arXiv:1712.06455v2 [] 24 Dec 2017

Lyubeznik Tables of Ideals of Cycle Graphs
Parvaneh Nadi, Farhad Rahmati and Majid Eghbali
Abstract. We compute the last column of the Lyubeznik table of k[x1 , ..., xn ]/ICn .
Mathematics Subject Classification (2010). 13D45; 13N10.
Keywords. Local cohomology, Lyubeznik number, Minimal vertex cover,
Cohomological dimension, Depth.

1. Introduction
Throughout this paper all rings are commutative and Noetherian. Let A
be a local ring which admits a surjective ring homomorphism π : S → A,
where (S, η, k) is a regular local ring of dimension n containing a field. Set
I = Ker(π). In 1993, Huneke and Sharp in characteristic p > 0 [18], using the Frobenius map, and Lyubeznik in characteristic zero [23], using Dmodules, showed that the local cohomology module Hηi (HIn−j (S)) is injective
and supported at η. In these cases Hηi (HIn−j (S)) is a finite direct sum of some
(= λi,j (A) known as the i, j Lyubeznik number of A) copies of the injective
hull E(S/η) of the residue field of S. In fact the Lyubeznik number λi,j (A),
i, j ≥ 0 is i-th Bass number of the local cohomology module HIn−j (S) as
follows:
λi,j (A) := µi (η, HIn−j (S)) = dimk ExtiS (k, HIn−j (S)) = µ0 (η, Hηi (HIn−j (S))).
Thus, it depends only on A, i and j.
Lyubeznik numbers carry some geometrical and topological information
as pointed out in [23]. Moreover, note that in the case of isolated singularities,
Lyubeznik numbers can be described in terms of certain singular cohomology
groups in characteristic zero [21], or étale cohomology groups in positive
characteristic [9, 10].
Lyubeznik numbers satisfy the following properties:
i) λi,j (A) = 0 if j > d,
ii) λi,j (A) = 0 if i > j and λd,d (A)
P6= 0, i−j
(−1) λi,j (A) = 1,
iii) Euler characteristic (cf.[1]):
0≤i,j≤d

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Nadi, Rahmati and Eghbali

where d = dim(A). Therefore, we collect all nonzero Lyubeznik numbers as
follows:


λ0,0 ... λ0,d

..  ,
..
Λ(A) = 
.
. 
λd,d
in the so-called Lyubeznik table. The Lyubeznik table of A is trivial if λd,d (A) =
1 and the rest of these invariants vanish. The highest Lyubeznik number
λd,d (A) can be described as the number of connected components of the socalled Hochster-Huneke graph G [24, 28].
However not much more is known about the Lyubeznik tables except
in some partial cases. See for instance [1, 8, 11]. Furthermore in the works of
Walther [26] and Kawasaki [20] the Lyubeznik table for rings of dimension
less than 3 are understood well. In the case where I ⊂ S is a monomial ideal
the Lyubeznik numbers of S/I have been studied in [2, 3, 7, 27].
To do so further, in the present paper, we are interested in examining
the Lyubeznik table of cycle graphs. In the following, let R = k[x1 , ..., xn ] be
a polynomial ring over a field k, m denotes its homogeneous maximal ideal
(x1 , ..., xn ) and G a graph with the vertex set V (G) = {x1 , ..., xn } and edge
set E(G). One can associate with G a monomial ideal of R which is generated
by {xi xj : {xi , xj } ∈ E(G)}, called the monomial edge ideal of G denoted
by IG . (See [25] for more details). The monomial edge ideals of graphs have
been studied by several authors. (See for example [4, 5, 6, 14, 15, 16, 17]).
In section 3, we compute the last column of the Lyubeznik table of
R/ICn where ICn is the monomial edge ideal of cycle graph Cn . For this purpose, in section 2, we determine the minimal vertex cover for Cn of minimum
cardinality.

2. Minimal Vertex Covers of Cycle Graphs
Let R = k[x1 , ..., xn ] be a polynomial ring over a field k, m denotes it’s homogeneous maximal ideal (x1 , ..., xn ). Let Cn , the n-cycle graph, be a graph on
the vertex set {x1 , ..., xn } and the edge set {{x1 , x2 }, {x2 , x3 }, ..., {xn−1 , xn },
{xn , x1 }}. Then the edge ideal of Cn is the ideal
ICn = (x1 x2 , x2 x3 , ..., xn−1 xn , xn x1 ).
Definition 2.1. A subset S ⊂ V (G) is a vertex cover for G if for all {x, y} ∈
E(G), x ∈ S or y ∈ S. The subset S is called a minimal vertex cover for G if
there is no proper subset of S which is a vertex cover.
Proposition 2.2. (cf.[25, Proposition 6.1.16]) A prime ideal q of R is a minimal prime of IG if and only if q is generated by a minimal vertex cover for
G.
First of all, we establish the minimal vertex covers for Cn .

Lyubeznik Tables of Ideals of Cycle Graphs

3

Lemma 2.3. Let n be an even integer. Then the only minimal vertex covers for Cn of cardinality n/2 are S1 = {x1 , x3 , x5 , ..., xn−3 , xn−1 } and S2 =
{x2 , x4 , ..., xn−2 , xn }.
Proof. It is clear by Definition 2.1, that S1 and S2 are minimal vertex covers
for Cn of cardinality n/2. Assume, by way of contradiction, that S ′ is another
minimal vertex cover for Cn of cardinality n/2. Let
S ′ ∩ S1 = S1 − {xi1 , ..., xij }, i1 < i2 < ... < ij .
Without loss of generality, we can assume that xi1 = x1 . Then by the definition C = {x2 , xi2 −1 , xi2 +1 , ..., xij −1 , xij +1 } ⊂ S ′ and C ∩ S1 = ∅. As
i1 < i2 < ... < ij , we must have
xi2 −1 = x2 , xi3 −1 = xi2 +1 ,...,xij −1 = xij−1 +1 .
Then we have S ′ = S1 − {x1 , x3 , ..., x2j−1 } ∪ {x2 , x4 , ..., x2j }. In the case
xn ∈
/ S ′ , one has x1 , xn ∈
/ S ′ , a contradiction.
′
For xn ∈ S we have xn = x2j . Thus j = n/2 concludes that S ′ ∩ S1 = ∅
and so S ′ = S2 which contradicts our assumption.

Example. Put n = 6. Then the ideal IC6 in the ring R can be retuned as
IC6 =
(x1 , x3 , x5 ) ∩ (x2 , x4 , x6 ) ∩ (x1 , x2 , x4 , x5 ) ∩ (x2 , x3 , x5 , x6 ) ∩ (x3 , x4 , x6 , x1 ).
So by Proposition 2.2, the minimal vertex covers for C6 of cardinality 3 are
S1 = {x1 , x3 , x5 }, S2 = {x2 , x4 , x6 }.
Lemma 2.4. Let s be a positive integer and n = 2s + 1, then the number of
the minimal vertex covers for Cn of cardinality s + 1 is n.
Proof. By definition the sets
Si = {x2r+1 : i − 1 ≤ r ≤ s} ∪ {x2r : 1 ≤ r ≤ i − 1},
for all 1 ≤ i ≤ s + 1 and
Rj = {x2r+1 : 0 ≤ r ≤ j − 1} ∪ {x2r : j ≤ r ≤ s},
for all 1 ≤ j are minimal vertex covers for Cn of cardinality s + 1. Each of
these sets contains two vertices of an edge of Cn . By way of contradiction,
let S ′ be another minimal vertex cover for Cn of cardinality s + 1. First
suppose that S ′ ∩ S1 = ∅. Since S ′ ⊂ S1 ∪ {x2 , x4 , ..., xn−3 , xn−1 }, we get
S ′ ⊂ {x2 , x4 , ..., xn−3 , xn−1 }. Thus s + 1 = |S ′ | ≤ s, which is a contradiction.
Next let S ′ ∩ S1 = S1 − {xi1 , ..., xij }, i1 < i2 < ... < ij . By the similar
argument as in the proof of Lemma 2.3, we have
xi1 = x1 , xi2 = x3 , ..., xij = x2j−1 ,
/ S ′ . Let xij = xn−t for some even integer t. Now
and xij 6= xn since x1 ∈
S ′ = {x2 , x4 , ..., xn−t+1 , xn−t+2 , ..., xn },

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Nadi, Rahmati and Eghbali

which there is only one edge of Cn with both of its vertices in S ′ . Hence
S ′ = Si for some 1 ≤ i ≤ s+1 or S ′ = Rj for some 1 ≤ j ≤ s, a contradiction.
So we get the desired conclusion.

Example. Put n = 7. Then the ideal IC7 in the ring R can be returned as
IC6 = (x1 , x3 , x5 , x7 ) ∩ (x2 , x3 , x5 , x7 ) ∩ (x2 , x4 , x5 , x7 ) ∩ (x2 , x4 , x6 , x7 ) ∩
(x1 , x2 , x4 , x6 ) ∩ (x1 , x3 , x4 , x6 ) ∩ (x1 , x3 , x5 , x6 ).
So by Proposition 2.2, the number of the minimal vertex covers for C7 of
cardinality 4 is 7.
Proposition 2.5. Let r be a positive integer. Let
Pj = (xj , xj+1 , ..., xj+h−1 ),
for 1 ≤ j ≤ r, be a monomial prime ideal of R of height h ≥ r. Let I =
∩rj=1 Pj , then
depth(R/I) = n − h.
Proof. We proceed by induction on r. Let r = 2. Then the Depth Lemma [25,
Lemma 1.3.9], implies depth(R/P1 ⊕ R/P2 ) = n − h, since depth(R/P1 ) =
depth(R/P2 ) = n − h.
Moreover, we know that P1 + P2 = (x1 , ..., xh , xh+1 ) and depth(R/(P1 +
P2 )) = n − h − 1 < n − h. By applying the Depth Lemma to the following
exact sequence
0 → R/(P1 ∩ P2 ) → R/P1 ⊕ R/P2 → R/(P1 + P2 ) → 0,
we have
depth(R/(P1 ∩ P2 )) = depth(R/(P1 + P2 )) + 1 = n − h.
Now, suppose that the statement is true for r − 1. We show that it is true
for r. Let J = ∩r−1
j=1 Pj , by using induction hypothesis, depth(R/J) = n − h.
It should be noted that Pj+1 + Pr ⊂ Pj + Pr , for every 1 ≤ j ≤ r − 2 because
{xj+1 , ..., xj+h−1 } ⊂ {xj , ..., xj+h−1 },
and xj+h ∈ Pr since r + j ≤ h + j < r + h − 1. On the other hand J + Pr =
∩r−1
j=1 (Pj + Pr ), by [19, Lemma 2.7]. Therefore J + Pr = Pr−1 + Pr . Since
Pr , Pr−1 differ by one generator, we get
depth(R/(J + Pr )) = n − h − 1.
Once again using the Depth Lemma to exact sequence,
0 → R/(Pr ∩ J) → R/Pr ⊕ R/J → R/(Pr + J) → 0,
we prove the claim.



Proposition 2.6. Keep the notations of Lemma 2.4. Let I1 = ∩s+1
i=1 pi , I2 =
∩sj=1 qj where pi , qj are the prime ideals of R generated by Si and Rj respectively. Then
I1 + I2 = (p1 + qs ) ∩ (q1 + ps+1 ).

Lyubeznik Tables of Ideals of Cycle Graphs

5

s
Proof. By [19, Lemma 2.7], we may have I1 + I2 = ∩s+1
i=1 pi + ∩j=1 qj =
s+1 s
∩i=1 ∩j=1 (pi + qj ). So it is enough to show that for all i, j, p1 + qs ⊆ pi + qj
or q1 + ps+1 ⊆ pi + qj . For the convenience of the reader we sort them below:
p1 = ({x2r+1 : 0 ≤ r ≤ s}),
q1 = ({x2r : 1 ≤ r ≤ s} ∪ {x1 }),
ps+1 = ({x2s+1 } ∪ {x2r : 1 ≤ r ≤ s}),
qs = ({x2r+1 : 0 ≤ r ≤ s − 1} ∪ {x2s }),
qj = ({x2r+1 : 0 ≤ r ≤ j − 1} ∪ {x2r : j ≤ r ≤ s}),
pi = ({x2r+1 : i − 1 ≤ r ≤ s} ∪ {x2r : 1 ≤ r ≤ i − 1}).
Let i < j. Then we show that p1 + qs ⊆ pi + qj . It is clear to see that

S1 − {x2r+1 : 0 ≤ r ≤ i − 1} ⊆ Si and {x2r+1 : 0 ≤ r ≤ i − 1} ⊆ Rj .
So p1 ⊆ pi +qj . Also Rs −{x2r } ⊆ S1 and x2r ∈ Rj therefore, p1 +qk ⊆ pi +qj .
For i ≥ j, q1 + ps+1 ⊆ pi + qj , as
Ss+1 − {x2r : i − 1 ≤ r ≤ s} ⊆ Si and {x2r : i − 1 ≤ r ≤ s} ⊆ Rj .
Hence ps+1 ⊆ pi + qj . Also q1 ⊆ pi + qj since R1 − {x1 } ⊆ Ss+1 and x1 ∈
Rj .

Example. Let n = 7. Using the same notation in Proposition 2.6 we can see
that IC7 = I1 ∩ I2 . Therefore, by the Proposition 2.5 and 2.6
depth(R/I1 ) = depth(R/I2 ) = 3
and
I1 + I2 = (x1 , x3 , x5 , x6 , x7 ) ∩ (x1 , x2 , x4 , x6 , x7 ).

3. Lyubeznik Numbers
We assume as in the previous section that R = k[x1 , ..., xn ] is a polynomial
ring over a field k, m denotes it’s homogeneous maximal ideal (x1 , ..., xn ).
The aim of this section is to describe the last column of the Lyubeznik table
of R/I, where I is the monomial edge ideal of the n-cycle graph Cn . It follows
from the definition, whenever n is an even integer, the minimum cardinality
of minimal vertex covers for Cn is n/2 and by Lemma 2.3, there exist only
two sets of this cardinality.
The next theorem is one of our main results:
Theorem 3.1. Let n be an even integer. Then the last column of the Lyubeznik
table of R/I is as follows:


0
 .. 
 . 

.
 0 
2
Proof. By Proposition 2.2, we may write
I = (x1 , x3 , x5 , ..., xn−3 , xn−1 ) ∩ (x2 , x4 , ..., xn−2 , xn ) ∩ J.

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Nadi, Rahmati and Eghbali

The dimension of R/J is (n/2) − 1, as {x1 , x2 , x4 , x5 , ..., xn−3 , xn−1 } is a
n/2
minimal vertex cover for Cn of cardinality (n/2)+1, it follows that HJ (R) =
0. Put
J ′ = (P1 = (x1 , x3 , x5 , ..., xn−3 , xn−1 )) ∩ (P2 = (x2 , x4 , ..., xn−2 , xn )).
As J, J ′ do not share any minimal primes, dim(R/(J + J ′ )) < dim(R/J) so
(n/2)+1
n/2
ht(J + J ′ ) > n/2 + 1 and consequently HJ+J ′ (R) = HJ+J ′ (R) = 0. By
virtue of the Mayer-Vietoris sequences
n/2

n/2

n/2

n/2

n/2

n/2

n/2

(n/2)+1

0 → HJ+J ′ (R) → HJ (R) ⊕ HJ ′ (R) → HJ∩J ′ (R) → HJ+J ′

(R) → ...

and
n/2

(n/2)+1

0 → HP1 +P2 (R) → HP1 (R) ⊕ HP2 (R) → HP1 ∩P2 (R) → HP1 +P2 (R) → ...
one has
n/2

HI

n/2

n/2

n/2

(R) ∼
= HJ ′ (R) ∼
= HP1 (R) ⊕ HP2 (R).

i
(−) to the above isomorphism, and taking account that,
By applying Hm
i
(−) commutes with direct sum, we observe that
Hm

λi,n/2 (R/I) = λi,n/2 (R/P1 ) + λi,n/2 (R/P2 ), for i ≥ 0.
As P1 and P2 are Cohen-Macaulay square-free monomial ideals, they have
trivial Lyubeznik tables [1]. Hence the last column of the Lyubeznik table of
R/I is as claimed.

Example. Consider IC4 = (P1 = (x1 , x3 )) ∩ (P2 = (x2 , x4 )) ⊂ R. Then we get

0 → HP21 +P2 (R) → HP21 (R) ⊕ HP22 (R) → HP21 ∩P2 (R) → HP31 +P2 (R) → 0 →
HP31 ∩P2 (R) → HP41 +P2 (R) → 0.
It follows that HP21 (R)⊕HP22 (R) ∼
= HP21 ∩P2 (R) and HP41 +P2 (R) ∼
= HP31 ∩P2 (R).
So the Lyubeznik table of R/IC4 has the form


0 1 0

0 0 .
2

Remark 3.2. Note that for n = 3, 5 we have IC3 = (x1 , x3 ) ∩ (x3 , x2 ) ∩
(x2 , x1 ) and IC5 = (x1 , x3 , x5 ) ∩ (x3 , x5 , x2 ) ∩ (x5 , x2 , x4 ) ∩ (x2 , x4 , x1 ) ∩
(x4 , x1 , x3 ). So by Propositions 2.5 and 2.6, depth(R/IC3 ) = dim(R/IC3 ) = 1
and depth(R/IC5 ) = dim(R/IC5 ) = 2. Thus IC3 and IC5 have the trivial
Lyubeznik tables. Hence we compute the last column of Lyubeznik table of
R/I for n = 2s + 1 where s ≥ 3.
Now is the time to determine the last column of the Lyubeznik table of
R/I for odd values of n.

Lyubeznik Tables of Ideals of Cycle Graphs

7

Theorem 3.3. Let s ≥ 3 be a positive integer and n = 2s+1. Then λs,s (R/I) =
λs−1,s (R/I) = 1 and the last column of Lyubeznik table of R/I is


0
 .. 
 . 


 0 .


 1 
1
Proof. By Lemma 2.4, the minimal vertex covers for Cn are Si , Rj , for all
1 ≤ i ≤ s + 1 , 1 ≤ j ≤ s. So by Proposition 2.2, we can write
s
I = (∩s+1
i=1 pi ) ∩ (∩j=1 qj ) ∩ J,

where pi and qj are the prime ideals generated by Si and Rj respectively. The
dimension of R/J is s − 1, because {x1 , x2 , x4 , x5 , ..., xn−2 , xn } is a minimal
vertex cover for Cn of cardinality s + 2. It follows that HJs+1 (R) = 0. Put
s
J ′ = (∩s+1
i=1 pi ) ∩ (∩j=1 qj ),

then dim(R/(J + J ′ )) < dim(R/J), as J, J ′ do not share any minimal primes.
s+1
s+2
For that reason HJ+J
′ (R) = HJ+J ′ (R) = 0. So by the following MayerVietoris sequence
s+1
s+1
s+1
s+2
0 → HJ+J
(R) ⊕ HJs+1
(R) → HJ∩J
′ (R) → HJ
′
′ (R) → HJ+J ′ (R) → ...

we have
HIs+1 (R) ∼
(R).
= HJs+1
′
i
By applying Hm
(−) to the above isomorphism we get

λi,s (R/I) = λi,s (R/J ′ ), for i ≥ 0.
Therefore, it is enough to obtain the last column of the Lyubeznik table of
s
R/J ′ . By using Proposition 2.5, for I1 = ∩s+1
i=1 pi and I2 = ∩j=1 qj one has
depth(R/I1 ) = depth(R/I2 ) = s = dim(R/I1 ) = dim(R/I2 ),
Furthermore, by virtue of [22] and [13, Corollary 4.2]
ht(I1 ) = ht(I2 ) = cd(R, I1 ) = cd(R, I2 ) = s + 1.
From Proposition 2.6, we observe that
I1 + I2 = (Q1 = (p1 + qs )) ∩ (Q2 = (q1 + ps+1 )),

√
Q1 + Q2 = m.

The exact sequence
s+3
s+2
s+2
s+2
s+2
(R) →
(R) → HQ
(R) → HQ
(R) ⊕ HQ
(R) → HQ
0 → HQ
1 +Q2
1 ∩Q2
2
1
1 +Q2
s+3
2s+1
s+3
s+3
2s
(R)
→0
(R)
→
H
HQ1 (R) ⊕ HQ2 (R) → HQ1 ∩Q2 (R) → ... → HQ
Q1 +Q2
1 ∩Q2

associated to Q1 , Q2 implies that

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Nadi, Rahmati and Eghbali
s+2
s+2
2s+1
HIs+2
(R) ∼
(R) ⊕ HQ
(R), HI2s
(R) ∼
(R).
= HQ
= Hm
1 +I2
1 +I2
1
2

i
Once again applying Hm
(−) to the above isomorphisms one has

λi,s−1 (R/(I1 + I2 )) = λi,s−1 (R/Q1 ) + λi,s−1 (R/Q2 ), for i ≥ 0
and
λi,1 (R/(I1 + I2 )) = λi,0 (R/m), for i ≥ 0.
As Q1 and Q2 are the Cohen-Macaulay
monomial ideals,
they have trivial


Lyubeznik tables. Also Λ(R/m) = λ0,0 (R/m) = 1 . Thus the Lyubeznik
table of R/(I1 + I2 ) is as follows:


0 1 ··· 0

.
 0 . . . .. 
.
Λ(R/(I1 + I2 )) = 


..

. 0
2
Using the Mayer-Vietoris sequence associated to I1 , I2 ,
(R) →
(R) → HIs+2
(R) → HIs+1
(R) ⊕ HIs+1
(R) → HIs+1
0 → HIs+1
1 +I2
1 ∩I2
2
1
1 +I2
s+3
s+3
s+3
s+2
s+2
(R)
⊕
H
(R)
→
H
(R)
→
H
(R)
→
H
(R)
⊕
H
HIs+2
I2 (R) →
I1
I1 +I2
I1 ∩I2
I2
1
n−1
n−2
(R)
→
0,
(R)
→
H
(R)
→
...
→
H
HIs+3
I1 +I2
I1 ∩I2
1 ∩I2
concludes that
∀2 ≤ i ≤ s − 1, HIs+i+1
(R).
(R) ∼
= HIs+i
1 ∩I2
1 +I2
So HI2s
(R), as a result λ0,2 (R/J ′ ) = 1.
(R) ∼
= HI2s−1
1 +I2
1 ∩I2
On the other hand by [12, Theorem 2.6], λ0,2 (R/J ′ ) = λs−1,s (R/J ′ ) =
1. Finally by the Euler characteristic formula, λs,s (R/J ′ ) = 1. Then the
Lyubeznik table of R/J ′ is


0 0 1 ··· 0

..
.. 
 0 ...
.
.
′
.

Λ(R/J ) = 

..

.
1
1
Thus the last column of the Lyubeznik table of R/I has the desired form. 
Example. Let I1 , I2 be as Example 2. So
depth(R/I1 ) = depth(R/I2 ) = dim(R/I1 ) = dim(R/I2 ) = 3.
It implies that
cd(R, I1 ) = cd(R, I2 ) = ht(I1 ) = ht(I2 ) = 4.

Lyubeznik Tables of Ideals of Cycle Graphs

9

Also I1 + I2 = (x1 , x3 , x5 , x6 , x7 ) ∩ (x1 , x2 , x4 , x6 , x7 ). Using Theorem 3.3, we
conclude that the Lyubeznik tables of R/(I1 + I2 ) and R/I have the forms


0 1 0
0 0 ,
Λ(R/(I1 + I2 )) = 
2


0 0 1 0

0 0 0
.
Λ(R/I) = 

0 1
1

Acknowledgments
The first author would like to thank Professor Dariush Kiani for some helpful
discussions.

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Parvaneh Nadi
1
Department of Mathematics and Computer Science, Amirkabir University of Technology
424 Hafez Av, Tehran, 1591634311
Iran
e-mail: nadi− p@aut.ac.ir
Farhad Rahmati
1
Department of Mathematics and Computer Science, Amirkabir University of Technology
424 Hafez Av, Tehran, 1591634311
Iran
e-mail: frahmati@aut.ac.ir

Lyubeznik Tables of Ideals of Cycle Graphs

11

Majid Eghbali
2
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)
P. O. Box: Tehran 19395-5746
Iran
e-mail: m.eghbali@yahoo.com

