The Rees Product of a Boolean Algebra and a
Chain
Jakob Jonsson
October 22, 2008
Abstract
For a simplicial complex Σ of dimension n−1, let Γ(Σ) be the partially
ordered set of nonempty injective words ω1 · · · ωr such that {ω1 , . . . , ωr }
is a face of Σ, the order being given by subword inclusion. We prove that
the order complex of Γ(Σ) has the same Euler characteristic as the order
complex of the Rees product of the face poset of Σ and an n-chain. In
particular, we settle a conjecture due to Björner and Welker, stating that
the two order complexes are homotopy equivalent. Shareshian and Wachs
discovered another proof of the conjecture independently of our work.
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Introduction
Let Σ be a simplicial complex. The purpose of this note is to prove that the order
complexes of two different partially ordered sets, soon to be defined rigorously,
have the same Euler characteristic, thereby settling a conjecture due to Björner
and Welker [2]. The first poset is the set of injective words ω1 · · · ωr such that
{ω1 , . . . , ωr } is a face of Σ. The second poset is the Rees product of the face poset
of Σ with a chain of length dim Σ + 1. By results due to on the one hand Farmer
[3] and Jonsson and Welker [6] and on the other hand Björner and Welker [2],
both posets are Cohen-Macaulay whenever Σ is Cohen-Macaulay. In particular,
the two complexes are homotopy equivalent whenever Σ is Cohen-Macaulay.
First, we define the poset of injective words; we refer the reader to Farmer
[3], Björner and Wachs [1], Hanlon and Hersh [5], Reiner and Webb [9], and
Jonsson and Welker [6] for more information and further references. A word ω
over a finite alphabet S is called injective if no letter appears more than once,
meaning that ω = ω1 · · · ωr for some ω1 , . . . , ωr ∈ S such that ωi 6= ωj for
1 ≤ i < j ≤ r. For n = |S| we denote by Γn the set of all injective words on
S. A word ω = ω1 · · · ωr with r letters is said to be of length r. A subword
of a word ω1 · · · ωr is a word ωj1 · · · ωjs such that 1 ≤ j1 < · · · < js ≤ r.
Clearly, a subword of an injective word is injective. We order Γn by saying that
ρ1 · · · ρs ω1 · · · ωr if and only if ρ1 · · · ρs is a subword of ω1 · · · ωr . We write
c(w) for the content {ω1 , . . . , ωr } of the word w = ω1 · · · ωr .
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For a simplicial complex Σ on the same vertex set as Γn , we define a subposet
Γ(Σ) of Γn by restricting to injective words w ∈ Γn such that the content c(w)
is a face of Σ. Note that Γ(Σ) is an order ideal in Γn .
Next, we define the Rees product of two posets, introduced by Björner and
Welker [2]. Let P and Q be posets of the same rank r. The Rees product of P
and Q consists of all pairs (p, q) such that the rank of p is at least the rank of
q, and we have a covering relation (p, q) < (p0 , q 0 ) exactly when p0 covers p in P
and either q = q 0 or q 0 covers q in Q.
For a simplicial complex Σ, let P (Σ) denote the face poset of Σ; the poset
P (Σ) consists of all nonempty faces of Σ ordered by inclusion. Let R(Σ) be the
Rees product of P (Σ) and the chain 0 < 1 < · · · < dim Σ. Note that elements
in R(Σ) are pairs (σ, k) such that σ ∈ Σ and 0 ≤ k ≤ |σ| − 1 = dim σ. For all
σ ∈ Σ \ {∅}, k < |σ|, and a ∈
/ σ such that σ ∪ {a} ∈ Σ, we have the two covering
relations (σ, k) < (σ ∪ {a}, k) and (σ, k) < (σ ∪ {a}, k + 1).
Throughout this note, homology computations are carried out with coefficients in a principal ideal domain R. For a poset P , let ∆(P ) be the order
complex of P . For simplicity, we write χ̃(P ) = χ̃(∆(P )) and H̃∗ (P ; R) =
H̃∗ (∆(P ); R).
We refer to the properties of being homotopically Cohen-Macaulay and
Cohen-Macaulay over a given principal ideal domain as Cohen-Macaulay properties.
Proposition 1.1 (Björner and Welker [2]) Let Σ be a simplicial complex.
If Σ has a given Cohen-Macaulay property, then so has R(Σ).
Proposition 1.2 (Jonsson and Welker [6]) Let Σ be a simplicial complex.
If Σ has a given Cohen-Macaulay property, then so has Γ(Σ).
The following theorem is the main result of this note.
Theorem 1.3 For any simplicial complex Σ, we have that
χ̃(Γ(Σ)) = χ̃(R(Σ)).
In particular, if Σ is Cohen-Macaulay over R, then the groups H̃dim Σ (Γ(Σ); R)
and H̃dim Σ (R(Σ); R) are isomorphic.
We prove Theorem 1.3 in Section 2. For an alternative proof discovered independently, see Shareshian and Wachs [7].
Let Σn be the full simplex on n vertices. Recall that Γn = Γ(Σn ), and define
Rn = R(Σn ).
Proposition 1.4 (Farmer [3]) ∆(Γn ) is homotopy equivalent to a wedge of
Dn spheres of dimension n − 1, where Dn is the derangement number; D0 = 1
and Dn = nDn−1 + (−1)n .
As an immediate consequence of Theorem 1.3 and Proposition 1.4, we obtain
the following result, which was conjectured by Björner and Welker [2].
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Corollary 1.5 We have that χ̃(Γn ) = χ̃(Rn ). In particular, ∆(Rn ) is homotopy
equivalent to a wedge of Dn spheres of dimension n − 1, where Dn is defined as
in Propostion 1.4.
One may examine the Rees product of other interesting posets with the chain.
For example, Muldoon and Readdy [8] considered the cubical lattice, whereas
Shareshian and Wachs [7] considered the face poset of the boundary of the
crosspolytope, which is the dual of the cubical lattice.
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Proof of Theorem 1.3
For a word x = x1 · · · xs , write c(x) = {x1 , . . . , xs } and d(x) = |{i : xi >
xi+1 , 1 ≤ i ≤ s − 1}|. Define f : Γ(Σ) → R(Σ) by mapping a word x = x1 · · · xs
to the pair (c(x), d(x)). To prove that the Euler characteristics coincide, it
suffices to show that the order complex of the lower fiber
Γ≤f
−1
(r)
(Σ) = f −1 ({r0 : r0 ≤ r})
has a vanishing reduced Euler characteristic for each r = (σ, k) ∈ R(Σ). This
approach was suggested by Björner and Welker [2] for the special case that Σ is
the full simplex.
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Write Γσ,k = Γ≤f (σ,k) (Σ); this poset does not depend on Σ. Note that
Γσ,k is the family of all nonempty injective words x on the set σ such that
|x| − |σ| + k ≤ d(x) ≤ k.
Moreover, Γσ,k is the face poset of a boolean complex Bσ,k . Obviously, the
Euler characteristic of Bσ,k coincides with that of ∆(Γσ,k ); the complexes are
homeomorphic.
To summarize, it suffices to show that the reduced Euler characteristic of
Bσ,k is zero. We prove this by defining a perfect matching on Bσ,k such that
each matched pair is of the form {xy, xay} for some words x and y and some
symbol a; let us refer to such a matching as an element matching. In terms
of discrete Morse theory [4], our matching is not acyclic in general. In fact,
it cannot be, as the complex Bσ,k is not contractible in general; see Björner
and Welker [2]. For example, B3,1 is not contractible, where Bm,k = B[m],k ;
132 + 312 − 213 − 231 is a homology cycle in the chain complex of B3,1 .
First, note that B1,0 = {φ, 1}, B2,0 = {φ, 1, 2, 12}, and B2,1 = {φ, 1, 2, 21};
these complexes clearly admit perfect element matchings.
Next, assume that |σ| ≥ 3. For simplicity, assume that σ = [m]. We divide
Bm,k into subfamilies as follows:
• Let A1 be the family of words x in Bm,k not containing m such that
d(x) = k. Let A2 be the family of words xm in Bm,k such that d(x) = k.
Note that d(xm) = d(x).
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• Let B1 be the family of words y in Bm,k not containing m such that
d(y) < k. Let B2 be the family of words my in Bm,k such that d(y) < k.
Note that d(my) = d(y) + 1.
• Let C be the family of words in Bm,k not contained in A1 ∪ A2 ∪ B1 ∪ B2 .
It is immediate that we have perfect element matchings on A = A1 ∪ A2 (match
x and xm) and B = B1 ∪ B2 (match y and my).
What remains is the family C. This family consists of all faces xmy ∈ Bm,k
such that x 6= φ and d(x) < k. Namely, d(x) = k would imply that y = φ and
hence that xm ∈ A2 ; x = φ would imply that my ∈ B2 , as d(y) < d(my) ≤ k.
Now, for each x such that d(x) < k and xm ∈ Bm,k , define
C(x) = {xmy : xmy ∈ C}.
For each xmy ∈ C(x), we have that [m − 1] \ c(x) 6= ∅, because if xm is a word
of full length m in Bm,k , then d(x) = d(xm) = k and hence xm ∈ A2 . Also,
note that d(xmy) = d(x) + d(y) + 1. We conclude that
C(x) = {xmy : y ∈ B[m−1]\c(x),k−d(x)−1 }.
By induction on m = |σ|, B[m−1]\c(x),k−d(x)−1 admits a perfect element matching, which implies that the same is true for C(x).
Taking the union of the matchings just defined, we obtain a perfect element
matching on C. The union of this matching and the above matchings on A and
B is a perfect element matching on Bm,k , which concludes the proof.
The last statement in the theorem is an immediate consequence of the first
statement and Propositions 1.1 and 1.2.
References
[1] A. Björner and M. Wachs, On lexicographically shellable posets, Trans.
Amer. Math. Soc. 277 (1983), 323–341.
[2] A. Björner and V. Welker, Segre and Rees products of posets, with ringtheoretic applications, J. Pure Appl. Algebra, 198 (2005), 43–55.
[3] F. D. Farmer, Cellular homology for posets, Math. Japan 23 (1978/79),
607–613.
[4] R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), 90–
145.
[5] P. Hanlon and P. Hersh. A Hodge Decomposition for the Complex of Injective Words. Pacific J. Math. 214 (2004) 109–125.
[6] J. Jonsson and V. Welker, Complexes of Injective Words and Their Commutation Classes, submitted.
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[7] J. Shareshian and M. L. Wachs, q-Eulerian Polynomials: Excedance Number and Major Index, preprint.
[8] P. Muldoon and M. A. Readdy, The Rees product of the cubical lattice with
the chain, Proceedings of the 20th Conference on Formal Power Series and
Algebraic Combinatorics, Vina del Mar, Chile, 2008.
[9] V. Reiner and P. Webb, The combinatorics of the bar resolution in group
cohomology, J. Pure Appl. Algebra 190 (2004) 291–327.
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