arXiv:1207.1789v1 [] 7 Jul 2012

Monomial ideals with 3-linear resolutions
Marcel Morales2 , Abbas Nasrollah Nejad1 ,
Ali Akbar Yazdan Pour1,2 , Rashid Zaare-Nahandi1
1

Institute for Advanced Studies in Basic Sciences, P. O. Box 45195-1159, Zanjan, Iran
2

Université de Grenoble I, Institut Fourier, Laboratoire de Mathématiques, France

Abstract
In this paper, we study Cstelnuovo-Mumford regularity of square-free monomial ideals generated
in degree 3. We define some operations on the clutters associated to such ideals and prove that the
regularity is conserved under these operations. We apply the operations to introduce some classes of
ideals with linear resolutions and also show that any clutter corresponding to a triangulation of the
sphere does not have linear resolution while any proper sub-clutter of it has a linear resolution.

1

Introduction

Let S = K[x1 , . . . , xn ] be the polynomial ring over a field K and I be a homogeneous ideal of S. Computing the Castenuovo-Mumford regularity of I or even proving that the ideal I has linear resolutions is
difficult in general. It is known that a monomial ideal has d-linear resolution if and only if its polarization, which is a square-free monomial ideal, has d-linear resolution. Therefore, classification of monomial
ideals with linear resolution is equal to classification of square-free monomial ideals. In this subject, one
of the fundamental results is the Eagon-Reiner theorem, which says that the Stanley-Reisner ideal of a
simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.
The problem of existing 2-linear resolution is completely solved by Fröberg [Fr] (See also [Mo]). An
ideals of S generated by square-free monomials of degree 2 can be assumed as edge ideal of a graph.
Fröberg proved that the edge ideal of a finite simple graph G has linear resolution if and only if the
complementary graph Ḡ of G is chordal, i.e., there is no any minimal induced cycle in G of length more
than 3.
Another combinatorial objects corresponding to square-free monomial ideals are clutters which are
special cases of hypergraphs. Let [n] = {1, . . . , n}. A clutter C on a vertex set [n] is a set of subsets of
[n] (called circuits of C) such that if e1 and e2 are distinct circuits, then e1 * e2 . A d-circuit is a circuit
with d vertices, and a clutter is called d-uniform if every circuit is a d-circuit. To a clutter C with circuits
{e1 , . . . , em } the ideal generated by Xej for all j = 1, . . . , m is corresponded which is called circuit ideal
of C and denoted by I(C). One says that a d-uniform clutter C has a linear resolution if the circuit ideal
of the complimentary clutter C¯ has d-linear resolution. Trying to generalize similar result of Fröberg for
d-uniform clutters (d > 2), several mathematicians including E. Emtander [E] and R. Woodroofe [W] have
defined notion of chordal clutters and proved that any d-uniform chordal clutter has a linear resolution.
These results are one-sided. That is, there are non-chordal d-uniform clutters with linear resolution.
In the present paper, we introduce some reduction processes on 3-uniform clutters which do not change
regularity of the minimal resolution. Then a class of 3-uniform clutters which have linear resolution and
a class of 3-uniform clutters which do not have linear resolution are constructed.
0 MSC 2010: Primary 13C14, 13D02; Secondary 13F55.
Key words: minimal free resolution, linear resolution, uniform clutter, monomial ideal, regularity.

1

Some of the results of this paper have been conjectured after explicit computations performed by the
computer algebra systems Singular [Si] and CoCoA [Co].

2

Preliminaries

Let K be a field, S = K[x1 , . . . , xn ] be the polynomial ring over K with the standard graduation and
m = (x1 , . . . , xn ) be the irredundant maximal ideal of S.
We quote the following well-known results that will be used in this paper.
Theorem 2.1 (Grothendieck, [St, Theorem 6.3]). Let M be a finitely generated S-module. Let t =
i
i
depth (M ) and d = dim(M ). Then Hm
(M ) 6= 0 for i = t and i = d, and Hm
(M ) = 0 for i < t and i > d.
i
Corollary 2.2. Let M be a finitely generated S-module. M is Cohen-Macaulay if and only if Hm
(M ) = 0
for i < dim M .

Lemma 2.3. Let S = K[x1 , . . . , xn , y1 , . . . , ym ] be the polynomial ring and I be an ideal in K[y1 , . . . , ym ].
Then,
S
S
depth
= depth .
(x1 · · · xn ) I
I
Definition 2.4 (Alexander duality). For a square-free monomial ideal I = (M1 , . . . , Mq ) ⊂ K[x1 , . . . , xn ],
the Alexander dual I ∨ , of I is defined to be
I ∨ = PM1 ∩ · · · ∩ PMq
where PMi is prime ideal generated by {xj : xj |Mi }.
Definition 2.5. Let I be a non-zero homogeneous ideal of S. For every i ∈ N one defines
S
tSi (I) = max{j : βi,j
(I) 6= 0}
S
where βi,j
(I) is the i, j-th graded Betti number of I as an S-module. The Castelnuovo-Mumford regularity
of I, is given by
reg (I) = sup{tSi (I) − i : i ∈ Z}.

We say that the ideal I has a d-linear resolution if I is generated by homogeneous polynomials of degree
S
d and βi,j
(I) = 0 for all j 6= i + d and i ≥ 0. For an ideal which has d-linear resolution, the CastelnuovoMumford regularity would be d.
Theorem 2.6 (Eagon-Reiner [ER, Theorem 3]). Let I be a square-free monomial ideal in S = K[x1 , . . . , xn ].
I has q-linear resolution if and only if S/I ∨ is Cohen-Macaulay of dimension n − q.
Theorem 2.7 ([T, Theorem 2.1]). Let I be a square-free monomial ideal in S = K[x1 , . . . , xn ] with
dim S/I ≤ n − 2. Then,
S
S
dim ∨ − depth ∨ = reg (I) − indeg(I).
I
I
Remark 2.8. Let I, J be square-free monomial ideals generated by elements of degree d ≥ 2 in S =
K[x1 , . . . , xn ]. By Theorem 2.7, we have
reg (I) = n − depth

S
,
I∨

reg (J) = n − depth

Therefore, reg (I) = reg (J) if and only if depth S/I ∨ = depth S/J ∨ .
2

S
.
J∨

Definition 2.9 (Clutter). Let [n] = 1, . . . , n. A clutter C on a vertex set [n] is a set of subsets of [n]
(called circuits of C) such that if e1 and e2 are distinct circuits of C then e1 * e2 . A d-circuit is a circuit
consisting of exactly d vertices, and a clutter is d-uniform if every circuit has exactly d vertices.
For a non-empty clutter C on vertex set [n], we define the ideal I(C), as follows:
I(C) = (xF :

F ∈ C)

and we define I(∅) = 0.
Let n, d be positive integers and d ≤ n. We define Cn,d , the maximal d-uniform clutter on [n] as
follow:
Cn,d = {F ⊂ [n] : |F | = d}.
¯ the complement of C, to be
If C is a d-uniform clutter on [n], we define C,
C¯ = Cn,d \ C = {F ⊂ [n] :

|F | = d, F ∈
/ C}.

¯ in
Frequently in this paper, we take a d-uniform clutter C and we consider the square-free ideal I = I(C)
the polynomial ring S = K[x1 , . . . , xn ]. The ideal I is called circuit ideal.
Definition 2.10 (Clique). Let C be a d-uniform clutter on [n]. A subset G ⊂ [n] is called a clique in C,
if all d-subset of G belongs to C.
¯ be the circuit ideal. If G is a clique in
Remark 2.11. Let C be a d-uniform clutter on [n] and I = I(C)
¯
C and F ∈ C, then ([n] \ G) ∩ F 6= ∅. So that x[n]\G ∈ PF . Hence
\
PF = I ∨ .
x[n]\G ∈
F ∈C̄

Example 2.12. We show that I(Cn,d ) has linear resolution. Let ∆ be a simplex on [n]. Then, clearly
I∆ = (0) and K[∆] = K[x1 , . . . , xn ] is Cohen-Macaulay. It follows from [BH, Exercise 5.1.23] that for
any r < n, ∆(r) = hF ⊂ [n] : |F | = r + 1i is Cohen-Macaulay. Note that


∨
(r) = (x :
I∆
=
I
∆
|F | = n − (r + 1))
(r)
F

which has linear resolution by Theorem 2.6. Using this argument for r = n − d − 1, one can say,
∨
I∆
(n−d−1) = I(Cn,d ) has linear resolution.
Definition 2.13 (Simplicial submaximal circuit). Let C be a d-uniform clutter on [n]. A (d − 1)-subset
e ⊂ [n] is called a submaximal circuit of C if there exists F ∈ C such
 that e ⊂ F . The set of all submaximal
circuits of C is denoted by E(C). For e ∈ E(C), let N[e] = e ∪ c ∈ [n] : e ∪ {c} ∈ C ⊂ [n]. We say that
e is a simplicial submaximal circuit if N[e] is a clique in C. In case of 3-uniform clutters, E(C) is called
the edge set and we say simplicial edge instead of simplicial submaximal circuit.

3

Operations on Clutters

In this section, for a clutter C, we introduce some operations as changing or removing circuits which do
not change linearity of resolution of the circuit ideal. We begin this section with the following well-known
results.
Lemma 3.1. Let M be an R-module. For any submodules A, B, C of M such that B ⊂ C, one has
(A + B) ∩ C = (A ∩ C) + B.
3

(1)

Theorem 3.2 (Mayer-Vietoris sequence). For any two ideals I1 , I2 in the commutative Noetherian local
ring (R, m), the short exact sequence
0 −→

R
R
R
R
−→
⊕
−→
−→ 0
I1 ∩ I2
I1
I2
I1 + I2

gives rise to the long exact sequence



 
 



i
i
i
i
i−1
R
R
R
R
R
→ Hm
→
· · · → Hm
I1 +I2
I1 ∩I2 −→ Hm I1 ⊕ Hm I2 −→ Hm I1 +I2


i+1
R
→ Hm
I1 +I2 → · · · .
Lemma 3.3. Let I1 , I2 be ideals in a commutative Noetherian local ring (R, m) such that
depth
Then, depth I1R
∩I2 = 1 + depth

R
I1

≥ depth

R
I2

R
> depth I1 +I
.
2

R
I1 +I2 .

Proof. Let r := 1 + depth R/(I1 + I2 ). Then, for all i < r,

 
 

i−1
i
i
R
R
R
Hm
=
H
=
H
m
m
I1 +I2
I1
I2 = 0.

Hence by the Mayer-Vietoris exact sequence,
 
 




 
 
i−1 R
i−1 R
i−1
i
i
i
R
R
R
R
· · · → Hm
⊕
H
→
H
→
H
→
H
⊕
H
→ ···
m
m
m I1 ∩I2
m I1
m I2
I1
I2
I1 +I2

i
we have Hm



R
I1 ∩I2



r
= 0 for all i < r, and Hm

depth



R
I1 ∩I2



6= 0. So that

R
R
= r = 1 + depth
.
I1 ∩ I2
I1 + I2

Lemma 3.4. Let I, I1 , I2 be ideals in a commutative Noetherian local ring (R, m) such that I = I1 + I2
and
R
R
r := depth
≤ depth .
I1 ∩ I2
I2
Then, for all i < r − 1 one has
 
 
R ∼ i R
i
.
Hm
= Hm
I1
I
Proof. For i < r − 1, our assumption implies that

 



i
i+1
i
R
R
R
=
H
=
H
Hm
m
m
I1 ∩I2
I2
I1 ∩I2 = 0.
Hence, from the Mayer-Vietoris exact sequence


 
 
i
i
i
i
R
R
R
· · · −→ Hm
−→
H
⊕
H
m I1
m I2 −→ Hm
I1 ∩I2
we have

i
Hm

as desired.



R
I1



i
∼
= Hm

 
R
,
I

4

R
I



i+1
−→ Hm

for all i < r − 1,



R
I1 ∩I2



−→ · · · .

′
Notation. Let for n > 3, T1,n , T1,n
⊂ S = K[x1 , . . . , xn ] denote the ideals

T1,n =

\

(x1 , xi , xj ),

′
T1,n
=

2≤i<j≤n

\

(xi , xj ).

2≤i<j≤n

Proposition 3.5. For n ≥ 3, let S = K[x1 , . . . , xn ] be the polynomial ring, then




Q
Q
Q
Q
′
xi .
xi , . . . ,
and T1,n = x1 ,
xi
(i) T1,n =
xi , . . . ,
2≤i≤n
i6=2

(ii)

2≤i≤n
i6=2

2≤i≤n
i6=n

2≤i≤n
i6=n

S
S
(res.
) is Cohen-Macaulay of dimension n − 2 (res. n − 3).
′
T1,n
T1,n

Proof. The assertion is well-known but one can find the a direct proof of the primary decomposition of
′
in [Mo, Example 7].
the Alexander dual of T1,n
Let C be a 3-uniform clutter on the vertex set [n]. Surely, one can consider C as a 3-uniform clutter on
¯ will be changed when we consider C either on [n] or on [m].
[m] for any m ≥ n. However, C¯ (and hence I(C))
To be more precise, when we pass from [n] to [n + 1], then the new generators {xn+1 xi xj : 1 ≤ i < j ≤ n}
¯ Below, we will show that the linearity does not change when we pass from [n] to
will be added to I(C).
[m].
Lemma 3.6. Let I ⊂ K[x1 , . . . , xn ] be a square-free monomial ideal generated in degree 3 such that
x1 xi xj ∈ I for all 1 < i < j ≤ n. If J = I ∩ K[x2 , . . . , xn ], then reg (I) = reg (J).
Proof. By our assumption, J is an ideal of K[x2 , . . . , xn ] and
I = J + (x1 xi xj : 1 < i < j ≤ n).
T
So that I ∨ = J ∨ T1,n . By Remark 2.8, it is enough to show that depth S/I ∨ = depth S/J ∨ .
The ideal J ∨ is intersection of Q
some primes P , such that the set of generators of P is a 3-subset of
xi ∈ J ∨ . Hence J ∨ + T1,n = (x1 , J ∨ ) by Proposition 3.5(i). In
{x1 , . . . , xn }. So that for all j,
1<i≤n−1
i6=j

particular

depth

J∨

S
S
= depth ∨ − 1.
+ T1,n
J

(2)

S
S
By Proposition 3.5 and (2), depth T1,n
≥ depth JS∨ > depth J ∨ +T
. Hence by Lemma 3.3 and (2), we
1,n
have
S
S
S
depth ∨ = 1 + depth ∨
= depth ∨ .
I
J + T1,n
J

Theorem 3.7. Let C =
6 Cn,d be a d-uniform clutter on [n] and e be a simplicial submaximal circuit. Let

C′ = C \ e = F ∈ C : e * F
¯ J = I(C¯′ ). Then, reg (I) = reg (J).
and I = I(C),

Proof. By Remark 2.8, it is enough to show that depth S/I ∨ = depth S/J ∨ . Without loss of generality,
we may assume that e = {1, . . . , d − 1} and N [e] = {1, . . . , r}.

5

Since e = {1, . . . , d − 1} is a simplicial submaximal circuit, by Remark 2.11 and Lemma 3.1, we have:


\
I ∨ = (x1 , . . . , xd−1 , xr+1 · · · xn ) ∩
PF
F ∈C̄
{1,...,d−1}*F



= (x1 , . . . , xd−1 ) ∩

\

PF

F ∈C̄
{1,...,d−1}*F

∨

J = (x1 , . . . , xd−1 , xd · · · xn ) ∩

= (x1 , . . . , xd−1 ) ∩



\




\

+ (xr+1 · · · xn ),

PF

F ∈C̄
{1,...,d−1}*F

F ∈C̄
{1,...,d−1}*F

PF






+ (xd · · · xn ).

Since
(x1 , . . . , xd−1 ) ∩

(x1 , . . . , xd−1 ) ∩




\

PF



∩ (xr+1 · · · xn ) = (x1 xr+1 · · · xn , . . . , xd−1 xr+1 · · · xn ),

PF



∩ (xd · · · xn ) = (x1 xd · · · xn , . . . , xd−1 xd · · · xn )

F ∈C̄
{1,...,d−1}*F

\

F ∈C̄
{1,...,d−1}*F

have depth equal to n − (d − 1), by Lemma 3.4 we have:
 


 
S
S
S ∼ i
i
i
∼
 = Hm
T
Hm
= Hm
∨
I∨
J
PF
(x1 , . . . , xd−1 ) ∩

for all i < n − d.

(3)

F ∈C̄
{1,...,d−1}*F

Since dim S/I ∨ = dim S/J ∨ = n − d, the above equation implies that depth S/I ∨ = depth S/J ∨ .
For a d-uniform clutter C, if there exist only one circuit F ∈ C which contains the submaximal circuit
e ∈ E(C), then clearly e is a simplicial submaximal circuit. Hence we have the following result.
¯ be the circuit ideal. If F is the only
Corollary 3.8. Let C be a d-uniform clutter on [n] and I = I(C)
circuit containing the submaximal circuit e, then reg (I) = reg (I + xF ).
Let C be 3-uniform clutter on [n] such that {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4} ∈ C. If there exist no
other circuit which contains e = {1, 2}, then e is a simplicial edge. Hence by Theorem 3.7 we have the
following corollary.
¯ be the circuit ideal of C.
¯ Assume
Theorem 3.9. Let C be 3-uniform clutter on [n] and I = I(C)
that {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4} ∈ C and there exist no other circuit which contains {1, 2}. If
J = I + (x1 x2 x3 , x1 x2 x4 ), then reg (I) = reg (J).
E. Emtander in [E] has introduced a generalized chordal clutter to be a d-uniform clutter, obtained
inductively as follows:
• Cn,d is a generalized chordal clutter.
• If G is generalized chordal clutter, then so is C = G ∪Ci,d Cn,d for all 0 ≤ i < n.
6

• If G is generalized chordal and V ⊂ V (G) is a finite set with |V | = d and at least one element of
{F ⊂ V : |F | = d − 1} is not a subset of any element of G, then G ∪ V is generalized chordal.
Also R. Woodroofe in [W] has defined a simplicial vertex in a d-uniform clutter to be a vertex v such
the if it belongs to two circuits e1 , e2 , then, there is another circuit in (e1 ∪ e2 ) \ {v}. He calls a clutter
chordal if any minor of the clutter has a simplicial vertex.
Remark 3.10. Let C be the class of 3-uniforms clutters which can be transformed to empty set after
a sequence of deletions of simplicial edges. Using Theorem 3.7, it is clear that if C ∈ C , then the
¯ has a linear resolution over any field K. It is easy to see that generalized 3-uniform chordal
ideal I(C)
clutters are contained in this class, so they have linear resolution over any field K. This generalizes
Theorem 5.1 of [E]. It is worth to say that C contains generalized chordal clutters strictly. For example,
C = {123, 124, 134, 234, 125, 126, 156, 256} is in C but it is not a generalized chordal clutter. Also it is
easy to see that any 3-uniform clutter which is chordal in sense of [W] has simplicial edges.
Definition 3.11 (Flip). Let C be 3-uniform clutter on [n]. Assume that {1, 2, 3}, {1, 
2, 4} ∈ C are the only
′
circuits
containing
{1,
2}
and
there
is
no
circuit
in
C
containing
{3,
4}.
Let
C
=
C
∪
{1, 3, 4}, {2, 3, 4} \

′
′
{1, 2, 3}, {1, 2, 4} . Then C is called a flip of C. Clearly, if C is a flip of C, then C is a flip of C ′ too (see
the following illustration).
3

3
2

1

←→

1

2

4

4

C

C′

¯ = reg I(C¯′ ).
Corollary 3.12. Let C be 3-uniform clutter on [n] and C ′ be a flip of C. Then, reg I(C)

Proof. With the same notation as in the above definition, let C ′′ = C ∪ {1, 3, 4}, {2, 3, 4} . Theorem 3.9
applied to {3, 4}, shows that reg I(C¯′′ ) = reg I(C¯′ ). Using Theorem 3.9 again applied to {1, 2}, we conclude
¯ So that reg I(C)
¯ = reg I(C¯′ ), as desired.
that reg I(C¯′′ ) = reg I(C).
For our next theorem, we use the following lemmas.
Lemma 3.13. Let n ≥ 4, S = K[x1 , . . . , xn ] be the polynomial ring and Tn be the ideal
Tn = (x4 · · · xn , x1 x2 x3 x̂4 · · · xn , . . . , x1 x2 x3 x4 · · · x̂n ).
Then, we have:
(i) Tn = (Tn−1 ∩ (xn )) + (x1 x2 x3 x4 · · · x̂n ).
(ii) depth

S
= n − 2.
Tn

Proof. (i) This is an easy computation.
(ii) The proof is on induction over n. For n = 4, every thing is clear. Let n > 4 and (ii) be true for
n − 1.
Clearly, (Tn−1 ∩ (xn )) ∩ (x1 x2 x3 x4 · · · x̂n ) = (x1 x2 x3 x4 · · · xn ) which has depth n − 1. So by Lemma 3.4,
2.3 and induction hypothesis, we have:
depth

S
S
= depth
= n − 2.
Tn
Tn−1

7

Lemma 3.14. Let C be a 3-uniform clutter on [n] such that F = {1, 2, 3} ∈ C and for all r > 3,

{1, 2, r}, {1, 3, r}, {2, 3, r} * C.

(4)

¯ I1 = I(C¯1 ). Then,
Let C1 = C \ F and I = I(C),
(i) depth

S
S
≥ depth ∨ − 1.
I ∨ + (x1 , x2 , x3 )
I

(ii) depth

S
S
≥ depth ∨ .
I1∨
I

Proof. Let t := depth S/I ∨ ≤ dim S/I ∨ = n − 3.
(i) One can easily check that condition (4), is equivalent to say that
for all r > 3, there exist F ∈ C¯ such that PF ⊂ (x1 , x2 , x3 , xr ).
So that
I∨ =

\

F ∈C̄



PF = 



=



\

PF  ∩ ((x1 , x2 , x3 , x4 ) ∩ · · · ∩ (x1 , x2 , x3 , xn ))

\

PF  ∩ (x1 , x2 , x3 , x4 · · · xn ) = I ∨ ∩ (x1 , x2 , x3 , x4 · · · xn ).

F ∈C̄

F ∈C̄



Clearly, x4 · · · xn ∈ I ∨ . So, from the Mayer-Vietoris long exact sequence





i−1
i−1
i
i−1 S
S
S
⊕
H
→
H
· · · → Hm
∨
∨
m
m
I
(x1 ,x2 ,x3 ,x4 ···xn )
I +(x1 ,x2 ,x3 ) → Hm
we have:

i−1
Hm



S
I ∨ + (x1 , x2 , x3 )



= 0,

S
I∨



→ ···

for all i < t ≤ n − 3.

(5)

This proves inequality (i).
(ii) Clearly, I1∨ = I ∨ ∩ (x1 , x2 , x3 ). So from Mayer-Vietoris long exact sequence

 




i
i
i
i−1
S
S
S
S
→
H
→
H
⊕
H
· · · → Hm
∨
∨
∨
m
m
m
I +(x1 ,x2 ,x3 )
I
I
(x1 ,x2 ,x3 ) → · · ·
1

and (5), we have:

i
Hm



S
I1∨



= 0,

for all i < t ≤ n − 3.

Theorem
3.15. Let C be a 3-uniform clutter on [n] such that F

 = {1, 2, 3} ∈ C and for all r > 3,
{1, 2, r}, {1, 3, r}, {2, 3, r} * C. Let C1 = C \ F , C ′ = C1 ∪ {0, 1, 2}, {0, 1, 3}, {0, 2, 3} and I =
¯ J = I(C¯′ ) be the circuit ideals in the polynomial ring S = K[x0 , x1 , . . . , xn ]. Then, reg (I) = reg (J).
I(C),
1

2
0

1

2

←→
3

3

8

Proof. By Remark 2.8, it is enough to show that depth S/I ∨ = depth S/J ∨ .
Let I1 = I(C¯1 ). Clearly, I1∨ = (x1 , x2 , x3 ) ∩ I ∨ and

! 
!
n
n
\
\
\
(x0 , x2 , xi ) ∩ 
(x0 , x1 , xi ) ∩
J ∨ = I1∨ ∩
(x0 , xi , xj )
3≤i<j≤n

i=4

i=4

= (x0 , x4 · · · xn , x1 x2 x3 x̂4 · · · xn , . . . , x1 x2 x3 x4 · · · x̂n ) ∩ I1∨ .

Let T be the ideal T = (x0 , x4 · · · xn , x1 x2 x3 x̂4 · · · xn , . . . , x1 x2 x3 x4 · · · x̂n ). Then, J ∨ = I1∨ ∩ T and by
Lemma 3.13, depth TS = n − 2. Moreover, our assumption implies that for all i > 4, there exist F ∈ C¯
such that PF ⊂ (x1 , x2 , x3 , xr ). So that




\
I1∨ + T = (x0 , x4 · · · xn , I1∨ ) = (x0 ) + x4 · · · xn , (x1 , x2 , x3 ) ∩ 
PF 
F ∈C̄





= (x0 ) + (x1 , x2 , x3 , x4 ) ∩ · · · ∩ (x1 , x2 , x3 , xn ) ∩ 

\

F ∈C̄





PF  = (x0 ) + 

\

F ∈C̄



PF  = (x0 , I ∨ ).

(6)

Hence, by Lemma 3.14(ii), depth I ∨S+T = depth IS∨ − 1 ≤ depth IS∨ − 1. Thus, depth TS ≥ depth IS∨ >
1

depth I ∨S+T . Using Lemma 3.3 and (6), depth
1

S
J∨

1

= 1 + depth I ∨S+T = depth
1

S
I∨ .

1

Lemma 3.16. Let T be a hexahedron. Then, the circuit ideal of T does not have linear resolution. If T′
be the hexahedron without one circuit, then the circuit ideal of T′ has a linear resolution.
4
2

1
3
5

Proof. Let I = I(T̄). We know that T̄ = {145, 245, 345, 123}. So that
I ∨ = (x1 x2 x3 , x4 , x5 ) ∩ (x1 , x2 , x3 ) ⊂ S := K[x1 , . . . , x5 ].

S
∨
∨
1
It follows from Theorem 3.2 that Hm
I ∨ 6= 0. Since dim S/I = 5 − 3 = 2, we conclude that S/I is
not Cohen-Macaulay. So that the ideal I does not have linear resolution by Theorem 2.6.
The second part of the theorem, is a direct conclusion of Theorem 3.8.
Let S 2 be a sphere in R3 . A triangulation of S 2 is a finite simple graph embedded on S 2 such that
each face is triangular and any two faces share at most one edge. Note that if C is a triangulation of
a surface, then C defines a 3-uniform clutter which we denote this again by C. Moreover, any proper
subclutter C ′ ⊂ C has an edge e ∈ E(C ′ ) such that e is contained in only one circuit of C ′ .
Corollary 3.17. Let S = K[x1 , . . . , xn ]. Let Pn be the clutter defined by a triangulation of the sphere
with n ≥ 5 vertices, and let I ⊂ S be the circuit ideal of Pn . Then,
(i) For any proper subclutter C1 ⊂ Pn , the ideal I(C¯1 ) has a linear resolution.
(ii) S/I does not have linear resolution.
9

Proof. (i) If C1 is a proper subclutter of Pn , then C1 has an edge e such that e is contained in only one
circuit of C1 and can be deleted without changing the regularity by Corollary 3.8. Continuing this process
proves the assertion.
(ii) The proof is by induction on n, the number of vertices. First step of induction is Lemma 3.16.
Let n > 5. If there is a vertex of degree 3 (the number of edges passing through the vertex in 3),
then by Theorem 3.15, we can remove the vertex and three circuits containing it and add a new circuit
instead. Then, we have a clutter with fewer vertices and by the induction hypothesis, S/I does not have
linear resolution. Now, assume that, there is no any vertex of degree 3 and take a vertex u of degree
> 3 and all circuits containing u (see the following illustrations). Using several flips and Corollary 3.12,
we can reduce our triangulation to another one such that there are only 3 circuits containing u. Now,
using Theorem 3.15, we get a triangulation of the sphere with n − 1 vertices which does not have linear
resolution by the induction hypothesis.

··

··

·

·

2

3

4

1

5

6

··

··

·

·

7
Remark 3.18. Let Pn be the 3-uniform clutter as in Corollary 3.17. Let I be the circuit ideal of P̄n and
∆ be a cimplicial complex such that the Stanley-Reisner ideal of ∆ is I. In this case, ∆∨ , the Alexander
dual of ∆, is a pure simplicial complex of dimension n − 4 which is not Cohen-Maculay, but adding any
more facet to ∆∨ makes it Cohen-Macaulay.

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10

[Si] W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann: Singular 3-1-3 — A computer algebra
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