Projective dimension and regularity of the path ideal of the line graph †

arXiv:1610.02958v2 [] 26 Oct 2016

Guangjun Zhu
Abstract: By generalizing the notion of the path ideal of a graph, we study some
algebraic properties of some path ideals associated to a line graph. We show that the
quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some
exact formulas for the projective dimension and the regularity of these ideals. As some
consequences, we give some exact formulas for the depth of these ideals.
Keywords: projective dimension, castelnuovo-Mumford regularity, path ideal, sequentially Cohen-Macaulay, line graph.
Mathematics Subject Classifications (2010): 13D02; 13F55; 13C15; 13D99.
1. Introduction
The path ideal of a graph was first introduced by Conca and De Negri [4]. Fix an
integer m ≥ 2, and suppose that Γ is a directed graph with vertex set V = {x1 , . . . , xn }, i.e.,
each edge has been assigned a direction. A sequence of m vertices xi1 , . . . , xim is said to be
a path of length m if there are m − 1 distinct edges e1 , . . . , em−1 such that ej = (xij , xij+1 )
is a directed edge from xij to xij+1 . By identifying the vertices with the variables in the
polynomial ring R = k[x1 , . . . , xn ] over a field k, the path ideal of Γ of length m is the
monomial ideal
Jm (Γ) = ({xi1 · · · xim | xi1 , . . . , xim is a path of length m in Γ})
Note that when m = 2, then J2 (Γ) is simply the edge ideal of Γ, which is defined by
Villarreal in [17]. Other higher dimensional analogues can be found in [7, 11], among
others. The underlying theme in all correspondences is to relate the algebraic properties to
the combinatorial properties, and vice versa. We mainly study the algebraic properties of
the path ideal.
Path ideals appeared in [4] as an example of a family of monomial ideals that are generated by M -sequences. Among other things, it is shown that when Γ is a directed tree,
the Rees algebra R(Jm (Γ)) is normal and Cohen-Macaulay. The path ideals of complete
bipartite graphs are shown to be normal in [14], while the path ideals of cycles are shown
to have linear type in [3]. In [12], He and Tuyl study Jm (Γ) in the special case that Γ is the
line graph Ln . The line graph Ln is a graph with vertex set V = {x1 , . . . , xn } and directed
edges ej = (xj , xj+1 ) for j = 1, . . . , n − 1. Thus, the graph Ln has the form

†
Supported by the National Natural Science Foundation of China (11271275) and by Foundation of
Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers
and Presidents and by Foundation of the Priority Academic Program Development of Jiangsu Higher
Education Institutions.
School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China, e-mail:
zhuguangjun@suda.edu.cn

1

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They prove that R/Jm (Ln ) is sequentially Cohen-Macaulay and also provide an exact
formula for the projective dimension of Jm (Ln ) in terms of m and n. They showed that:
Theorem 1.1. (Theorem 4.1) Let p, m, n, d be integers such that n = p(m + 1) + d, where
p ≥ 0, 0 ≤ d ≤ m and 2 ≤ m ≤ n. Then the projective dimension of Jm (Ln ) is given by

2p − 1 d 6= m;
pd (Jm (Ln )) =
2p
d = m.
In [1], using purely combinatorial arguments, Alilooee and Faridi also gave the above
formula for projective dimension of Jm (Ln ). Furthermore, they gave an explicit formula
for Castelnuovo-Mumford regularity of Jm (Ln ) in terms of m and n. They showed that:
Theorem 1.2. (Corollary 4.14) Let p, m, n, d be integers such that n = p(m + 1) + d, where
p ≥ 0, 0 ≤ d ≤ m and 2 ≤ m ≤ n. Then the regularity of Jm (Ln ) is given by

p(m − 1) + 1 d 6= m;
reg (Jm (Ln )) =
p(m − 1) + m d = m.
We generalize the notion of the path ideal as the following: Let Γ be a directed graph
with vertex set V = {x1 , . . . , xn }, the path ideal of Γ of length m is the monomial ideal
Im,k (Γ) = (u1 , . . . , uk ), where u1 , . . . , uk are some paths of length m in Γ.
When u1 , . . . , uk are all paths of length m in Γ, Im,k (Γ) = Jm (Γ).
To the best of our knowledge, little is known about these ideals. It is, therefore, of interest
to determine algebraic properties of the ideals Im,k (Γ). In this article we shall focus on the
case that Γ is the line graph Ln and Im,k (Ln ) = (u1 , . . . , uk ), where for any 1 ≤ i ≤ k,
m
Q
x(i−1)(m−l)+j is a path of length m in Ln and 1 ≤ l ≤ m is an integer. we shall
ui =
j=1

abuse notation and write Im,k (Ln ) for Im,l,k . In Section 2, we study algebraic properties
of the ideal Im,l,k and show that R/Im,l,k is sequentially Cohen-Macaulay. In Section 3,
using the notion of a Betti-splitting, as defined in [9], we derive some exact formulas for the
projective dimension and regularity of the ideal Im,l,k (see Theorems 3.5, 3.7 and 3.10). As
some consequences, we give some exact formulas for the depth of these ideals.
2. Preliminaries
In this section, we will show that the ideal Im,l,k can be viewed as the facet ideal of
the simplicial complex ∆m,l,k or the edge ideal of the clutter Cm,l,k . By proving Cm,l,k
has the free vertex property, we can obtain that the quotient ring R/Im,l,k is sequentially
Cohen-Macaulay. We recall the relevant definitions.
Definition 2.1. A simplicial complex ∆ on the vertex set V is a collection of subsets of
V with the property that if F ∈ ∆ then all subsets of F are also in ∆. An element of ∆ is
called a face, the dimension of a face F is |F | − 1, and the dimension of ∆ is the largest
dimension of faces of ∆. The maximal faces of ∆ under inclusion are called facets, and the
set of facets of ∆ is denoted by Facets (∆). Simplicial complex ∆ is called pure if all of its
facets have the same dimension, otherwise ∆ is non-pure. If Facets (∆) = {F1 , . . . , Fq } we
write ∆ = hF1 , . . . , Fq i.
Definition 2.2. A clutter C on vertex set V is a family of subsets of V , called edges, none
of which is included in another. The set of vertices and edges of C are denoted by VC and
EC respectively.

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Given a clutter C = (VC , EC ), we can associate to C the simplicial complex ∆ = {F ⊆
VC | F ⊆ Ei for some Ei ∈ EC }. Conversely, given a simplicial complex ∆ with vertex set
V and set of facets Facets (∆), we can associate to ∆ the clutter C = (V, Facets (∆)).
Let I be a squarefree monomial ideal of R with minimal generators xv1 , . . . , xvq . We
use xa as an abbreviation for xa1 1 · · · xann , where a = (a1 , . . . , an ) ∈ Nn . Note that the
entries of each vi are in {0, 1}. We associate to the ideal I a clutter C by taking the set
of indeterminates VC = {x1 , . . . , xn } as the vertex set and EC = {S1 , . . . , Sq } as the edge
set, where Si = supp (xvi ) is the support of xvi , i.e., Si is the set of variables that occur
in xvi . For this reason I is called the edge ideal of C and is denoted I = I(C). Edge ideal
of a clutter is also called facet ideal because {S1 , . . . , Sq } is exactly the set of facets of the
simplicial complex ∆ = hS1 , . . . , Sq i generated by S1 , . . . , Sq .
Let ∆ be a simplicial complex and σ ∈ ∆, the deletion of σ from ∆ is the simplicial
complex defined by ∆ \ σ = {τ ∈ ∆| σ 6⊆ τ }, when σ = {x}, we shall abuse notation and
write ∆ \ x for ∆ \ {x}. If ∆ = hF1 , . . . , Fq i, the simplicial complex obtained by removing
the facet Fi from ∆ is the simplicial complex ∆ \ hFi i = hF1 , . . . , F̂i , . . . , Fq i.
The following definition of shellable is due to Björner and Wachs [2] and is usually referred
to as nonpure shellable, here we drop the adjective “nonpure”.
Definition 2.3. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered
F1 , . . . , Fs such that for all 1 ≤ i < j ≤ s, there exists some x ∈ Fj \ Fi and some
l ∈ {1, . . . , j − 1} with Fj \ Fl = {x}. We call F1 , . . . , Fs a shelling of ∆ when the facets
have been ordered with respect to the shellable definition.
If the simplicial complex ∆ is pure and satisfies the above definition of shellable, we will
say ∆ is pure shellable.
Definition 2.4. Let M be a graded R-module. M is called sequentially Cohen-Macaulay if
there exists a filtration of graded R-submodules of M
0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr = M
such that each quotient Mi /Mi−1 is Cohen-Macaulay and the Krull dimensions of the quotients are increasing, i.e., dim (M1 /M0 ) < dim (M2 /M1 ) < · · · < dim (Mr /Mr−1 ).
A simplicial complex is said to be sequentially Cohen-Macaulay if its Stanley-Reisner
ideal has a sequentially Cohen-Macaulay quotient.
It is well known that shellable implies sequentially Cohen-Macaulay.
Let C be a clutter with vertex set V . A vertex cover of C is a subset A of V that intersects
every edge of C. If A is a minimal element (under inclusion) of the set of vertex covers of
C, it is called a minimal vertex cover. To a squarefree monomial ideal I = I(C), it also
corresponds to a simplicial complex via the Stanley-Reisner correspondence [16]. We let
∆C represent this simplicial complex. Note that F is a facet of ∆C if and only if X \ F is a
minimal vertex cover of C. As for clutters, we may say that the clutter C is shellable if ∆C
is shellable.
Definition 2.5. Let I ′ ( I be two ideals of R, I ′ is called a minor of I if there is a
subset V ′ = {xi1 , . . . , xir , xj1 , . . . , xjs } of the set of variables V = {x1 , . . . , xn } such that
I ′ is a proper ideal of R′ = k[V \ V ′ ] that can be obtained from the generator set of I by
setting xik = 0 and xjl = 1 for all k, l. A minor of C is a clutter C ′ on the vertex set

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VC ′ = V \ V ′ that corresponds to a minor (0) ( I ′ ( R′ . The edges of C ′ are obtained from
I ′ by considering the unique set of squarefree monomials of R′ that minimally generate I ′ .
For use below we say xi is a free variable (resp. free vertex) of I (resp. C) if xi only appears
in one of the monomials xv1 , . . . , xvq (resp. in one of the edges of C). If all the minors of
C have free vertices, we say that C has the free vertex property. Note that if C has the free
vertex property, then so do all of its minors.
Tuyl and Villarreal in [15](also see in [18]) showed that the clutter with the free vertex
property is shellable.
Theorem 2.6. If a clutter C with the free vertex property, then ∆C is shellable.
The squarefree monomial ideal Im,l,k corresponds to a clutter (resp. simplicial complex),
say Cm,l,k (resp. ∆m,l,k ), its edges are precisely some such paths of length m in the line
graph Ln . That is, ECm,l,k = {{x1 , . . . , xm }, {x(m−l)+1 , . . . , x2(m−l)+l }, . . . ,
{x(k−1)(m−l)+1 , . . . , xk(m−l)+l }}. Throughout this paper, we will assume that the clutter
with edge set ECm,l,k = {{x1 , . . . , xm }, {x(m−l)+1 , . . . , x2(m−l)+l }, . . . , {x(k−1)(m−l)+1
, . . . , xk(m−l)+l }} where l is an integer such that 1 ≤ l ≤ m. This set corresponds to a
squarefree monomial ideal Im,l,k , which is the path ideal of the line graph Ln , i.e., Im,l,k =
I(Cm,l,k ).
Combining Definition 2.5 and Theorem 2.6, we then get the following proposition.
Proposition 2.7. Let k, l, m be positive integers, Cm,l,k be a clutter with edge set ECm,l,k =
{E1 , . . . , Ek } where Ei = {x(i−1)(m−l)+1 , x(i−1)(m−l)+2 , . . . , x(i−1)(m−l)+m } for i = 1, . . . , k
and Im,l,k = I(ECm,l,k ) be the edge ideal of the clutter Cm,l,k . Then the quotient ring R/Im,l,k
is sequentially Cohen-Macaulay.
Proof . By theorem 2.6, it is enough to prove that Cm,l,k has the free vertex property. Let
V = {xi1 , . . . , xir , xj1 , . . . , xjs } be any subset of the set of variables {x1 , . . . , xk(m−l)+l } and
c
c
c
R′ = k[x1 , . . . , xc
js , . . . , xk(m−l)+l ].
j1 , . . . , x
ir , . . . , x
i1 , . . . , x
One can assume that I ′ is an ideal of R′ minimally generated by monomials u′l1 , . . . , u′lt
with l1 < l2 < · · · < lt , and xia ∤ u′lb for any 1 ≤ a ≤ r, 1 ≤ b ≤ t, and for any
m
Q
x(lb −1)(m−l)+j by the product of all
1 ≤ b ≤ t, u′lb is obtained by dividing ulb =
j=1

the xjc such that c ∈ {1, . . . , s}. Set aj = (l1 − 1)(m − l) + j for j = 1, . . . , m and
d = min{aj | j ∈ {1, . . . , m} and aj ∈
/ {j1 , . . . , js }}. It is obvious that xd is a free variable
of I ′ and the proof is completed.
✷
3. Projective dimension and regularity of the ideal Im,l,k
In this section, we will provide some formulas for computing the projective dimension
and the regularity of Im,l,k . As some consequences, we also give some exact formulas for
the depth of Im,l,k .
Let M be a graded R-module where R = K[x1 , . . . , xn ]. Associated to M is a minimal
graded free resolution of the form
L
L
L
R(−j)β0,j (M) → M → 0, where
R(−j)βp−1,j (M) → · · · →
R(−j)βp,j (M) →
0→
j

j

j

the maps are exact, p ≤ n, and R(−j) is the R-module obtained by shifting the degrees of
R by j. The number βi,j (M ), the (i, j)-th graded Betti number of M , is an invariant of M
that equals the number of minimal generators of degree j in the ith syzygy module of M .

5

Of particular interest are the following invariants which measure the size of the minimal
graded free resolution of I. The projective dimension of I, denoted pd (I), is defined to be
pd (I) := max{i | βi,j (I) 6= 0}.
The regularity of I, denoted reg (I), is defined by
reg (I) := max{j − i | βi,j (I) 6= 0}.
We now derive some formulas for pd (Im,l,k ) and reg (Im,l,k ) in some special cases by
using some tools developed in [9]. We let G(I) denote the unique set of minimal generators
of a monomial ideal I.
Definition 3.1. Let I be a monomial ideal, and suppose that there exists monomial ideals
J and K such that G(I) is the disjoint union of G(J) and G(K). Then I = J + K is a Betti
splitting if
βi,j (I) = βi,j (J) + βi,j (K) + βi−1,j (J ∩ K) for all i, j ≥ 0,
where βi−1,j (J ∩ K) = 0 if i = 0.
This formula was first obtained for the total Betti numbers by Eliahou and Kervaire [5]
and extended to the graded case by Fatabbi [8]. In the article [9], the authors describe a
number of sufficient conditions for an ideal I to have a Betti splitting. We shall require the
following such condition.
Theorem 3.2. ([9, Corollary 2.7]). Suppose that I = J + K where G(J) contains all
the generators of I divisible by the variable xi and G(K) is a nonempty set containing the
remaining generators of I. If J has a linear resolution, then I = J + K is a Betti splitting.
When I = J + K is a Betti splitting ideal, Definition 3.1 implies the following result:
Corollary 3.3. If I = J + K is a Betti splitting, then
(i) reg (I) = max{reg (J), reg (K), reg (J ∩ K) − 1},
(ii) pd (I) = max{pd (J), pd (K), pd (J ∩ K) + 1}.
We need the following Lemma:
Lemma 3.4. Let R1 = k[x1 , . . . , xm ] and R2 = k[xm+1 , . . . , xn ] be two polynomial rings,
I ⊆ R1 and J ⊆ R2 be two nonzero homogeneous ideals. Then
(1) pd (I + J) = pd (I) + pd (J) + 1,
(2) reg (I + J) = reg (I) + reg (J) − 1,
(3) reg (IJ) = reg (I) + reg (J).
Proof . Let R = k[x1 , . . . , xn ]. Then, by Proposition 2.2.20 of [16], we have that R/I + J ∼
=
R1 /I ⊗k R2 /J. Hence we get that pd (R/I + J) = pd (R1 /I) + pd (R2 /J). It follows that
pd (I + J)

= pd (R/I + J) − 1 = pd (R1 /I) + pd (R2 /J) − 1
= (pd (I) + 1) + (pd (J) + 1) − 1 = pd (I) + pd (J) + 1,

As for the second and the third assertion, by Lemma 3.2 of [13], we obtain that reg (R/I +
J) = reg (R1 /I) + reg (R2 /J) and reg (R/IJ) = reg (R1 /I) + reg (R2 /J) + 1. Therefore, we
can conclude that
reg (I + J) =
=

reg (R/I + J) + 1 = reg (R1 /I) + reg (R2 /J) + 1
(reg (I) − 1) + (reg (J) − 1) + 1 = reg (I) + reg (J) − 1,

6

and
reg (IJ)

= reg (R/IJ) + 1 = reg (R1 /I) + reg (R2 /J) + 2
= (reg (I) − 1) + (reg (J) − 1) + 2 = reg (I) + reg (J).

We finished the proof.

✷

Now, we prove some main results of this section.
Theorem 3.5. Let k, l, m, n be integers such that n = k(m − l) + l where k ≥ 1, m ≥ 2
m
m
and l < ⌈ m
2 ⌉, here ⌈ 2 ⌉ denotes the smallest integer ≥ 2 . Let Im,l,k = (u1 , . . . , uk ) with
m
Q
x(i−1)(m−l)+j for any 1 ≤ i ≤ k. Then pd (Im,l,k ) = k − 1, reg (Im,l,k ) = (k −
ui =
j=1

1)(m − l − 1) + m.

Proof . We first claim that m − 2l − 1 ≥ 0. In fact, if m = 2s + 1, then ⌈ m
2 ⌉ = s + 1. By
⌉
−
1)
+
1
=
2s
+
1
=
m.
On
the
other hand,
the hypothesis, we have that 2l + 1 ≤ 2(⌈ m
2
m
if m = 2s, then ⌈ m
⌉
=
s.
Thus
2l
+
1
≤
2(⌈
⌉
−
1)
+
1
=
2s
−
1
<
m.
This
proves the
2
2
claim. We prove these assertions by induction on k. It is clear for k = 1. If k = 2, we
let J1 = Im,l,1 and K1 = (u2 ), which contains all the generators of Im,l,2 divisible by the
variable x2m−l . Because K1 has a linear resolution, Im,l,2 = J1 + K1 is a Betti splitting
m−l
Q
xj ). Note that J1 , K1 and J1 ∩ K1 are principal
by Theorem 3.2 and J1 ∩ K1 = K1 (
j=1

ideals, which implies that pd (J1 ) = pd (K1 ) = pd (J1 ∩ K1 ) = 0. Thus, by Corollary 3.3,
we obtain that
pd (Im,l,2 ) = max{pd (J1 ), pd (K1 ), pd (J1 ∩ K1 ) + 1} = 1.
m−l
Q

Because the variables that appear in K1 and (

xj ) are different, reg (J1 ∩K1 ) = reg (J1 )+

j=1

reg (K1 ) = m + (m − l) by Lemma 3.4. Therefore, by Corollary 3.3, we can conclude that
reg (Im,l,2 ) =
=

max{reg (J1 ), reg (K1 ), reg (J1 ∩ K1 ) − 1}
max{m, m, m + (m − l) − 1} = m + (m − l − 1).

This settles the case k = 2.
Suppose that k ≥ 3 and that the statement holds for all Im,l,t with t < k. We consider
m−l
Q
x(k−i−1)(m−l)+j ) for any 1 ≤ i ≤ k − 2. We
the ideals L0 = Im,l,k and Li = Im,l,k−i−1 + (
j=1

m−l
Q

denote Ji = Im,l,k−i for 1 ≤ i ≤ k−1, K1 = (uk ), Ki = (

x(k−i)(m−l)+j ) for 2 ≤ i ≤ k−1.

j=1

Similar to the case k = 2, we get that, for 1 ≤ i ≤ k−2, Li = Ji+1 +Ki+1 is a Betti splitting.
m−l
Q
xj ) and the
Notice that Ji ∩ Ki = Ki Li , for any 1 ≤ i ≤ k − 2, Jk−1 ∩ Kk−1 = Kk−1 (
j=1

fact that the variables that appear in Ki and Li are different and none of the variables
m−l
Q
xj , we obtain that, for 1 ≤ i ≤ k − 2,
that divide Kk−1 divide any generator of
j=1

pd (Ji ∩ Ki ) = pd (Li ) = max{pd (Ji+1 ), pd (Ki+1 ), pd (Ji+1 ∩ Ki+1 ) + 1},
reg (Li ) = max{reg (Ji+1 ), reg (Ki+1 ), reg (Ji+1 ∩ Ki+1 ) − 1},
reg (Ji ∩ Ki ) = reg (Ki Li ) = reg (Ki ) + reg (Li ) ≥ reg (Ki ) + 1,
(1)
m−l
Q
xj )) = 2(m − l).
reg (Jk−1 ∩ Kk−1 ) = reg (Kk−1 ) + reg ((
j=1

Since Jk−1 ∩ Kk−1 and Ki are principal ideals, pd (Jk−1 ∩ Kk−1 ) = pd (Ki ) = 0 for
1 ≤ i ≤ k − 1. By repeated use of the above equalities (1), the induction assumption

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pd (Ji ) = k − i − 1, reg (Ji ) = (k − i − 1)(m − l − 1) + m and m − 2l − 1 ≥ 0, we obtain that
pd (J1 ∩ K1 ) = pd (L1 ) = k − 2 and reg (J1 ∩ K1 ) = (k − 1)(m − l − 1) + m + 1. It follows
that
pd (L0 ) = max{pd (J1 ), pd (K1 ), pd (J1 ∩ K1 ) + 1}
= max{k − 2, 0, k − 1} = k − 1,
reg (L0 ) = max{reg (J1 ), reg (K1 ), reg (J1 ∩ K1 ) − 1}
= max{(k − 2)(m − l − 1) + m, m, (k − 1)(m − l − 1) + m + 1 − 1}
= (k − 1)(m − l − 1) + m,
✷
As a consequence of the above theorem, we have:
Corollary 3.6. Let k, l, m, n and Im,l,k be as in Theorem 3.5, Then
depth (Im,l,k ) = n − k + 1.
Proof . By Auslander-Buchsbaum formula, it follows that
depth (Im,l,k ) = n − pd (Im,l,k ) = n − k + 1.
✷
The following theorem generalizes Theorem 4.1 of [12] and Corollary 4.14 of [1].
Theorem 3.7. Let k, l, m, n be integers such that n = k(m − l) + l where k ≥ 1, m ≥ 2
m
Q
x(i−1)(m−l)+j for any 1 ≤ i ≤ k.
and ⌈ m
2 ⌉ ≤ l < m. Let Im,l,k = (u1 , . . . , uk ) with ui =
j=1

If m ≡ 0 (mod (m − l)) and we can write n as n = p(2m − l) + d where 0 ≤ d < 2m − l, then

2p − 1 if d 6= m;
(1) pd (Im,l,k ) =
if d = m.
 2p
p(2m − l − 2) + 1 if d 6= m;
(2) reg (Im,l,k ) =
p(2m − l − 2) + m if d = m.
Proof . Let t = 2m−l
m−l , then t > 2. In fact, if t = 2, then l = 0, contradicting the assumption
m
that l ≥ ⌈ 2 ⌉. We prove these assertions by induction on k.
The cases k = 1, 2 are from Theorem 3.5. Suppose that k ≥ 3 and that the statements
hold for all Im,l,s with s < k. If 3 ≤ k ≤ t, then n = (2m−l)+d with d = (k−2)(m−l) < m.
m−l
Q
x(k−2)(m−l)+j ). Thus
Set J1 = Im,l,k−1 and K1 = (uk ), we get that J1 ∩ K1 = K1 (
j=1

pd (J1 ∩ K1 ) = 0 and reg (J1 ∩ K1 ) = m + (m − l) = 2m − l. As the number of the variables
that appear in J1 is (2m − l) + d − (m − l), using the induction hypothesis, pd (J1 ) = 1 and
reg (J1 ) = 2m− l − 1. It follows that pd (Im,l,k ) = max{pd (J1 ), pd (K1 ), pd (J1 ∩K1 )+ 1} =
1, and reg (Im,l,k ) = max{reg (J1 ), reg (K1 ), reg (J1 ∩ K1 ) − 1} = max{2m − l − 1, m, 2m −
l − 1} = 2m − l − 1. This proves the assertion for 3 ≤ k ≤ t.
If k ≥ qt + 1 with q ≥ 1. Set J1 = Im,l,k−1 and K1 = (uk ). By similar arguments
as in Theorem 3.5, we get that Im,l,k = J1 + K1 is a Betti splitting and J1 ∩ K1 =
m−l
Q
x(k−2)(m−l)+j )). Notice that the variables that appear in K1 , Im,l,k−t
K1 (Im,l,k−t + (
j=1

8
m−l
Q

and (

x(k−2)(m−l)+j ) are different, it follows that

j=1

pd (J1 ∩ K1 ) =

m−l
Y

x(k−2)(m−l)+j ))

pd (Im,l,k−t + (

j=1

=

m−l
Y

pd (Im,l,k−t ) + pd (

x(k−2)(m−l)+j ) + 1

j=1

=

pd (Im,l,k−t ) + 1.

where the second equality follows from Lemma 3.4 (1). We distinguish three cases:
(1) If k − 1 = qt with q ≥ 1, then the numbers of the variables that appear in J1 and
Im,l,k−t are p(2m − l) + l and (p − 1)(2m − l) + m, respectively. By inductive assumption,
we get that pd (J1 ) = 2p − 1, pd (Im,l,k−t ) = 2(p − 1), reg (J1 ) = p(2m − l − 2) + 1 and
reg (Im,l,k−t ) = (p − 1)(2m − l − 2) + m. Thus
pd (J1 ∩ K1 )
pd (Im,l,k )
reg (J1 ∩ K1 )

= pd (Im,l,k−t ) + 1 = 2p − 1,
= max{pd (J1 ), pd (K1 ), pd (J1 ∩ K1 ) + 1}
= max{2p − 1, 0, (2p − 1) + 1} = 2p;
m−l
Q
x(k−2)(m−l)+j ))
= reg (K1 ) + reg (Im,l,k−t + (
j=1

m−l
Q

= reg (K1 ) + reg (Im,l,k−t ) + reg (

x(k−2)(m−l)+j ) − 1

j=1

reg (Im,l,k )

=
=
=
=
=

m + [(p − 1)(2m − l − 2) + m] + (m − l) − 1
p(2m − l − 2) + m + 1,
max{reg (J1 ), reg (K1 ), reg (J1 ∩ K1 ) − 1}
max{p(2m − l − 2) + 1, m, p(2m − l − 2) + m}
p(2m − l − 2) + m.

(2) If k − 1 = qt + 1 with q ≥ 1, then the numbers of the variables that appear in J1 and
Im,l,k−t are (p − 1)(2m − l) + m and (p − 1)(2m − l), respectively. Thus by induction, we
have that pd (J1 ) = 2(p − 1), pd (Im,l,k−t ) = 2(p − 1) − 1, reg (J1 ) = (p − 1)(2m − l − 2) + m
and reg (Im,l,k−t ) = (p − 1)(2m − l − 2) + 1. Therefore, similar to the above assertions, we
obtain that pd (Im,l,k ) = 2p − 1 and reg (Im,l,k ) = p(2m − l − 2) + 1.
(3) If k − 1 = qt + c with q ≥ 1 and 2 ≤ c < t, then the numbers of the variables that
appear in J1 and Im,l,k−t are p(2m − l) + (c − 2)(m − l) and (p − 1)(2m − l) + (c − 1)(m − l),
respectively. Thus by induction, we have that pd (J1 ) = 2p − 1, pd (Im,l,k−t ) = 2(p − 1) − 1,
reg (J1 ) = p(2m − l − 2) + 1 and reg (Im,l,k−t ) = (p − 1)(2m − l − 2) + 1. Similarly, we can
conclude that pd (Im,l,k ) = 2p − 1 and reg (Im,l,k ) = p(2m − l − 2) + 1. We completed the
proof.
✷
Remark 3.8. Theorem 4.1 of [12] and Corollary 4.14 of [1] are some corollaries of the
above theorem by specializing to the case that l = m − 1.
As another corollary, we obtain the following result:
Corollary 3.9. Let k, l, m, n and Im,l,k be as in Theorem 3.7, Then
depth (Im,l,k ) = n + 2 − ⌈

n + (m − l)
n + (m − l)
⌉−⌊
⌋.
2m − l
2m − l

Proof . Let k − 1 = qt + c, where q ≥ 0 and 0 ≤ c < t. From the proof of the theorem, we
get that if c = 0, then d = m, otherwise, d = (c − 1)(m − l). Thus, by some straightforward
n+(m−l)
computations, we have that if c = 0, then ⌈ n+(m−l)
2m−l ⌉ = ⌊ 2m−l ⌋ = p + 1, otherwise,

9
n+(m−l)
⌈ n+(m−l)
2m−l ⌉ = p + 1 and ⌊ 2m−l ⌋ = p. By Auslander-Buchsbaum formula, we obtain that
depth (Im,l,k ) = n − pd (Im,l,k ), the desired conclusion follows.
✷

Theorem 3.10. Let k, l, m, n be integers such that n = k(m − l) + l where k ≥ 1, m ≥ 2
m
Q
x(i−1)(m−l)+j for any 1 ≤ i ≤ k.
and ⌈ m
2 ⌉ ≤ l < m. Let Im,l,k = (u1 , . . . , uk ) with ui =
j=1

If m ≡ s (mod (m − l)) with 1 ≤ s < m − l and we can write n as n = p(2m − l − s) + d
where 0 ≤ d < 2m − l − s, then

2p − 1 d 6= m;
pd (Im,l,k ) =
2p
d = m.

Proof . Let t = 2m−l−s
m−l , then t > 2 by similar arguments as in Theorem 3.5. We prove
these conclusions by induction on k.
The cases k = 1, 2 are from Theorem 3.5. Suppose that k ≥ 3 and that the statements
hold for all Im,l,s with s < k. If 3 ≤ k ≤ t, then n = (2m−l−s)+d with s+(m−l) ≤ d < m.
m−l
Q
x(k−2)(m−l)+j ). Thus
Set J1 = Im,l,k−1 and K1 = (uk ), we get that J1 ∩ K1 = K1 (
j=1

pd (J1 ∩ K1 ) = 0. As the number of the variables that appear in J1 is (2m − l − s) +
d − (m − l), using the induction hypothesis, pd (J1 ) = 1. It follows that pd (Im,l,k ) =
max{pd (J1 ), pd (K1 ), pd (J1 ∩ K1 ) + 1} = 1. This proves the assertion for 3 ≤ k ≤ t.
If k ≥ qt + 1 with q ≥ 1. We consider the ideals L0 = Im,l,k , J1 = Im,l,k−1 , K1 = (uk ),
m−l
m−l
Q
Q
x[k−(q−1)t−2](m−l)+j ), and
x(k−2)(m−l)+j ), J2q = Im,l,k−qt , K2q = (
L1 = Im,l,k−t + (
j=1

j=1

for 1 ≤ i ≤ q − 1,

J2i = Im,l,k−it (Γ),
J2i+1 = Im,l,k−it−1 (Γ),
m−l
Q
x[k−(i−1)t−2](m−l)+j ),
K2i = (
j=1
(t−1)(m−l)
Q

K2i+1 = (

x(k−it−1)(m−l)+j ),

j=1

(t−1)(m−l)
Q

L2i = Im,l,k−it−1 (Γ) + (

j=1
m−l
Q

L2i+1 = Im,l,k−(i+1)t (Γ) + (

x(k−it−1)(m−l)+j ),

x(k−it−2)(m−l)+j ).

j=1

By similar arguments as in Theorem 3.5, we obtain that, for any 1 ≤ i ≤ 2q, we get that
Li = Ji+1 + Ki+1 is a Betti splitting and Ji ∩ Ki = Ki Li . Notice that the variables that
appear in Ki and Li are different, we obtain that, for any 1 ≤ i ≤ 2q − 1,
pd (Ji ∩ Ki ) = pd (Li ) = max{pd (Ji+1 ), pd (Ki+1 ), pd (Ji+1 ∩ Ki+1 ) + 1}.
There are three cases to consider:
(1) If k−1 = qt for some q ≥ 1, then n = k(m−l)+l = (qt+1)(m−l)+l = qt(m−l)+m =
q(2m − l − s) + m. By comparing this with the equality n = p(2m − l − s) + d, we have that
q = p and d = m. The numbers of the variables that appear in J1 and J2q are p(2m−l−s)+l
and m, respectively. Similarly, for any 1 ≤ i ≤ q − 1, the numbers of the variables that
appear in J2i+1 and J2i are (p − i)(2m − l − s) + l and (p − i)(2m − l − s) + m, respectively.
Hence, by inductive assumption, pd (J1 ) = 2p − 1, pd (J2q ) = 0, pd (J2i+1 ) = 2(p − i) − 1
(t−1)(m−l)
Q
and pd (J2i ) = 2(p − i) for 1 ≤ i ≤ q − 1. Note that J2q ∩ K2q = K2q (
xj ) and Ki
j=1

for 1 ≤ i ≤ 2q are principal ideals, we get that pd (J2q ∩ K2q ) = pd (Ki ) = 0. By repeated

10

use of the equality pd (Ji ∩ Ki ) = max{pd (Ji+1 ), pd (Ki+1 ), pd (Ji+1 ∩ Ki+1 ) + 1} for i =
2q − 1, 2q − 2, . . . , 1, we obtain that pd (J1 ∩ K1 ) = 2p − 1. Therefore
pd (Im,l,k ) = max{pd (J1 ), pd (K1 ), pd (J1 ∩ K1 ) + 1}
= max{2p − 1, 0, (2p − 1) + 1} = 2p.
This settles the case k − 1 = qt for some q ≥ 1.
(2) If k − 1 = qt + 1 for some q ≥ 1, then, similar to the case (1), we have that q = p + 1
and d = s. In this case, the numbers of the variables that appear in J1 and J2q are
(p − 1)(2m − l − s) + m and 1 · (2m − l − s) + s, respectively. Similarly, for any 1 ≤ i ≤ q − 1,
the numbers of the variables that appear in J2i+1 and J2i are (p − i − 1)(2m − l − s) + m and
(p − i)(2m − l − s) + s, respectively. Hence, by inductive assumption, pd (J1 ) = 2(p − 1),
pd (J2q ) = 1, pd (J2i+1 ) = 2(p − i − 1) and pd (J2i ) = 2(p − i) − 1 for 1 ≤ i ≤ q − 1. Let
(t−1)(m−l)
Q
L2q = Im,l,k−qt−1 + (
x(k−qt−1)(m−l)+j ), J2q+1 = Im,l,k−qt−1 = Im,l,1 , K2q+1 =
j=1

(t−1)(m−l)
Q

(

x(m−l)+j ), then L2q = J2q+1 +K2q+1 is a Betti splitting and J2q ∩K2q = K2q L2q .

j=1

m−l
Q

Note that J2q+1 ∩ K2q+1 = K2q+1 (

xj ) and Ki for 1 ≤ i ≤ 2q + 1 are principal ideals, we

j=1

get that pd (J2q+1 ∩ K2q+1 ) = pd (Ki ) = 0. By repeated use of the equality pd (Ji ∩ Ki ) =
max{pd (Ji+1 ), pd (Ki+1 ), pd (Ji+1 ∩ Ki+1 ) + 1} for i = 2q, 2q − 1, . . . , 1, we obtain that
pd (J1 ∩ K1 ) = 2(p − 1). Therefore
pd (Im,l,k ) = max{pd (J1 ), pd (K1 ), pd (J1 ∩ K1 ) + 1}
= max{2(p − 1), 0, 2(p − 1) + 1} = 2p − 1.
This settles the case k − 1 = qt + 1 for some q ≥ 1.
(3) If k − 1 = qt + c for some q ≥ 1 and 2 ≤ c < t, then, similar to the case (1), we have
that p = q + 1 and d = s + (c − 1)(m − l).
We claim: d 6= m. If d = m, then c − 1 = m−s
m−l = t − 1. This implies that c = t,
contradicting the assumption that c < t. This implies s+(m−l) ≤ d < m−l+(c−1)(m−l) <
t(m − l).
In this situation, the numbers of the variables that appear in J1 and J2q are p(2m − l −
s) + s + (c − 2)(m − l) and 1 · (2m − l − s) + s + (c − 1)(m − l), respectively. Similarly,
for any 1 ≤ i ≤ q − 1, the numbers of the variables that appear in J2i+1 and J2i are
(p − i)(2m − l − s) + s + (c − 2)(m − l) and (p − i)(2m − l − s) + s + (c − 1)(m − l), respectively.
Hence, by inductive assumption, pd (J1 ) = 2p−1, pd (J2q ) = 1, pd (J2i+1 ) = 2(p−i)−1 and
(t−1)(m−l)
Q
pd (J2i ) = 2(p−i)−1 for 1 ≤ i ≤ q−1. Let L2q = Im,l,k−qt−1 +(
x(k−qt−1)(m−l)+j ),
j=1

(t−1)(m−l)
Q

J2q+1 = Im,l,k−qt−1 = Im,l,c , K2q+1 = (

xc(m−l)+j ), then L2q = J2q+1 + K2q+1 is a

j=1

m−l
Q

Betti splitting, J2q ∩K2q = K2q L2q and J2q+1 ∩K2q+1 = K2q+1 (

xj ). Similar to the above

j=1

case (2), we get that pd (J2q+1 ∩ K2q+1 ) = pd (Ki ) = 0. By repeated use of the equality
pd (Ji ∩ Ki ) = max{pd (Ji+1 ), pd (Ki+1 ), pd (Ji+1 ∩ Ki+1 ) + 1} for i = 2q, 2q − 1, . . . , 1, we
can conclude that pd (J1 ∩ K1 ) = 2(p − 1). Therefore
pd (Im,l,k ) = max{pd (J1 ), pd (K1 ), pd (J1 ∩ K1 ) + 1}
= max{2p − 1, 0, 2(p − 1) + 1} = 2p − 1.
The proof is completed.

✷

11

An immediate consequence of the above theorem is the following:
Corollary 3.11. Let k, l, m, n, s and Im,l,k be as in Theorem 3.10. Then
n+m−l−s
n+m−l−s
depth (Im,l,k ) = n + 2 − ⌈
⌉−⌊
⌋.
2m − l − s
2m − l − s
Proof . Let k−1 = qt+c, where q ≥ 0 and 0 ≤ c < t. From the proof of the theorem, we get
that if c = 0, then d = m, otherwise, d = s + (c − 1)(m − l). Thus, by some straightforward
n+(m−l−s)
computations, we have that if c = 0, then ⌈ n+(m−l−s)
2m−l−s ⌉ = ⌊ 2m−l−s ⌋ = p + 1, otherwise,
n+(m−l−s)
⌈ n+(m−l−s)
2m−l−s ⌉ = p + 1 and ⌊ 2m−l−s ⌋ = p. By Auslander-Buchsbaum formula, we obtain
that depth (Im,l,k ) = n − pd (Im,l,k ), the desired conclusion follows.
✷

To conclude, we ask the following open question.
Problem 3.12. Let k, l, m, n, s and Im,l,k be as in Theorem 3.10. Does there exist some
methods to compute the regularity of the ideal Im,l,k ?

Acknowledgments
The author would like to thank Sara Saeedi Madani who point out some mistakes in
my the first version.
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