ON THE h-TRIANGLES OF SEQUENTIALLY (S_{r}) SIMPLICIAL COMPLEXES VIA ALGEBRAIC SHIFTING

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

Dedicated with gratitude to our teacher and friend Jürgen Herzog on the occasion of his 70th birthday

Abstract. Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (S_{r}) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (S_{r}). Let Δ be a (d − 1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (h_{i,j}(Δ))_{0≤j≤i≤d} be the h-triangle of Δ and (h_{i,j}(Γ(Δ)))_{0≤j≤i≤d} be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (S_{r}) and for every i and j with 0 ≤ j ≤ i ≤ r − 1, the equality h_{i,j}(Δ) = h_{i,j}(Γ(Δ)) holds true.

1. Introduction and preliminaries

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra in various combinatorial contexts. 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 last several decades. Throughout the paper we deal with algebraic shifting and h-triangles of simplicial complexes. The study of these concepts has long been a topic of interest both in combinatorics and combinatorial commutative algebra. In this section, we recall some definitions and notation concerning combinatorics and combinatorial commutative algebra for later use. We refer the reader to the books by Stanley [13], Bruns and Herzog [2], Miller and Sturmfels [11], as well as Herzog and Hibi [8] as general references in the subject.

1.1. Notions from combinatorics. 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}. The degree of a face F ∈ Δ, denoted deg F, is defined to be deg F = max{|G| | F ⊆ G and G ∈ Δ}. Also, the degree of Δ, denoted deg Δ, is defined to be deg Δ = min{deg F | F ∈ Δ}. A facet of Δ is a maximal face of Δ with respect to inclusion. Let F(Δ) denote the set of facets of Δ. It is clear that F(Δ) determines Δ. When F(Δ) = {F_{1}, …, F_{m}}, we write Δ = ⟨F_{1}, …, F_{m}⟩. We say that Δ is pure if all facets of Δ have the same cardinality. Note that Δ is pure if and only if all of its faces have the same degree. A nonface of Δ is a subset F of [n] with F ∉ Δ. Let N (Δ) denote the set of minimal nonfaces of Δ with respect to inclusion. 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}. Let Δ and Δ′ be two simplicial complexes on disjoint vertex sets V and W, respectively. The join Δ ∗ Δ′ is the simplicial complex on the vertex set V ∪ W with faces F ∪ G, where F ∈ Δ and G ∈ Δ′.

2000 Mathematics Subject Classification. Primary: 05E45, 13F55; Secondary: 13H10.

Key words and phrases. Simplicial complex, Simplicial complex with (S_{r}) property, Sequentially (S_{r}) simplicial complex, Algebraic shifting, h-Triangle.

The research of M. R. Pournaki and S. Yassemi was in part supported by a grant from IPM (No.

90130073 and No. 90130214).

Let Δ be a (d − 1)-dimensional simplicial complex and let −1 ≤ i, j ≤ d − 1. In 1996, Björner and Wachs (see [1, Definition 2.8]) defined Δ^{(i,j)} as

Δ^{(i,j)} = {F ∈ Δ | deg F ≥ i + 1 and dim F ≤ j}.

One can extend this definition to the case i > d−1, by defining Δ^{(i,j)} to be the empty simplicial complex. Clearly, Δ^{(i,j)} is a subcomplex of Δ. Throughout, we consider the following special cases: (i) the simplicial complex Δ^{(i)} := Δ^{(−1,i)}, called the ith skeleton of Δ; (ii) the simplicial complex Δ^{⟨i⟩} := Δ^{(i,d−1)}, called ith sequential layer of Δ; (iii) the simplicial complex Δ^{[i]} := Δ^{(i,i)}, called the ith pure skeleton of Δ. Note that Δ^{⟨i⟩} is the subcomplex of Δ generated by all facets whose dimension is at least i. That is, in fact, the subcomplex of all faces of Δ whose degree is at least i + 1. Also, Δ^{[i]} is the pure subcomplex of Δ generated by all i-dimensional faces. The notation Δ^{[i]} is due to Wachs [14].

1.2. Notions from combinatorial commutative algebra. 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 is the ideal I_{Δ} of R which is generated by those squarefree monomials x_{F} with F ∉ Δ. In other words, I_{Δ} = ⟨x_{F} | F ∈ N (Δ)⟩. The Stanley–Reisner ring of Δ, denoted by 𝕂[Δ], is defined to be 𝕂[Δ] = R/I_{Δ}.

h-TRIANGLES OF SEQUENTIALLY (S_{r}) SIMPLICIAL COMPLEXES 3

We now recall two notions related to commutative algebra. Let M be a nonzero finitely generated R-module. We say that M is Cohen–Macaulay, if for every 𝔭 ∈ Spec(R), the equality depth M_{𝔭} = dim M_{𝔭} holds true. Also M is said to satisfy Serre’s condition (S_{r}), or simply M is an (S_{r}) module, if for every 𝔭 ∈ Spec(R), the inequality depth M_{𝔭} ≥ min{r, dim M_{𝔭}} holds true. It is easy to see that M is Cohen–Macaulay if and only if it is an (S_{r}) module for all r ≥ 1. We say that a simplicial complex Δ is Cohen–Macaulay if the Stanley–Reisner ring of Δ is Cohen–Macaulay. Also Δ is said to satisfy Serre’s condition (S_{r}), or simply Δ is an (S_{r}) simplicial complex, if its Stanley–Reisner ring satisfies Serre’s condition (S_{r}). Since every simplicial complex satisfies Serre’s condition (S_{1}), we assume that r ≥ 2. It is well known that if Δ is an (S_{r}) simplicial complex, then Δ is pure (see [12, Lemma 2.6]).

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

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

One can easily check that

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

and

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

It is well known that if h(Δ) = (h_{0}, h_{1}, …, h_{d}) is the h-vector of Δ and H_{𝕂[Δ]}(t) is the Hilbert series of 𝕂[Δ], then we have

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

where d = dim Δ + 1 is the Krull dimension of 𝕂[Δ].

2. Simplicial complexes satisfying the (d_{r}) property

Algebraic shifting is a procedure that to every simplicial complex Δ corresponds a new simplicial complex Γ(Δ), called the shifted complex of Δ, with the same h-vector as Δ and a nice combinatorial structure. Additionally, algebraic shifting preserves many algebraic and topological properties of the original complex, including Cohen– Macaulayness: a simplicial complex Δ is Cohen–Macaulay if and only if Γ(Δ) is Cohen–Macaulay, which, in turn, holds if and only if Γ(Δ) is pure (see Proposition 2.3).

Notation and Remarks 2.1. If S = {s_{1} < ⋯ < s_{j}} and T = {t_{1} < ⋯ < t_{j}} are two j-subsets of integers, then S ≤_{P} T under the standard partial order if s_{p} ≤ t_{p} for all p. Also S <_{L} T under the lexicographic order if there is a q such that s_{q} < t_{q} and s_{p} = t_{p} for all p with p < q. A collection C of two j-subsets S and T of integers is shifted if S ≤_{P} T and T ∈ C together imply that S ∈ C. A simplicial complex Δ is shifted if the set of j-dimensional faces of Δ is shifted for every j. In other words, a simplicial complex Δ on [n] is shifted if, for F ∈ Δ, i ∈ F and j ∈ [n] with j < i, one has (F \ {i}) ∪ {j} ∈ Δ.

Definitions and Remarks 2.2. Let Δ be a simplicial complex with the set of vertices V = {e_{1}, …, e_{n}} linearly ordered e_{1} < ⋯ < e_{n}. For a given field 𝕂, let ⋀ 𝕂V denote the exterior algebra of the vector space 𝕂V; it has a 𝕂-vector space basis consisting of all the monomials e_{S} := e_{i_{1}} ∧ ⋯ ∧ e_{i_{j}}, where S = {e_{i_{1}} < ⋯ < e_{i_{j}}} ⊆ V and e_{∅} = 1. Let I(Δ) be the ideal of ⋀ 𝕂V generated by the set {e_{S} | S ∉ Δ}, and let x̃ denote the image of x ∈ 𝕂V modulo I(Δ). Let {f_{1}, …, f_{n}} be a generic basis of 𝕂V, i.e., f_{i} = ∑_{j=1}^{n} α_{i_{j}}e_{j}, where the α_{i_{j}}’s are n^{2} transcendentals, algebraically independent over 𝕂. Define f_{S} := f_{i_{1}} ∧ ⋯ ∧ f_{i_{k}} for S = {i_{1} < ⋯ < i_{k}} and set f_{∅} = 1. Let

Γ(Δ) := {S ⊆ [n] | f̃_{S} ∉ Span{f̃_{R} | R <_{L} S}}

be the algebraically shifted complex obtained from Δ. As the name implies, Γ(Δ) is a shifted simplicial complex, and it is independent of the numbering of the vertices of Δ or the choices of α_{i_{j}}.

As is often the case with algebraic shifting, we do not use the definition directly, but rather some theorems that characterize the results of algebraic shifting.

We continue this section by stating the following result. This result is just a translation of a purity result given by Kalai for algebraic shifting of Cohen–Macaulay simplicial complexes in terms of degree.

Proposition 2.3 (Kalai). Let Δ be a (d − 1)-dimensional simplicial complex and let Γ(Δ) be its algebraic shifting. Then Δ is Cohen–Macaulay if and only if deg Γ(Δ) = d.

Proof. It is known that Δ is Cohen–Macaulay if and only if Γ(Δ) is pure (see [10]). Therefore, Δ is Cohen–Macaulay if and only if Γ(Δ) has no facet of dimension less than d − 1, which holds if and only if deg Γ(Δ) = d, as requested. □

We now give the following definition to obtain an extension of Proposition 2.3.

Definition 2.4. Let 𝕂 be a field, Δ be a simplicial complex and let 𝕂[Δ] be its Stanley–Reisner ring. Then we say that Δ is a (d_{r}) simplicial complex (over 𝕂), provided depth 𝕂[Δ] ≥ r.

By using Exercise 5.1.23 of [2] and Corollary 4.5 of [3] we may conclude the following proposition.

Proposition 2.5. Let Δ be a simplicial complex and let Γ(Δ) be its algebraic shifting. Then Δ is (d_{r}) if and only if deg Γ(Δ) ≥ r.

Remark 2.6. Note that Proposition 2.5 is a generalization of Proposition 2.3. In order to see this, let 𝕂 be a field, Δ be a (d − 1)-dimensional simplicial complex and 𝕂[Δ] be its Stanley–Reisner ring. Then Δ is Cohen–Macaulay if and only if depth 𝕂[Δ] ≥ d, which holds if and only if Δ is (d_{d}). Therefore, by Proposition 2.5, Δ is Cohen–Macaulay if and only if deg Γ(Δ) ≥ d. But by the definition of degree, deg Γ(Δ) is always less than or equal to d, and therefore, we conclude that Δ is Cohen–Macaulay if and only if deg Γ(Δ) = d, which is Proposition 2.3.

We now state and prove the following theorem which gives us a necessary and sufficient condition for a simplicial complex to be (d_{r}) in terms of reduced homology. For the definition of reduced homology we refer the reader to [2].

Theorem 2.7. Let Δ be a (d−1)-dimensional simplicial complex and let 𝕂 be a field. Then Δ is (d_{r}) if and only if H̃_{i}(lk_{Δ}(σ); 𝕂) = 0 for all σ ∈ Δ with |σ| ≤ r and for all i with i < dim lk_{Δ^{(r−1)}}(σ).

Proof. By using Exercise 5.1.23 of [2] we conclude that Δ is (d_{r}) if and only if Δ^{(r−1)} is Cohen–Macaulay. But Reisner’s criterion (see [2, Corolarry 5.3.9]) implies that the latter is equivalent to the vanishing of H̃_{i}(lk_{Δ^{(r−1)}}(σ); 𝕂) for all σ ∈ Δ^{(r−1)} and all i with i < dim lk_{Δ^{(r−1)}}(σ). Since σ ∈ Δ^{(r−1)} is equivalent to |σ| ≤ r, we obtain that Δ is (d_{r}) if and only if H̃_{i}(lk_{Δ^{(r−1)}}(σ); 𝕂) = 0 for all σ ∈ Δ with |σ| ≤ r and all i with i < dim lk_{Δ^{(r−1)}}(σ). But for every σ with |σ| ≤ r we have

lk_{Δ^{(r−1)}}(σ) = (lk_{Δ}(σ))^{(r−|σ|−1)},

and so for every i with i < dim lk_{Δ^{(r−1)}}(σ) we conclude that

H̃_{i}(lk_{Δ^{(r−1)}}(σ); 𝕂) = H̃_{i}(lk_{Δ}(σ); 𝕂).

This implies that Δ is (d_{r}) if and only if H̃_{i}(lk_{Δ}(σ); 𝕂) = 0 for all σ ∈ Δ with |σ| ≤ r and all i with i < dim lk_{Δ^{(r−1)}}(σ), as requested. □

In the next proposition we collect some basic properties of (d_{r}) simplicial complexes.

Proposition 2.8. Let Δ and Δ′ be two simplicial complexes. Then the following statements hold :

(1) if Δ is (d_{r}), then Δ^{(i)} is (d_{s}) for all i, where s = min{r, i + 1};

(2) if Δ is (d_{r}) and Δ′ is (d_{r′}), then Δ ∗ Δ′ is (d_{r+r′});

(3) if Δ is (d_{r}), then lk_{Δ}(v) is (d_{r−1}) for every vertex v of Δ.

Proof. We easily conclude (1) by using [2, Exercise 5.1.23]. Also (2) is proved in [5, Lemmas 1 and 2]. For proving (3), we may assume that r ≥ 2; otherwise, there is nothing to prove. Note that by Theorem 2.7, it is enough to prove that for a given field 𝕂, H̃_{i}(lk_{lk_{Δ}(v)}(σ); 𝕂) = 0 for all σ ∈ lk_{Δ}(v) with |σ| ≤ r − 1 and all i with i < dim lk_{(lk_{Δ}(v))^{(r−2)}}(σ). This is also equivalent to showing that for every σ ∈ lk_{Δ}(v) with |σ| ≤ r − 1 and every i with i < dim lk_{(lk_{Δ}(v))^{(r−2)}}(σ), H̃_{i}(lk_{Δ}(σ ∪ {v}); 𝕂) = 0. But i < dim lk_{(lk_{Δ}(v))^{(r−2)}}(σ) implies that i < dim lk_{Δ^{(r−1)}}(σ ∪ {v}), and so Theorem 2.7 completes the proof (see also [12, Lemma 4.3] for another proof). □

We now state and prove the following lemma for later use.

Lemma 2.9. Let Δ be an (S_{r}) simplicial complex and let Δ^{(i)} be its i-th skeleton. Then Δ^{(i)} is Cohen–Macaulay for every i with 1 ≤ i ≤ r − 1.

Proof. Since Δ is an (S_{r}) simplicial complex, Proposition 2.3 of [7] implies that Δ^{(r−1)} is also (S_{r}). On the other hand, dim Δ^{(r−1)} = r − 1. Therefore, we conclude that Δ^{(r−1)} is Cohen–Macaulay. For every i with 1 ≤ i < r − 1, we have Δ^{(i)} = (Δ^{(r−1)})^{(i)}. Also, by Corollary 24.4 of [9], every skeleton of a Cohen–Macaulay simplicial complex is again Cohen–Macaulay. Hence, we conclude that Δ^{(i)} is Cohen–Macaulay. All in all, we obtain that for every i with 1 ≤ i ≤ r − 1, Δ^{(i)} is Cohen–Macaulay, as requested. □

Let 𝕂 be a field, Δ be a simplicial complex and let 𝕂[Δ] be its Stanley–Reisner ring. By using Exercise 5.1.23 of [2] we conclude that

depth 𝕂[Δ] = max {i + 1 | Δ^{(i)} is Cohen − Macaulay}.

Therefore, Lemma 2.9 implies that if Δ is (S_{r}), then depth 𝕂[Δ] ≥ r and so Δ is (d_{r}). This means that

(S_{r}) ⟹ (d_{r}).

We now apply the above observation to one implication of Proposition 2.5 to get the following proposition, which in turn is another generalization of Proposition 2.3.

Proposition 2.10. Let Δ be a simplicial complex and let Γ(Δ) be its algebraic shifting. If Δ is (S_{r}), then deg Γ(Δ) ≥ r.

Note that the converse of Proposition 2.10 is not true in general. In order to see this, let Δ be an (S_{r}) simplicial complex which is not Cohen–Macaulay. Then Γ(Δ) is not pure, while by Proposition 2.10 we have deg Γ(Δ) ≥ r. Hence Γ(Γ(Δ)) = Γ(Δ) has degree grater than or equal to r and is not (S_{r}).

Therefore it is natural to ask the following question.

Question 2.11. What is the characterization of (S_{r}) simplicial complexes via algebraic shifting?

3. On the h-triangle of a sequentially (S_{r}) simplicial complex

The h-vector of a simplicial complex plays an important role in the theory of Cohen– Macaulay complexes, including pure shellable complexes (see, for example, [13]). To extend part of this to the nonpure case we are led to introduce doubly indexed hnumbers. In order to do this, let Δ be a (d − 1)-dimensional simplicial complex. For every i and j with 0 ≤ j ≤ i ≤ d, let f_{i,j} denote the number of faces of Δ of degree i and size j and consider

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

Then the triangular integer arrays f = (f_{i,j})_{0≤j≤i≤d} and h = (h_{i,j})_{0≤j≤i≤d} are called the f-triangle and h-triangle of Δ, respectively. In order to show that f_{i,j} (resp. h_{i,j}) is the (i, j)th array of f (resp. h), we sometimes write f_{i,j}(Δ) (resp. h_{i,j}(Δ)) instead of f_{i,j} (resp. h_{i,j}).

Before stating the results of this section, we define the notion of a sequentially (S_{r}) simplicial complex which was introduced in [7].

Definition 3.1. Let M be a finitely generated ℤ-graded module over a standard graded 𝕂-algebra R, where 𝕂 is a field. For a positive integer r, we say that M is sequentially (S_{r}) if there exists a finite filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{t} = M of M by graded submodules M_{i} satisfying the following two conditions:

(a) every quotient M_{i}/M_{i−1} satisfies the (S_{r}) condition of Serre;

(b) dim(M_{1}/M_{0}) < dim(M_{2}/M_{1}) < ⋯ < dim(M_{t}/M_{t−1}).

We say that a simplicial complex Δ on [n] = {1, …, n} is sequentially (S_{r}) (over a field 𝕂) if its Stanley–Reisner ring 𝕂[Δ] = 𝕂[x_{1}, …, x_{n}]/I_{Δ}, as a module over R = 𝕂[x_{1}, …, x_{n}] is sequentially (S_{r}).

Duval [3] has shown that algebraic shifting preserves the h-triangle of a simplicial complex Δ, provided Δ is sequentially Cohen–Macaulay. The analogue of Duval’s result is given in the following theorem.

Theorem 3.2. Let Δ be a sequentially (S_{r}) simplicial complex and let Γ(Δ) be its algebraic shifting. Then for every i and j with 0 ≤ j ≤ i ≤ r − 1, we have h_{i,j}(Δ) = h_{i,j}(Γ(Δ)).

Proof. Let Δ be a sequentially (S_{r}) simplicial complex. By Theorem 2.6 of [7], the simplicial complex Δ^{[i]} for i ≤ r − 1 is (S_{r}) and since its dimension is equal to i, we conclude that Δ^{[i]} = (Δ^{⟨i⟩})^{(i)} is Cohen–Macaulay. Now by Corollary 4.5 of [3], deg Γ(Δ^{⟨i⟩}) ≥ i+1 and so Theorem 4.6 of [3] implies that Γ(Δ^{⟨i⟩}) = Γ(Δ)^{⟨i⟩}. Therefore, for every j and for every i with i ≤ r − 1, we have f_{j}(Γ(Δ^{⟨i⟩})) = f_{j}(Γ(Δ)^{⟨i⟩}), where f_{j} denotes the jth component of the f-vector. By using [3] we obtain that

f_{i,j}(Δ) = f_{j−1}(Δ^{⟨i−1⟩}) − f_{j−1}(Δ^{⟨i⟩}),
f_{i,j}(Γ(Δ)) = f_{j−1}(Γ(Δ)^{⟨i−1⟩}) − f_{j−1}(Γ(Δ)^{⟨i⟩}).

Therefore, for every i and j with 0 ≤ j ≤ i ≤ r − 1, we have f_{i,j}(Γ(Δ)) = f_{i,j}(Δ). Now the relation between f-triangles and h-triangles completes the proof. □

We now state and prove our two next results. But first, let us to recall the following definition (see [1, Definition 2.1]).

Definition 3.3. A simplicial complex Δ is called shellable (in nonpure sense) if there exists a linear order F_{1}, …, F_{r} of the facets of Δ such that for every i with 2 ≤ i ≤ r, the simplicial complex ⟨F_{i}⟩ ∩ ⟨F_{1}, …, F_{i−1}⟩ is pure of dimension dim F_{i} − 1.

Corollary 3.4. Let Δ be a sequentially (S_{r}) simplicial complex. Then for every i and j with 0 ≤ j ≤ i ≤ r − 1, we have h_{i,j}(Δ) ≥ 0.

Proof. By Theorem 3.2, for every i and j with 0 ≤ j ≤ i ≤ r − 1, we have h_{i,j}(Δ) = h_{i,j}(Γ(Δ)). But shifted complexes are shellable and so by Theorem 3.4 of [1] their h-triangles are nonnegative. □

Corollary 3.5. Let h = (h_{i,j})_{0≤j≤i≤d} be an array of integers. Then the following conditions are equivalent :

(1) there exists a sequentially (S_{r}) simplicial complex Δ such that h_{i,j}(Δ) = h_{i,j} for every i and j with 0 ≤ j ≤ i ≤ r − 1;

(2) there exists a shifted simplicial complex Δ such that h_{i,j}(Δ) = h_{i,j} for every i and j with 0 ≤ j ≤ i ≤ r − 1;

(3) there exists a shellable simplicial complex Δ such that h_{i,j}(Δ) = h_{i,j} for every i and j with 0 ≤ j ≤ i ≤ r − 1;

(4) there exists a sequentially Cohen–Macaulay simplicial complex Δ such that h_{i,j}(Δ) = h_{i,j} for every i and j with 0 ≤ j ≤ i ≤ r − 1.

Proof. (1)⇒(2) Consider the algebraic shifting of Δ and apply Theorem 3.2. (2)⇒(3) Every shifted simplicial complex is shellable. (3)⇒(4) Every shellable simplicial complex is sequentially Cohen–Macaulay. (4)⇒(1) Every sequentially Cohen–Macaulay simplicial complex is sequentially

(S_{r}). □

We now state and prove the following proposition. We recall that a sequence of integers 𝔥 = (h_{0}, h_{1}, …, h_{r}) is an M-vector if there exists a Cohen–Macaulay (r − 1)- dimensional simplicial complex Δ′ with h(Δ′) = 𝔥.

Proposition 3.6. Let Δ be a (d−1)-dimensional sequentially (S_{r}) simplicial complex. Then for every c with 1 ≤ c ≤ r, the sequence (h′_{c,0}, h′_{c,1}, …, h′_{c,c}) is an M-vector, where for every j with 0 ≤ j ≤ c,

h′_{c,j} = h_{c,j} + ∑_{s=c+1}^{d} ∑_{i=0}^{j} \binom{s − c − 1 + j − i}{j − i}.

Proof. By Theorem 2.6 of [7], for every c with 1 ≤ c ≤ r the simplicial complex Δ^{[c−1]} = Δ^{(c−1,c−1)} is (S_{r}) and since its dimension is less than r, it is Cohen–Macaulay. Therefore its h-vector is an M-vector. Now Lemma 3.3(v) of [1] completes the proof.

□

Proposition 3.7. Let Δ be a (d−1)-dimensional sequentially (S_{r}) simplicial complex. Then for every c with r + 1 ≤ c ≤ d the following conditions hold, where for every j with 0 ≤ j ≤ c,

h′_{c,j} = h_{c,j} + ∑_{s=c+1}^{d} ∑_{i=0}^{j} \binom{s − c − 1 + j − i}{j − i}.

(1) The sequence (h′_{c,0}, h′_{c,1}, …, h′_{c,r}) is an M-vector; (2) For every i with 0 ≤ i ≤ r, we have

\binom{i}{i}h′_{c,r} + \binom{i + 1}{i}h′_{c,r+1} + ⋯ + \binom{c + i − r}{i}h′_{c,c} ≥ 0.

Proof. By Theorem 2.6 of [7], for every c with r + 1 ≤ c ≤ d, the simplicial complex Δ^{[c−1]} = Δ^{(c−1,c−1)} is (S_{r}) and by Lemma 3.3(v) of [1] the sequence

(h′_{c,0}, h′_{c,1}, …, h′_{c,c})

is the h-vector of Δ^{(c−1,c−1)}. Now, Theorem 3.1 of [6] completes the proof. □

Iterated Betti numbers, introduced in [4], are a nonpure generalization of the reduced homology Betti numbers β̃_{i−1}(Δ) = dim_{𝕂} H̃_{i−1}(Δ). Although they can be defined as the Betti numbers of a certain chain complex (see [4, Section 4]), we will use the following equivalent formulation.

Notation and Remarks 3.8. Let Δ be a simplicial complex. For a set F of positive integers, let init(F) = max{r | {1, …, r} ⊆ F }. Therefore, init(F) measures the largest “initial segment” in F, and is 0 if there is no initial segment, i.e., if 1 does not belong to F. Now, by Theorem 4.1 of [4], the ith iterated Betti number of Δ is

β_{j−1}[i](Δ) = |{F ∈ F(Γ(Δ)) | dim F = j − 1 and init(F) = i}|.

A special case is i = 0. In this case, β_{j}[0](Δ) = β̃_{j}(Δ) is the ordinary Betti number of reduced homology. Björner and Wachs [1, Theorem 4.1] have shown that if Δ is nonpure and shellable, then

β̃_{j−1}(Δ) = h_{j,j}(Δ)

for every j with 0 ≤ j ≤ d. This equation was generalized in Theorem 1.2 of [4] to

β_{j−1}[i](Δ) = h_{j,j−i}(Δ)

for nonpure shellable Δ. This algebraic interpretation of the h-triangle of nonpure shellable complexes was part of the motivation for iterated Betti numbers. Theorem 3.2 allows us to generalize even further, by weakening the assumption on Δ from being nonpure shellable to being merely sequentially (S_{r}).

Corollary 3.9. Let Δ be a sequentially (S_{r}) simplicial complex. Then for every i with i ≤ r − 1, we have β_{j−1}[i](Δ) = h_{j,j−i}(Δ).

Proof. By Theorem 5.4 of [4], β_{j−1}[i](Δ) = h_{j,j−i}(Γ(Δ)) for every simplicial complex Δ. We now apply Theorem 3.2 to obtain the result. □

References

[1] A. Björner, M. L. Wachs, Shellable nonpure complexes and posets I, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327.

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

[3] A. M. Duval, Algebraic shifting and sequentially Cohen–Macaulay simplicial complexes, Electron. J. Comb. 3 (1996), no. 1, Research Paper 21, approx. 14 pp.

[4] A. M. Duval, L. L. Rose, Iterated homology of simplicial complexes, J. Algebraic. Combin. 12 (2000), no. 3, 279–294.

[5] R. Fröberg, A note on the Stanley–Reisner ring of a join and of a suspension, Manuscripta Math. 60 (1988), no. 1, 89–91.

[6] A. Goodarzi, M. R. Pournaki, S. A. Seyed Fakhari, S. Yassemi, On the h-vector of a simplicial complex with Serre’s condition, J. Pure Appl. Algebra 216 (2012), no. 1, 91–94.

[7] H. Haghighi, N. Terai, S. Yassemi, R. Zaare-Nahandi, Sequentially S_{r} simplicial complexes and sequentially S_{2} graphs, Proc. Amer. Math. Soc. 139 (2011), no. 6, 1993–2005.

[8] J. Herzog, T. Hibi, Monomial Ideals, Springer-Verlag, London, Ltd., London, 2011.

[9] T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Publicatin, 1992.

[10] G. Kalai, Algebraic shifting, Adv. Stud. Pure Math. 33 (2001), 121–163.

[11] E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, Springer-Verlag, New York, 2005.

[12] S. Murai, N. Terai, h-vectors of simplicial complexes with Serre’s conditions, Math. Res. Lett. 16 (2009), no. 6, 1015–1028.

[13] R. P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Birkhäuser Boston, Inc., Boston, MA, 1996.

[14] M. L. Wachs, Whitney homology of semipure shellable posets, J. Algebraic Combin. 9 (1999), no. 2, 173–207.

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, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, 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/