arXiv:1201.3325v2 [] 14 Aug 2012

SIMPLICIAL COMPLEXES WITH RIGID DEPTH
ADNAN ASLAM AND VIVIANA ENE
Abstract. We extend a result of Minh and Trung [8] to get criteria for
√
depth I = depth I where I is an unmixed monomial ideal of the polynomial ring S = K[x1 , . . . , xn ]. As an application we characterize all the pure
simplicial complexes ∆ which have rigid depth, that is, which satisfy the con√
dition that for every unmixed monomial ideal I ⊂ S with I = I∆ one has
depth(I) = depth(I∆ ).

Introduction
Let S = K[x1 , . . . , xn ] be the polynomial ring over a ﬁeld K and I ⊂ S a monomial ideal. In [5], the authors compare the properties
of I with the properties of its
√
radical by using the inequality βi (I) ≥ βi ( I). In particular, from the inequality
√
between the Betti numbers, one gets the inequality depth(S/I)√≤ depth(S/ I),
which implies, for instance, that S/I is Cohen-Macaulay if S/ I is so. In [8],
the authors presented criteria for the Cohen-Macaulayness of a monomial ideal in
terms of its primary decomposition. We extend their criteria to characterize
the
√
unmixed monomial ideals for which the equality depth(S/I) = depth(S/ I) holds.
We recall that an ideal I ⊂ S is unmixed if the associated prime ideals of S/I are
the minimal prime ideals of I.
Let ∆ be a pure simplicial complex with the facet set denoted, as usual, by F (∆),
T
and let I∆ = F ∈F (∆) PF be its Stanley-Reisner ideal. For any subset F ⊂ [n],
we denoted by PF the monomial prime ideal generated by √
the variables xi with
i∈
/ F . Let I ⊂ S be an unmixed monomial ideal such that I = I∆ and assume
T
that I = F ∈F (∆) IF where IF is the PF -primary component of I. Following [8],
for every a ∈ Nn , a = (a1 , . . . , an ), we set xa = xa1 1 · · · xann and denote by ∆a the
simplicial complex on the set [n] with the facet set F (∆a ) = {F ∈ F (∆) | xa ∈
/ IF }.
Moreover, for every simplicial complex Γ with F (Γ) ⊆ F (∆), we set
[
\
IG }.
IF \
LΓ (I) = {a ∈ Nn | xa ∈
F ∈F (∆)\F (Γ)

G∈F (Γ)

In Section 1, we prove the following theorem which is a natural extension of
Theorem 1.6 in [8].

1991 Mathematics Subject Classification. Primary 13C15, Secondary 13F55,13D45.
Key words and phrases. Monomial ideals, Simplicial complexes, Stanley-Reisner rings, Depth.
The second author was supported by the grant UEFISCDI, PN-II-ID-PCE- 2011-3-1023.
1

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ADNAN ASLAM AND VIVIANA ENE

Theorem 1. Let ∆ be a pure simplicial
√ complex with depth K[∆] = t. Let I ⊂ S
be an unmixed monomial ideal with I = I∆ . Then the following conditions are
equivalent:
√
(a) depth(S/I) = depth(S/ I),
(b) depth K[∆a ] ≥ t for all a ∈ Nn ,
(c) LΓ (I) = ∅ for every simplicial complex Γ with F (Γ) ⊆ F (∆) and depth K[Γ] <
t.
As a main application of the above theorem we study in Section 2 a special class of simplicial complexes. We say that a pure simplicial
complex has
√
rigid depth if for every unmixed monomial ideal I ⊂ S with I = I∆ one has
depth(S/I) = depth(S/I∆ ). In Theorem 2.3 which generalizes [5, Theorem 3.2], we
give necessary and suﬃcient conditions for ∆ to have rigid depth. In particular,
from this characterization, it follows that if a pure simplicial complex has rigid
depth over a ﬁeld of characteristic 0, then it has rigid depth over any ﬁeld. In the
last part we discuss the behavior of rigid depth in connection to the skeletons of
the simplicial complex.
√
1. Criteria for depth(S/I) = depth(S/ I)
Let S = K[x1 , . . . , xn ] be the polynomial
ring over a ﬁeld K. Let I ⊂ S be an
√
unmixed monomial ideal such that I = I∆ where ∆ is a pure simplicial complex
T
/ F ) for
with the facet set F (∆). Then I∆ = F ∈F (∆) PF , where PF = (xi | i ∈
T
every F ∈ F (∆). Let I = F ∈F (∆) IF where IF is the PF -primary component of
I.
In order to prove the main result of this section we need to recall some facts from
[8, Section 1]. For a = (a1 , . . . , an ) ∈ Zn , let Ga = {i | ai < 0}. We denote by ∆a
the simplicial complex on [n] of all the sets of the form F \ Ga where Ga ⊂ F ⊂ [n]
and such that F satisﬁes the condition xa ∈
/ ISF where SF = S[x−1
| i ∈ F ]. It is
i
shown in [8, Section 1] that if ∆a is non-empty, then ∆a is a pure subcomplex of
∆ of dim ∆a = dim ∆ − |Ga |.
For every simplicial subcomplex Γ of ∆ with F (Γ) ⊂ F (∆) we set
[
\
IG }.
IF \
LΓ (I) = {a ∈ Nn | xa ∈
F ∈F (∆)\F (Γ)

G∈F (Γ)

By [8, Lemma 1.5], we have
(1)

∆a = Γ if and only if a ∈ LΓ (I).

For the proof of the next theorem we also need to recall Takayama’s formula [9].
i
For every degree a ∈ Zn we denote by Hm
(S/I)a the a-component of the ith local
cohomology module of S/I with respect to the homogeneous maximal ideal of S.
For 1 ≤ j ≤ n, let
ρj (I) = max{νj (u) | u is a minimal generator of I},
where by νj (u) we mean the exponent of the variable xj in u. If xj does not divide
u, then we use the usual convention, νj (u) = 0.

SIMPLICIAL COMPLEXES WITH RIGID DEPTH

3

Theorem 1.1 (Takayama’s formula).

 dimK H̃i−|Ga |−1 (∆a , K), if Ga ∈ ∆ and
i
dimK Hm
(S/I)a =
aj < ρj (I) for 1 ≤ j ≤ n,

0,
else.
The next theorem is a natural extension of [8, Theorem 1.6].

Theorem 1.2. Let ∆ be a pure simplicial
√ complex with depth K[∆] = t. Let I ⊂
S be an unmixed monomial ideal with I = I∆ . The following conditions are
equivalent:
(a) depth(S/I) = t,
(b) depth K[∆a ] ≥ t for all a ∈ Nn with ∆a 6= ∅,
(c) LΓ (I) = ∅ for every simplicial complex Γ with F (Γ) ⊆ F (∆) and depth K[Γ] <
t.
Proof. The proof of this theorem follows closely the ideas of the proof of [8, Theorem 1.6]. For the equivalence (a) ⇔ (b) we need to recall some known facts about
local cohomology; see [2, Section A. 7]. For any ﬁnitely generated graded S-module
i
M we have depth M ≥ t if and only if Hm
(M√
) = 0 for all i < t. Therefore, in our
hypothesis, and since depth(S/I) ≤ depth(S/ I) = t, we get
(2)

i
depth(S/I) = t ⇔ Hm
(S/I) = 0 for i < t.

In addition, for every a ∈ Nn , we get
(3)

i
depth(K[∆a ]) ≥ t ⇔ Hm
(K[∆a ]) = 0 for i < t.

For b ∈ Zn , we set Gb = {i | bi < 0} and Hb = {i | bi > 0}. By using [2,
Theorem A.7.3], for every b ∈ Zn , we obtain
i
dimK Hm
(K[∆a ])b = dimK H̃i−|Gb |−1 (linkstar Hb Gb ; K).

Here we denoted by star Hb the star of Hb in ∆a , and by linkstar Hb Gb the link of
Gb in the complex star Hb . We recall that if Γ is a simplicial complex and F is a face
of Γ, then starΓ F = {G | F ∪G ∈ Γ} and linkΓ F = {G | F ∪G ∈ Γ and F ∩G = ∅}.
Therefore, the equivalence (3) my be written
depth K[∆a ] ≥ t
(4)

⇔ H̃i−|Gb |−1 (linkstar Hb Gb ; K) = 0 for i < t and for every b ∈ Zn .

Since linkstar Hb Gb is acyclic for Hb 6= ∅ and star Hb = ∆a if Hb = ∅, we get
depth K[∆a ] ≥ t
(5)

⇔ H̃i−|Gb |−1 (link∆a Gb ; K) = 0 for i < t and for every b ∈ Zn .

By Takayama’s formula, the equivalence (2) may be rewritten
depth(S/I) = t
(6)

⇔ dimK H̃i−|Gb |−1 (∆b ; K) = 0 for i < t and for every b ∈ Zn .

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ADNAN ASLAM AND VIVIANA ENE

Now, the equivalence (a)⇔ (b) follows by relations (5) and (6) if we notice that,
by the proof of (i) ⇒ (ii) in [8, Theorem 1.6], we have link∆a Gb = ∆b for any
Gb ∈ ∆a .
For the rest of the proof we only need to use (1). Indeed, for (b) ⇒ (c), let
us assume that LΓ (I) 6= ∅ for some subcomplex Γ of ∆ with F (Γ) ⊂ F (∆) and
such that depth(K[Γ]) < t. Then there exists a ∈ LΓ (I), hence Γ = ∆a . But this
equality is impossible since depth(K[∆a ]) ≥ t. For (c) ⇒ (b), let us assume that
there exists a ∈ Nn such that depth K[∆a ] < t. Then, for Γ = ∆a we get LΓ (I) 6= ∅,
a contradiction.

Obviously, for t = dim K[∆] in the above theorem we recover Theorem 1.6 in
[8].
The above theorem is especially useful in the situation when I is either an intersection of monomial prime ideal powers or an intersection of irreducible monomial
ideals. The ﬁrst class of ideals may be studied with completely similar arguments to
those used in [8, Section 1]. In the sequel we discuss ideals which are intersections
of irreducible monomial ideals.
Tr
Let F (∆) = {F1 , . . . , Fr } and I = i=1 IFi be an intersection of irreducible
a
monomial ideals, that is, for every 1 ≤ i ≤ r, IFi = (xj ij | j 6∈ Fi ) for some
positive exponents aij . As a consequence
√ of the above theorem, one may express
the condition depth(S/I) = depth(S/ I) in terms of linear inequalities on the
exponents aij .
Proposition
√ 1.3. The set of exponents (aij ) for which the equality depth(S/I) =
depth(S/ I) holds consists of all points of positive integer coordinates in a finite
union of rational cones in Rr(n−d) .
Proof. Let Γ be a subcomplex of ∆ with depth(K[Γ]) < t and F (∆) \ F (Γ) =
{Fi1 , . . . , Fis } where 1 ≤ i1 < · · · < is ≤ r. The condition LΓ (I) = ∅ gives
s
\

aiq j

(xj

q=1

:j∈
/ Fiq ) ⊆

[

IFk .

k∈{i
/ 1 ,...,is }

This implies that the following conditions must hold
ai

j

ai

j

a

lcm(xj1 1 1 , xj2 2 2 , . . . , xjsis js ) ∈

[

IFk

k∈{i
/ 1 ,...,is }

for all s-tuples (j1 , j2 , . . . , js ), with jq ∈
/ Fiq for 1 ≤ q ≤ s. This is equivalent to
saying that for every s-tuple (j1 , j2 , . . . , js ), with jq ∈
/ Fiq for 1 ≤ q ≤ s, there exists
1 ≤ q ≤ s such that
aiq jq ≥ min{akjq : k 6= i1 , i2 , . . . , is , }.

In the following example we consider tetrahedral type ideals.

SIMPLICIAL COMPLEXES WITH RIGID DEPTH

5

Example 1.4. Let ∆ be the 4-cycle, that is, I∆ = (x1 , x2 ) ∩ (x1 , x4 ) ∩ (x2 , x3 ) ∩
(x3 , x4 ). Note that S/I∆ is Cohen-Macaulay, hence depth(S/I∆ ) = 2.
Let I = (xa1 1 , xa2 2 ) ∩ (xa1 3 , xa4 4 ) ∩ (xa2 5 , xa3 6 ) ∩ (xa3 7 , xa4 8 ). Then depth(S/I) =
depth(S/I∆ ), that is, I is a Cohen-Macaulay ideal, if and only if one of the following
condition holds:
(1)
(2)
(3)
(4)

a3
a2
a5
a1

≤ a1 ,
≤ a5 ,
≤ a2 ,
≤ a3 ,

a2
a6
a1
a4

= a5 ,
= a7 ,
= a3 ,
= a8 ,

a7
a4
a8
a6

≤ a6 .
≤ a8 .
≤ a4 .
≤ a7 .

In order to prove the above claim, we ﬁrst notice that any subcomplex Γ of ∆
which has depth(K[Γ]) < 2 corresponds to a disconnected subgraph of ∆. But
∆ has two disconnected subgraphs which correspond to the pair of disjoint edges



{1, 2}, {3, 4} and {1, 4}, {2, 3} . Let Γ be the subgraph {1, 2}, {3, 4} . Then
the inequalities of the proof of Proposition 1.3 give
(a1 ≤ a3 or a2 ≤ a5 ) and (a1 ≤ a3 or a7 ≤ a6 )
and (a8 ≤ a4 or a2 ≤ a5 ) and (a8 ≤ a4 or a7 ≤ a6 ),

which is equivalent to
(7)

(a1 ≤ a3 and a8 ≤ a4 ) or

(a2 ≤ a5 and a7 ≤ a6 ).

Now we consider the other disconnected subgraph which corresponds to the pair

of disjoint edges {1, 4}, {2, 3} and get, similarly,
(8)

(a3 ≤ a1 and a5 ≤ a2 ) or

(a6 ≤ a7 and a4 ≤ a8 ).

By intersecting conditions (7) and (8), we get the desired relations.
Note that in this example the union of the four rational cones deﬁned by the
set of the linear inequalities (1) − (4) is not a convex set. Indeed, if we take
the exponent vectors a = (3, 5, 1, 3, 5, 9, 7, 9) and a′ = (1, 3, 1, 1, 7, 11, 11, 1), then
the corresponding ideals are both Cohen-Macaulay. However, for the vector b =
(a + a′ )/2 = (2, 4, 1, 2, 6, 10, 9, 5), the corresponding ideal is not Cohen-Macaulay.
2. Rigid depth
Definition 2.1. Let ∆ be a pure simplicial complex.
√ We say that ∆ has rigid depth
if for every unmixed monomial ideal I ⊂ S with I = I∆ one has depth(S/I) =
depth(S/I∆ ).
For example, any pure simplicial complex ∆ with depth(K[∆]) = 1 has rigid
depth. In this section we characterize all the pure simplicial complexes which have
rigid depth.
In the next theorem we will use the formula given in the following proposition
for computing the depth of a Stanley-Reisner ring. We recall that the ith skeleton
of a simplicial complex ∆ is deﬁned as ∆(i) = {F ∈ ∆ | dim F ≤ i}.
Proposition 2.2. [6] Let ∆ be a simplicial complex of dimension d − 1. Then:
depth(K[∆]) = max{i | ∆(i) is Cohen-Macaulay} + 1.

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ADNAN ASLAM AND VIVIANA ENE

The following theorem generalizes [5, Theorem 3.2].
Theorem 2.3. Let ∆ be a pure simplicial complex with depth(K[∆]) = t and
T
I∆ = F ∈F (∆) PF . The following statements are equivalent:
(a) ∆ has rigid depth.
T
(b) depth(S/I) = t for every ideal I = F ∈F (∆) IF where IF are irreducible
√
monomial ideals with IF = PF for all F ∈ F (∆).
T
(c) depth(S/I) = t for every ideal I = F ∈F (∆) PFmF where mF are positive
integers.
(d) depth(K[Γ]) ≥ t for every subcomplex Γ of ∆ with F (Γ) ⊂ F (∆).
(e) For every subcomplex Γ of ∆ with F (Γ) ⊂ F (∆), the skeleton Γ(t−1) is
Cohen-Macaulay.
(f) Let F (∆) = {F1 , . . . , Fr }. Then, for every 1 ≤ k ≤ min{r, t} and for any
indices 1 ≤ i1 < · · · < ik ≤ r, we have |Fi1 ∩ · · · ∩ Fik | ≥ t − k + 1.

Proof. (a) ⇒ (b) and (a) ⇒ (c) are trivial.
(b) ⇒ (d): Let Γ be a subcompex of ∆ with F (Γ) ⊂ F (∆). We have to show
that depth(K[Γ]) ≥ t. For every F ∈ F (Γ), let IF = (x2i | i ∈
/ F ), and for every
T
F ∈ F (∆) \ F (Γ) let IF = PF = (xi | i ∈
/ F ). Let I = F ∈F (∆) IF . By assumption,
depth(S/I) = t. Let S ′ ⊂ K[x1 , . . . , xn , y1 , . . . , yn ] be the polynomial ring over K
in all the variables which are needed for the polarization of I, and let I p ⊂ S ′ be
T
the polarization of I. We have I p = F ∈F (∆) IFp , where

(xi yi | i ∈
/ F ), if F ∈ F (Γ),
p
IF =
PF ,
if F ∈ F (∆) \ F (Γ).

Then proj dim(S ′ /I p ) = proj dim(S/I). Let N be the multiplicative set generated
T
p
/ F ) and
by all the variables xi . Then IN
= F ∈F (Γ) (yi | i ∈
proj dim(S ′ /I p )N ≤ proj dim(S ′ /I p ) = proj dim(S/I).

This inequality implies that depth(K[Γ]) ≥ depth(S/I) = t.
(d) ⇔ (e) follows immediately by applying the criterion given in Proposition 2.2.
(d) ⇒ (f): We proceed by induction on k. The initial inductive step is trivial.
Let k > 1 and assume that |Fi1 ∩ · · · ∩ Fiℓ | ≥ t − ℓ + 1 for 1 ≤ ℓ < k and for any
1 ≤ i1 < · · · < iℓ ≤ r. Obviously, it is enough to show that |F1 ∩ · · ·∩ Fk | ≥ t − k + 1.
By [3, Theorem 1.1], we have the following exact sequence of S-modules:
(9)

0 → Tk

S

i=1 PFi

→

k
M
S
→
P
Fi
i=1

M
1≤i<j≤k

S
S
→ ··· →
→ 0.
PFi + PFj
PF1 + · · · + PFk

Tk

By assumption, depth(S/ i=1 PFi ) ≥ t. We decompose the above sequence in
k − 1 short exact sequences as follows:
0 → Tk

S

i=1

0 → U1 →

PFi
M

→

1≤i<j≤k

k
M
S
→ U1 → 0,
P
i=1 Fi

S
→ U2 → 0,
PFi + PFj

SIMPLICIAL COMPLEXES WITH RIGID DEPTH

7

..
.
0 → Uk−2 →

M

1≤j1 <···<jk−1 ≤k

PFj1

S
S
→ 0.
→
+ · · · PFjk−1
PF1 + · · · + PFk

Note that, for all ℓ and any 1 ≤ j1 < · · · < jℓ ≤ k, we have
PFj1 + · · · + PFjℓ = PFj1 ∩···∩Fjℓ .
In particular, S/(PFj1 + · · · + PFjℓ ) is Cohen-Macaulay of depth equal to |Fj1 ∩ · · · ∩
Fjℓ |. Therefore,
M
depth(
S/(PFj1 + · · · PFjℓ )) ≥ t − ℓ + 1
1≤j1 <···<jℓ ≤k

for every 1 ≤ ℓ < k and any 1 ≤ j1 < · · · < jℓ ≤ k. Now, by using the inductive
hypothesis and by applying Depth Lemma in the ﬁrst k − 2 above short exact sequences from top to bottom, step by step, we obtain depth(U1 ) ≥ t−1, depth(U2 ) ≥
t − 2, . . . , depth(Uk−2 ) ≥ t − k + 2. Finally, by applying Depth Lemma in the last
short exact sequence, since the depth of the middle term is ≥ t − k + 2, we get
depth(S/(PF1 + · · · + PFk )) = |F1 ∩ · · · ∩ Fk | ≥ t − k + 1.
(f)⇒(d): Let Γ be a subcomplex of ∆ with F (Γ) = {Fj1 , . . . , Fjk } ⊂ F (∆). We
have to show that depth(K[Γ]) ≥ t. We may obviously assume that k < r and the
facets of Γ are F1 , . . . , Fk . If k ≤ t, then we use the short exact sequences derived
from (9) in the proof of (d) ⇒ (f) and, by applying successively Depth Lemma from
bottom to the top, we get, step by step, depth(Uk−2 ) ≥ t − k + 2, . . . , depth(U2 ) ≥
t− 2, depth(U1 ) ≥ t− 1, and, ﬁnally, from the ﬁrst exact sequence, depth(K[Γ]) ≥ t.
If t < k, we use only the ﬁrst t short exact sequences, that is, we stop at
M
S
→ Ut → 0.
0 → Ut−1 →
PFj1 + · · · + PFjt
1≤j1 <···<jt ≤k

Since the middle term in this short exact sequence has depth ≥ 1, we get depth(Ut−1 ) ≥
1. Next, by using the same arguments as before, we get depth(Ut−2 ) ≥ 2, . . . , depth(U1 ) ≥
t − 1, and, ﬁnally, depth(K[∆]) ≥ t, as desired.
The implication (d) ⇒ (a) follows by Theorem 1.2.
Finally, the implication (c) ⇒ (e) follows similarly to the proof of Corollary 1.9
in [8].

In order to state the ﬁrst consequence of the above theorem, we need to know
the behavior of the depth of a Stanley-Reisner ring over a ﬁeld when passing from
characteristic 0 to characteristic p > 0. We show in the next lemma that the Betti
numbers of the Stanley-Reisner ring can only go up when passing from characteristic
0 to a positive characteristic which, in particular, implies that the depth does not
increase. This result is certainly known. However we include here its proof since we
could not ﬁnd any precise reference. The argument of the proof was communicated
to the second author by Ezra Miller.
Lemma 2.4. Let ∆ be a simplicial complex on the vertex set [n] and let K, L be
two fields with char K = 0, char L = p > 0. Then βi (K[∆]) ≤ βi (L[∆]) for all i.

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ADNAN ASLAM AND VIVIANA ENE

Proof. Any ﬁeld is ﬂat over its prime ﬁeld. Therefore, since char K = 0, we have
βi (K[∆]) = βi (Q[∆]) for all i, and since char L = p, we have βi (K[∆]) = βi (Fp [∆])
for all i, where Fp is the prime ﬁeld of characteristic p. In other words, the Betti
numbers depend only on the characteristic of the base ﬁeld. Let Zp be the local
ring of the integers at the prime p. The ring Zp [X] is *local ([1, Section 1.5])
and the Stanley-Reisner ideal I∆ ⊂ Zp [X] is *homogeneous. Let F be a minimal
free resolution of Zp [∆] over Zp [x1 , . . . , xn ]. Since p is a nonzerodivisor on Zp [∆],
by [7, Lemma 8.27], the quotient F /pF is a minimal free resolution of Fp [∆] over
Fp [x1 , . . . , xn ]. On the other hand, the localization F [p−1 ] by inverting p is a free
resolution, not necessarily minimal, of Q[∆] over Q[x1 , . . . , xn ]. Since the modules
in F /pF and F [p−1 ] have the same ranks, it follows that βi (Q[∆]) ≤ βi (Fp [∆]) for
all i which leads to the desired inequalities.

Corollary 2.5. Let ∆ be a pure simplicial complex with rigid depth over a field of
characteristic 0. Then ∆ has rigid depth over any field.
Proof. Let K be a ﬁeld of characteristic 0 and L a ﬁeld of characteristic p > 0.
The above lemma implies that proj dim K[∆] ≤ proj dim L[∆]. By AuslanderBuchsbaum formula, it follows that depth K[∆] ≥ depth L[∆]. Therefore, the desired statement follows by applying the combinatorial condition (f) of Theorem 2.3.

Example 2.6. Let ∆ be the six-vertex triangulation of the real projective plane;
see [1, Section 5.3]. If char K 6= 2, then ∆ is Cohen-Macaulay over K, hence
depth(K[∆]) = 2, and, by condition (f) of Theorem 2.3, it follows that ∆ does
not have rigid depth over K. But if char K = 2, then depth(K[∆]) = 1, and,
consequently, ∆ has rigid depth over K.
The simplicial complexes with one or two facets have rigid depth.
Lemma 2.7. Let ∆ be a pure simplicial complex with at most two facets. Then ∆
has rigid depth.
Proof. We only need to consider the case of simplicial complexes with two facets
since the other case is obvious. Let dim ∆ = d − 1 and F (∆) = {F, G}. We show
that depth(K[∆]) = t if and only if |F ∩ G| = t − 1. Then the claim follows by
condition (f) in Theorem 2.3. We consider the exact sequence
0 → K[∆] → (S/PF ) ⊕ (S/PG ) → S/(PF + PG ) ∼
= K[xi | i ∈ F ∩ G] → 0.
As (S/PF ) ⊕ (S/PG ) and S/(PF + PG ) are Cohen-Macaulay of dimensions d and,
respectively, |F ∩ G|, it follows that depth(K[∆]) = t if and only if |F ∩ G| = t − 1.

Example 2.8. Let ∆ and Γ be the simplicial complexes with F (∆) = {{1, 2, 3},
{1, 4, 5}} and F (Γ) = {{1, 2, 3}, {1, 3, 4}}. Obviously, by Lemma 2.7, ∆ is nonCohen-Macaulay of rigid depth 2, while Γ is Cohen-Macaulay of rigid depth.
In the sequel we investigate whether the rigid depth property is preserved by the
skeletons of the simplicial complexes with rigid depth. The next example shows
that this is not the case.

SIMPLICIAL COMPLEXES WITH RIGID DEPTH

9

Example 2.9. Let ∆ be the simplicial complex on the vertex set [8] with F (∆) =
{F, G} where F = {1, 2, 3, 4, 5} and G = {1, 2, 6, 7, 8}. Then, by Lemma 2.7 and
its proof, it follows that depth(K[∆]) = 3 and ∆ has rigid depth. Let ∆(3) be
the 3-dimensional skeleton of ∆ and Γ the subcomplex of ∆(3) with the facets
G1 = {1, 2, 3, 5} and G2 = {2, 6, 7, 8}. Then, again by the proof the above lemma,
we get depth(K[Γ]) = 2. But depth K[∆(3) ] = 3, thus the skeleton ∆(3) of ∆ does
not have rigid depth since it does not satisfy condition (d) in Theorem 2.3.
However, as an application of Theorem 2.3, we prove the following
Proposition 2.10. Let ∆ be a pure simplicial complex with rigid depth and let
t = depth(K[∆]). If ∆(i) has rigid depth for some i ≥ t − 1, then ∆(j) has rigid
depth for every j ≥ i.
Proof. By [4], we know that depth(K[∆(i) ]) = t for i ≥ t − 1. It is enough to show
that if ∆(i) has rigid depth for some i ≥ t − 1, then ∆(i+1) has the same property.
Let Γ ⊂ ∆(i+1) be a subcomplex with F (Γ) ⊂ F (∆(i+1) ). Then Γ(i) is a subcomplex of ∆(i) and F (Γ(i) ) ⊂ F (∆(i) ). By our assumption and by using condition
(e) in Theorem 2.3, it follows that Γ(t−1) is Cohen-Macaulay. Therefore, ∆(i+1)
satisﬁes condition (e) in Theorem 2.3, which ends our proof.

Acknowledgment
We thank Jürgen Herzog for helpful discussions on the subject of this paper and
Ezra Miller for the proof of Lemma 2.4. We would also like to thank the referee for
his valuable suggestions to improve our paper.
References
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[6] T. Hibi, Quotient algebras of Stanley-Reisner rings and local cohomology, J. Algebra, 140,
(1991), 336–343.
[7] E. Miller, B. Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics
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[8] N. C. Minh, N. V. Trung, Cohen-Macaulayness of monomial ideals and symbolic powers of
Stanley-Reisner ideals, Adv. Math. 226 (2011), 1285–1306.
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[10] R. H. Villarreal, Monomial algebras, Marcel Dekker, 2001.
Abdus Salam School of Mathematical Sciences (ASSMS), GC University, Lahore, Pakistan.
E-mail address: adnanaslam15@yahoo.com

10

ADNAN ASLAM AND VIVIANA ENE

Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124,
900527 Constanta, Romania
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700, Buchaest,
Romania
E-mail address: vivian@univ-ovidius.ro

