arXiv:1609.00830v1 [] 3 Sep 2016

COHEN-MACAULAY LEXSEGMENT COMPLEXES IN
ARBITRARY CODIMENSION
HASSAN HAGHIGHI, SIAMAK YASSEMI, AND RAHIM ZAARE-NAHANDI
Abstract. We characterize pure lexsegment complexes which are CohenMacaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen-Macaulay if and only if it is pure and its one dimensional links are connected, and, a lexsegment flag complex is Cohen-Macaulay
if and only if it is pure and connected. We show that any non-Cohen-Macaulay
lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For t ≥ 2, a lexsegment complex is strictly CohenMacaulay in codimension t if and only if it is the join of a lexsegment pure disconnected flag complex with a (t − 2)-dimensional simplex. When the StanleyReisner ideal of a pure lexsegment complex is not quadratic, the complex is
Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension.
Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai.

1. Introduction
Primary significance of lexsegment ideals comes from Macaulay’s result that for
any monomial ideal there is a unique lexsegment monomial ideal with the same
Hilbert function (see [10] and [2]). Recent studies on the topic began with the work
of Bigatti [3] and Hulett [6] on extremal properties of lexsegment monomial ideals.
Aramova, Herzog and Hibi showed that for any squarefree monomial ideal there
exists a squarefree lexsegment monomial ideal with the same Hilbert function [1].
In this direction, some characterizations of pure, Cohen-Macaulay and Buchsbaum
complexes associated with squarefree lexsegment ideals was given by Bonansinga,
Sorrenti and Terai in [5]. As a generalization of Cohen-Macaulay and Buchsbaum
complexes, CMt complexes, were studied in [8]. These are pure simplicial complexes which are Cohen-Macaulay in codimension t. Naturally, one may ask for
a characterization of CMt lexsegment complexes. In this paper, using the behavior of CMt property under the operation of join of complexes [9], we first provide
some modifications of the results of Bonanzinga, Sorrenti and Terai in [5]. Then,
we characterize CMt lexsegment simplicial complexes. Our characterizations are
mostly in terms of purity and connectedness of certain subcomplexes. In particular, it turns out a lexsegment complex is Cohen-Macaulay if and only it is pure
and its one dimensional links are connected while for lexsegment flag complexes,
2000 Mathematics Subject Classification. 13H10, 13D02.
Key words and phrases. Squarefree lexsegment ideal, Cohen-Macaulay complex, Buchsbaum
complex, flag complex, CMt complex.
H. Haghighi was supported in part by a grant from K. N. Toosi University of Technology.
S. Yassemi and R. Zaare-Nahandi were supported in part by a grant from the University of
Tehran.
Emails: haghighi@kntu.ac.ir, yassemi@ipm.ir, rahimzn@ut.ac.ir.
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HASSAN HAGHIGHI, SIAMAK YASSEMI, AND RAHIM ZAARE-NAHANDI

the Cohen-Macaulay property is equivalent to purity and connectedness of the simplicial complex. The Buchsbaum property is equivalent to being Cohen-Macaulay
or a pure flag complex. A non-Buchsbaum complex is CMt , t ≥ 2, if and only
if it is the join of a Buchsbaum complex with a (t − 2)-simplex. It also appears
that any CMt lexsegment complex for which the associated Stanley-Reisner ideal
is generated in degree d ≥ 3, is indeed Cohen-Macaulay. Our proofs are heavily
based on the results in [4] and particularly, on results in [5].
2. Preliminaries and notations
Let R = k[x1 , · · · , xn ] be the ring of polynomials in n variables over a field k
with standard grading. Let Md be the set of all squarefree monomials of degree d
in R. Consider the lexicographic ordering of monomials in R induced by the order
x1 > x2 > · · · > xn . A squarefree lexsegment monomial ideal in degree d is an ideal
generated by a lexsegment L(u, v) = {w ∈ Md : u ≥ w ≥ v} for some u, v ∈ Md
with u ≥ v.
Let ∆ be a simplicial complex on [n] = {1, · · · , n} with the Stanley-Reisner ring
k[∆]. Recall that for any face F ∈ ∆, the link of F in ∆ is defined as follows:
lk ∆ (F ) = {G ∈ ∆|G ∪ F ∈ ∆, G ∩ F = ∅}.
In the sequel by a complex we will always mean a simplicial complex. When
a complex has a quadratic Stanley-Reisner ideal, that is, it is the independence
complex of a graph,then it is called a flag complex.
A complex is said to satisfy the S2 condition of Serre if k[∆] satisfies the S2
condition. Using [13, Lemma 3.2.1] and Hochster’s formula on local cohomology
modules, a pure (d−1)-dimensional complex ∆ satisfies the S2 condition if and only
e 0 (link∆ (F ); k) = 0 for all F ∈ ∆ with #F ≤ d − 2 (see [14, page 4]). Therefore,
if H
∆ is S2 if and only if it is pure and link∆ (F ) is connected whenever F ∈ ∆ and
dim(link∆ (F )) ≥ 1.
Let t be an integer 0 ≤ t ≤ dim(∆) − 1. A pure complex ∆ is called CMt , or
Cohen-Macaulay in codimension t, over k if the complex lk ∆ (F ) is Cohen-Macaulay
over k for all F ∈ ∆ with #F ≥ t. It is clear that for any j ≥ i, CMi implies CMj .
For t ≥ 1, a CMt complex is said to be strictly CMt if it is not CMt−1 . A squarefree
monomial ideal is called CMt if the associated simplicial complex is CMt . Note that
from the results by Reisner [11] and Schenzel [13] it follows that CM0 is the same
as Cohen-Macaulayness and CM1 is identical with the Buchsbaum property.
A complex ∆ is said to be lexsegment if the associated Stanley-Reisner ideal
I∆ is a lexsegment ideal. Therefore, ∆ is a CMt lexsegment complex if I∆ is a
lexsegment ideal and ∆ is CMt .
3. CMt lexsegment complexes
The following result plays a significant role in the study of CMt lexsegment
complexes.
Theorem 3.1. [9, Theorem 3.1] Let ∆1 and ∆2 be two complexes of dimensions
r1 − 1 and r2 − 1, respectively. Then

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(i) The join complex ∆1 ∗ ∆2 is Cohen-Macaulay if and only if ∆1 and ∆2 are
both Cohen-Macaulay.
(ii) If ∆1 is Cohen-Macaulay and ∆2 is CMt for some t ≥ 1, then ∆1 ∗ ∆2 is
CMr1 +t (independent of r2 ). This is sharp, i.e., if ∆2 is strictly CMt , then
∆1 ∗ ∆2 is strictly CMr+t .
Let d ≥ 2 be an integer and let u ≥ v be in Md , u = xi xi2 · · · xid with i < i2 <
· · · < id . Let ∆ be a complex on [n] such that I∆ = (L(u, v)) ⊂ k[x1 , · · · , xn ].
Let ∆1 be the simplex on [i − 1] = {1, · · · , i − 1}, and let ∆2 be the complex of
the lexsegment ideal generated by L(u, v) as an ideal in k[xi , · · · , xn ]. Then it is
immediate that ∆ = ∆1 ∗ ∆2 . Observe that, ∆ is disconnected if and only if ∆1 = ∅
and ∆2 is disconnected. Similarly, ∆ is pure if and only if ∆2 is pure. Furthermore,
by Theorem 3.1 we have the following corollary.
Corollary 3.2. With the notation and assumption as above the following statements hold:
(i) The complex ∆ is Cohen-Macaulay if and only if ∆2 is Cohen-Macaulay.
In other words, to check the Cohen-Macaulay property of ∆ one may always
assume i = 1.
(ii) Assume that ∆2 is strictly CMt for some t ≥ 1. Then ∆ is strictly CMt+i .
In particular, a characterization of CMt squarefree lexsegment ideals with
i = 1 uniquely provides a characterization of CMt lexsegment ideals.
Remark 3.3. Corollary 3.2 substantially simplifies the statements and proofs of
[4] and [5].
Based on Corollary 3.2, unless explicitly specified, we will assume that i = 1.
Bonanzinga, Sorrenti and Terai [5] have given a characterization of CohenMacaulay squarefree lexsegment ideals in degree d ≥ 2. We give an improved
version of their result. Our proof is extracted from their proof.
Theorem 3.4. [5, An improved version of Theorem 3.4] Let u > v be in Md ,
I∆ = (L(u, v)). Then the following statements are equivalent:
(i) ∆ is shellable;
(ii) ∆ is Cohen-Macaulay;
(iii) ∆ is S2 ;
(iv) ∆ is pure and lk ∆ (F ) is connected for all F ∈ ∆ with dim(lk ∆ (F )) = 1.
Proof. Clearly, (iii)⇒(iv). Thus, by [5, Theorem 3.4] of Bonanzinga, Sorrenti and
Terai, we only need to prove (iv) ⇒ (i). As mentioned above, we may assume i = 1.
Checking all cases from the proof of (iii) ⇒ (iv) in [5, Theorem 3.4], it reveals that
when ∆ is pure, if u and v are not in the list (1),...,(7) in their theorem, then one
of the following cases (a) (with d ≥ 3), (b), or (c) specified in the their proof, may
occur: (a) lk ∆ ([n] \ {1, 2, d + 1, d + 2}) = h{1, 2}, {d + 1, d + 2}i,
(b) lk ∆ ({n − d + 1, · · · , k̂, k[
+ 1, · · · , n}) = h{1, 2}, {1, 3}, · · · , {1, n − d}, {k, k + 1}i,
where n − d + 1 ≤ k ≤ n − 1,
(c) lk ∆ ({n − d + 3, · · · , n}) = h{1, 2}, · · · , {1, n − d − 1}, {n − d, n − d + 1}, {n −
d, n − d + 2}, {n − d + 1, n − d + 2}i).

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In all these cases, lk ∆ (F ) is disconnected for some F ∈ ∆ with dim(lk ∆ (F )) = 1.
Therefore, assuming (iv) above, u and v will be in the list (1),...,(7) in their theorem.
Hence, by the proof of (iv) ⇒ (i) in their theorem, ∆ is shellable.

Remark 3.5. It is known that for some flag complexes including the independence
complex of a bipartite graph or a chordal graph, the conditions S2 and CohenMacaulay-ness are equivalent [7]. Nevertheless, the condition (d) is in general
weaker than the S2 property. For example, if ∆ = h{1, 2, 3, 4}, {1, 5, 6.7}i, then
∆ satisfies the condition (d) but does not satisfy the S2 property. Indeed, any onedimensional face has a connected one-dimensional link, but lk ∆ ({1}) is disconnected
of dimension 2.
For d = 2, the statement in Theorem 3.4(iv) could be relaxed as follows. The
assumption i = 1 is still in order.
Theorem 3.6. Let u > v be in M2 , I∆ = (L(u, v)). Then the following statements
are equivalent:
(i) ∆ is shellable;
(ii) ∆ is Cohen-Macaulay;
(iii) ∆ is S2 ;
(iv) ∆ is pure and connected.
Proof. We only need to check (iv) ⇒ (i). Once again, checking the proof of [5,
Theorem 3.4], the case (a) could not occur for d = 2. In case (b), it follows
that u = x1 xn−1 , v = xn−2 xn with n > 3. Then, ∆ = h{1, 2}, {n − 1, n}i is
disconnected. In case (c), it turns out that u = x1 xn−2 , v = xn−2 xn−1 with n > 4.
Then, ∆ = h{1, 2}, {n−2, n}, {n−1, n}i is again disconnected. Therefore, assuming
purity and connectedness of ∆, u and v will be in the list (1),...,(7) of [5, Theorem
3.4]. Hence, by the proof of (iv) ⇒ (i) of the same theorem, ∆ is shellable.

Let u > v be in Md , I∆ = (L(u, v)). Bonanzinga, Sorrenti and Terai in [5] have
shown that for d ≥ 3, ∆ is Buchsbaum if and only if it is Cohen-Macaulay. The
same proof implies that this is the case for CMt lexsegment complexes.
Proposition 3.7. Let u > v be in Md with d ≥ 3 and I∆ = (L(u, v)). Then for
any t ≥ 0, ∆ is CMt if and only if ∆ is Cohen-Macaulay.
Proof. Clearly any Cohen-Macaulay complex is CMt . Assume that ∆ is CMt . Then
as noticed in the proof [5, Thorem 4.1], for d ≥ 3, depthk[∆] ≥ 2. Hence ∆ is S2 .
Therefore, by Theorem 3.6, ∆ is Cohen-Macaulay.

By Proposition 3.7 to check the CMt property with t ≥ 1, for squarefree lexsegment ideals we should restrict to the case d = 2.
We now drop the assumption i = 1 and assume that u > v are in M2 , u = xi xj ,
v = xr xs with i < j and r < s. Let I∆ = (L(u, v)) ⊂ k[x1 , · · · , xn ]. Recall that if
∆1 is the simplex on [i−1], and ∆2 is the complex of the lexsegment ideal generated
by L(u, v) as an ideal in k[xi , · · · , xn ]. Then ∆ = ∆1 ∗ ∆2 .
Theorem 3.8. Let u > v be in M2 , I∆ = (L(u, v)). Assume that ∆ is not CohenMacaulay. Then the following statements are equivalent:
(1) ∆ is Buchsbaum;

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(2) One of the following conditions hold;
(a) u = x1 xn−2 , v = xn−2 xn−1 , n > 4;
(b) u = x1 xn−1 , v = xn−2 xn , n > 3.
(3) ∆1 = ∅ and ∆2 is pure.
Furthermore, in either of these equivalent cases, ∆ is disconnected.
Proof. The equivalence of (1) and (2) is given in [4, Theorem 2.1]. Assuming (2),
then ∆1 = ∅, and as shown in the proof of Theorem 3.6, ∆ = ∆2 is pure in both
cases (a) and (b). Thus (2) ⇒ (3). Now if ∆1 = ∅ and ∆2 is pure, as it was
observed in the proof of Theorem 3.6, the only cases where ∆ is pure but not
Cohen-Macaulay are the cases (a) and (b) above, which settles (3) ⇒ (2). For
the last statement, observe that if ∆ = ∆2 also happens to be connected, then by
Theorem 3.6, ∆ is Cohen-Macaulay. But this is contrary to the assumption. Hence
∆ is disconnected.

Theorem 3.9. Let u > v be in M2 , I∆ = (L(u, v)). Let ∆ = ∆1 ∗ ∆2 be as above.
Let t ≥ 2. Assume that ∆ is not CMt−1 . Then the following is equivalent:
(i) ∆ is CMt ;
(ii) One of the following conditions hold;
(a) u = xt xn−2 , v = xn−2 xn−1 , n > 4;
(b) u = xt xn−1 , v = xn−2 xn , n > 3.
(iii) ∆1 is of dimension t − 2 and ∆2 is pure.
(iv) ∆1 is of dimension t − 2 and ∆2 is Buchsbaum.
Proof. The equivalence of (ii), (iii) and (iv) follows by applying Theorem 3.8 to
∆2 . Assuming (iv), the statement (i) follows by Corollary 3.2. Now assume (i).
Let u = xi xj , v = xr xs . Then since ∆ is CMt , it is pure, and hence ∆2 is pure. But
since ∆ is not Cohen-Macaulay, ∆2 can not be Cohen-Macaulay. Thus by Theorem
3.8, ∆2 is Buchsbaum but not Cohen-Macaulay. Now since t ≥ 2 and ∆ is not
CMt−1 , it follows that ∆ 6= ∅ and i ≥ 2. Hence by 3.2 ∆ = ∆1 ∗ ∆2 is CMi but not
CMi−1 . Therefore, i = t and ∆1 is of dimension t − 2.

Acknowledgments
Part of this note was prepared during a visit of Institut de Mathématiques de
Jussieu, Université Pierre et Marie Curie by the second and the third author. This
visit was supported by Center for International Studies & Collaborations(CISSC)
and French Embassy in Tehran in the framework of the Gundishapur project
27462PL on the Homological and Combinatorial Aspects of Commutative Algebra. The third author has been supported by research grant no. 4/1/6103011 of
University of Tehran.
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Hassan Haghighi, Department of Mathematics, K. N. Toosi University of Technology,
Tehran, Iran.
Siamak Yassemi, School of Mathematics, Statistics & Computer Science, University
of Tehran, Tehran Iran.
Rahim Zaare-Nahandi, School of Mathematics, Statistics & Computer Science, University of Tehran, Tehran, Iran.

