arXiv:1301.1602v1 [] 8 Jan 2013

SYMBOLIC POWERS OF MONOMIAL IDEALS WHICH ARE GENERICALLY
COMPLETE INTERSECTIONS
ADNAN ASLAM
Abstract. We classify all unmixed monomial ideals I of codimension 2 which are generically
a complete intersection and which have the property that the symbolic power algebra A(I) is
standard graded. We give a lower bound for the highest degree of a generator of A(I) in the
case that I is a modification of the vertex cover ideal of a bipartite graph, and show that this
highest degree can be any given number. We furthermore give an upper bound for the highest
degree of a generator of the integral closure of A(I) in the case that I is a monomial ideal which
is generically a complete intersection.

Introduction
The aim of the paper is to study the symbolic power algebra of monomial ideals. Classically one
defines the symbolic power algebra of prime ideal P in a Noetherian commutative ring R as the
L
graded algebra k≥0 P (k) tk where P (k) is the kth symbolic power of P . In general, this algebra is
not finitely generated, even though R is Noetherian. Examples of this bad behavior have been given
by Cowsik [1], Roberts [2] and others. On the other hand, it has been shown in [5] the symbolic
power algebra of a monomial ideal is always finitely generated. In the first section of this paper
we recall the definition of symbolic powers for monomial ideals. In the case, that the monomial
ideal I is squarefree, it well understood when its symbolic power algebra A(I) is standard graded,
in other words, when the ordinary powers of I coincide with the symbolic powers of I, see [5],[8],
and [9].
The interest in symbolic powers of monomial ideals is partly due to the fact that symbolic powers
of squarefree monomial ideals may be interpreted as vertex cover ideals of graphs and hypergraphs.
This point of view has been stressed in the papers [5],[6] and [7]. On the other hand, not much
is known about the generation of symbolic power algebras in the case that the monomial ideal is
not squarefree. This paper is an attempt to better understand symbolic power algebras of nonsquarefree monomial ideals. The situation in this more general case is much more complicated.
While the symbolic power algebra of an unmixed squarefree monomial ideal of codimension 2 is
generated in degree at most 2, the highest degree d(A(I)) of a generator of the algebra A(I), where
I is an unmixed monomial ideal of codimension 2 can be any given number, as is shown in a family
of examples in Section 2. In Section 1 we introduced some concepts and first recall a few facts
about vertex cover algebras which will be needed in the following sections. The main result of
Section 2 is Theorem 2.3 in which it is shown that the symbolic power algebra of monomial ideal I
of codimension 2 which is generically a complete intersection is standard graded if and only if I is
a trivial modification of a vertex cover ideal of a bipartite graph, which means that it is obtained
from the vertex cover ideal simply by variable substitutions. In Section 3 we give a general lower
bound for the number d(A(I)) in the case that I is a modification of a vertex cover ideal of a
bipartite graph. To give an upper bound for d(A(I)) is much harder. However, in Section 4, an
explicit upper bound for the highest degree of a generator of the integral closure A(I) of A(I) is
1991 Mathematics Subject Classification. 13C14, 13D02, 13D25, 13P10.
1

given when I is a monomial ideal which is an arbitrary generically complete intersection with no
condition on the codimension. It remains an open question how the numbers d(A(I)) and d(A(I))
are related to each other.
1. A review on symbolic powers of monomial ideals
Let K be a field and S = K[x1 , . . . , xn ] the polynomial ring in n indeterminates over K, and
let I ⊂ S be a monomial ideal. Then I has a unique irredundant presentation
m
\
Ci .
I=
i=1

as an intersection of irreducible monomial ideals, see [4]. The irreducible monomial ideals Ci are all
of the form (xai11 , . . . , xaikk ). In particular, they are monomial complete intersections. One obtains
from this presentation a canonical presentation of I as an intersection of primary ideals
r
\
(1)
Qi .
I=
i=1

p
where each Qi is Pi -primary and is defined to be the intersection of all Cj with Cj = Pi . For
example, if I = (x31 , x32 , x21 x23 , x1 x2 x23 , x22 x23 ) then I has the irredundant presentation as intersection
of irreducible ideals
I = (x31 , x32 , x23 ) ∩ (x21 , x2 ) ∩ (x1 , x22 ).
We have Ass(x21 , x2 ) = (x1 , x2 ) = Ass(x1 , x22 ). Intersecting (x21 , x2 ) and (x1 , x22 ) we obtain (x1 , x2 )primary ideal (x21 , x1 x2 , x2 ) and hence the irredundant primary decomposition
I = (x31 , x32 , x23 ) ∩ (x21 , x1 x2 , x22 ).
Given a primary decomposition as in (1) one defines the kth symbolic power of I to be
I (k) =

r
\

Qki .

i=1
k

(k)

Obviously one has I ⊂ I for all k. Note that there is no ambiguity in the definition of I (k) since
the presentation of I as an intersection of primary ideals as given in (1) is uniquely determined
by I. Furthermore, if I is generically complete intersection, as will be assumed in the following
sections, this definition of I (k) coincide with one given in [5]
One of the basic questions related to symbolic powers is to find conditions when I k = I (k) for
all k. Since I (k) I (l) ⊂ I (k+l) for all k and l, one may consider the so-called symbolic power algebra
M
A(I) =
I (k) tk ⊂ S[t].
k≥0

This algebra contains the Rees algebra R(I) =
alent:

L

k≥0

I k tk , and the following statements are equiv-

(a) I k = I (k) for all k.
(b) R(I) = A(I).
(c) A(I) is a standard graded S-algebra, i.e., A(I) is generated over S by elements of degree
1.
It is known that A(I) is always finitely generated, see [5, Theorem 3.2]. For squarefree monomial
ideals this has first been shown in [3, proposition1]. In case that I is a squarefree monomial ideal it
is well understood when A(I) is standard graded, see [5, Theorem 5.1]. In particular, the following
results will be of importance for our considerations. Assume that I is a squarefree monomial ideal
which is obtained as an intersection of ideals of the form (xi , xj ). To such an ideal we can associate
2

a graph G on the vertex set [n] for which {i, j} is an edge of G if and only if (xi , xj ) appears in
the intersection of I. Thus we can write
\
(xi , xj ).
I=
{i,j}∈E(G)

The monomial generators of I correspond to the vertex covers of G. Indeed, u = xi1 xi2 · · · xik is a
generator of I if and only if the set C = {i1 , i2 , . . . , ik } is a vertex cover of G, that is, C ∩ {i, j} =
6 ∅
for all edges {i, j} ∈ E(G). One therefore calls I the vertex cover ideal of G and denotes it by IG .
The vertex cover ideal is in fact the Alexander dual of the edge ideal associated to the graph G.
In the next section we will use the following result that can be found in [5, Theorem 5.1(b)],
Theorem 1.1. The following conditions are equivalent:
(a) R(IG ) = A(IG ).
(2)
2
(b) IG
= IG .
(c) G is a bipartite graph.
2. Unmixed monomial ideals of codimension 2 which are generically a complete
intersection
A monomial ideal I ⊂ S = K[x1 , . . . , xn ] of codimension 2 is unmixed and generically a complete
intersection if and only if
\
a
a
I=
(xi ij , xj ji ),
1≤i<j≤n

where A = (aij ) is an n × n-matrix of non-negative integers. Whenever aij = 0 or aji = 0, we have
a
a
(xi ij , xj ji ) = S, and we may omit this ideal in the intersection. Thus we may rewrite the above
intersection as
\
a
a
(xi ij , xj ji ),
I=
{i,j}∈E(G)

where G is a simple graph on the vertex set [n] and where for each edge {i, j} of G the numbers
a
a are positive integers. The graph G introduced is uniquely determined by I, and
√ij and
T ji
I = {i,j}∈E(G) (xi , xj ) is the vertex cover ideal IG of G.
T
T
a
a
b
b
We say that J = 1≤i<j≤n (xi ij , xjji ) is a trivial modification of I = 1≤i<j≤n (xi ij , xj ji ), if
there exist integers ci > 0 such that bij = ci aij for all i and j. In particular, I is a trivial
T
modification of {i,j}∈E(G) (xi , xj ), if akl does not depend on l (and only on k).
We begin with an example which shows that for a non-squarefree monomial ideal I, the algebra
A(I) may have arbitrarily high degree generators.
Example 2.1. Given an integer n. Let R be the polynomial ring Q[a, b, c] and I = (an , ab, bc).
Then the highest degree of an algebra generator of A(I) is precisely n. In particular, the highest
degree of a generator of the symbolic power of a monomial ideal of height 2 may be arbitrarily
large. For the proof we note first that
I = (an , b) ∩ (a, c).

Hence I is a monomial ideal of codimension 2. We claim that the kth symbolic power I (k) of I is
minimally generated by the monomials
(2)

ai bk−⌊i/n⌋ ck−i
nj k−j

a b

with

with 0 ≤ i ≤ k,
⌊k/n⌋ < j ≤ k.

Indeed, we have that
I (k) = (an , b)k ∩ (a, c)k .
3

Therefore I (k) is generated by the monomials
(3)

amax (nj,i) bk−j ck−i ,

0 ≤ i, j ≤ k.

Therefore, we have the generators
(4)

ai bk−j ck−i

with

0 ≤ i, j ≤ k

and nj ≤ i,

and
(5)

anj bk−j ck−i

with 0 ≤ i, j ≤ k

and nj > i.

Among the generators (4) it is enough to take the generators
ai bk−⌊i/n⌋ ck−i

with 0 ≤ i ≤ k.

And among the generators (5) it is enough to take the generators
anj bk−j

⌊k/n⌋ < j ≤ k.

with

This implies the assertion about the generators of I (k) .
Next we claim that the element an bn−1 ∈ I (n) is a minimal generator of the algebra. Suppose
this is not the case. Then, since an bn−1 is a minimal generator of I (n) , there exist integers k1 , k2 > 0
with k1 + k2 = n, a minimal generators u of I (k1 ) and a minimal generator v of I (k2 ) such that
an bn−1 = uv. Since an bn−1 does not contain the factor c, it follows from (2) that u = ak1 bk1 −⌊k1 /n⌋
and v = ak2 bk2 −⌊k2 /n⌋ with ki < n, k1 + k2 = n and k1 + k2 − ⌊k1 /n⌋ − ⌊k2 /n⌋ = n − 1. Since
⌊ki /n⌋ = 0 for i = 1, 2, this is impossible.
It remains to be shown that for k > n, the monomials in (2) can be presented as products of
generators of the symbolic powers I (l) with 0 < l < k. In fact,
• for 0 ≤ i < k we have ai bk−⌊i/n⌋ ck−i = (ai bk−1−⌊i/n⌋ ck−1−i )(bc) with
ai bk−1−⌊i/n⌋ ck−1−i ∈ I (k−1) and bc ∈ I
• ak bk−⌊k/n⌋ = (ak−1 bk−1−⌊k/n⌋ )(ab) with ak−1 bk−1−⌊k/n⌋ ∈ I (k−1) and ab ∈ I.
• for ⌊k/n⌋ < j < k we have anj bk−j = (an(j−1) bk−1−j )(ab) with
an(j−1) bk−1−j ∈ I (k−1) and ab ∈ I.
• ank = an(k−1) an with an(k−1) ∈ I k−1 and an ∈ I.
Next we classify all unmixed monomial ideals I of codimension 2 which are generically a complete
intersection and which have the property that A(I) is standard graded.
Let A be finitely generated graded S-algebra. We denote by d(A) the highest degree of the
minimal generators of A. By using this notation we have that A(I) is standard graded if and only
if d(A(I)) = 1.
Before proving the main result of this section we observe
Proposition 2.2. Let I be an arbitrary monomial ideal and J a trivial modification of I. Then
d(A(J)) = d(A(I)).
Proof. Let c1 , . . . , cn be positive integers such that J arises from I by the substitution xi 7→ xci i
for i = 1, . . . , n, and let ϕ : S → S be the K-algebra homomorphism with ϕ(xi ) = xci i . For an
ideal J = (f1 , . . . , fm ) ⊂ S we denote by ϕ(J) the ideal in S generated by the elements ϕ(fi ),
i = 1, . . . , m. Then we have ϕ(I k ) = ϕ(I)k for all k. We claim that for any monomial ideal I we
also have ϕ(I (k) ) = ϕ(I)(k) for all k. To show this it suffices to prove that whenever I and J are
monomial ideals, then ϕ(I ∩ J) = ϕ(I) ∩ ϕ(J). Indeed, if I = (u1 , . . . , ur ) and J = (v1 , . . . , vs ),
then I ∩ J = ({lcm(ui , vj )} i=1,...,r ). Thus the claim follows from the fact that
j=1,...,s

ϕ(lcm(u, v)) = lcm(ϕ(u), ϕ(v))
4

for all monomials u, v.

Let t be an indeterminate over S. Then ϕ induces a graded K-algebra homomorphism ϕ̂ : S[t] →
L
P
P
S[t] with ϕ̂( k fk tk ) = k ϕ(fk )tk . Restricting ϕ̂ to A(I) = k I (k) tk we obtain
M
M
M
ϕ̂(A(I)) =
ϕ(I (k) )tk =
ϕ(I)(k) tk =
J (k) tk = A(J).
k

k

k

This shows that ϕ̂ : A(I) → A(J) is surjective. Since ϕ̂ is injective, it follows that A(I) and A(J)
are isomorphic, and in particular, d(A(I)) = d(A(J)), as desired.

Theorem 2.3. Let I be an unmixed monomial ideal of codimension 2 which is generically a
complete intersection. Then the following conditions are equivalent:
(a) I k = I (k) for all k.
(b) I 2 = I (2) .
(c) I is a trivial modification of a vertex cover ideal of a bipartite graph.
Proof. The implication (a) ⇒ (b) is trivial. Next we show that (c) ⇒ (a): The vertex cover ideal
J of a bipartite graph is standard graded, see Theorem 1.1. In other words, d(A(J)) = 1. Since I
is a trivial modification of J, Proposition 2.2 implies that d(A(I)))
√ =1
It remains to be shown that (b) ⇒ (c). Assume first that I = IG for a bipartite graph G
and I is a not trivial modification of vertex cover ideal of a bipartite graph G. It is convenient to
identify the edges of G with variables in the polynomial ring. Thus let the vertex set V (G) of G
be {x1 , . . . , xr } ∪ {y1 , . . . , ys } with the edge set E(G) ⊂ {x1 , . . . , xr } × {y1 , . . . , ys }. Then
\
b
a
(xi ij , yj ij ).
I=
{xi ,yj }∈E(G)

Since we assume that I is not a trivial modification of IG there exits an integer i, say i = 1, such
that
\
a
b
(x1 1j , yj 1j )
I1 =
{x1 ,yj }∈E(G)

T

is not a trivial modification of {x1 ,yj }∈E(G) (x1 , yj ). Let G1 be the subgraph of G with E(G1 ) =
{{x1 , yj } : {x1 , yj } ∈ E(G)}. Then I = I1 ∩ I2 , where
\
b
a
(xi ij , yj ij ),
I2 =
{xi ,yj }∈E(G)
i6=1

and I1 is not a trivial modification of IG1 .
Let f = x2 · · · xr . Then for the localization with respect to f we have (I12 )f = (I 2 )f and
(2)
(2)
(2)
(I1 )f = (I (2) )f . Assume we know already I12 6= I1 . If (I12 )f 6= (I1 )f , then (I 2 )f 6= (I (2) )f . But
then also I 2 6= I (2) , as we want to show. Thus we have reduced the problem to show that for an
ideal of the type
I = (xa1 1 , y1b1 ) ∩ · · · ∩ (xa1 r , yrbr )
with a1 ≥ a2 ≥ · · · ≥ ar and one of the inequalities is strict. Since I is a trivial modification of the
ideal (xa1 1 , y1 ) ∩ · · · ∩ (xa1 r , yr ), we may as well assume that all bi = 1, see Proposition 2.2. Take
the smallest i such that a1 > ai . Then for a1 ≥ 2ai ,
xa1 1 y1 y2 · · · yi−1 ∈ I (2)

and for a1 < 2ai ,
We have

(2)
i
x2a
1 y1 y2 · · · yi−1 ∈ I

I = (xa1 1 , xa1 2 y1 , xa1 3 y1 y2 , . . . , xa1 i y1 y2 · · · yi−1 , . . . , y1 · · · yr ).
5

It is obvious that xa1 1 y1 y2 · · · yi−1 6∈ I 2 . On the other hand, the possible product of two generators
ai
a1
i
of I to get x2a
1 y1 y2 · · · yi−1 would be x1 (x1 y1 y2 · · · yi−1 ). But this is also not possible, because
a1 + ai > 2ai .
Assume now that G is not bipartite. Then G contains an odd cycle. If this cycle has a chord,
then this chord decomposes this cycle into two smaller cycles, where one of them is again an odd
cycle. This argument shows that G contains an odd cycle with no chord. Suppose first that this
cycle has length > 3. By a localization argument as in the first part of the proof (c) ⇒ (b) we
may assume that G coincides with this cycle. Say, the edges of G are {1, n} and {i, i + 1} for
i = 1, . . . , n − 1 with n > 3. Localizing I at f = x3 x5 · · · xn−1 we are reduced to show that for the
ideal J = (xa1 1 , xb21 ) ∩ (xa1 2 , xbn2 ) we have that J 2 6= J (2) . Without loss of generality we may assume
that a1 > a2 and b1 = 1 = b2 see Proposition 2.2. If a1 ≥ 2a2 then xa1 1 x2 ∈ J (2) \ J 2 , if a1 < 2a2
(2)
2
then x2a
\ J 2 , as desired.
1 x2 ∈ J
Finally we assume that E(G) = {{1, 2}, {2, 3}, {1, 3}}. Then
I = (xa1 1 , xb21 ) ∩ (xb22 , xc31 ) ∩ (xc32 , xa1 2 ).
We want so show that I (2) 6= I 2 . Notice that
I

=

max(a1 ,a2 ) b2
max(a1 ,a2 ) c1
max(c1 ,c2 )
max(b1 ,b2 )
max(b1 ,b2 ) c2
x2 , xa1 1 xb22 xc32 , x1
x3 , xa1 1 x3
, xa1 2 x2
, x2
x3 ,
a2 b1 c1
b1 max(c1 ,c2 )
x1 x2 x3 , x2 x3
).

(x1

Assume that a1 ≥ a2 then
I

max(c1 ,c2 )

= (xa1 1 xb22 , xa1 1 xb22 xc32 , xa1 1 xc31 , xa1 1 x3
max(c1 ,c2 )

xb21 x3

max(b1 ,b2 )

, xa1 2 x2

max(b1 ,b2 ) c2
x3 , xa1 2 xb21 xc31 ,

, x2

).

Then it follows that
max (b1 ,b2 ) c1
x3

xa1 1 x2

∈ I (2) \ I 2 ,

if

a1 ≥ 2a2 ,

and
max (b1 ,b2 ) max (c1 ,c2 )
x3

xa1 1 x2

∈ I (2) \ I 2 ,

if

a2 ≤ a1 < 2a2 .


3. Bounds for the generators of the algebra A(I)
In this section we want to give a lower bound for d(A(I)) in the case that I is a modification of
the vertex cover ideal of a bipartite graph.
Theorem 3.1. Let I ⊂ S is an ideal, S = K[x1 , . . . , xr , y1 , . . . , ys ] and aij and bij are nonnegative integers with 1 ≤ i ≤ r and 1 ≤ j ≤ s. Let G be a bipartite graph with vertex set
V (G) = {x1 , . . . , xr , y1 , . . . , ys } and edge set E(G) ⊂ {x1 , . . . , xr } × {y1 , . . . , ys }, and let
\
b
a
(xi ij , yj ij )
I=
{xi ,yj }∈E(G)

Then

n
d(A(I)) ≥ max max
i,j,l

n
o
o
bij
aij
, max
.
gcd(aij , ail ) i,j,l gcd(bij , bil )

Proof. Let f ∈ S = A0 (I) be a squarefree monomial, then A(I)f is again naturally graded because
f is an element of degree 0 in A(I), and the generators of A(I) are also generators of A(I)f . Hence
it follows that d(A(I)f ) ≤ d(A(I)). Thus any lower bound for d(A(I)f ) is also lower bound for
d(A(I)).
6

Notice that
a

\

(I (k) )f =

b

(xi ij , yj ij )k Sf

{xi ,yj }∈E(G)
{xi ,yj }∩supp(f )=∅

Therefore
(If )(k) = (I (k) )f .
This implies that A(I)f = A(If ). Let
\
b
a
J=
/ supp(f )}]
(xi ij , yj ij ) ⊂ K[{xi , yj : xi , yj ∈
{xi ,yj }∈E(G)
{xi ,yj }∩supp(f )=∅

−1
Then d(A(If )) = d(A(J)), because A(If ) = A(J) ⊗k K[{xi , x−1
i , yj , yj : xi , yj ∈ supp(f )}].
We may assume that

m = max{aij / gcd(aij , ail )} ≥ max{bij / gcd(bij , bil )}.
i,j,l

i,j,l

Let i be an integer for which there exist j and l such that aij / gcd(aij , ail ) takes the maximal value
m. We may assume that i = 1 and j = 1. If m = 1, the assertion of the theorem is trivial. Thus
we may assume that m > 1. This then implies that l 6= j and aij > ail . We may assume that
l = 2. Localizing at f = x2 · · · xr y3 · · · ys and applying the above considerations, we may assume
that
I = (x1a11 , y1b11 ) ∩ (xa1 12 , y2b12 ) with a11 > a12 .

Now I is a trivial modification of the ideal (xa1 , y1 )∩(xb1 , y2 ) with a > b, where a = a11 / gcd(a11 , a12 )
and b = a12 / gcd(a11 , a12 ).
Thus, according to the Proposition 2.2, it is enough to be shown that d(A(I)) ≥ a for I =
(xa1 , y1 ) ∩ (xb1 , y2 ) with a > b and gcd(a, b) = 1.
In order to prove this, we first claim that the kth symbolic power I (k) of I is minimally generated
by the monomials
(6)

k−i k−j
xai
1 y1 y2

with 0 ≤ i, j ≤ k

and j ≤ ⌊ai/b⌋.

k−i k−j
xjb
1 y1 y2

with 0 ≤ i, j ≤ k

and j > ⌊ai/b⌋.

and the monomials
(7)

Indeed, we have that
I (k) = (xa1 , y1 )k ∩ (xb1 , y2 )k .

Therefore I (k) is generated by the monomials

max (ai,bj) k−i k−j
y1 y2 ,

x1

0 ≤ i, j ≤ k.

Therefore, we have the generators
(8)

k−i k−j
xai
,
1 y1 y2

with 0 ≤ i, j ≤ k

and ai ≥ bj,

k−i k−j
xbj
,
1 y1 y2

with

0 ≤ i, j ≤ k

and ai < bj.

and
(9)

Thus we obtain exactly the monomials listed in (6) and (7).
a−b
∈ I (a) belongs to the minimal set of generators of the
Next we claim that the element xab
1 y1
a−b a
algebra. This will then show that d(A(I)) ≥ a. Suppose xab
t is not a minimal generator
1 y1
a−b
is a minimal generator of I (a) , there exist integers k1 , k2 > 0
of A(I). Then, since xab
1 y1
with k1 + k2 = a, a minimal generator u of I (k1 ) and a minimal generator v of I (k2 ) such that
a−b
a−b
xab
= uv. Since xab
does not contain the factor y2 , it follows from (6) and (7) that
1 y1
1 y1
7

c−i
c−i
the monomials u and v are of the form xai
or xbc
where c = k1 or c = k2 . We have to
1 y1
1 y1
distinguish several cases:
1 k1 −i1
2 k2 −i2
(i) If u = xai
with ai1 ≥ bk1 and v = xai
with ai2 ≥ bk2 , then we have
1 y1
1 y1

a(i1 +i2 ) k1 +k2 −(i1 +i2 )
y1

a−b
xab
= uv = x1
1 y1

a(i1 +i2 ) a−(i1 +i2 )
y1
.

= x1

This implies that i1 +i2 = b. Adding the inequalities ai1 ≥ bk1 and ai2 ≥ bk2 , and using k1 +k2 = a
and i1 + i2 = b, we get
ab = ai1 + ai2 ≥ bk1 + bk2 = ba,
which implies that ai1 = bk1 and ai2 = bk2 . Since gcd(a, b) = 1, we see that i1 = bc and k1 = ac
for some integer c > 0. This is not possible, because k1 < a.
1 k1 −i1
2 k2 −i2
(ii) If u = xai
with ai1 ≥ bk1 and v = xbk
with ai2 < bk2 , then we have
1 y1
1 y1

k +k2 −(i1 +i2 )

a−b
1 +bk2
xab
= uv = xai
y1 1
1 y1
1

a−(i1 +i2 )

1 +bk2
= xai
y1
1

.

This implies that ai1 + bk2 = ab and i1 + i2 = b. Using ai2 < bk2 in the equation ai1 + bk2 = ab,
we get a(i1 + i2 ) < ab which contradicts the fact that i1 + i2 = b.
1 k1 −i1
2 k2 −i2
(iii) If u = xbk
with ai1 < bk1 and v = xbk
with ai2 < bk2 , then we have
1 y1
1 y1

b(k1 +k2 ) k1 +k2 −(i1 +i2 )
y1

a−b
xab
= uv = x1
1 y1

a−(i1 +i2 )

2
= x2bk
y1
1

.

This implies that b(k1 + k2 ) = ab and i1 + i2 = b. Using ai1 < bk1 and ai2 < bk2 in the equation
b(k1 + k2 ) = ab, we get a(i1 + i2 ) < ab which contradicts the fact that i1 + i2 = b.

One would like to obtain a similar lower bound for ideals which are modifications of the vertex
cover ideals of an arbitrary graph. The first non-trivial case to be considered would be
I = (xa , y b ) ∩ (y c , z d ) ∩ (xe , z f ) with

gcd(a, e) = 1, gcd(b, c) = 1

and

gcd(d, f ) = 1.

We conjecture that d(A(I)) ≥ max{a, b, c, d, e, f }. However, in general this lower bound, if correct,
is not strict as the following examples shows: let I = (a2 , b) ∩ (b, c) ∩ (a, c). Then a2 b2 ct3 ∈ I (3) t3
is a minimal generator of A(I).
4. The integral closure of A(I) for a monomial ideal I which is generically a
complete intersection
It seems to be pretty hard to find a general upper bound for the number d(A(I)). However
for the integral closure A(I) of A(I) where I is a monomial ideal which is generically a complete
intersection such bound can be given, with respect to the number of indeterminates and degrees of
the powers of the variables which appears in the irredundant irreducible decomposition of I. Since
I is generically a complete intersection it is of the form
I=

m
\

mal ,

l=1

where for a nonzero vector a = (a1 , . . . , an ) ∈ Nn we set ma = ({xai i }i∈supp(a) ).
Lemma 4.1. The integral closure of A(I) is spanned as a K-vector space by all monomials
xc11 · · · xcnn tk for which the exponent vector (c1 , . . . , cn , k) satisfies the following inequalities:
X
ci
− k ≥ 0, 1 ≤ l ≤ m.
c1 ≥ 0, . . . , cn ≥ 0, k ≥ 0 and
ali
i∈supp(al )
8

Proof. . The integral closure A(I) of A(I) is generated as a K-vector space by all monomials
rcn
1
u = xc11 · · · xcnn tk ∈ S[t] such that ur ∈ I (rk) trk . Now ur ∈ I (rk) trk if and only if xrc
∈
1 · · · xn
al1
rc
a
aln rk
rcn
aln rk
1
l1
(x1 , . . . , xn ) for all l = 1, . . . , m. Note that x1 · · · xn ∈ (x1 , . . . , xn ) if and only if there
exists non-negative integers j1 , . . . , jn with j1 + · · · + jn = rk and such that rc1 ≥ j1 al1 , . . . , rcn ≥
P
jn aln . These inequalities are satisfied for c1 , . . . , cn if and only if i∈supp(al ) ci /ali ≥ k. This yields
the desired conclusion.

Theorem 4.2. Let I ⊂ S = K[x1 , . . . , xn ] be a monomial ideal which is generically a complete
Tm
intersection, and suppose that I = l=1 mal . Then
d(A(I)) ≤

(n + 1)! n(n−1)
d
,
2

where

d = max{ail }.
i,l

Proof. The proof of the theorem follows the line of arguments in the proof of Theorem 5.6 in [5].
The inequalities given in Lemma 4.1 describe a positive cone C ⊂ Rn+1 . If q = (c1 , . . . , cn , k) is
an integral vector in C, we set deg q = k and call this number the degree of q. Let E be a set of
integral vectors spanning the extremal arrays. Then the maximal degree of a generator of A(I) is
bounded by the maximum of the degree of the integral vectors of the form a1 q1 + · · · + an+1 qn+1
with 0 ≤ aj ≤ 1 and a1 + · · · + an+1 < n + 1, and such that each of the integral vectors qj spans
an extremal array. Hence if F is a set of integral vectors in C which contains the integral vectors
spanning the extremal arrays of C, we obtain the bound
d(A(I)) ≤ (n + 1) max{deg q : q ∈ F }.

(10)

Note that each integral vector q spanning an extremal array of C is given as an integral solution
of n linear equations describing supporting hyperplanes of C. Thus there exist numbers l1 < l2 <
· · · < lr with r ≤ n and numbers k1 < k2 < · · · < kn−r such that q is an integral solution of the
following system of n homogeneous linear equations
Y
X
Y
alt ,j y = 0 for t = 1, . . . , r
alt ,j zi −
i∈supp(alt )

j∈supp(al )
t
j6=i

j∈supp(alt )

zks

=

0 for

s = 1, . . . , n − r.

which arise from the linear inequalities by clearing denominators.
The coefficient matrix of this linear equation is an n × n + 1 matrix B. Thus an integral solution
of this system is the vector q whose ith component is (−1)i det Bi where Bi is obtained from B by
skipping the ith column of B. In particular it follows that deg q = | det Bn+1 |. Obviously, det Bn+1
is equal to a suitable minor of the matrix A whose rows are a1 , . . . , am .
If follows that each such minor is a sum of terms ±g where each g is a suitable product of the
aij . If we add up in this sum only the positive terms, then this will give an upper bound for this
minor. We denote this upper bound of g by g+ . Thus it follows from (10) that the desired upper
bound for d(A(I)) is given by (n+ 1)f where f is bound for the maximal possible g+ . The maximal
possible g+ we obtain if we consider n minors. Each of these n-minors has n!/2 positive terms
where each of this terms is a product of n(n − 1) entries ali . Thus dn(n−1) is a bound for each of
these entries. Combining all this, we obtain the desired upper bound for d(A(I)).


9

References
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Adnan Aslam, Abdus Salam School of Mathematical Sciences (ASSMS), GC University, Lahore, Pakistan.
E-mail address: adnanaslam15@yahoo.com

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