arXiv:1405.5450v2 [] 30 May 2014

TWO LOWER BOUNDS FOR THE STANLEY DEPTH OF
MONOMIAL IDEALS
L. KATTHÄN AND S. A. SEYED FAKHARI
Abstract. Let J I be two monomial ideals of the polynomial ring S = K[x1 , . . . , xn ].
In this paper, we provide two lower bounds for the Stanley depth of I/J. On the
one hand, we introduce the notion of lcm number of I/J, denoted by l(I/J), and
prove that the inequality sdepth(I/J) ≥ n − l(I/J) + 1 hold. On the other hand, we
show that sdepth(I/J) ≥ n−dim LI/J , where dim LI/J denotes the order dimension
of the lcm lattice of I/J. We show that I and S/I satisfy Stanley’s conjecture, if
either the lcm number of I or the order dimension of the lcm lattice of I is small
enough. Among other results, we also prove that the Stanley–Reisner ideal of a
vertex decomposable simplicial complex satisfies Stanley’s conjecture.

1. Introduction
Let K be a field and S = K[x1 , . . . , xn ] be the polynomial ring in n variables over
the field K. Let M be a nonzero finitely generated Zn -graded S-module. Let u ∈ M
be a homogeneous element and Z ⊆ {x1 , . . . , xn }. The K-subspace uK[Z] generated
by all elements uv with v ∈ K[Z] is called a Stanley space of dimension |Z|, if it
is a free K[Z]-module. Here, as usual, |Z| denotes the number of elements of Z.
A decomposition D of M as a finite direct sum of Stanley spaces is called a Stanley
decomposition of M. The minimum dimension of a Stanley space in D is called Stanley
depth of D and is denoted by sdepth(D). The quantity

sdepth(M) := max sdepth(D) | D is a Stanley decomposition of M
is called Stanley depth of M. Stanley [19] conjectured that
depth(M) ≤ sdepth(M)
for all Zn -graded S-modules M. For a reader friendly introduction to Stanley decomposition, we refer to [15] and for a nice survey on this topic we refer to [5]. In this
paper we prove Stanley’s conjecture for some classes of monomial ideals.
Before stating the main results of this paper, we mention that for the monomials
u1 , . . . , uk ∈ S, we denote their least common multiple by lcm(u1, u2 , . . . , uk ). Also,
2000 Mathematics Subject Classification. Primary: 13C15, 05E99; Secondary: 13C13.
Key words and phrases. Monomial ideal, Stanley depth, lcm number, lcm lattice, Order dimension,
Simplicial complex.
The first author was partially supported by the German Research Council DFG-GRK 1916. The
second author was in part supported by a grant from IPM (No. 93130422).
1

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L. KATTHÄN AND S. A. SEYED FAKHARI

for a monomial ideal I, we denote by G(I) the set of minimal monomial generators
of I.
Definition 1.1. Let J I ⊆ S = K[x1 , . . . , xn ] be monomial ideals. The lcm number
of I/J, denoted by l(I/J) is the maximum integer t, for which there exist monomials
u1 , . . . , ut ∈ G(I) ∪ G(J) such that
u1 6= lcm(u1 , u2) 6= . . . 6= lcm(u1 , u2, . . . , ut ).
Remark 1.2. We mention that the lcm number of monomial ideals was first considered by Terai to determine an upper bound for the arithmetical rank of squarefree
monomial ideals (see [11, Corollary 4]).
Let J
I be two monomial ideals. In Section 2, we determine lower bounds for
the Stanley depth of I/J. More explicit, we prove that sdepth(I/J) ≥ n − l(I/J) + 1.
This, in particular, implies that
sdepth(I) ≥ n − l(I) + 1 and sdepth(S/I) ≥ n − l(I).
Definition 1.3. Let J I ⊆ S = K[x1 , . . . , xn ] be monomial ideals. The lcm lattice
of I/J, denoted by LI/J , is the set of all least common multiples of non-empty subsets
of G(I) ∪ G(J), ordered by divisibility and augmented with an additional minimal
element 0̂. Moreover, we set LI := LI/0 .
Remark 1.4. The lcm-lattice of a monomial ideal was introduced by Gasharov, Peeva
and Welker in [3]. Note that the lcm number l(I/J) is length of LI/J , i.e. one less
than the maximal number of elements of a maximal chain in LI/J .
Definitions 1.5.
(1) Let P and P ′ be finite posets. An embedding is a map
′
j : P → P between two posets such that p ≤ q if and only if j(p) ≤ j(q) for
p, q ∈ P .
(2) The order dimension of a poset, dim P , is the minimal d ∈ N, such that there
exists an embedding P → Nd .
Note that an embedding is necessarily injective and monotonic. Even if P and P ′
are lattices we do not require an embedding to respect the join. We refer the reader
to [20] for background information about the dimension of posets.
Let J
I be two monomial ideals. In Section 3, we give a lower bound for the
Stanley depth of I/J. Namely, we prove that sdepth(I/J) ≥ n − dim LI/J and that
sdepth(I) ≥ n − dim LI + 1.
Remark 1.6. Both lower bounds for the Stanley depth are known to be bounds for
the usual depth, in the case I = S. Indeed, let J ⊂ S be a monomial ideal. By [3]
the projective dimension of S/J can be computed from the homology of the order
complex of lower intervals in LJ . It is easy to see that the dimension of these order
complexes is bound above by the lcm number l(S/J). Hence [3, Theorem 2.1] implies
that
depth(S/J) ≥ n − l(S/J)

TWO LOWER BOUNDS FOR THE STANLEY DEPTH

3

Moreover, it follows easily from Theorem 1 of [17] that
depth(S/J) ≥ n − dim LJ .
We provide proofs of both bounds on the general case below.
In Section 4, we show that the Stanley–Reisner ideal of a vertex decomposable
simplicial complex satisfies Stanley’s conjecture (see Theorem 4.2). Using this result
and the above inequalities, we prove that I and S/I satisfy Stanley’s conjecture
provided that l(I) ≤ 3 or dim LI ≤ 3. (see Theorem 4.4).
2. A lower bound for the Stanley depth
Let J
I be two monomial ideals. In this section, we prove the first main result
of this paper. Indeed, in Theorem 2.4, we determine a lower bound for the Stanley
depth of I/J. In [18], the author provides linear algebraic lower bonds for the Stanley
depth of I and the Stanley depth of S/I, where I is squarefree monomial ideal. The
bound which will be proven in Theorem 2.4 is stronger than these mentioned lower
bounds, given in [18]. On the other hand, we do not focus on squarefree monomial
ideals and consider a general monomial ideal.
To prove the main result, we need a couple of lemmas. The following lemma shows
that the lcm number of a monomial ideal does not increase under the colon operation
with respect to an arbitrary variable.
Lemma 2.1. Let J I be two monomial ideals of S. Then for every 1 ≤ i ≤ n, we
have l((I : xi )/(J : xi )) ≤ l(I/J).
Proof. Assume without loss of generality that i = 1. We note that
u
(I : x1 ) = h
: u ∈ G(I)i
gcd(u, x1 )
and
u
(J : x1 ) = h
: u ∈ G(J)i,
gcd(u, x1 )
where gcd(u, x1 ) denotes the greatest common divisor of u and x1 . Set l((I : xi )/(J :
xi )) = t and suppose that v1 , . . . , vt are monomials in G((I : x1 )) ∪ G((J : x1 )) such
that
v1 6= lcm(v1 , v2 ) 6= . . . 6= lcm(v1 , v2 , . . . , vt ).
For every 1 ≤ j ≤ t, set uj = vj , if vj ∈ G(I)∪G(J) and uj = x1 vj if vj ∈
/ G(I)∪G(J).
It is clear that in both cases uj ∈ G(I) ∪ G(J). We claim that for every 1 ≤ k ≤ t − 1,
lcm(u1 , u2 , . . . , uk ) 6= lcm(u1, u2 , . . . , uk+1).
Indeed, if vj ∈ G(I) ∪ G(J), for every 1 ≤ j ≤ k, then vj = uj , for every 1 ≤ j ≤ k
and thus
lcm(u1, u2 , . . . , uk ) = lcm(v1 , v2 , . . . , vk ) 6= lcm(v1 , v2 , . . . , vk+1 )
and since lcm(v1 , v2 , . . . , vk+1 ) divides lcm(u1, u2 , . . . , uk+1), it follows that
lcm(u1 , u2 , . . . , uk ) 6= lcm(u1, u2 , . . . , uk+1).

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L. KATTHÄN AND S. A. SEYED FAKHARI

Now assume that vj ∈
/ G(I) ∪ G(J), for some 1 ≤ j ≤ k. Then uj = x1 vj and hence
lcm(u1 , u2 , . . . , uk ) = x1 lcm(v1 , v2 , . . . , vk ) 6=
x1 lcm(v1 , v2 , . . . , vk+1 ) = lcm(u1, u2 , . . . , uk+1).
This proves the claim and shows that
u1 6= lcm(u1 , u2) 6= . . . 6= lcm(u1 , u2, . . . , ut ).
Therefore, l(I) ≥ t.



In the following lemma, we consider the behavior of the lcm number of monomial
ideals under the elimination of a variable. As usual, for every monomial u, the support
of u, denoted by Supp(u), is the set of variables which divide u.
Lemma 2.2. Let J

I be two monomial ideals of S = K[x1 , . . . , xn ], such that
[
x1 ∈
Supp(u).
u∈G(I)∪G(J)

Let S ′ = K[x2 , . . . , xn ] be the polynomial ring obtained from S by deleting the variable
x1 and consider the ideals I ′ = I ∩ S ′ and J ′ = J ∩ S ′ . Then l(I ′ /J ′ ) + 1 ≤ l(I/J).
Proof. Assume that l(I ′ /J ′ ) = t. Suppose that u1 , . . . , ut are monomials in G(I ′ ) ∪
G(J ′ ) such that
u1 6= lcm(u1 , u2) 6= . . . 6= lcm(u1 , u2, . . . , ut ).
It is obvious that uj ∈ G(I) ∪ G(J), for every 1 ≤ j ≤ t. By assumption, there exists
a monomial, say ut+1 ∈ G(I) ∪ G(J), such that x1 divides ut+1 . Since u1 , . . . , ut do
not divide x1 , it follows that for every 1 ≤ j ≤ t, ut+1 6= uj and
u1 6= lcm(u1, u2 ) 6= . . . 6= lcm(u1 , u2, . . . , ut ) 6= lcm(u1 , u2 . . . , ut , ut+1 ).
This shows that l(I/J) ≥ t + 1.



Remark 2.3. It is completely clear from the proof of the Lemma 2.2, that one can
consider any arbitrary variable instead of x1 .
In the following theorem we determine a lower bound for the Stanley depth of I/J.
We believe this bound is known to be a lower bound for depth also. But we did not
find a reference and hence for the sake of completeness we provide a proof.
Theorem 2.4. Let J
I be two monomial ideals of S = K[x1 , . . . , xn ]. Then
depth(I/J) ≥ n − l(I/J) + 1 and sdepth(I/J) ≥ n − l(I/J) + 1.
Proof. We prove the assertions by induction on n and
X
deg(u).
u∈G(I)∪G(J)

The assertion can be checked easily, when n = 1 or
X
deg(u) = 1.
u∈G(I)∪G(J)

TWO LOWER BOUNDS FOR THE STANLEY DEPTH

5

We now assume that n ≥ 2 and
X

deg(u) ≥ 2.

u∈G(I)∪G(J)
′

Let S = K[x2 , . . . , xn ] be the polynomial ring obtained from S by deleting the variable
x1 and consider the ideals I ′ = I ∩ S ′ , J ′ = J ∩ S ′ , I ′′ = (I : x1 ) and J ′′ = (J : x1 ). If
[
x1 ∈
/
Supp(u),
u∈G(I)∪G(J)

then trivially depth(I/J) = depthS ′ (I ′ /J ′ ) + 1 and by [8, Lemma 3.6], we
clude that sdepth(I/J) = sdepthS ′ (I ′ /J ′ ) + 1. On the other hand it is clear
l(I/J) = l(I ′ /J ′ ). Therefore, using the induction hypothesis on n we conclude
depth(I/J) ≥ n − l(I/J) + 1 and sdepth(I/J) ≥ n − l(I/J) + 1. Therefore, we
assume that
[
x1 ∈
Supp(u).
′

′

′

′

′′

conthat
that
may

u∈G(I)∪G(J)
′′

Now I/J = (I S /J S ) ⊕ x1 (I S/J S) and therefore by definition of the Stanley
depth we have
(1)

sdepth(I/J) ≥ min{sdepthS ′ (I ′ S ′ /J ′ S), sdepthS (I ′′ /J ′′ )},

On the other hand, by applying the depth lemma on the exact sequence
.x

1
0 −→ I ′′ /J ′′ −→
I/J −→ I/(x1 I ′′ + J) −→ 0

we conclude that
(2)

depth(I/J) ≥ min{depthS (I ′′ /J ′′ ), depthS (I/(x1 I ′′ + J)).

We note that every I ′ S ′ /J ′ S ′ -regular sequence in S ′ is also a regular sequence for
I/(x1 I ′′ + J). This shows that depthS (I/(x1 I ′′ + J)) ≥ sdepthS ′ (I ′ S ′ /J ′ S). Hence it
follows from inequality (2) that
(3)

sdepth(I/J) ≥ min{sdepthS ′ (I ′ S ′ /J ′ S), sdepthS (I ′′ /J ′′ )},

Using Lemma 2.1 we conclude that that l(I ′′ /J ′′ ) ≤ l(I/J). Hence our induction
hypothesis on
X
deg(u)
u∈G(I)∪G(J)

implies that
sdepthS (I ′′ /J ′′ ) ≥ n − l(I ′′ /J ′′ ) + 1 ≥ n − l(I) + 1
and similarly depthS (I ′′ /J ′′ ) ≥ n − l(I) + 1.
On the other hand, since
[
x1 ∈
Supp(u),
u∈G(I)∪G(J)

using Lemma 2.2 we conclude that l(I ′ S ′ /J ′ S) ≤ l(I/J) − 1 and therefore by the
induction hypothesis on n we conclude that

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L. KATTHÄN AND S. A. SEYED FAKHARI

sdepthS ′ (I ′ S ′ /J ′ S) ≥ (n − 1) − l(I ′ S ′ /J ′ S) + 1 ≥ (n − 1) − (l(I/J) − 1) + 1
= n − l(I/J) + 1
′

′

′

and similarly depthS ′ (I S /J S) ≥ n − l(I/J) + 1. Now the assertions follow from
inequalities (1) and (3).

As an immediate consequence of Theorem 2.4 we obtain the following result.
Corollary 2.5. Let I be a monomial ideal of S = K[x1 , . . . , xn ]. Then sdepth(I) ≥
n − l(I) + 1 and sdepth(S/I) ≥ n − l(I).
For every vector a = (a1 , . . . , an ) of non-negative integers, we denote the monomial
xa11 . . . xann by xa . Let I ⊆ S be a monomial ideal and G(I) = {xa1 , . . . , xam } be the
set of minimal monomial generators of I. The rank of I, denoted by rank(I) is the
cardinality of the largest Q-linearly independent subset of {a1 , . . . , am }, where Q is the
set of rational numbers. In [18], the author proves that for every squarefree monomial
ideal of S the inequalities sdepth(I) ≥ n − rank(I) + 1 and sdepth(S/I) ≥ n − rank(I)
hold. We note that Corollary 2.5 implies this result.
Corollary 2.6. ([18, Thorem 3.3]) Let I be a squarefree monomial ideal of S =
K[x1 , . . . , xn ]. Then sdepth(I) ≥ n − rank(I) + 1 and sdepth(S/I) ≥ n − rank(I).
Proof. Assume that l(I) = t. Suppose that u1 , . . . , ut are monomials in the set of
minimal monomial generators of I such that
(∗)

u1 6= lcm(u1 , u2) 6= . . . 6= lcm(u1 , u2, . . . , ut ).

Since u1 , . . . , ut are squarefree, inequalities (∗) imply that
Supp(u1 ) $

2
[
i=1

Supp(ui ) $ . . . $

t
[

Supp(ui ).

i=1

This shows that u1 , . . . , ut are Q-linearly independent and thus rank(I) ≥ t. Now
Corollary 2.5 completes the proof.

Let I be a monomial ideal and assume that G(I) is the set of minimal monomial
generators of I. The initial degree of I, denote by indeg(I) is the minimum degree of
the monomials belonging to G(I). The following proposition provides an upper bound
for the lcm number of a squarefree monomial ideal in terms of its initial degree.
Proposition 2.7. Let I be a squarefree monomial ideal of S = K[x1 , . . . , xn ]. Then
l(I) ≤ n − indeg(I) + 1.
Proof. Similar to the proof of Corollary 2.6, one can see that if u1 , . . . , ut are monomials in the set of minimal monomial generators of I such that
u1 6= lcm(u1 , u2) 6= . . . 6= lcm(u1 , u2, . . . , ut ),

TWO LOWER BOUNDS FOR THE STANLEY DEPTH

7

then
(∗∗)

Supp(u1 ) $

S2

i=1

Supp(ui ) $ . . . $

St

i=1

Supp(ui ).

Since u1 , . . . , ut are squarefree monomials, the cardinality of S
Supp(u1 ) is greater than
or equal to indeg(I). On the other hand, the cardinality of ti=1 Supp(ui ) is at most
n. Hence, the inclusions (∗∗) show that
l(I) ≤|

t
[

Supp(ui ) | − | Supp(u1 ) | +1 ≤ n − indeg(I) + 1.

i=1


The following corollary is an immediate consequence of Corollary 2.5 and Corollary
2.7.
Corollary 2.8. Let I be a squarefree monomial ideal of S = K[x1 , . . . , xn ]. Then
sdepth(I) ≥ indeg(I) and sdepth(S/I) ≥ indeg(I) − 1.
Remark 2.9. Let I be a squarefree monomial ideal. We mention that the inequality
sdepth(I) ≥ indeg(I) was known by [8, Proposition 3.1].
Let I be a monomial ideal of S = K[x1 , . . . , xn ] and G(I) be the set of minimal
monomial generators of I. Assume that | G(I) |= m. Cimpoeaş [2] proves that
sdepth(S/I) ≥ n − m (see [2, Proposition 1.2]). It is completely clear that the bound
given in Corollary 2.5 for the Stanley depth of S/I is better than the bound given
by Cimpoeaş. Indeed, there are examples (see Example 2.10), for which m − l(I)
is large enough and the inequality sdepth(S/I) ≥ n − l(I) is sharp for them, i.e.,
sdepth(S/I) = n − l(I).
Example 2.10. Let
I = (xi xj : 1 ≤ i < j ≤ n)
be a monomial ideal of S = K[x1 , . . . , xn ]. Then l(I) = n − 1 and thus
(n − 1)(n − 2)
m − l(I) =
,
2
where m is the cardinality of the set of minimal monomial generators of I. This shows
that by choosing a suitable n, the number m−l(I) can be larger than any given integer.
On the other hand, the height of every associated prime of I is equal to n − 1. Thus,
it follows from [8, Proposition 1.3] and Corollary 2.5 that sdepth(S/I) = 1 = n − l(I).
3. Stanley depth and order dimension
In this section, we give the proof of our second main result. Let us recall some
definitions of lattice theory. For a comprehensive treatment of this subject we refer
the reader to [4]. Recall that a join-semilattice is a poset in which every two elements
have a least upper bound, called their join. We call a subset L′ of a finite joinsemilattice L a join-subsemilattice if it is a join-semilattice with the induced joinoperation from L. It is well-known that every finite join-semilattice with a minimal

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L. KATTHÄN AND S. A. SEYED FAKHARI

element is in fact a lattice. However, as we will never consider the meet, it is more
convenient to work in the category of join-semilattices. An element m ∈ L is called
join-irreducible if it cannot be written as the join of two elements different from m.
Note that every element m in a finite join-semilattice is the join of the set of all
join-irreducible elements less than or equal to m.
The following is a convenient characterization of the dimension of a finite joinsemilattice.
Lemma 3.1. Let L be a finite join-semilattice and let d ∈ N. Then the following are
equivalent:
(1) There exists a surjective join-preserving map φ : L′ → L for a finite joinsubsemilattice L′ of Nd .
(2) dim L ≤ d.
W
Proof. 1) ⇒ 2): Consider the map φ† : L → Nd defined by φ† (a) := φ−1 (a). It is
an embedding of L into Nd by [10, Lemma 4.1], hence dim L ≤ d.
2) ⇒ 1): Let j : L → Nd be an embedding and set L′ ⊂ Nd to be the joinsubsemilattice of Nd generated by the image of j, i.e.Wthe set of all joins of subsets
of the image of L. Define φ : L′ → L by φ(x′ ) := {x ∈ L : j(x) ≤ x′ }. This
map is clearly monotonic. Moreover, monotonicity implies that j(x ∨ y) ≥ j(x) ∨ j(y)
and φ(x′ ∨ y ′) ≥ φ(x′ ) ∨ φ(y ′). So it remains to show that φ is surjective and that
φ(x′ ∨ y ′ ) ≤ φ(x′ ) ∨ φ(y ′ ) for x′ , y ′ ∈ L′ .
For the first claim, we show that φ ◦ j = IdL . Indeed,
_
_
φ(j(x)) = {y ∈ L : j(y) ≤ j(x)} = {y ∈ L : y ≤ x} = x
for every x ∈ L. Moreover, we claim that for every x′ ∈ L′ we have
j(φ(x′ )) ≥ x′ .

(†)

To see this we compute
_
_
j(φ(x′ )) = j( {y ∈ L : j(y) ≤ x′ }) ≥ {j(y) : y ∈ L, j(y) ≤ x′ }
_
≥ {y ′ ∈ L′ : y ′ join-irreducible, y ′ ≤ x′ } = x′
Here we used that j is surjective onto the join-irreducible elements of L′ . Now †
implies that
x′ ∨ y ′ ≤ x′ ∨ j(φ(y ′)) ≤ j(φ(x′ )) ∨ j(φ(y ′))
and thus
φ(x′ ∨ y ′ ) ≤ φ(j(φ(x′ )) ∨ j(φ(y ′))) ≤ φ(j(φ(x′ ) ∨ φ(y ′))) = φ(x′ ) ∨ φ(y ′ ).

Theorem 3.2. Let J ( I ⊂ S be two monomial ideals. Then
sdepthS (I/J) ≥ n − dim LI/J

TWO LOWER BOUNDS FOR THE STANLEY DEPTH

9

and
sdepthS (I) ≥ n − dim LI + 1.
Proof. Let d := dim LI/J . Consider the join-semilattice L′ ⊂ Nd of the preceding
lemma and the corresponding surjective join-preserving map φ : L′ → LI/J . Let
moreover L′′ ⊂ L′ be the join-subsemilattice corresponding to LJ , i.e. the joinsubsemilattice generated by the images of LJ in Nd . By construction φ maps L′′ onto
LJ . We interpret the elements of L′ as exponent vectors to see that L′ and L′′ are
lcm lattices of I ′ /J ′ and J ′ , for two monomial ideals J ′ ( I ′ ⊂ S ′ = K[x1 , . . . , xd ]
in d variables. Now it follows from [10, Theorem 4.9] that n − sdepthS (I/J) ≤
d − sdepthS ′ (I ′ /J ′ ) ≤ d and thus
sdepthS (I/J) ≥ n − d.
Moreover, by the same argument n − sdepthS (I) ≤ d − sdepthS ′ (I ′ ) ≤ d − 1 and
hence
sdepthS (I) ≥ n − d + 1.

Remark 3.3. It also holds that
depthS (I/J) ≥ n − dim LI/J
and
depthS (I) ≥ n − dim LI + 1.
This is proven by the same argument, using [10, Theorem 4.11] instead of [10, Theorem
4.9].
We present two examples to show that in general there is no inequality between the
lcm number of I and the order dimension of LI .
Examples 3.4.
(1) Consider the ideal I = (x2 , xy, y 2) ⊂ S = K[x, y, z]. It is easy
to see that l(I) = 3, so Theorem 2.5 gives the bound sdepthS S/I ≥ 3 − 3 = 0.
On the other hand, dim LI = 2 (the exponent vectors give an embedding into
N2 ), so Theorem 3.2 gives the better bound sdepthS S/I ≥ 3 − 2 = 1.
(2) Let I ⊂ S = K[x1 , . . . , x5 ] be the ideal generated by all squarefree monomials
of degree 3. Again, we have that l(I) = 3, so Theorem 2.5 gives the bound
sdepthS S/I ≥ 5 − 3 = 2. We computationally verified that dim LI = 4, so in
this case Theorem 3.2 gives the worse bound sdepthS S/I ≥ 5 − 4 = 1.
4. Monomial ideals with small lcm number and order dimension
In this section, we prove that the Stanley–Reisner ideal of a vertex decomposable
simplicial complex satisfies Stanley’s Conjecture (see Theorem 4.2). Using this result,
Corollary 2.5 and Theorem 3.2, we prove that I and S/I satisfy Stanley’s conjecture
if
(i) l(I) ≤ 3 or
(ii) dim LI ≤ 3 or

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L. KATTHÄN AND S. A. SEYED FAKHARI

(iii) l(I) ≤ 4 and S/I is Gorenstein or
(iv) dim LI ≤ 4 and S/I is Gorenstein.
To state and prove the next results, we need to introduce some notation and wellknown facts from combinatorial commutative algebra.
A simplicial complex ∆ on the set of vertices [n] := {1, . . . , n} is a collection of
subsets of [n] which is closed under taking subsets; that is, if F ∈ ∆ and F ′ ⊆ F , then
also F ′ ∈ ∆. Every element F ∈ ∆ is called a face of ∆, the size of a face F is defined
to be |F | and its dimension is defined to be |F | − 1. (As usual, for a given finite set
X, the number of elements of X is denoted by |X|.) The dimension of ∆ which is
denoted by dim ∆, is defined to be d − 1, where d = max{|F | | F ∈ ∆}. A facet of ∆
is a maximal face of ∆ with respect to inclusion. Let F (∆) denote the set of facets
of ∆. It is clear that F (∆) determines ∆. When F (∆) = {F1 , . . . , Fm }, we write
∆ = hF1 , . . . , Fm i. We say that ∆ is pure if all facets of ∆ have the same cardinality.
The link of ∆ with respect to a face F ∈ ∆, denoted by lk∆ (F ), is the simplicial
complex lk∆ (F ) = {G ⊆ [n] \ F | G ∪ F ∈ ∆} and the deletion of F , denoted by
del∆ (F ), is the simplicial complex del∆ (F ) = {G ⊆ [n] \ F | G ∈ ∆}. When F = {x}
is a single vertex, we abuse notation and write lk∆ (x) and del∆ (x).
Let S = K[x1 , . . . , xnQ
] and let ∆ be a simplicial complex on [n]. For every subset
F ⊆ [n], we set x F = i∈F xi . The Stanley–Reisner ideal of ∆ over K is the ideal
I∆ of S which is generated by those squarefree monomials xF with F ∈
/ ∆. In other
words, I∆ = hxF | F ∈ N (∆)i, where N (∆) denotes the set of minimal nonfaces
of ∆ with respect to inclusion. The Stanley–Reisner ring of ∆ over K, denoted by
K[∆], is defined to be K[∆] = S/I∆ . Let I ⊆ S be an arbitrary squarefree monomial
ideal. Then there is a unique simplicial complex ∆ such that I = I∆ . A simplicial
complex ∆ is said to be Cohen–Macaulay if K[∆] is Cohen–Macaulay. For every
integer 0 ≤ i ≤ dim∆ the simplicial complex ∆[i] := hF ∈ ∆ | dim F = ii is called
the i-pure skeleton of ∆. A simplicial complex ∆ is said to be sequentially Cohen–
Macaulay if ∆[i] is Cohen–Macaulay, for every 0 ≤ i ≤ dim∆.
Let ∆ be a simplicial complex on the vertex set [n]. Then we say that ∆ is vertex
decomposable if either
(1) ∆ is a simplex, i.e., a simplicial complex with only one facet or
(2) there exists k ∈ [n] such that del∆ (k) and lk∆ (k) are vertex decomposable and
every facet of del∆ (k) is a facet of ∆.
It is know that every vertex decomposable simplicial complex is sequentially Cohen–
Macaulay (see for example [21]).
Remark 4.1. It follows from [13, Corollary 3.33] that for every vertex decomposable
simplical complex ∆, we have

depth(K[∆]) = min | F |: F is a facet of ∆ .
In the next theorem, we prove that the Stanley–Reisner ideal of vertex decomposable simplicial complexes satisfy Stanley’s conjecture.

TWO LOWER BOUNDS FOR THE STANLEY DEPTH

11

Theorem 4.2. Let ∆ be a vertex decomposable simplicial complex. Then I∆ satisfies
Stanley’s conjecture.
Proof. We prove the assertion by induction on n. If ∆ is a simplex, then I∆ = 0 and
there is nothing to prove. Thus, assume that ∆ is not a simplex. Therefore, there
exists a vertex k ∈ [n] such that del∆ (k) and lk∆ (k) are vertex decomposable and
every facet of del∆ (k) is a facet of ∆. Let S ′ = K[x1 , . . . , xk−1 , xk+1 , . . . , xn ] be the
polynomial ring obtained from S by deleting the variable xk and consider the ideals
I ′ = I ∩ S ′ and I ′′ = (I : xk ). Now I = I ′ S ′ ⊕ xk I ′′ S and therefore by definition of
the Stanley depth we have
(‡)

sdepth(I) ≥ min{sdepthS ′ (I ′ S ′ ), sdepthS (I ′′ )}.

Note that I ′′ = (I : xk ) is the Stanley–Reisner ideal of lk∆ (k), considered as a
simplicial complex on [n] \ {k}. Since lk∆ (k) is a vertex decomposable simplicial
complex, it follows from [8, Lemma 3.6], Remark 4.1 and the induction hypothesis
that
sdepthS (I ′′ ) = sdepthS ′ (I ′′ ) + 1 ≥ depthS ′ (I ′′ ) + 1
≥ (depthS (I) − 1) + 1 = depthS (I).
′
On the other hand, I = I ∩ S ′ is the Stanley–Reisner ideal of del∆ (k), considered
as a simplicial complex on [n] \ {k}. Since del∆ (k) is a vertex decomposable simplicial
complex, it follows from Remark 4.1 and the induction hypothesis that
sdepthS ′ (I ′ S ′ ) ≥ depthS ′ (I ′ S ′ ) ≥ depthS (I).
Now inequality (‡) completes the proof.



Let I be a monomial ideal. In [7], the authors prove that S/I satisfies Stanley’s
conjecture, provided that depth(S/I) ≥ n − 1 (see [7, Corollary 2.3]). The following
lemma is an extension of this result.
Lemma 4.3. Let I be a monomial ideal of S = K[x1 , . . . , xn ] and assume that
depth(S/I) ≥ n − 2. Then I and S/I satisfy Stanley’s conjecture.
P
Proof. We use induction on u∈G(I) deg(u), where G(I) is the set of minimal monomial generators of I. If
X
deg(u) = 1,
u∈G(I)

then I is a principal ideal. Therefore, it follows from [16, Theorem 1.1] that sdepth(S/I) =
n − 1. On the other hand, it is clear that sdepth(I) = n. Thus, I and S/I satisfy
Stanley’s conjecture. Now, we consider the following cases.
Case 1. dim(S/I) = depth(S/I) = n − 2. In this case S/I is Cohen–Macaulay and
the height of I is equal to 2. Thus S/I satisfies Stanley’s conjecture by [7, Proposition
2.4]. To prove that I satisfies Stanley’s conjecture, let I p denote the polarization of I
which is considered in a new polynomial ring, say T (see [6] for the definition of polarization). Then by [6, Corollary 1.6.3], we conclude that T /I p is Cohen–Macaulay and
the height of I p is equal to 2. It follows from [1, Theorem 2.3] (see also [12]) that I p is

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L. KATTHÄN AND S. A. SEYED FAKHARI

the Stanley–Reisner ideal of a vertex decomposable simplicial complex and therefore,
I p satisfies Stanley’s conjecture by Theorem 4.2. Now [9, Corollary 4.5] implies that
I satisfies Stanley’s conjecture.
Case 2. dim(S/I) = depth(S/I) = n − 1. In this case, the height of every associated prime of I is equal to one. Thus, I is a principal ideal. Therefore, it follows
from [16, Theorem 1.1] that sdepth(S/I) = n − 1. On the other hand, it is clear that
sdepth(I) = n. Thus, I and S/I satisfy Stanley’s conjecture.
Case 3. dim(S/I) = n − 1 and depth(S/I) = n − 2. In this case, the height of at
least one of the associated primes of I is equal to one. Hence, there exists a variable,
say xk , such that I ⊂ (xk ). Thus, I = xk (I : xk ). This shows that I and (I : xk ) are
isomorphic (as Zn -graded S-module). Thus depth(I) = depth((I : xk )), which implies
that depth(S/I) = depth(S/(I : xk )). On the other hand, it follows from [2, Theorem
1.1] that sdepth(I) = sdepth((I : xk )) and sdepth(S/I) = sdepth(S/(I : xk )). Hence,
the induction hypothesis implies
sdepth(I) = sdepth((I : xk )) ≥ depth((I : xk )) = depth(I).
Similarly, sdepth(S/I) ≥ depth(S/I). Therefore, I and S/I satisfy Stanley’s conjecture.

Let I be a monomial ideal. In the following theorem, we prove that I and S/I
satisfies Stanley’s conjecture, if the lcm number of I or the order diension of LI is
small.
Theorem 4.4. Let I be a monomial ideal of S = K[x1 , . . . , xn ]. If l(I) ≤ 3 or
dim LI ≤ 3 holds, then I and S/I satisfy Stanley’s conjecture.
Proof. It follows from Corollary 2.5 resp. Theorem 3.2 that sdepth(S/I) ≥ n − 3 and
sdepth(I) ≥ n − 2. This implies that if depth(S/I) ≤ n − 3, then I and S/I satisfy
Stanley’s conjecture. Otherwise, the assertions follow from Lemma 4.3.

In the following corollary, we consider the Gorenstein monomial ideals with lcm
number or order dimension at most four.
Corollary 4.5. Let I be a monomial ideal of S = K[x1 , . . . , xn ] such that S/I is
Gorenstein. If l(I) ≤ 4 or dim LI ≤ 4, then I and S/I satisfy Stanley’s conjecture.
Proof. By Corollary 2.5 resp. Theorem 3.2, we conclude that sdepth(S/I) ≥ n − 4
and sdepth(I) ≥ n − 3. Thus, there is nothing to prove, if depth(S/I) ≤ n − 4. If
depth(S/I) ≥ n − 2, then the assertions follow from Lemma 4.3. Thus, we assume
that depth(S/I) = n − 3. In this case, the height of I is equal to 3 and it follows
from [7, Theorem 3.1] that S/I satisfies Stanley’s conjecture. In order to prove that
I satisfies Stanley’s conjecture, we use the machinery of Polarization. Let I p denote
the polarization of I which is considered in a new polynomial ring, say T . Then by [6,

TWO LOWER BOUNDS FOR THE STANLEY DEPTH

13

Corollary 1.6.3], we conclude that T /I p is Gorenstein and the height of I p is equal to
3. Using [1, Theorem 2.5], we conclude that I is the Stanley–Reisner ideal of a vertex
decomposable simplicial complex and therefore, I p satisfies Stanley’s conjecture by
Theorem 4.2. Now [9, Corollary 4.5] implies that I satisfies Stanley’s conjecture. 
As an immediate consequence of Proposition 2.7, Theorem 4.4 and Corollary 4.5
we obtain the following result.
Corollary 4.6. Let I be a squarefree monomial ideal of S = K[x1 , . . . , xn ]. Then I
and S/I satisfy Stanley’s conjecture if
(i) indeg(I) ≥ n − 2 or
(ii) S/I is Gorenstein and indeg(I) ≥ n − 3
A simplicial complex ∆ is called doubly Cohen–Macaulay if ∆ is Cohen–Macaulay
and for every vertex x of ∆, the simplicial complex del∆ (x) is Cohen–Macaulay of
the same dimension as ∆. The following corollary shows that I∆ and S/I∆ satisfy
Stanley’s conjecture if ∆ is a doubly Cohen–Macaulay simplicial complex and the
initial degree of I∆ is large enough.
Corollary 4.7. Let I be the Stanley–Reisner ideal of a doubly Cohen–Macaulay simplicial complex and assume that indeg(I∆ ) ≥ n − 3. Then I and S/I satisfy Stanley’s
conjecture.
Proof. By Corollary 2.5, we conclude that sdepth(S/I) ≥ n−4 and sdepth(I) ≥ n−3.
Thus, the assertion is true, if depth(S/I) ≤ n − 4. If depth(S/I) = n − 3, then the
height of I is equal to 3. In this case, S/I satisfies Stanley’s conjecture by [14, Theorem
4.2]. On the other hand, it follows from [1, Theorem 2.13] that I is the Stanley–Reisner
ideal of a vertex decomposable simplicial complex and therefore, I satisfies Stanley’s
conjecture by Theorem 4.2. The remaining case (depth(S/I) ≥ n − 2) follows from
Lemma 4.3.

Acknowledgment
The authors wish to thank Christian Bey for pointing out the reference [17] to us.
References
[1] S. M. Ajdani, A. Soleyman Jahan, vertex decomposability of 2–CM and Gorenstein simplicial
complexes of codimension 3, Bull. Malays. Math. Sci. Soc., to appear.
[2] M. Cimpoeaş, Stanley depth of monomial ideals with small number of generators, Central
European Journal of Mathematics, 7 (2009), 629–634.
[3] V. Gasharov, I. Peeva, V. Welker, The lcm-lattice in monomial resolutions, Math. Res. Lett. 6
(1999), no. 5-6, 521–532.
[4] G. Grätzer, Lattice Theory: Foundation, Birkhäuser-Verlag, 2011.
[5] J. Herzog, A survey on Stanley depth. In ”Monomial Ideals, Computations and Applications”,
A. Bigatti, P. Giménez, E. Sáenz-de-Cabezón (Eds.), Proceedings of MONICA 2011. Lecture
Notes in Math. 2083, Springer (2013).
[6] J. Herzog, T. Hibi, Monomial Ideals, Springer-Verlag, 2011.

14

L. KATTHÄN AND S. A. SEYED FAKHARI

[7] J. Herzog, A. Soleyman Jahan, S. Yassemi, Stanley decompositions and partitionable simplicial
complexes, J. Algebraic Combin. 27 (2008), 113–125.
[8] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J.
Algebra 322 (2009), no. 9, 3151–3169.
[9] B. Ichim, L. Katthän, J. J. Moyano–Fernández, The behavior of Stanley depth under polarization, preprint.
[10] B. Ichim, L. Katthän, J. J. Moyano–Fernández, The behavior of depth and Stanley depth under
maps of the lcm-lattice, preprint.
[11] K. Kimura, Lyubeznik resolutions and the arithmetical rank of monomial ideals, Proc. Amer.
Math. Soc. 137 (2009), 3627–3635.
[12] F. Mohammadi, Powers of the vertex cover ideal of a chordal graph, Comm. Algebra 39 (2011),
3753–3764.
[13] S. Morey, R. H. Villarreal, Edge ideals: Algebraic and combinatorial properties, Francisco,
Christopher (ed.) et al., Progress in commutative algebra 1. Combinatorics and Homology.
Berlin: Walter de Gruyter. De Gruyter Proceedings in Mathematics, 85–126 (2012)
[14] U. Nagel, T. Römer, Glicci simplicial complexes, J. Pure Appl. Algebra 212 (2008), 2250–2258.
[15] M. R. Pournaki, S. A. Seyed Fakhari, M. Tousi, S. Yassemi, What is . . . Stanley depth? Notices
Amer. Math. Soc. 56 (2009), no. 9, 1106–1108.
[16] A. Rauf, Stanley decompositions, pretty clean filtrations and reductions modulo regular elements, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 50(98) (2007), no. 4, 347–354.
[17] V. Reiner, V. Welker, A homological lower bound for order dimension of lattices, Order 16
(1999) no. 2, 165–170.
[18] S. A. Seyed Fakhari, Stanley depth of weakly polymatroidal ideals and squarefree monomial
ideals, Illinois J. Math., to appear.
[19] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982),
no. 2, 175–193.
[20] W. T. Trotter, Combinatorics and partially ordered sets: Dimension theory, Johns Hopkins
University press, 1992.
[21] R. Woodroofe, Vertex decomposable graphs and obstructions to shellability, Proc. Amer. Math.
Soc. 137 (2009), no. 10, 3235–3246.
L. Katthän, Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück,
Germany
E-mail address: lkatthaen@uos.de
S. A. Seyed Fakhari, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.
E-mail address: fakhari@ipm.ir
URL: http://math.ipm.ac.ir/fakhari/

