A GENERALIZATION OF THE SWARTZ EQUALITY

M. R. POURNAKI, S. A. SEYED FAKHARI, AND S. YASSEMI

Abstract. For a given (d − 1)-dimensional simplicial complex Γ, we denote its hvector by h(Γ) = (h_{0}(Γ), h_{1}(Γ), …, h_{d}(Γ)) and set h_{−1}(Γ) = 0. The known Swartz equality implies that if Δ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ih_{i}(Δ)+(d−i+1)h_{i−1}(Δ) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25 (1996), no. 1, 137–148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18 (2004/05), no. 3, 647–661), which in turn leads to a generalization of the abovementioned inequalities for Cohen–Macaulay simplicial complexes in co-dimension t.

1. Introduction and preliminaries

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra in various combinatorial contexts and vice versa. One of the fastest developing subfields within algebraic combinatorics is combinatorial commutative algebra. It has evolved into one of the most active areas of mathematics during the past several decades. We refer the reader to the works of Bruns and Herzog [1] and Stanley [3] as general references on the subject.

Let us start with some preliminaries. A simplicial complex Δ 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 ∈ Δ and F′ ⊆ F, then also F′ ∈ Δ. Every element F ∈ Δ is called a face of Δ, 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 Δ, which is denoted by dim Δ, is defined to be d − 1, where d = max{|F | | F ∈ Δ}. A facet of Δ is a maximal face of Δ with respect to inclusion. We say that Δ is pure if all facets of Δ have the same cardinality. The link of Δ with respect to a face F ∈ Δ, denoted by lk_{Δ}(F), is the simplicial complex lk_{Δ}(F) = {G ⊆ [n] \ F | G ∪ F ∈ Δ}.

2000 Mathematics Subject Classification. Primary: 13F55, 05E45; Secondary: 13C14. Key words and phrases. Simplicial complex, h-Vector, Swartz equality, CM_{t} simplicial complex. The research of M. R. Pournaki, S. A. Seyed Fakhari, and S. Yassemi was in part supported by a

grant from IPM (No. 91130115, No. 92130422, and No. 91130214). This work has been also partially supported by the Center for International Scientific Studies and Collaboration (CISSC), Iran, and Égide – Programme Gundishapur 2012, Hubert Curien Partnership, France (No. 27462PL).

One of the connections between combinatorics and commutative algebra is via rings constructed from the combinatorial objects. Let R = 𝕂[x_{1}, …, x_{n}] be the polynomial ring in n variables over a field 𝕂, and let Δ be a simplicial complex on [n]. For every subset F ⊆ [n], we set x_{F} = ∏_{i∈F} x_{i}. The Stanley–Reisner ideal of Δ over 𝕂 is the ideal I_{Δ} of R which is generated by square-free monomials x_{F} with F ∉ Δ. The Stanley–Reisner ring of Δ over 𝕂, denoted by 𝕂[Δ], is defined as 𝕂[Δ] = R/I_{Δ}. A simplicial complex Δ is called Cohen–Macaulay over 𝕂 (resp. Buchsbaum over 𝕂), if its Stanley–Reisner ring 𝕂[Δ] is a Cohen–Macaulay ring (resp. a Buchsbaum ring).

We now record the numerical data associated with a (d − 1)-dimensional simplicial complex Δ. Let f_{i}(Δ) denote the number of faces of Δ of dimension i. The sequence f(Δ) = (f_{0}(Δ), f_{1}(Δ), …, f_{d−1}(Δ)) is called the f-vector of Δ. Also, to find out more about f(Δ), we need to study the h-vector of Δ. Although it is seemingly complicated, often it is an elegant way to record the face numbers. Letting f_{−1}(Δ) = 1, we define the h-vector h(Δ) = (h_{0}(Δ), h_{1}(Δ), …, h_{d}(Δ)) of Δ by the formula

∑_{i=0}^{d} h_{i}(Δ)t^{d−i} = ∑_{i=0}^{d} f_{i−1}(Δ)(t − 1)^{d−i},

and for every i < 0, we set h_{i}(Δ) = 0. One can easily check that

f_{i−1}(Δ) = ∑_{j=0}^{i} \binom{d − j}{i − j}h_{j}(Δ)

and

h_{i}(Δ) = ∑_{j=0}^{i}(−1)^{i−j}\binom{d − j}{i − j}f_{j−1}(Δ).

It is well known that

H_{𝕂[Δ]}(t) = \frac{∑_{i=0}^{d} h_{i}(Δ)t^{i}}{(1 − t)^{d}} ,

where H_{𝕂[Δ]}(t) is the Hilbert series of 𝕂[Δ] and d = dim Δ + 1 is its Krull dimension.

The study of h-vectors of simplicial complexes has long been a topic of interest both in combinatorics and combinatorial commutative algebra. In this direction, let us state an equality, called the Swartz equality.

Proposition 1.1 ([4], Proposition 2.3). Let Δ be a pure (d−1)-dimensional simplicial complex. Then for every 0 ≤ i ≤ d, we have

ih_{i}(Δ) + (d − i + 1)h_{i−1}(Δ) = ∑_{v∈[n]} h_{i−1}(lk_{Δ}(v)).

It is well known that Δ is Buchsbaum over a field 𝕂 if and only if it is pure and for every non-empty face F of Δ, the link of Δ with respect to F is Cohen–Macaulay over 𝕂 (see [3, Theorem 8.1, Page 73]). Also, by a classical result of Stanley, if Δ is Cohen–Macaulay over 𝕂 with dimension d − 1, then for every 0 ≤ i ≤ d, we have h_{i}(Δ) ≥ 0 (see [3, Theorem 3.3, Page 59]). By combining these facts with Proposition 1.1, the Swartz equality, we conclude the following corollary.

Corollary 1.2. Let Δ be a (d − 1)-dimensional Buchsbaum simplicial complex over a field 𝕂. Then for every 0 ≤ i ≤ d, we have

ih_{i}(Δ) + (d − i + 1)h_{i−1}(Δ) ≥ 0.

Terai in [5] has proved some inequalities for the h-vectors of Buchsbaum simplicial complexes. In Proposition 2.1, by using Corollary 1.2, we give a simple proof for Terai’s result, which shows that this corollary is stronger than his result. This is a motivation for us to see whether we can extend Corollary 1.2 for a larger class of simplicial complexes than that of Buchsbaum. Recently, Haghighi et al. [2] have defined the notion of CM_{t} simplicial complexes as a generalization of Cohen–Macaulay and Buchsbaum simplicial complexes (see Definition 2.2). This notion is different from that of the so-called t-CM, which may be found in, for example, [4]. The notion of CM_{0} simplicial complexes agrees with the notion of Cohen–Macaulay simplicial complexes, and the notion of CM_{1} simplicial complexes agrees with that of Buchsbaum simplicial complexes. In Proposition 2.4, we give a generalization of Corollary 1.2 for CM_{t} simplicial complexes. In order to do this, we need to generalize Proposition 1.1, the Swartz equality, which we state and prove in Theorem 2.3.

2. The results

We start this section by stating the following proposition, which has been already proved by Terai [5]. Here we give a simple proof for this proposition by using Corollary 1.2.

Proposition 2.1 ([5], Theorem 2.1). Let 𝕂 be a field and Δ be a (d − 1)-dimensional Buchsbaum simplicial complex over 𝕂. Then the following inequalities hold true.

(1) dh_{d}(Δ) + h_{d−1}(Δ) ≥ 0,

(2) \binom{d}{2}h_{d}(Δ) + (d − 1)h_{d−1}(Δ) + h_{d−2}(Δ) ≥ 0,

(3) \binom{d}{3}h_{d}(Δ) + \binom{d − 1}{2}h_{d−1}(Δ) + (d − 2)h_{d−2}(Δ) + h_{d−3}(Δ) ≥ 0,

(4) ∑_{k=0}^{i}\binom{d + i − k − 3}{i − k}h_{d−k}(Δ) ≥ 0, if i ≥ 4.

Proof. By Corollary 1.2, for every 0 ≤ i ≤ d, we have ih_{i}(Δ) + (d − i + 1)h_{i−1}(Δ) ≥ 0. This implies that for every 1 ≤ i ≤ d, we have

0 ≤ \frac{1}{i} ∑_{k=1}^{i} \binom{d − k}{i − k}((d − k + 1)h_{d−k+1}(Δ) + kh_{d−k}(Δ))

= \frac{d}{i}\binom{d − 1}{i − 1}h_{d}(Δ) + \frac{1}{i} ∑_{k=1}^{i−1} \{k\binom{d − k}{i − k} + (d − k)\binom{d − k − 1}{i − k − 1}\}h_{d−k}(Δ)

+h_{d−i}(Δ),

and so we conclude that

(†)  ∑_{k=0}^{i} \binom{d − k}{i − k}h_{d−k}(Δ) ≥ 0.

Now the proof of (1), (2) and (3) follows from inequality (†) by substituting i = 1, i = 2 and i = 3 respectively. For proof of (4), by inequality (†), for every i ≥ 4, we have

0 ≤ ∑_{j=3}^{i} ∑_{k=0}^{j} \binom{i − 3}{i − j}\binom{d − k}{j − k}h_{d−k}(Δ)

= ∑_{j=3}^{i} \binom{i − 3}{i − j}\binom{d}{j}h_{d}(Δ) + ∑_{j=3}^{i} \binom{i − 3}{i − j}\binom{d − 1}{j − 1}h_{d−1}(Δ)

+ ∑_{j=3}^{i} \binom{i − 3}{i − j}\binom{d − 2}{j − 2}h_{d−2}(Δ) + ∑_{k=3}^{i} ∑_{j=k}^{i} \binom{i − 3}{i − j}\binom{d − k}{j − k}h_{d−k}(Δ)

= \binom{d + i − 3}{i}h_{d}(Δ) + \binom{d + i − 4}{i − 1}h_{d−1}(Δ) + \binom{d + i − 5}{i − 2}h_{d−2}(Δ)

+ ∑_{k=3}^{i} \binom{d + i − k − 3}{i − k}h_{d−k}(Δ)

= ∑_{k=0}^{i} \binom{d + i − k − 3}{i − k}h_{d−k}(Δ),

as required. □

Let us define the notion of CM_{t} simplicial complexes.

Definition 2.2. Let t be a non-negative integer, 𝕂 a field and Δ a pure simplicial complex. We say that Δ is a CM_{t} simplicial complex over 𝕂 if for every F ∈ Δ with |F | ≥ t, the link of Δ with respect to F is Cohen–Macaulay over 𝕂.

We now generalize Corollary 1.2 for CM_{t} complexes. In order to do this, we state and prove the following theorem, which is a generalization of Proposition 1.1, the Swartz equality.

Theorem 2.3. Let Δ be a pure (d − 1)-dimensional simplicial complex. Then for every 1 ≤ t ≤ d and every 0 ≤ i ≤ d, we have

∑_{\begin{matrix}F ∈ Δ \\ |F | = t \end{matrix}} h_{i−t}(lk_{Δ}(F)) = ∑_{k=0}^{t} \binom{i − k}{t − k}\binom{d − i + k}{k}h_{i−k}(Δ).

Proof. We prove the theorem by induction on t. The case t = 1 follows from Proposition 1.1. We now assume that t ≥ 2. Suppose that the assertion is true for t − 1. Therefore, we have

∑_{\begin{matrix}F ∈ Δ \\ |F | = t − 1 \end{matrix}} h_{i−t+1}(lk_{Δ}(F)) = ∑_{k=0}^{t−1} \binom{i − k}{t − k − 1}\binom{d − i + k}{k}h_{i−k}(Δ).

By using Proposition 1.1, we conclude that

∑_{\begin{matrix}F ∈ Δ \\ |F | = t \end{matrix}} h_{i−t}(lk_{Δ}(F)) = \frac{1}{t} ∑_{\begin{matrix}F ∈ Δ \\ |F | = t − 1 \end{matrix}} ∑_{u∈lk_{Δ}(F)} h_{i−t}(lk_{lk_{Δ}(F)}(u))

= \frac{1}{t} ∑_{\begin{matrix}F ∈ Δ \\ |F | = t − 1 \end{matrix}} \{(i − t + 1)h_{i−t+1}(lk_{Δ}(F))

+(d − t + 1 − i + t)h_{i−t}(lk_{Δ}(F))}

= \frac{1}{t}(i − t + 1) ∑_{\begin{matrix}F ∈ Δ \\ |F | = t − 1 \end{matrix}} h_{i−t+1}(lk_{Δ}(F))

+\frac{1}{t}(d − i + 1) ∑_{\begin{matrix}F ∈ Δ \\ |F | = t − 1 \end{matrix}} h_{i−t}(lk_{Δ}(F)).

Now, the induction hypothesis implies that

∑_{\begin{matrix}F ∈ Δ \\ |F | = t \end{matrix}} h_{i−t}(lk_{Δ}(F)) = \frac{1}{t}(i − t + 1) ∑_{k=0}^{t−1} \binom{i − k}{t − k − 1}\binom{d − i + k}{k}h_{i−k}(Δ)

+\frac{1}{t}(d − i + 1) ∑_{k=0}^{t−1} \binom{i − k − 1}{t − k − 1}\binom{d − i + k + 1}{k}h_{i−k−1}(Δ)

= \binom{i}{t}h_{i}(Δ) + ∑_{k=1}^{t−1} \{\frac{1}{t}(i − t + 1)\binom{i − k}{t − k − 1}\binom{d − i + k}{k}

+\frac{1}{t}(d − i + 1)\binom{i − k}{t − k}\binom{d − i + k}{k − 1}\}h_{i−k}(Δ)

+\binom{d − i + t}{t}h_{i−t}(Δ)

= \binom{i}{t}h_{i}(Δ) + ∑_{k=1}^{t−1} \binom{i − k}{t − k}\binom{d − i + k}{k}h_{i−k}(Δ)

+\binom{d − i + t}{t}h_{i−t}(Δ)

= ∑_{k=0}^{t} \binom{i − k}{t − k}\binom{d − i + k}{k}h_{i−k}(Δ),

as required. □

Since every component of the h-vector of a Cohen–Macaulay simplicial complex over a field 𝕂 is non-negative, Theorem 2.3 implies the following generalization of Corollary 1.2.

Proposition 2.4. Let t be a non-negative integer and 𝕂 a field. Let Δ be a (d − 1)- dimensional CM_{t} simplicial complex over 𝕂. Then for every 0 ≤ i ≤ d, we have

∑_{k=0}^{t} \binom{i − k}{t − k}\binom{d − i + k}{k}h_{i−k}(Δ) ≥ 0.

Acknowledgment

The authors would like to thank the referee for the comments and suggestions.

References

[1] W. Bruns, J. Herzog, Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.

[2] H. Haghighi, S. Yassemi, R. Zaare-Nahandi, A generalization of k-Cohen–Macaulay simplicial complexes, Ark. Mat. 50 (2012), no. 2, 279–290.

[3] R. P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Progress in Mathematics, 41, Birkhäuser Boston, Inc., Boston, MA, 1996.

[4] E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18 (2004/05), no. 3, 647–661.

[5] N. Terai, On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25 (1996), no. 1, 137–148.

M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@ipm.ir
URL: http://math.ipm.ac.ir/pournaki/

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/

S. Yassemi, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

E-mail address: yassemi@ipm.ir
URL: http://math.ipm.ac.ir/yassemi/