALGEBRAIC PROPERTIES OF THE BINOMIAL EDGE IDEAL OF A
COMPLETE BIPARTITE GRAPH
PETER SCHENZEL, SOHAIL ZAFAR
Abstract. Let J denote the binomial edge ideal of a connected undirected graph on
G
n vertices. This is the ideal generated by the binomials x y −x y ,1 ≤ i < j ≤ n, in
i j j i
3
1 the polynomialringS =K[x1,...,xn,y1,...,yn]where{i,j}isanedgeofG. We study
the arithmetic properties of S/J for G, the complete bipartite graph. In particular we
0 G
2 compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and
multiplicities of them. As main results we give an explicit description of the modules of
n
a deficiencies,the duals oflocalcohomologymodules, andprovethe purityofthe minimal
J free resolution of S/J .
G

4

]
C
A
1. Introduction
.
h
t The main intention of the present paper is the study of the binomial edge ideal of the
a
m completebipartitegraph. LetGdenoteaconnectedundirectedgraphonnverticeslabeled
by [n] = {1,2,...,n}. Foranarbitraryfield K let S = K[x ,...,x ,y ,...,y ] denote the
[ 1 n 1 n

polynomial ring in 2n variables. To the graph G one can associate the binomial edge ideal
1
v JG ⊂ S generatedbybinomialsxiyj−xjyi,i < j,suchthat{i,j}isanedgeofG. Thisisan
9 extension of the edge ideal (generated by the monomials) as it was studied for instance in
8
[12]. The binomial edge ideals of a graph G might have applications in algebraic statistics
7
0 (see [6]). By the work of Herzog et al. (see [6]) the minimal primary decomposition of
.
1 JG is known. Besides of that not so much is known about the arithmetic properties of
0 S/J . If G denotes the complete graph on n vertices, then S/J is the coordinate ring
G G
3
of the Segre embedding P1 × Pn . This is a variety of minimal degree. Therefore S/J
1 K K G
: is a Cohen-Macaulay ring with a linear resolution. In the paper Ene, Herzog and Hibi
v
i (see [4]) they studied Cohen-Macaulayness property for some special classes of graphs.
X
By view of the primary decomposition of J (see [6]) it follows that S/J is not so often
r G G
a a Cohen-Macaulay ring. As a certain generalization of the Cohen-Macaulay property the
second author has studied approximately Cohen-Macaulay rings (see [13]). In the present
paper we investigate the binomial edge ideal of another important class of graphs, namely
the complete bipartite graph G = K (see the definitions in Section 3).
m,n
As the main result of our investigations we prove (among others) the following results:
Theorem 1.1. With the previous notation let J ⊂ S denote the binomial edge ideal
G
associated to the complete bipartite graph G = K .
m,n
2010 Mathematics Subject Classification. 05E40,13H10, 13D45.
Key words and phrases. binomail edge ideal, complete bipartite graph, pure resolution.
This research was partially supported by the Higher Education Commission, Pakistan.
1
2 PETER SCHENZEL,SOHAILZAFAR
(a) dimS/J = max{n+m+1,2m} and
G
m+2, if n = 1,
depthS/J =
G n+2, if m ≥ n > 1.
(cid:26)
(b) There is an explicit expression of the Hilbert series and the multiplicity equals
1, if m > n+1 or n = 1 and m > 2,
e(S/J ) =
G
2m, otherwise.
(
(c) The Castelnuovo-Mumford regularity is regS/J = 2 and S/J admits a pure
G G
minimal free resolution.
(d) There are at most 5 non-vanishing local cohomology modules Hi (S/J ). The
S+ G
modules of deficiencies ωi(S/J ) = Hom (Hi (S/J ),K) are either Cohen-
G K S+ G
Macaulay modules or the direct sum of two Cohen-Macaulay modules.
(e) S/J is a Cohen-Macaulay canonically ring in the sense of [10].
G
For the details on the modules of deficiencies we refer to Section 4 of the paper. This
is - at least for us - the first time in the literature that there is a complete description of
the structure of the modules of deficiencies besides of sequentially Cohen-Macaulay rings
or Buchsbaum rings. Our analysis is based on the primary decomposition of J as shown
G
in [6].
In Section 2 we start with preliminary and auxiliary results needed in the rest of the
paper. In particular we give a short overview on the modules of deficiencies. In Section
3 we study some of the properties of the the binomial edge ideal J ⊂ S associated to
G
a complete bipartite graph. In Section 4 we give a complete list of all the modules of
deficiencies of the complete bipartite graphs. In the final Section 5 we prove the purity of
the minimal free resolution of S/J . This is the heart of our investigations. It gives in a
G
natural way some non-Cohen-Macaulay rings with pure resolutions. We might relate our
investigations as a better understanding of general binomial edge ideals.
2. Preliminaries and auxiliary results
First of all we will introduce the notation used in the sequel. Moreover we summarize
a few auxiliary results that we need.
We denote by G a connected undirected graph on n vertices labeled by [n] =
{1,2,...,n}. For an arbitrary field K let S = K[x ,...,x ,y ,...,y ] denote the poly-
1 n 1 n
nomial ring in the 2n variables x ,...,x ,y ,...,y . To the graph G one can associate
1 n 1 n
an ideal J ⊂ S generated by all binomials x y − x y for all 1 ≤ i < j ≤ n such that
G i j j i
{i,j} is an edge of G. This construction was invented by Herzog et al. in [6] and [4]. At
first let us recall some of their definitions.
˜
Definition 2.1. Fix the previous notation. For a set T ⊂ [n] let G denote the complete
T
graph on the vertex set T. Moreover let G denote the graph obtained by deleting all
[n]\T
vertices of G that belong to T.
Let c = c(T) denote the number of connected components of G . Let G ,...,G
[n]\T 1 c
denote the connected components of G . Then define
[n]\T
P (G) = (∪ {x ,y },J ,...,J ),
T i∈T i i G˜1 G˜C(T)
BINOMIAL EDGE IDEAL OF A COMPLETE BIPARTITE GRAPH 3
where G˜ ,i = 1,...,c, denotes the complete graph on the vertex set of the connected
i
component G ,i = 1,...,c.
i
The following result is important for the understanding of the binomial edge ideal of
G.
Lemma 2.2. With the previous notation the following holds:
(a) P (G) ⊂ S is a prime ideal of height n−c+|T|, where |T| denotes the number of
T
elements of T.
(b) J = ∩ P (G).
G T⊆[n] T
(c) J ⊂ P (G) is a minimal prime if and only if either T = ∅ or T 6= ∅ and
G T
c(T \{i}) < c(T) each i ∈ T.
(cid:3)
Proof. For the proof we refer to [6].
Therefore J is the intersection of prime ideals. That is, S/J is a reduced ring.
G G
Moreover, weremarkthatJ isanidealgeneratedbyquadricsandthereforehomogeneous,
G
so that S/J is a graded ring with natural grading induced by the N-grading of S. As a
G
technical tool we shall need the following result.
Proposition 2.3. Let I ⊂ S denote an ideal. Let f = f ,...,f denote an S/I-regular
1 r
sequence. Then fS ∩I = fI.
Proof. It is easy to see that TorS(S/fS,S/I) ∼= fS ∩I/fI. Moreover
1
TorS(S/fS,S/I) ∼= H (f;S/I),
1 1
where H (f;S/I) denotes the Koszul homology of f with respect to S/I. But these
i
homology modules vanish for i > 0. (cid:3)
Let M denote a finitely generated graded S-module. In the sequel we shall use also the
local cohomology modules of M with respect to S , denoted by Hi(M),i ∈ Z. Note that
+
they are graded Artinian S-modules. We refer to the textbook of Brodmann and Sharp
(see [1]) for the basics on it. In particular the Castelnuovo-Mumford regularity regM of
M is defined as
reg(M) := max{e(Hi(M))+i|depth(M) ≤ i ≤ dim(M)},
where e(Hi(M)) is the least integer m such that, for all k > m, the degree k part of the
i-th local cohomology module of M is zero. For our investigations we need the following
definition.
Definition 2.4. Let M denote a finitely generated graded S-module and d = dimM.
For an integer i ∈ Z put
ωi(M) = Ext2n−i(M,S(−2n))
S
and call it thei-th module of deficiency. Moreover we define ω(M) = ωd(M) the canonical
module of M. We write also ω (M) = ω(ω(M)). These modules have been introduced
2×
and studied in [8].
4 PETER SCHENZEL,SOHAILZAFAR
Note that by the graded version of Local Duality (see e.g. [1]) there is the natural
graded isomorphism ωi(M) ∼= Hom (Hi(M),K) for all i ∈ Z. For a finitely generated
K
graded S-module M and an integer i ∈ N we set
(AssM) = {p ∈ AssM|dimS/p = i}.
i
In the following we shall summarize a few properties on the modules of deficiencies.
Proposition 2.5. Let M denote a finitely generated graded S-module and d = dimM.
(a) dimωi(M) ≤ i and dimωd(M) = d.
(b) (Assωi(M)) = (AssM) for all 0 ≤ i ≤ d.
i i
(c) M satisfies the Serre condition S if and onlyif dimωi(M) ≤ i−2 forall 0 ≤ i < d.
2
(d) There is a natural homomorphism M → ωd(ωd(M)). It is an isomorphism if and
only if M satisfies the Serre condition S .
2
(e) For a homogeneous ideal I ⊂ S there is a natural isomorphism ωd(ωd(S/I)) ∼=
Hom (ωd(S/I),ωd(S/I)),d = dimS/I, and it admits the structure of a commuta-
S
tive Noetherian ring, the S -fication of S/I.
2
(f) The natural map S/I → Hom (ωd(S/I),ωd(S/I)),d = dimS/I, sends the unit
S
element to the identity map. Therefore it is a ring homomorphism.
Proof. The results are shown in [8] and [10]. The proofs in the graded case follow the
(cid:3)
same line of arguments.
A decreasing sequence {M } of a d-dimensional S-module M is called dimension
i 0≤i≤d
filtration of M, if M /M is either zero or of dimension i for all i = 0,...,d, where
i i−1
M = 0. It was shown (see [9]) that the dimension filtration exists and is uniquely
−1
determined.
Definition 2.6. An S-module M is called sequentially Cohen-Macaulay if the dimension
filtration {M } has the property that M /M is either zero or an i-dimensional
i 0≤i≤d i i−1
Cohen-Macaulay module for all i = 0,...,d, (see [9]). Note that in [9] this notion was
originally called Cohen-Macaulay filtered.
Note that a sequentially Cohen-Macaulay S-module M with depthM ≥ dimM − 1
was studied by Goˆto (see [5]) under the name approximately Cohen-Macaulay. For our
purposes here we need the following characterization of sequentially Cohen-Macaulay
modules.
Theorem 2.7. Let M be a finitely generated graded S-module with d = dimM. Then the
following conditions are equivalent:
(i) M is a sequentially Cohen-Macaulay.
(ii) For all 0 ≤ i < d the module of deficiency ωi(M) is either zero or an i-dimensional
Cohen-Macaulay module.
(iii) For all 0 ≤ i ≤ d the modules ωi(M) are either zero or i-dimensional Cohen-
Macaulay modules.
Proof. In the case of a local ring admitting a dualizing complex this result was shown in
[9, Theorem 5.5]. Similar arguments work also in the case of a finitely generated graded
S-module M. Note that the equivalence of (i) and (iii) was announced (without proof)
(cid:3)
in [11].
BINOMIAL EDGE IDEAL OF A COMPLETE BIPARTITE GRAPH 5
3. Complete bipartite graphs
A bipartite graph is a graph whose vertices can be divided into two disjoint sets V and
1
V such every edge of G connects a vertex in V to one in V . Now the complete bipartite
2 1 2
graph is a bipartite graph G such that for any two vertices, v ∈ V and v ∈ V , v v is
1 1 2 2 1 2
an edge in G. If |V | = n and |V | = m then it is usually denoted by K . To simplifying
1 2 n,m
notations we denote it often by G.
Definition 3.1. For a sequence of integers 1 ≤ i < i < ... < i ≤ n + m let
1 2 k
I(i ,i ,...,i ) denote the ideal generated by the 2×2 minors of the matrix
1 2 k
x x ··· x
i1 i2 ik .
y y ··· y
(cid:18) i1 i2 ik(cid:19)
NotethatI(i ,i ,...,i )istheidealofthecompletegraphonthevertexset{i ,i ,...,i }.
1 2 k 1 2 k
Let J be the binomial edge ideal of complete bipartite graph on [n+m] vertices and
G
J be the binomial edge ideal of complete graph on [n + m] vertices. We begin with a
G˜
lemma concerning the dimension of S/J .
G
Lemma 3.2. Let G = K ,m ≥ n, denote the complete bipartite graph. Let G˜ denote
m,n
the complete graph on [n + m]. Let A = (x ,...,x ,y ,...,y ) for n ≥ 1 and B =
n 1 n 1 n m
(x ,...,x ,y ,...,y ) for n ≥ 2 and B = S for n = 1.
n+1 n+m n+1 n+m m
(a) J = J ∩A ∩B is the minimal primary decomposition of J .
G G˜ n m G
(b) dimS/J = max{n+m+1,2m}.
G
(c) (J ∩A ,J ∩B ) = J .
G˜ n G˜ m G˜
Proof. Westart with theproofof (a). We usethe statement proved inLemma 2.2. At first
consider the case m > n = 1. By view of Lemma 2.2 we have to find all ∅ =6 T ⊆ [1+m]
such thatc(T\{i}) < c(T). ClearlyT = {1}satisfytheconditionbecausec(T ) = m > 1.
0 0
Let T denote T ⊂ [1+m] a subset different of T . Then If 1 ∈ T then c(T) = m+1−|T|
0
and c(T \{i}) = m+2−|T| for i 6= 1 and if 1 6∈ T then c(T) = 1 and c(T \{i}) = 1 for
all i ∈ T. Hence we have the above primary decomposition.
Now consider the case of m ≥ n ≥ 2. As above we have to find all ∅ 6= T ⊆ [n+ m]
such that c(T \{i}) < c(T) for all i ∈ T. T = {1,2,...,n} satisfy the above condition
1
because c(T) = m and c(T\{i}) = 1 for all i ∈ T. Similarly T = {n+1,n+2,...,n+m}
2
also satisfies the above condition.
Our claim is that no other T ⊆ [n+m] satisfies this condition. If T 6⊆ T and T 6⊆ T
1 2
then c(T) = 1 so in this case T does not satisfy the above condition. Now suppose that
T ( T then c(T) = m−|T \T | and c(T \{i}) = m+1−|T \T | if i ∈ T \T . The same
1 1 1 1
argument works if T ( T. Hence we have J = J ∩A ∩B .
2 G G˜ n m
Then the statement on the dimension in (b) is a consequence of the reduced primary
decomposition shown in (a). To this end recall that dimS/A = 2m,dimS/B = 2n and
n m
dimS/J = n+m+1.
G˜
For the proof of (c) we use the notation of the Definition 3.1. Then it follows that
J ∩A = (I(1,...,n,n+i),i = 1,...,m,I(n+1,...,n+m)∩A ).
G˜ n n
Now A consists of an S/I(n+1,...,n+m)-regular sequence and
n
I(n+1,...,n+m)∩A = A I(n+1,...,n+m)
n n
6 PETER SCHENZEL,SOHAILZAFAR
by Proposition 2.3. Therefore we get
J ∩A = (I(1,...,n,n+i),i = 1,...,m,A I(n+1,...,n+m))
G˜ n n
and similarly
J ∩B = (I(j,n+1,...,n+m),j = 1,...,n,B I(1,...,n)).
G˜ m m
But this clearly implies that (J ∩ A ,J ∩ B ) = J which proves the statement in
G˜ n G˜ m G˜
(cid:3)
(c).
For the further computations we use the previous Lemma 3.2. In particular we use
three exact sequences shown in the next statement.
Corollary 3.3. With the previous notation we have the following three exact sequences.
(1) 0 → S/J → S/J ∩A ⊕S/J ∩B → S/J → 0.
G G˜ n G˜ m G˜
(2) 0 → S/J ∩A → S/J ⊕S/A → S/(J ,A ) → 0.
G˜ n G˜ n G˜ n
(3) 0 → S/J ∩B → S/J ⊕S/B → S/(J ,B ) → 0.
G˜ m G˜ m G˜ m
Proof. Theproofisaneasyconsequence oftheprimarydecompositionasshown inLemma
(cid:3)
3.2. We omit the details.
Note that in case of n = 1 we have B = S therefore it is enough to consider the exact
m
sequence (2) as (1) and (3) gives no information.
Corollary 3.4. With the previous notation we have that
m+2, if n = 1 ;
depthS/J =
G n+2, if m ≥ n > 1
(cid:26)
and regS/J ≤ 2.
G
Proof. The statement is an easy consequence of the short exact sequences shown in Corol-
lary 3.3. To this end note that S/J ,S/(J ,A ) and S/(J ,B ) are Cohen-Macaulay
G˜ G˜ n G˜ m
rings of dimension n + m + 1,m + 1 and n + 1 respectively. Moreover regS/J =
G˜
regS/(J ,A ) = regS/(J ,B ) = 1. By using the exact sequences it provides the
G˜ n G˜ m
statement on the regularity. For the behaviour of the depth respectively the regularity in
(cid:3)
short exact sequences see [2, Proposition 1.2.9] respectively [3, Corollary 20.19].
4. The modules of deficiency
The goal of this section is to describe all the local cohomology modules Hi(S/J ) of the
G
binomial edge ideal of a complete bipartite graphG. We do this by describing their Matlis
dualswhichbyLocalDualityarethemodulesofdeficiencies. Moreover, forahomogeneous
ideal J ⊂ S let H(S/J,t) denote the Hilbert series, i.e. H(S/J,t) = (dim [S/J] )ti.
i≥0 K i
We start our investigations with the so-called star graph. That is complete bipartite
P
graph K with n = 1. For m ≤ 2 the ideal J is a complete intersection generated by
m,n G
one respectively two quadrics so let us assume that m > 2.
Theorem 4.1. Let G denote the star graph K . Then the binomial edge ideal J ⊂ S
m,1 G
has the following properties:
(a) regS/J = 2.
G
(b) ωi(S/J ) = 0 if and only if i 6∈ {m+2,2m}.
G
BINOMIAL EDGE IDEAL OF A COMPLETE BIPARTITE GRAPH 7
(c) ω2m(S/J ) ∼= S/A (−2m)
G 1
(d) ωm+2(S/J ) is a (m + 2)-dimensional Cohen-Macaulay module and there is an
G
isomorphism ωm+2(ωm+2(S/J )) ∼= (J ,A )/J .
G G˜ 1 G˜
Proof. We use the short exact sequence of Corollary 3.3 (2). It induces a short exact
sequence
0 → Hm+1(S/(J ,A )) → Hm+2(S/J ) → Hm+2(S/J ) → 0
G˜ 1 G G˜
and an isomorphism H2m(S/J ) ∼= H2m(S/A ). Moreover the Cohen-Macaulayness of
G 1
S/J ,S/A and S/(J ,A ) of dimensions m+2, 2m and m+1 respectively imply that
G˜ 1 G˜ 1
Hi(S/J ) = 0 if i 6∈ {m+2,2m}.
G
The short exact sequence on local cohomology induces the following exact sequence
0 → ωm+2(S/J ) → ωm+2(S/J ) → ωm+1(S/(J ,A )) → 0
G˜ G G˜ 1
by Local Duality. Now we apply again local cohomology and take into account that both
ωm+2(S/J ) and ωm+1(S/(J ,A )) are Cohen-Macaulay modules of dimension m+2 and
G˜ G˜ 1
m+1 respectively. Then depthωm+2(S/J ) ≥ m+1. By applying local cohomology and
G
dualizing again it induces the following exact sequence
0 → ωm+2(ωm+2(S/J )) → S/J →f S/(J ,A ) → ωm+1(ωm+2(S/J )) → 0.
G G˜ G˜ 1 G
The homomorphism f is induced by the commutative diagram
S/J → S/(J ,A )
G˜ G˜ 1
↓ ↓
ω (S/J ) → ω (S/J ,A ).
2× G˜ 2× G˜ 1
Note that the vertical maps are isomorphisms (see Proposition 2.5). Since the up-
per horizontal map is surjective the lower horizontal map is surjective too. Therefore
ωm+1(ωm+2(S/J )) = 0. That is depthωm+2(S/J ) = m+2 and it is a Cohen-Macaulay
G G
module. Moreover ωm+2(ωm+2(S/J )) ∼= (J ,A )/J . This finally proves the statements
G G˜ 1 G˜
in (b), (c) and (d).
It is well known that regS/J = regS/(J ,A ) = 1 and regS/A = 0. Then an
G G˜ 1 1
inspection with the short exact sequence of Corollary 3.3 shows that regS/J = 2. (cid:3)
G
In the next result we will consider the modules of deficiencies of the complete bipartite
graph G = K ,n ≥ 2.
m,n
Theorem 4.2. Let m ≥ n > 1 and assume that the pair (m,n) is different from (n+1,n)
and (2n−2,n). Then
(a) regS/J = 2.
G
(b) ωi(S/J ) = 0 if and only if i 6∈ {n+2,m+2,2n,m+n+1,2m} and there are the
G
following isomorphisms and integers
i ωi(S/J ) depthωi(S/J ) dimωi(S/J )
G G G
n+2 ωn+1(S/(J ,B )) n+1 n+1
G˜ m
m+2 ωm+1(S/(J ,A )) m+1 m+1
G˜ n
2n S/B (−2n) 2n 2n
m
n+m+1 ωm+n+1(S/J ) n+m+1 n+m+1
G˜
2m S/A (−2m) 2m 2m
n
8 PETER SCHENZEL,SOHAILZAFAR
Proof. Under the assumption of n+1 < m < 2n−2 it follows that 2m > m+n+1 > 2n >
m+2 > n+2. Then the short exact sequences (see Corollary 3.3) induce the following
isomorphisms:
(1) Hn+2(S/J ) ∼= Hn+2(S/J ∩B ) ∼= Hn+1(S/(J ,B )),
G G˜ m G˜ m
(2) Hm+2(S/J ) ∼= Hm+2(S/J ∩A ) ∼= Hm+1(S/(J ,A )),
G G˜ n G˜ n
(3) H2n(S/J ) ∼= H2n(S/B ) and
G m
(4) H2m(S/J ) ∼= H2m(S/A ).
G n
Moreover there is the following short exact sequence
0 → Hm+n+1(S/J ) → Hm+n+1(S/J ∩A )⊕Hm+n+1(S/J ∩B ) → Hm+n+1(S/J ) → 0
G G˜ n G˜ m G˜
and Hi(S/J ) = 0 if i 6∈ {n+2,m+2,2n,m+n+1,2m}.
G
Because of the short exact sequences in Corollary 3.3 there are isomorphisms
Hm+n+1(S/J ∩B ) ∼= Hm+n+1(S/J ) ∼= Hm+n+1(S/J ∩A ).
G˜ m G˜ G˜ n
So by Local Duality we get a short exact sequence
0 → ω(S/J ) → ω(S/J )⊕ω(S/J ) → ωm+n+1(S/J ) → 0.
G˜ G˜ G˜ G
This implies depthω(S/J ) ≥ n+m. Moreover by applying local cohomology and again
G
the Local Duality we get the following commutative diagram with exact rows
0 → S/J → S/J ∩A ⊕S/J ∩B → S/J → 0
G G˜ n G˜ m G˜
↓ ↓ k
0 → Ω(S/J ) → S/J ⊕S/J →f S/J → ωm+n(ωm+n+1(S/J )) → 0,
G G˜ G˜ G˜ G
where Ω(S/J ) = ωm+n+1(ωm+n+1(S/J )). Now we show that ωm+n(ωm+n+1(S/J )) = 0.
G G G
This follows since f is easily seen to be surjective. That is, ωm+n+1(S/J ) is a (m+n+1)-
G
dimensional Cohen-Macaulay module. Moreover f is a split homomorphism and therefore
Ω(S/J ) ≃ S/J By duality this implies that ωm+n+1(S/J ) ∼= ω(S/J ). This completes
G G˜ G G˜
theproofofthestatementsin(b). Bysimilarargumentstheothercasesfor(m,n)different
of (n+1,n) and (2n−2,n) can be proved. We omit the details. Clearly regS/J = 2 as
G
(cid:3)
follows by (b).
As a next sample of our considerations let us consider the case of the complete bipartite
graph K with (m,n) = (n+1,n).
m,n
Theorem 4.3. Let m = n+1 and n > 3. Then:
(a) regS/J = 2.
G
(b) ωi(S/J ) = 0 if and only if i 6∈ {n+2,n+3,2n,2n+2} and there are the following
G
isomorphisms and integers
i ωi(S/J ) depthωi(S/J ) dimωi(S/J )
G G G
n+2 ωn+1(S/(J ,B )) n+1 n+1
G˜ n+1
n+3 ωn+2(S/(J ,A )) n+2 n+2
G˜ n
2n S/B (−2n) 2n 2n
n+1
2n+2 ω(S/J )⊕S/A (−2n−2) 2n+2 2n+2.
G˜ n
Proof. By applying the local cohomology functors H·(−) to the exact sequence (2) in
Corollary 3.3 we get the following:
BINOMIAL EDGE IDEAL OF A COMPLETE BIPARTITE GRAPH 9
(1) Hn+3(S/J ∩A ) ∼= Hn+2(S/(J ,A )),
G˜ n G˜ n
(2) H2n+2(S/J ∩A ) ∼= H2n+2(S/J )⊕H2n+2(S/A ) and
G˜ n G˜ n
(3) Hi(S/J ∩A ) = 0 for i 6= n+2,2n+2.
G˜ n
Similarly, if we apply H·(−) to the exact sequence (3) in Corollary 3.3 we get
(4) Hn+2(S/J ∩B ) ∼= Hn+1(S/(J ,B )).
G˜ n+1 G˜ n+1
(5) H2n(S/J ∩B ) ∼= H2n(S/B ).
G˜ n+1 n+1
(6) H2n+2(S/J ∩B ) ∼= H2n+2(S/J ).
G˜ n+1 G˜
(7) Hi(S/J ∩B ) = 0 for i 6= n+2,2n,2n+2.
G˜ n+1
With these results in mind the short exact sequence (1) of Corollary 3.3 provides (by
applying the local cohomology functor) the vanishing Hi(S/J ) = 0 for all i 6= n+2,n+
G
3,2n,2n+2. Moreover it induces isomorphisms
Hn+2(S/J ) ∼= Hn+1(S/(J ,B )) and H2n(S/J ) ∼= H2n(S/B )
G G˜ n+1 G n+1
and as n > 3 so 2n > n+3 the isomorphism Hn+3(S/J ) ∼= Hn+2(S/(J ,A )). Moreover
G G˜ n
we obtain the following short exact sequence
0 → H2n+2(S/J ) → H2n+2(S/J )⊕H2n+2(S/A )⊕H2n+2(S/J ) → H2n+2(S/J ) → 0.
G G˜ n G˜ G˜
By Local Duality this proves the first three rows in the table of statement (b). By Local
Duality we get also the the following short exact sequence
0 → ω(S/J ) → ω(S/J )⊕ω(S/A )⊕ω(S/J ) → ω(S/J ) → 0.
G˜ G˜ n G˜ G
Note that we may write ω instead of ω2n+2 because all modules above are canonical
modules. First of all the short exact sequence provides that depthω(S/J ) ≥ 2n+1. By
G
applying local cohomology and dualizing again we get the following exact sequence
0 → ω (S/J ) → S/J ⊕S/A ⊕S/J →f S/J → ω2n+1(ω(S/J )) → 0.
2× G G˜ n G˜ G˜ G
As in the proof of Theorem 4.1 we see that f is surjective. Therefore ω2n+1(ω(S/J )) = 0
G
and depthω(S/J ) = 2n+2. Whence ω(S/J ) is a (2n+2)-dimensional Cohen-Macaulay
G G
∼
module. Then f is a split surjection and ω (S/J ) = S/J ⊕ S/A . This implies the
2× G G˜ n
∼
isomorphism ω(S/J ) = ω(S/J ) ⊕ω(S/A ) and this finishes the proof of (b). Clearly
G G˜ n
regS/J = 2. (cid:3)
G
Theorem 4.4. Let m > 3 and n = 2. Then:
(a) regS/J = 2.
G
(b) ωi(S/J ) = 0 if and only if i 6∈ {4,m+2,m+3,2m} and there are the following
G
isomorphisms and integers
i ωi(S/J ) depthωi(S/J ) dimωi(S/J )
G G G
4 ω4((J ,B )/B ) 4 4
G˜ m m
m+2 ωm+1(S/(J ,A )) m+1 m+1
G˜ 2
m+3 ω(S/J ) m+3 m+3
G˜
2m S/A (−2m) 2m 2m.
2
Moreover there is an isomorphism ω4(ω4(S/J ) ∼= (J ,B )/B .
G G˜ m m
Proof. By applying the local cohomology functors H·(−) to the exact sequence (2) in
Corollary 3.3 we get the following:
10 PETER SCHENZEL,SOHAILZAFAR
(1) Hm+2(S/J ∩A ) ∼= Hm+1(S/(J ,A )),
G˜ 2 G˜ 2
(2) Hm+3(S/J ∩A ) ∼= Hm+3(S/J ),
G˜ 2 G˜
(3) H2m(S/J ∩A ) ∼= H2m(S/A ) and
G˜ 2 2
(4) Hi(S/J ∩A ) = 0 for i 6= m+2,m+3,2m.
G˜ 2
Similarly, if we apply H·(−) to the exact sequence (3) in Corollary 3.3 we get the isomor-
phism Hm+3(S/J ∩B ) ∼= Hm+3(S/J ) and the exact sequence
G˜ m G˜
0 → H3(S/(J ,B )) → H4(S/J ∩B ) → H4(S/B ) → 0.
G˜ m G˜ m m
The short exact sequence on local cohomology induces the following exact sequence
0 → ω4(S/B ) → ω4(S/J ∩B ) → ω3(S/(J ,B )) → 0
m G˜ m G˜ m
by Local Duality. Now we apply again local cohomology and taking into account that
both ω4(S/B ) and ω3(S/(J ,B )) are Cohen-Macaulay modules of dimension 4 and
m G˜ m
3 respectively. Then depthω4(S/J ∩ B )) ≥ 3. By applying local cohomology and
G˜ m
dualizing again it induces the following exact sequence
0 → ω4(ω4(S/J ∩B )) → S/B →f S/(J ,B ) → ω3(ω4(S/J ∩B )) → 0.
G˜ m m G˜ m G˜ m
Now the homomorphism f is an epimorphism. ω3(ω4(S/J ∩ B )) = 0. That is
G˜ m
depthω4(S/J ∩B ) = 4 and it is a Cohen-Macaulay module. Moreover ω4(ω4(S/J ∩
G˜ m G˜
∼
B )) = (J ,B )/B . With these results in mind the short exact sequence (1) of Corol-
m G˜ m m
lary 3.3 provides (by applying the local cohomology functor) the vanishing Hi(S/J ) = 0
G
for all i 6= 4,m+2,m+3,2m. Moreover it induces isomorphisms
H4(S/J ) ∼= H4(S/J ∩B ),Hm+2(S/J ) ∼= Hm+1(S/(J ,A ))
G G˜ m G G˜ 2
and H2m(S/J ) ∼= H2m(S/A ). Moreover we obtain the following short exact sequence
G 2
0 → Hm+3(S/J ) → Hm+3(S/J )⊕Hm+3(S/J ) → Hm+3(S/J ) → 0.
G G˜ G˜ G˜
This implies the isomorphism ωm+3(S/J ) ∼= ω(S/J ). (cid:3)
G G˜
As a final step we shall consider the case of the complete bipartite graph K with
m,n
2n = m+ 2. In all of the previous examples we have the phenomenon that ωi(S/J ) is
G
either zero or a Cohen-Macaulay module with i − 1 ≤ dimωi(S/J ) ≤ i for all i ∈ Z.
G
and the canonical module ω(S/J ) = ωd(S/J ),d = dimS/J , is a d-dimensional Cohen-
G G G
Macaulay module. For 2n = m+2 this is no longer true.
Theorem 4.5. Let m+2 = 2n and m > n+1. Then:
(a) regS/J = 2.
G
(b) ωi(S/J ) = 0 if and only if i 6∈ {n+2,m+2 = 2n,m+n+1,2m} and there are
G
the following isomorphisms and integers
i ωi(S/J ) depthωi(S/J ) dimωi(S/J )
G G G
n+2 ωn+1(S/(J ,B )) n+1 n+1
G˜ m
m+2 ωm+1(S/(J ,A ))⊕S/B (−2n) m+1 m+2
G˜ n m
m+n+1 ω(S/J ) m+n+1 m+n+1
G˜
2m S/A (−2m) 2m 2m.
n