REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

arXiv:1205.3127v2 [] 18 Jan 2013

LOUIZA FOULI AND KUEI-NUAN LIN
A BSTRACT. We study the defining equations of the Rees algebras of square-free monomial ideals in
a polynomial ring over a field. We propose the construction of a graph, namely the generator graph
of a monomial ideal, where the monomial generators serve as vertices for the graph. When I is a
square-free monomial ideal such that each connected component of the generator graph of I has at
most 5 vertices then I has relation type at most 3. In general, we establish the defining equations
of the Rees algebra in this case and give a combinatorial interpretation of them. Furthermore, we
provide new classes of ideals of linear type. We show that when I is a square-free monomial ideal
and the generator graph of I is the graph of a disjoint union of trees and graphs with a unique odd
cycle, then I is an ideal of linear type.

1. I NTRODUCTION
The main problem of interest in this article is the question of determining the defining equations
of the Rees algebra of a square-free monomial ideal in a polynomial ring over a field. Let R be a
Noetherian ring and let I be an ideal. The Rees algebra, R (I), is defined to be the graded algebra
R (I) = R[It] = ⊕ I it i ⊂ R[t], where t is an indeterminate. The Rees algebra of an ideal encodes
i≥0

many of the analytic properties of the variety defined by I and it is the algebraic realization of the
blowup of a variety along a subvariety. The blowup of Spec(R) along V (I) is the projective spectrum
of the Rees algebra, R (I), of I. This important construction is the main tool in the resolution of
singularities of an algebraic variety.
From the algebraic point of view the Rees algebra of an ideal facilitates the study of the asymptotic behavior of the ideal and it is essential in computing the integral closure of powers of ideals.
The Rees algebra of an ideal I in a Noetherian ring can be realized as a quotient of a polynomial
ring and hence once the defining ideal of R (I) is determined, it is straightforward to compute the
integral closure, R (I). It is well known that R (I) = ⊕ I it i and one obtains I i = [R (I)]i .
i≥0

We consider the following construction for the Rees algebra. Let R be a Noetherian ring and let
I be an R-ideal. Let f1 , . . . , fn be a minimal generating set for I. Consider the polynomial ring S =
R[T1 , . . . , Tn ], where T1 , . . . , Tn are indeterminates. Then there is a natural map φ : S −→ R (I) = R[It]
that sends Ti to fit. Let J = ker φ be the defining ideal of R (I). Then R (I) ≃ S/J and J =

∞
L

Ji

i=1

is a graded ideal. A minimal generating set for J is often referred to as the defining equations of
the Rees algebra. Also, J1 is known as the ideal of linear relations as it is the defining ideal of the
symmetric algebra, Sym(I), of I and it is generated by linear forms in the Ti . The generators of J1
2010 Mathematics Subject Classification. 13A30, 13A02, 13A15.
Key words and phrases. Rees algebras, square-free monomial ideals, relation type, ideals of linear type, graph.
1

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L. FOULI AND K. N. LIN

arise from the first syzygies of I. When J = J1 then I is called an ideal of linear type. In this case
R (I) ≃ Sym(I) and the defining ideal of R (I) is well understood.
The problem of finding an explicit description of the defining equations of R (I) has been addressed by many authors, see for example [2, 3, 4, 5, 6, 7, 8, 9, 16, 19, 20, 21, 25]. In general,
finding the generators for the defining ideal of the Rees algebra of an ideal is a difficult problem to
address.
In Section 2, we develop a series of lemmas that allow us to determine conditions under which
an element in the defining ideal of the Rees algebra becomes redundant and hence not needed in
the defining equations. One of our essential tools is the result of Taylor, where she determines a
non-minimal generating set for the defining ideal of the Rees algebra of a monomial ideal [22]. One
of our main results in this section give an explicit description of the defining equations of the Rees
algebra of an ideal generated by n ≤ 5 square-free monomials, Theorems 2.12, 2.15. Recall that
the relation type of an ideal I is defined to be rt(I) = min{s | J =

s
L

Ji }, where J is the defining

i=1

ideal of R (I). In other words, the relation type of I is the largest degree (in the Ti ) of any minimal
generator of J. In particular, when I is of linear type it is of relation type 1. The relation type of an
ideal has been explored in various articles, see for instance [1, 10, 17, 23, 27]. In Theorem 2.12 we
establish that for a square-free monomial ideal generated by n ≤ 5 monomials, the relation type is at
most n − 2. We also provide a class of examples of square-free monomial ideals generated by n > 4
monomials, Example 2.13, for which the relation type is at least 2n − 7. Notice that when n = 5
then 2n − 7 = n − 2, and hence showing that the bound in Theorem 2.12 is attained. We remark
that we do not impose any restrictions on the degrees of the monomial generators of the square-free
monomial ideals under consideration.
We construct the generator graph associated to monomial ideals, which is also known as the line
graph of the hypergraph. The monomial generators are the vertices for this graph and the edges are
determined by the greatest common divisor among the generators, see Construction 2.16. This construction enables us to extend Theorem 2.12 and allow more freedom on the number of generators
of the ideal. We show that when I is a square-free monomial ideal such that its generator graph is
the disjoint union of graphs with at most 5 vertices, then rt(I) ≤ 3, Theorem 2.20. Furthermore, we
show that the non-linear part of the defining equations of the Rees algebra, R (I), of I arise from
even closed walks of the generator graph, Theorem 2.20.
Recall that for a Noetherian local ring (R, m) and I an R-ideal, the special fiber ring, F (I), of
I is defined to be F (I) = grR (I) ⊗ R/m, where grR (I) = R[It]/IR[It] =

∞
L
I i /I i+1 . When R is a

i=0

polynomial ring over a field k and I is an R-ideal then F (I) is defined to be k[ f1 , . . . , fn ], where
f1 , . . . , fn is a minimal generating set for I. Therefore, there is a homomorphism ψ : k[T1 , . . . , Tn ] →
F (I) that sends Ti to fi . Let H = ker ψ and F (I) ≃ k[T1 , . . . , Tn ]/H. Let S = R[T1 , . . . , Tn ] and let
J be the defining ideal of the Rees algebra, R (I) ≃ S/J, of I. An ideal I is called an ideal of fiber
type if the defining ideal J of R (I) is obtained by the linear relations and the defining equations
for the special fiber ring, i.e. J = SJ1 + SH. In the case of edge ideals, Villarreal showed that they

REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

3

are all ideals of fiber type, [26, Theorem 3.1]. Moreover, Villarreal gave an explicit description of
the defining equations of the Rees algebra of any square-free monomial ideal generated in degree
2, [26, Theorem 3.1]. It is also worth noting that Villarreal exhibited an example to show that his
techniques do not extend for monomial ideals generated in degree higher than 2, [26, Example 3.1].
His example is a square-free monomial ideal generated in degree 3 that is not of fiber type. We
remark that in general, a square-free monomial ideal generated in degree greater or equal to3 is
not of fiber type. In Example 2.22 we give a concrete explanation of how one obtains the defining
equations for the ideal in [26, Example 3.1].
In Section 3, we concentrate on the special class of ideals of linear type. The first well known
class of ideals of linear type are complete intersection ideals, [18]. Huneke and Valla showed
that ideals generated by d-sequences are also ideals of linear type, [14], [24]. Later on, HerzogSimis-Vasconcelos and Herzog-Vasconcelos-Villarreal used strongly Cohen-Macaulay ideals and
certain conditions on local numbers of generators to describe other classes of ideals of linear type,
[11, 12, 13]. Huneke also showed that the ideal of all (n − 1) × (n − 1) minors of a generic matrix in
R = Z[Xi, j ] is also an ideal of linear type, [15]. In the case of square–free monomial ideals generated
in degree 2, Villarreal gave a complete characterization of ideals of linear type. More precisely, he
showed that such an ideal I is of linear type if and only if the graph of I is the disjoint union of
graphs of trees or graphs that have a unique odd cycle, [26, Corollary 3.2]. In this work we find
new classes of square-free monomial ideals that are of linear type. We prove that when I is an ideal
generated by square-free monomials and the generator graph of I is a disjoint union of graphs of
trees and graphs with a unique odd cycle then I is an ideal of linear type, Theorem 3.4. Our results
have a similar flavor as those of Villarreal in [26]. In some sense these results extend the work of
Villarreal, even though we do not fully recover his results. Nonetheless, our techniques allow us to
consider square-free monomial ideals without any restrictions on the degrees of the generators.
2. T HE

DEFINING EQUATIONS OF THE

R EES

ALGEBRA

Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let f1 , . . . , fn be a
minimal monomial generating set of I and let R (I) denote the Rees algebra of I. Then R (I) =
R[ f1t, . . . , fnt] ⊂ R[t] and there is an epimorphism
φ : S = R[T1 , . . . , Tn ] −→ R (I)
induced by φ(Ti ) = fit. Let J = ker φ. We note that J =

∞
L

Ji is a graded ideal of S. As mentioned

i=1

in the Introduction, a minimal generating set for the ideal J is referred to as the defining equations
of the Rees algebra of I.
Definition 2.1. Let I be a monomial ideal in a polynomial ring R over a field k. Let f1 , . . . , fn be
a minimal monomial generating set of I. Let Is denote the set of all non-decreasing sequences of
integers α = (i1 , . . . , is ) ⊂ {1, . . . ., n} of length s. Then fα = fi1 · · · fis is the corresponding product

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L. FOULI AND K. N. LIN

of monomials in I. We also let Tα = Ti1 · · · Tis be the corresponding product of indeterminates in S =
fβ
fα
Tα −
T.
R[T1 , . . . , Tn ]. For every α, β ∈ Is we consider the binomial Tα,β =
gcd( fα , fβ )
gcd( fα , fβ ) β
The following is in [22] and we cite it here for ease of reference.
Theorem 2.2. ([22]) Let R be a polynomial ring over a field and let I be a monomial ideal in R.
Let f1 , . . . , fn be a minimal monomial generating set of I and let R (I) be the Rees algebra of I.
Then R (I) ≃ S/J, where S = R[T1 , . . . , Tn ] is a polynomial ring, T1 , . . . , Tn are indeterminates, and
J = SJ1 + S · (

∞
S

i=2

Ji ) such that Js = {Tα,β | for α, β ∈ Is }.

The following remark states some properties for the greatest common divisor among square-free
monomials. The proofs are omitted as they are elementary.
Remark 2.3. Let a, b, c, d, b1 , . . . , bs be square-free monomials in a polynomial ring R over a field.
Let n, m, l, r be positive integers. Then
(a) gcd(an , bm ) = gcd(a, b)min{m,n} ,
(b) gcd(an , b1 · · · bs ) = gcd(as , b1 · · · bs ) for all n ≥ s,
gcd(a, b)min{n,m} gcd(a, c)min{n,l}
, for some C ∈ R.
(c) gcd(an , bm cl ) =
C
The main goal in this section is to determine conditions under which for sequences α, β of length
s−1
S

s ≥ 2 we have Tα,β ∈ SJ1 + S · (

Ji ). In other words, we are interested in finding conditions on the

i=2

sequences α, β such that the generator Tα,β is redundant in the defining ideal of the Rees algebra.
The following two lemmas follow directly from Definition 2.1.
Lemma 2.4. Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let α =
(a1 , . . . , as ), β = (b1 , . . . , bs ) ∈ Is be two sequences of length s ≥ 2, where Is is as in Definition 2.1.
s−1
S

Suppose that ai = b j for some i and some j. Then Tα,β ∈ SJ1 + S · (

Ji ).

i=2

Proof. Without loss of generality we may assume that a1 = b1 . Let α1 = (a2 , . . . , as ) and β1 =
(b2 , . . . , bs ). Then gcd( fα , fβ ) = fa1 gcd( fα1 , fβ1 ) = fb1 gcd( fα1 , fβ1 ). Notice that fa1 = fb1 and Ta1 =
Tb1 . Then
Tα,β =

fβ
fβ1
fα
f α1
Tα −
Tβ = Ta1 [
Tα −
T ]
gcd( fα , fβ )
gcd( fα , fβ )
gcd( fα1 , fβ1 ) 1 gcd( fα1 , fβ1 ) β1
s−1
[

= Ta1 [Tα1 ,β1 ] ∈ SJ1 + S · (

Ji ).



i=2

Lemma 2.5. Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let α, β ∈ Is
be two sequences of length s ≥ 2, where Is is as in Definition 2.1. Suppose that α = (α1 , . . . , α1 )
and β = (β1 , . . . , β1 ), where α1 , β1 ∈ Im and m ≤ s. Then Tα,β ∈ SJ1 + S · (

m
S

i=2

Ji ).

REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

5

Proof. First we observe that s is a multiple of m. Let l be an integer such that s = lm. Notice
that fα = fαl 1 and fβ = fβl 1 . Also, by Remark 2.3 (a) we have that gcd( fα , fβ ) = gcd( fα1 , fβ1 )l . Let
fα1 Tβ1
fβ1 Tα1
,b=
, and let A = (al−1 + al−2 b + . . . + abl−2 + bl−1 ). Notice that
a=
gcd( fα1 , fβ1 )
gcd( fα1 , fβ1 )
a, b ∈ S and thus A ∈ S. Then
Tα,β =

fβ
fα
Tα −
T = al − bl
gcd( fα , fβ )
gcd( fα , fβ ) β

= (a − b)(al−1 + al−2 b + . . . + abl−2 + bl−1 ) = Ta1 ,b1 A ∈ SJ1 + S · (

m
[

Ji ). 

i=2

We observe the following properties for greatest common divisors among monomials. Again, we
omit the proofs as they are elementary.
Remark 2.6. Let a, b, c, d, e, f be monomials in a polynomial ring R over a field.
gcd(a, b)gcd(a, c)
, for some C ∈ R.
(a) gcd(a, bc) =
C
(b) Suppose that gcd(a, c) = 1. Then gcd(ab, c) = gcd(b, c).
The following Lemma plays an important role in the rest of this article as it allows us to show
that certain expressions Tα,β are redundant in the defining ideal of the Rees algebra.
Lemma 2.7. Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let α,
β ∈ Is be two sequences of length s ≥ 2, where Is is as in Definition 2.1. Suppose α = (α1 , . . . , αm )
and β = (β1 , . . . , βm ) with αi , βi ∈ Isi and s1 + . . . + sm = s. Suppose that for all 1 ≤ i ≤ m, there
exist Ci ∈ R such that
i−1

m

m

j=1

k=i

k=i+1

gcd( ∏ fα j , fβ ) gcd( ∏ fαk , ∏ fβk ) gcd( fαi , fβi )
gcd( fα , fβ ) =

where empty products are taken to be 1. Then
"
m

Tα,β = ∑ Ai
i=1

,

Ci
i−1

m

∏

j=i+1

#

Tα j ∏ Tβk Tαi ,βi ∈ SJ1 + S · (
k=1

r
[

Ji ),

i=2

where r = max{s1 , . . . , sm } and Ai ∈ R for all i.
m

m

i=1

i=1

Proof. Notice that fα = ∏ fαi and fβ = ∏ fβi . Then we have
fβ
fα
Tα −
T
gcd( fα , fβ )
gcd( fα , fβ ) β


i−1
m
m
i−1
T
T
f
f
C

α j ∏ βk 
i ∏ αj ∏
βk ∏
m 
fβi
f αi
j=i+1
j=1
k=1
k=i+1


T
−
T
= ∑
αi
βi .
m
m
i−1
gcd(
f
,
f
)
gcd(
f
,
f
)


αi βi
αi βi
i=1 gcd(
∏ fα j , fβ ) gcd( ∏ fαk , ∏ fβk )

Tα,β =

j=1

k=i

k=i+1

6

L. FOULI AND K. N. LIN
i−1

m

j=1

k=i+1

Ci ∏ fα j ∏ fβk
Finally, we note that Ai =

i−1

m

m

j=1

k=i

k=i+1

∈ R.



gcd( ∏ fα j , fβ ) gcd( ∏ fαk , ∏ fβk )
Next we give a list of conditions that when satisfied by two sequences α, β then the generator
Tα,β is not a minimal generator in the defining ideal of the Rees algebra of the corresponding ideal.
Proposition 2.8. Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let α,
β ∈ Is be two sequences of length s ≥ 2, where Is is as in Definition 2.1. Let α = (a1 , . . . , as ), β =
(b1 , . . . , bs ). Suppose that after some reordering there exist integers k, l with 1 ≤ k, l ≤ s− 1 such that
gcd( fai , fb j ) = 1 for every 1 ≤ i ≤ l and every k +1 ≤ j ≤ s. We further assume that gcd( fau , fbv ) = 1,
for every l + 1 ≤ u ≤ s and every 1 ≤ v ≤ k. Then Tα,β ∈ SJ1 + S · (

r
S

Ji ), where r = max{k, |k −

i=2

l|, s − k, s − l}.
Proof. Suppose that l = k and write α = (α1 , α2 ) and β = (β1 , β2 ), where α1 = (a1 , . . . , ak ), α2 =
(ak+1 , . . . , as ), β1 = (b1 , . . . , bk ), and β2 = (bk+1 , . . . , bs ). By our assumptions, we have gcd( fαi , fβ j ) =
1 for i 6= j. Hence by Remark 2.6, we have
gcd( fα , fβ1 ) gcd( fα , fβ2 ) gcd( fα1 , fβ1 ) gcd( fα , fβ2 )
gcd( fα , fβ ) =
=
,
C1
C1
gcd( fα1 , fβ ) gcd( fα2 , fβ ) gcd( fα1 , fβ ) gcd( fα2 , fβ2 )
gcd( fα , fβ ) =
=
.
C2
C2
The result follows from Lemma 2.7 with m = 2.
Without loss of generality we may now assume that l < k. We write α = (α1 , α2 , α3 ) and β =
(β1 , β2 , β3 ), with α1 = (a1 , . . . , al ), α2 = (al+1 , . . . , ak ), α3 = (ak+1 , . . . , as ), β1 = (b1 , . . . , bl ), β2 =
(bl+1 , . . . , bk ), and β3 = (bk+1 , . . . , bs ). Then gcd( fα1 , fβ3 ) = 1 and gcd( fαi , fβ j ) = 1 for i = 2, 3 and
j = 1, 2. Hence by Remark 2.6 we have
gcd( fα , fβ1 ) gcd( fα , fβ2 fβ3 ) gcd( fα1 , fβ1 ) gcd( fα , fβ2 fβ3 )
=
,
C1
C1
gcd( fα1 , fβ ) gcd( fα2 fα3 , fβ )
gcd( fα , fβ ) =
C2
gcd( fα1 , fβ ) gcd( fα2 fα3 , fβ3 ) gcd( fα2 fα3 , fβ1 fβ2 )
=
C2′
gcd( fα1 , fβ ) gcd( fα2 fα3 , fβ3 ) gcd( fα2 fα3 , fβ2 )
=
C2′
gcd( fα1 , fβ ) gcd( fα2 fα3 , fβ3 ) gcd( fα2 , fβ2 )
, and
=
C2′
gcd( fα1 fα2 , fβ ) gcd( fα3 , fβ ) gcd( fα1 fα2 , fβ ) gcd( fα3 , fβ3 )
gcd( fα , fβ ) =
=
.
C3
C3
Therefore we may apply Lemma 2.7 with m = 3.
gcd( fα , fβ ) =



REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

7

The next three Lemmas deal with possible repetitions in the sequences α, β ∈ Is and how that
affects the term Tα,β .
Lemma 2.9. Let R be a polynomial ring over a field and let I be a square-free monomial ideal in
R. Let α, β ∈ Is be two sequences of length s ≥ 2, where Is is as in Definition 2.1. Suppose that
s−1
S

α = (a1 , . . . , a1 ) and β = (b1 , . . . , bs ). Then Tα,β ∈ SJ1 + S · (

Ji ).

i=2

Proof. Notice that fα = fas1 and fβ = fb1 · · · fbs . Let β1 = (b2 , . . . , bs ). We also note that gcd( fas1 , fb1 ) =
gcd( fa1 , fb1 ), by Remark 2.3 (a). Then
gcd( fα , fβ ) =

, fβ1 )
gcd( fas1 , fb1 ) gcd( fas1 , fβ1 ) gcd( fa1 , fb1 ) gcd( fas−1
1
=
,
C
C
s−1
S

for some C ∈ R, by Remark 2.3 (b) and (c) . Hence Tα,β ∈ SJ1 + S · (

Ji ), by Lemma 2.7.

i=2



Lemma 2.10. Let R be a polynomial ring over a field and let I be a square-free monomial ideal in
R. Let α, β ∈ Is be two sequences of length s ≥ 2, where Is is as in Definition 2.1. Suppose that
α = (a1 , . . . , a1 , a2 , . . . , a2 ) and β = (b1 , . . . , b1 , b2 , . . . , b2 ). Then Tα,β ∈ SJ1 + SJ2 .
Proof. We will proceed by induction on s. If s = 2 then there is nothing to show. Suppose that s > 2.
Suppose that there are li distinct copies of ai in α and ki distinct copies of bi in β for i = 1, 2. Then
fα = fal11 fal22 and fβ = fbk11 fbk22 . For simplicity we write α = (al11 , al22 ) and β = (bk11 , bk22 ). If k1 = k2 and
l1 = l2 then the result follows from Lemma 2.5. Thus we may assume without loss of generality that
k1 > k2 and l1 ≥ l2 .
We will show
gcd( fal11−1 fal22 , fbk11 fbk22 ) gcd( fa1 , fb1 )
gcd( fα , fβ ) =
G
gcd( fal11−1 fal22 , fbk11 −1 fbk22 ) gcd( fal11 fal22 , fb1 )
=
H
for some G, H ∈ R. Then Tα,β ∈ SJ1 ∪ SJ2 , by Lemma 2.7 and the induction.
Notice that we have the following equalities:
gcd( fα , fβ ) =
=
gcd( fα , fβ ) =

gcd( fal11−1 fal22 , fbk11 fbk22 ) gcd( fa1 , fbk11 fbk22 )
C
k1 k2
l
−1
l
1
2
gcd( fa1 fa2 , fb1 fb2 ) gcd( fa1 , fbk11 −1 fbk22 ) gcd( fa1 , fb1 )
,
CD
gcd( fal11 fal22 , fbk11 −1 fbk22 ) gcd( fal11 fal22 , fb1 )
E
gcd( fal11−1 fal22 , fbk11 −1 fbk22 ) gcd( fa1 , fbk11 −1 fbk22 ) gcd( fal11 fal22 , fb1 )

,
EF
for some C, D, E, F ∈ R by Remark 2.6 (a). We will show that for any integer t and any variable x ∈ R
if xt | gcd( fa1 , fbk11 fbk22 ), then xt | CD and xt | EF. This is equivalent to showing that for any variable x ∈
R and any integer u, if x | gcd( fa1 , fbk11 fbk22 ) and xu | gcd( fα , fβ ) then xu | gcd( fal11−1 fal22 , fbk11 fbk22 ) gcd( fa1 , fb1 )
=

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L. FOULI AND K. N. LIN

and xu | gcd( fal11−1 fal22 , fbk11 −1 fbk22 ) gcd( fal11 fal22 , fb1 ). This follows immediately since s = l1 + l2 = k1 +

k2 > l1 and l1 ≥ 2s > k2 .
Lemma 2.11. Let R be a polynomial ring over a field and let I be a square-free monomial ideal in
R. Let α, β ∈ Is be two sequences of length s ≥ 4, where Is is as in Definition 2.1. Suppose that
α = (a1 , . . . , a1 , a2 , . . . , a2 , a3 , . . . , a3 ) and β = (b1 , . . . , b1 , b2 , . . . , b2 ). Then Tα,β ∈ SJ1 + S · (J2 ∪ J3 ).
s−1
S

Proof. We will show Tα,β ∈ SJ1 + S · (

i=2

Ji ) and by induction on s ≥ 4, we conclude that Tα,β ∈

SJ1 + S · (J2 ∪ J3 ). Suppose that there are li distinct copies of ai in α and ki distinct copies of bi in β.
Then l1 + l2 + l3 = k1 + k2 = s ≥ 4. Without lost of generality, we assume that l1 ≥ l2 ≥ l3 ≥ 1 and
k1 ≥ k2 ≥ 1. Since s ≥ 4, we have l1 ≥ 2 and k1 ≥ 2. We write α = (al11 , al22 , al33 ) and β = (bk11 , bk22 ).
We then have three possible scenarios. We claim the following:
(i) If l1 + l2 > k1 , then
gcd( fα , fβ ) = gcd( fa1 fa2 , fb1 fb2 ) gcd( fα1 , fβ )/A
= gcd( fα , fb1 fb2 ) gcd( fα1 , fβ1 )/B,
for some A, B ∈ R, and α1 = (al11 −1 , al22 −1 , al33 ) and β1 = (bk11 −1 , bk22 −1 ).
(ii) If l1 > k2 , then
gcd( fα , fβ ) = gcd( fa1 , fb1 ) gcd( fα1 , fβ )/C
= gcd( fα , fb1 ) gcd( fα1 , fβ1 )/D,
for some C, D ∈ R, and α1 = (al11 −1 , al22 , al33 ) and β1 = (bk11 −1 , bk22 ).
(iii) If l1 + l2 ≤ k1 and l1 ≤ k2 , then
2s/3 s/3
fb2 )
2
fb1 fb2 ))s/3 .

s/3 s/3 s/3

gcd( fα , fβ ) = gcd( fa1 fa2 fa3 , fb1
= (gcd( fa1 fa2 fa3 ,

Once we obtain the above claims then the result will follow by Lemma 2.7.
To establish claim (i) we notice that we have the following equalities:
gcd( fα , fβ ) = gcd( fa1 fa2 , fβ ) gcd( fα1 , fβ )/E
= gcd( fa1 fa2 , fb1 fb2 ) gcd( fa1 fa2 , fβ1 ) gcd( fα1 , fβ )/EF,
= gcd( fα , fb1 fb2 ) gcd( fα , fβ1 )/G
= gcd( fα , fb1 fb2 ) gcd( fα1 , fβ1 ) gcd( fa1 fa2 , fβ1 )/GH,
for some E, F, G, H ∈ R. It is enough to show that for any variable x ∈ R and any positive integer t if
xt | gcd( fa1 fa2 , fβ1 ), then xt | EF and xt | GH. This is equivalent to showing that if xu | gcd( fα , fβ ) and
x | gcd( fa1 fa2 , fβ1 ), then xu | gcd( fa1 fa2 , fb1 fb2 ) gcd( fα1 , fβ ) and xu | gcd( fα , fb1 fb2 ) gcd( fα1 , fβ1 ) for
any positive integer u. But this follows immediately since l1 + l2 > k1 ≥ k2 and l1 + l2 < l1 + l2 + l3 =
s = k1 + k2 .

REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

9

For claim (ii) we notice that we have the following equalities:
gcd( fα , fβ ) = gcd( fa1 , fβ ) gcd( fα1 , fβ )/K
= gcd( fa1 , fb1 ) gcd( fa1 , fβ1 ) gcd( fα1 , fβ )/KL,
= gcd( fα , fb1 ) gcd( fα , fβ1 )/M
= gcd( fα , fb1 ) gcd( fα1 , fβ1 ) gcd( fa1 , fβ1 )/MN,
for some K, L, M, N ∈ R. The claim follows since l1 > k2 .
Finally to establish the last claim we note that the assumption l1 ≤ k2 is equivalent to l2 + l3 ≥ k1
and thus l2 + l3 ≥ k1 ≥ l1 + l2 . Hence l3 ≥ l1 ≥ l2 ≥ l3 which implies li = s/3 and 2s/3 ≤ k1 .
Furthermore, s/3 ≥ k2 ≥ s/3 and hence k2 = s/3 and k1 = 2s/3.

The next Theorem is one of the main results of this section. We will use all the information we
obtained about how various conditions on two sequences α, β ∈ Is affect the term Tα,β to obtain a
bound on the relation type of I as well as a description of the defining equations of the Rees algebra.
Theorem 2.12. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
generated by n square-free monomials in R. When n ≤ 5, then R (I) = S/J, where S = R[T1 , . . . , Tn ]
n−2
S

and J = SJ1 + S · (

Ji ). In particular, rt(I) ≤ n − 2.

i=2

Proof. By Theorem 2.2, it suffices to show that given two sequences α, β ∈ Is of length s > n − 2
n−2
S

then Tα,β ∈ SJ1 + S · (

Ji ). Suppose that there are li distinct copies of ai in α and ki distinct copies

i=2

of bi in β and write α = (al11 , . . . , almm ), and β = (bk11 , . . . , btkt ).
By Lemma 2.4 we may assume ai 6= b j for all i, j. Also by Lemma 2.9, we may assume 1 < m ≤ 3
and 1 < t ≤ 2, since n ≤ 5. We are now left with the following cases:
(i) Suppose that α = (al11 , al22 ) and β = (bk11 , bk22 ), where l1 + l2 = k1 + k2 ≥ 3. The result follows
from Lemma 2.10.
(ii) Suppose that α = (al11 , al22 , al33 ) and β = (bk11 , bk22 ), where l1 + l2 + l3 = k1 + k2 ≥ 4. The result
follows from Lemma 2.11.

The following example provides a class of square-free monomial ideals generated by n > 4
square-free monomials for which the relation type is at least 2n − 7. Notice when n = 5 one has
2n − 7 = n − 2. In particular, this establishes that the bound given in Theorem 2.12 is sharp. We
also note that the ideals in the following example are not of fiber type.
Example 2.13. Let R = k[y, x2 , . . . , xn−2 , z, u2 , . . . , un−2 ] be a polynomial ring over a field k with n a
n−2

n−2

i=2

j=2, j6=i

positive integer such that n > 4. Let I = ( f1 , . . . , fn ), where f1 = ( ∏ xi )z, fi = xi y( ∏ u j ) for all
n−2

n−2

i = 2, . . . , n − 2, fn−1 = ( ∏ xi )y, and fn = ( ∏ ui )z. Consider
i=2

i=2

n−2

n−2

i=2

i=2

n−5
n−3 n−4
Tn .
Tn and G = zT1n−5 ∏ Ti − yTn−1
F = T1n−4 ∏ Ti − Tn−1

10

L. FOULI AND K. N. LIN

It is clear that F and G are in the defining ideal of the Rees algebra of I. Moreover, F and G are
2n−8
S

irreducible and thus F ∈ SJ2n−7 \ S · (

2n−9
S

Ji ) and G ∈ SJ2n−8 \ S · (

Ji ). Hence the relation type

i=1

i=1

of I is at least 2n − 7. Finally, we note that G is not in the defining ideal of the special fiber of I and
therefore I is not an ideal of fiber type.
The next lemma establishes conditions for when various generators of the defining ideal of the
Rees algebra as in Theorem 2.12 are irredundant.
Lemma 2.14. Let R be a polynomial ring over a field and let I be a square-free monomial ideal in
R. Let Let α, β ∈ Is be two sequences of length s ≥ 4, where Is is as in Definition 2.1. Suppose that
α = (a1 , . . . , as ), β = (b1 , . . . , b1 , b2 ), where ai 6= b j for all i, j. Suppose that for all i there exists a
variable xi ∈ R such that xi | fa j for all j 6= i and xi | fb1 , xi ∤ fai , and xi ∤ fb2 . Furthermore, suppose
that for all i there exists a variable zi ∈ R such that zi | gcd( fai , fb2 ), zi ∤ fa j for all j 6= i and zi ∤ fb1 .
s−1
S

Then Tα,β ∈ SJs \ S · (

Ji ).

i=1

s

s

s

j=1, j6=i

i=1

i=1

Proof. We write fai = zi hi ( ∏ x j ) for all i = 1, . . . , s, fb1 = k1 ( ∏ xi ), and fb2 = k2 ( ∏ zi ), for
some square-free monomials h1 , . . . , hs , k1 , k2 ∈ R such that xi ∤ h j , xi ∤ kl , zi ∤ h j , and zi ∤ kl for all i,
s

s

i=1

i=1

s

s

s

i=1

i=1

i=1

j, l. Then we have fα = ∏ xis−1 ∏ zi ∏ hi and fβ = k1s−1 k2 ( ∏ xis−1 ∏ zi ). Let
s

fα
M=
=
gcd( fα , fβ )

∏ hi

i=1
s

gcd( ∏ hi , k1s−1 k2 )

and N =

fβ
=
gcd( fα , fβ )

i=1

One can observe immediately that xi ∤ M and zi ∤ N for all i.

k1s−1 k2
s

.

gcd( ∏ hi , k1s−1 k2 )
i=1

Let α′

and β′

be any proper subsequences
f α′
f α′
of α and β. Then there exists either xi or z j such that xi |
or z j |
by
gcd( fα′ , fβ′ )
gcd( fα′ , fβ′ )
fβ′
f α′
∤ M, and similarly,
∤ N. This shows that
our assumptions. Therefore
gcd( fα′ , fβ′ )
gcd( fα′ , fβ′ )
s−1
S

Tα,β ∈ SJs \ S · (

Ji ).



i=1

The following Theorem that establishes precisely the defining equations for the Rees algebra of
a square-free monomial ideal with up to five generators.
Theorem 2.15. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
generated by n square-free monomials in R. Suppose that n ≤ 5. Suppose that there does not
exist a pair of sequences α, β ∈ Is of length s ≥ 2 as in Definition 2.1, such that the conditions of
Lemma 2.14 are satisfied. Then I is an ideal of linear type.
Proof. Consider a pair of sequences α, β as in Definition 2.1 of length s ≥ 2 such that α =
(a1 , . . . , as ), β = (b1 , . . . , b1 , b2 ), where ai 6= b j for all i, j. Suppose that one of the following conditions is not satisfied: for all i, there exists a variable xi ∈ R such that xi | fa j for all j 6= i and xi | fb1 ,

REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

11

xi ∤ fai , and xi ∤ fb2 , and for all i there exists a variable zi ∈ R such that zi | gcd( fai , fb2 ), zi ∤ fa j for all
s−1
S

j 6= i and zi ∤ fb1 . We will show that Tα,β ∈ SJ1 + S · (

i=1

Ji ) and by induction Tα,β ∈ SJ1 .

By the Theorem 2.12 and Lemmas 2.4, 2.9 it is enough to consider sequences of length 2 or 3,
i.e. either α = (a1 , a2 ) and β = (b1 , b2 ) or α = (a1 , a2 , a3 ) and β = (b1 , b1 , b2 ) . The proof for the
first case can be treated as a special case of the second and thus we will only consider the case with
α = (a1 , a2 , a3 ) and β = (b1 , b1 , b2 ). Without lost of generality, we will show that if there does not
exist a variable x ∈ R such that x | gcd( fa2 , fa3 ), x | fb1 , x ∤ fa1 , and x ∤ fb2 then
gcd( fa1 fa2 fa3 , fb21 fb2 ) =
=

gcd( fa1 , fb1 ) gcd( fα , fb1 fb2 )
A
gcd( fa1 , fβ ) gcd( fa2 fa3 , fb1 fb2 )
,
B

for some A, B ∈ R.
Similarly, if there does not exist a variable z ∈ R such that z | gcd( fa1 , fb2 ), z ∤ fa j for j = 2, 3, and
z ∤ fb1 then
gcd( fa1 , fb1 ) gcd( fa2 fa3 , fβ )
C
gcd( fα , fb1 ) gcd( fa2 fa3 , fb1 fb2 )
,
=
D
for some C, D ∈ R. Hence we may apply Lemma 2.7 to obtain the result.
For the first part, we notice that
gcd( fa1 fa2 fa3 , fb21 fb2 ) =

gcd( fa1 fa2 fa3 , fb21 fb2 ) =
=

gcd( fα , fb1 ) gcd( fα , fb1 fb2 )
E
gcd( fa1 , fb1 ) gcd( fa2 fa3 , fb1 ) gcd( fα , fb1 fb2 )
,
EF

gcd( fa1 , fβ ) gcd( fa2 fa3 , fβ )
G
gcd( fa1 , fβ ) gcd( fa2 fa3 , fb1 ) gcd( fa2 fa3 , fb1 fb2 )
=
,
GH
for some E, F, G, H ∈ R. It is enough to show for any variable x ∈ R if x | gcd( fa2 fa3 , fb1 ), then
x | EF and x | GH. This is equivalent to showing that for any integer u if xu | gcd( fα , fβ ) then
xu | gcd( fa1 , fb1 ) gcd( fα , fb1 fb2 ) and xu | gcd( fa1 , fβ ) gcd( fa2 fa3 , fb1 fb2 ). This is straightforward to
verify and the only case that is not trivial is when x ∤ fa1 and x | gcd( fa2 , fa3 ). Then x | fb2 by
assumption. Hence x2 | gcd( fα , fb1 fb2 ) and x2 | gcd( fa2 fa3 , fb1 fb2 ).
We can use a similar argument for the second part and the only case that is not trivial is when
x | gcd( fa1 , fb1 fb2 ) and x ∤ fb1 . Then x | fb2 and x | fa2 or x | fa3 by assumption. Hence x2 ∤ gcd( fα , fβ ),

x | gcd( fa2 fa3 , fβ ), and x | gcd( fa2 fa3 , fb1 fb2 ).
We now introduce a construction that is also known as the 1-skeleton or the line graph of the
hypergraph of a monomial ideal.
gcd( fa1 fa2 fa3 , fb21 fb2 ) =

12

L. FOULI AND K. N. LIN

Construction 2.16. Let R be a polynomial ring over a field and let I be a monomial ideal in R . Let
e
f1 , . . . , fn be a minimal monomial generating set of I. We construct the following graph G(I):
For
each fi we associate a vertex yi to it. The edges of this graph are {yi , y j }, where gcd( fi , f j ) 6= 1. We
e the generator graph of I.
call the graph G(I)
The purpose of introducing the generator graph of a monomial ideal is to utilize the graph structure in order to determine combinatorial conditions that determine the defining equations of the
Rees algebra. We first observe the following.

Remark 2.17. With the same assumptions as in Lemma 2.14 the sequences α, β correspond to an
e
even closed walk in G(I).
Indeed, the following
{yb1 , {yb1 , ya1 }, ya1 , {ya1 , yb1 }, yb1 , {yb1 , ya2 }, ya2 , {ya2 , yb1 }, yb1 , . . . ,

yb1 , {yb1 , yas−1 }, yas−1 , {yas−1 , yb2 }, yb2 , {yb2 , yas }, yas , {yas , yb1 }, yb1 }
s−1
S

is an even closed walk and Tα,β ∈ SJs \ S · (

Ji ).

i=1

We now introduce the notion of a subgraph induced by two sequences.
Definition 2.18. Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let α,
β ∈ Is be two sequences of length s ≥ 2, where Is is as in Definition 2.1. Let α = (al11 , al22 , . . . , almm )
and β = (br11 , . . . , btrt ), where li and ri are the number of distinct copies of ai and bi , respectively. Let
α1 = (a1 , . . . , am ) and β1 = (bi1 , . . . , bir ), where {bi1 , . . . , bir } = {b1 , . . . , bt } \ {a1 , . . . , am }. The last
condition ensures that ai 6= bik for all i, ik . Consider K = ( fa1 , . . . , fam , fbi1 , . . . , fbir ) and the generator
e
e
e
graph G(K)
of K. Notice that K ⊂ I and that G(K)
is a subgraph of G(I),
since ai 6= bik for all i, ik .
e
We call G(K) the graph induced by α and β.

Remark 2.19. Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let
α, β ∈ Is be two sequences of length s ≥ 2, where Is is as in Definition 2.1. Let G be the graph
s−1
S

induced by α and β. If G is a disconnected graph then Tα,β ∈ SJ1 + S · (

Ji ), by Proposition 2.8.

i=2

The following Theorem is an extension of Theorem 2.12. We use the generator graph to give a
description for the defining equations of the Rees algebra.
Theorem 2.20. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
generated by n square-free monomials in R. Suppose that the generator graph of I is the disjoint
union of graphs with at most 5 vertices. Let R (I) = S/J, where S = R[T1 , . . . , Tn ]. Then J = SJ1 +
S · (J2 ∪ J3 ) and in particular, rt(I) ≤ 3. Furthermore, if Tα,β ∈ S · (J2 ∪ J3 ) \ SJ1 , then the subgraph
e
induced by α, β is an even closed walk of G(I).

e is connected. Then the first part follows imProof. By Remark 2.19 we may assume that G(I)
mediately by Theorem 2.12. The last part follows from Theorem 2.15, Lemma 2.14, and Remark 2.17.

The next example illustrates the result of Theorem 2.20.

REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

13

Example 2.21. Let R = k[x1 , . . . , x7 ] be a polynomial ring over a field k. Let I = ( f1 , . . . , f5 ), where
f1 = x1 x2 x3 , f2 = x1 x2 x4 x7 , f3 = x2 x3 x6 , f4 = x4 x5 x6 , and f5 = x1 x3 x5 . Let R (I) = S/J, where
S = R[T1 , . . . , T5 ]. Let α = (1, 1, 4) and β = (2, 3, 5). The subgraph induced by α, β is the even
closed walk
{y1 , {y1 , y2 }, y2 , {y1 , y2 }, y1 , {y1 , y3 }, y3 , {y3 , y4 }, y4 , {y4 , y5 }, y5 , {y1 , y5 }}
of the generator graph of I shown below:
y2 = x1 x2 x4 x7
y3 = x2 x3 x6

y1 = x1 x2 x3

y4 = x4 x5 x6

y5 = x1 x3 x5

Then by Theorem 2.20 we have that Tα,β = T2 T3 T5 − x7 T12 T4 ∈ SJ3 \ (SJ1 + SJ2 ), since it is irreducible.
The following example comes from [26, Example 3.1]. Villarreal uses this example to show that
his methods do not extend in the case of square-free monomial ideals generated in degree higher
than 2. In light of Theorem 2.20 we now give an explicit description of the defining equations of
the Rees algebra for this example.
Example 2.22. Let R = k[x1 , . . . , x7 ], where k is a field and let I be an ideal of R generated by
f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 , f4 = x3 x6 x7 . Then the defining ideal of R (I) is minimally
generated by the binomials:
x3 T3 − x5 T4 , x6 x7 T1 − x1 x2 T4 , x6 x7 T2 − x2 x4 T3 , x4 x5 T1 − x1 x3 T2 , x4 T1 T3 − x1 T2 T4 .
Notice that the only binomial that is not linear is x4 T1 T3 − x1 T2 T4 and it comes from the unique even
e
cycle of G(I),
which in this case is the graph of a square.
3. S QUARE - FREE

MONOMIAL IDEALS OF LINEAR TYPE

In this section we turn our attention to ideals of linear type. Villarreal showed that when I is the
edge ideal of a graph then I is of linear type if and only if the graph of I is the disjoint union of
graphs of trees or graphs that have a unique odd cycle, [26, Corollary 3.2]. Inspired by this result
we use the generator graph associated to any monomial ideal as in Construction 2.16 in order to
obtain similar results as Villarreal for any square-free monomial ideal without restrictions on the
degrees of the generators. A natural first class to consider is ideals with generator graph the graph
of a forest.
Proposition 3.1. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
e is the graph of a forest. Then I is an ideal of
in R. We further assume that the generator graph G(I)
linear type.

14

L. FOULI AND K. N. LIN

Proof. Let f1 , . . . , fn be a minimal monomial generating set for I. We show that for all sequences
s−1
S

α, β ∈ Is of length s ≥ 2, where Is is as in Definition 2.1, we have Tα,β ∈ SJ1 + S · (

Ji ) and by

i=2

induction it follows that Tα,β ∈ SJ1 .
e is the graph of a forest then every subgraph of G(I)
e is also a forest. By Proposition 2.8
Since G(I)
e is connected, i.e. it is the graph of a tree. Notice that for every graph of a
we may assume that G(I)
tree there exists a vertex that is only connected to one other vertex.
Suppose that s ≥ 2. Let α = (al11 , al22 , . . . , almm ) and β = (br11 , . . . , btrt ), where li and ri are the number
of distinct copies of ai and bi , respectively. Notice that if ai = b j for some i and some j then the
result follows by Lemma 2.4 and the induction hypothesis. Hence we may assume that ai 6= b j
for all i, j. Let α1 = (a1 , . . . , am ) and β1 = (b1 , . . . , bt ). Then the graph G′ induced by α1 , β1 is a
e
subgraph of G(I),
by Definition 2.18. Hence G′ is the graph of a forest. If G′ is disconnected then
the result follows by Proposition 2.8 and the induction hypothesis. Therefore, we may assume that
G′ is connected and hence the graph of a tree. Thus without loss of generality we may assume that
ya1 is only connected to yb1 .
s−1
S

If l1 ≥ r1 , then gcd( fa1 , fbr22 · · · fbrtt ) = 1 and hence Tα,β ∈ SJ1 + S · (
we may assume that l1 < r1 . Then we claim that

Ji ), by Proposition 2.8. So

i=2

(a) gcd( fα , fβ ) = gcd( fal11 , fbl11 ) gcd( fal22 · · · falmm , fβ ),
gcd( fα , fbl11 ) gcd( fal22 · · · falkm , fbr11 −l1 fbr22 · · · fbrtt )
,
(b) gcd( fα , fβ ) =
A
s−1
S

for some A ∈ R. Notice that by the claim and Lemma 2.7 it follows that Tα,β ∈ SJ1 + S · (

Ji ).

i=2

Therefore, it remains to prove the claim.
For part (a) notice that gcd( fα , fβ ) = gcd( fal11 , fβ ) gcd( fal22 · · · falmm , fβ ) = ( fal11 , fbl11 ) gcd( fal22 · · · falmm , fβ ),
where the equalities follow from the fact that gcd( fa1 , fai ) = gcd( fa1 , fb j ) = 1 for any i > 1, j > 1
and Remark 2.6 (b).
For part (b), notice that
gcd( fα , fβ ) = gcd( fα , fbl11 ) gcd( fα , fbr11 −l1 fbr22 · · · fbrtt )/B
= gcd( fα , fbl11 ) gcd( fal11 , fbr11 −l1 fbr22 · · · fbrtt ) gcd( fal22 · · · falmm , fbr11 −l1 fbr22 · · · fbrtt )/BC
= gcd( fα , fbl11 ) gcd( fal11 , fbr11 −l1 ) gcd( fal22 · · · falmm , fbr22 · · · fbrtt )/BC,
for some B,C ∈ R, since gcd( fal11 , fbr22 · · · fbrtt ) = 1. It is enough to show that gcd( fal11 , fbr11 −l1 ) | BC, i.e.
for any variable x ∈ R and any integer v if xv | gcd( fal11 , fbr11 −l1 ), then xv | BC. This is equivalent to
showing that for any variable x ∈ R if x | gcd( fal11 , fbr11 −l1 ) and xu | gcd( fα , fβ ) for some integer u,
then xu | gcd( fα , fbl11 ) gcd( fal22 · · · falmm , fbr11 −l1 fbr22 · · · fbrtt ). If x | gcd( fal11 , fbr11 −l1 ) and xu | gcd( fα , fβ ), then

u ≤ l1 by the fact that ya1 is only connected to yb1 and l1 < r1 . Therefore xu | gcd( fα , fbl11 ).

REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

15

The following lemma allows us to handle the induction step in the case of odd cycles, in order to
prove that when the generator graph of a square-free monomial ideal is the graph of an odd cycle,
then the ideal is of linear type.
Lemma 3.2. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
in R. Let α, β ∈ Is be two sequences of length s ≥ 4, where Is is as in Definition 2.1. Let α =
(al11 , al22 , . . . , almm ) and β = (br11 , . . . , btrt ), where li and ri are the number of distinct copies of ai and bi ,
e is the graph of an odd cycle of
respectively. Suppose that l1 < r1 , l2 < r2 , and that the graph G(I)
length at least 5. We further assume that ya1 is connected to yb1 and ya2 only, and ya2 is connected
s−1
S

to ya1 and yb2 only. Then Tα,β ∈ SJ1 + S · (

Ji ).

i=2

Proof. Let α′ = (al33 , . . . , almm ) and β′ = (br11 −l1 , br22 −l2 , br33 , . . . , btrt ). Using Remark 2.6 and Remark 2.3
we have that
gcd( fα , fβ ) = gcd( fal11 fal22 , fβ ) gcd( fα′ , fβ )
= gcd( fal11 fal22 , fbl11 fbl22 ) gcd( fα′ , fβ ),
since l1 < r1 and l2 < r2 .

gcd( fα , fbl11 fbl22 )( fα′ , fβ′ )
, for some C ∈ R. Then the result will follow
We claim that gcd( fα , fβ ) =
C
by Lemma 2.7. Notice that
gcd( fα , fβ ) =
=

gcd( fα , fbl11 fbl22 ) gcd( fα , fβ′ )
D
gcd( fα , fbl11 fbl22 ) gcd( fα′ , fβ′ ) gcd( fal11 fal22 , fβ′ )

DE
gcd( fα , fbl11 fbl22 ) gcd( fα′ , fβ′ ) gcd( fal11 fal22 , fbr11 −l1 fbr22 −l2 )
=
,
DE
where D, E ∈ R and the third equality follows from the fact that ya1 and ya2 are only connected to
yb1 and yb2 , respectively. It is enough to show that for any variable x ∈ R and any integer v if xv |
gcd( fal11 fal22 , fbr11 −l1 fbr22 −l2 ), then xv | DE. This is equivalent to showing that if x | gcd( fal11 fal22 , fbr11 −l1 fbr22 −l2 )
and xu | gcd( fα , fβ ), then xu | gcd( fα , fbl11 fbl22 ) gcd( fα′ , fβ′ ). If x | gcd( fal11 fal22 , fbr11 −l1 fbr22 −l2 ), then x can
not divide both fa1 and fa2 , since otherwise {ya1 , ya2 , yb1 } or {ya1 , ya2 , yb2 } will form a 3-cycle in
e
G(I).
Similarly, x can not divide both fb1 and fb2 . Therefore, without loss of generality we may
assume x | fa1 and x | fb1 . Since l1 < r1 and ya1 is not connected to yai for all i > 2, we obtain that if
xu | gcd( fα , fβ ), then u ≤ l1 . Therefore xu | gcd( fα , fbl11 fbl22 ).

In the next Proposition we are able to handle the case where each of the connected components
of the generator graph has a unique odd cycle.
Proposition 3.3. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
e of I is the disjoint union of graphs with a unique
in R. We further assume that generator graph G(I)
odd cycle. Then I is an ideal of linear type.

16

L. FOULI AND K. N. LIN

e
Proof. By Remark 2.19 it suffices to consider the connected components of G(I).
Hence we may
e
assume that G(I) is connected and it has a unique odd cycle. We show that for all sequences α, β ∈ Is
s−1
S

of length s ≥ 2, where Is is as in Definition 2.1, we have Tα,β ∈ SJ1 + S · (

Ji ). By induction it will

i=2

follow that Tα,β ∈ SJ1 .
Let α = (al11 , al22 , . . . , almm ) and β = (br11 , . . . , btrt ), where li and ri are the number of distinct copies
of ai and bi , respectively. Using Lemma 2.4 we may assume without loss of generality that ai 6= b j
for all i and j. Let α1 = (a1 , . . . , am ) and β1 = (b1 , . . . , bt ). Then the graph G′ induced by α1
g by Definition 2.18. By Proposition 2.8 we may assume that G′ is a
and β1 is a subgraph of G(I),
connected graph. Notice that if there exists a vertex yi that is only connected to one of the y j then
s−1
S

Tα,β ∈ SJ1 + S · (

Ji ) by the proof of Proposition 3.1. Hence we may assume that G′ is the graph

i=2

of an odd cycle. Therefore every vertex is only connected to two other vertices.
Let l = m + t. Since G′ is the graph of an odd cycle then l is odd and hence m 6= t. We proceed by
e is the graph of a triangle and the result follows from Lemma 2.4
induction on l. If l = 3 then G(I)
and Lemma 2.9. Suppose that l ≥ 5. Using Lemma 2.9 we may assume that m ≥ 2 and t ≥ 2. Notice
that ya1 is connected to two other vertices. Without loss of generality we may assume that we have
the following possible cases:
(i) ya1 is connected only to ya2 and ya3 ,
(ii) ya1 is connected only to ya2 and yb1 ,
(iii) ya1 is connected only to yb1 and yb2 .
We handle each case separately. For case (i) we note that since ya1 is connected only to ya2 and ya3 ,
then using k = l1 in Proposition 2.8 and induction we obtain the result.
For case (ii) suppose that ya1 is connected only to ya2 and yb1 . If l1 ≥ r1 then using k = l1 in
Proposition 2.8 and induction we obtain the result. If l1 < r1 then we consider ya2 . Notice that ya2
can not be connected to yb1 since then {ya1 , ya2 , yb1 } will form a triangle, which is a contradiction
since l ≥ 5. Hence either ya2 is connected to ya3 , or ya2 is connected to ybi for some i ≥ 2. If ya2 is
connected to ya3 then case (i) yields the result.
Suppose that ya2 is connected to ybi for some i ≥ 2. Without loss of generality suppose that i = 2.
If l2 ≥ r2 then we may use k = l2 in Proposition 2.8 and induction. If l2 < r2 and since l1 < r1 we
can use Lemma 3.2 and induction.
Finally for case (iii) suppose that ya1 is connected only to yb1 and yb2 . If every yai is connected to
ybi1 and ybi2 for i1 , i2 6= 1, 2 then either m = t, which is impossible since l is odd or there exist i 6= j
such that ybi is connected to yb j . Then by switching the role of α and β we are in either case (i) or
case (ii).

We are now ready to state the main theorem of this section.
Theorem 3.4. Let R be a polynomial ring over a field and let I be a square-free monomial ideal in
e
R. We further assume that the generator graph, G(I),
is the disjoint union of graphs of trees and
graphs that with a unique odd cycle. Then I is an ideal of linear type.

REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS

17

Proof. The result follows immediately by Proposition 3.1, Proposition 3.3, and Proposition 2.8. 
We conclude this article with the following remark.
Remark 3.5. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
generated by 3 square-free monomials in R. Then the fact that I is of linear type is already known,
but it also follows immediately from Proposition 3.1 and Proposition 3.3.

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18

L. FOULI AND K. N. LIN

D EPARTMENT OF M ATHEMATICAL S CIENCES , N EW M EXICO S TATE U NIVERSITY, L AS C RUCES , N EW M EXICO
88003, USA
E-mail address: lfouli@math.nmsu.edu
D EPARTMENT OF M ATHEMATICS , U NIVERSITY
CA 92521, USA
E-mail address: linkuei@ucr.edu

SIDE ,

OF

C ALIFORNIA , R IVERSIDE , 900 U NIVERSITY AVE ., R IVER -

