THE COXETER COMPLEX AND THE EULER CHARACTERISTIC OF A HECKE ALGEBRA

T. TERRAGNI AND TH. WEIGEL

Abstract. For any Hecke algebra H = H_{q}(W, S) associated to a Coxeter group (W, S) and a distinguished element q ∈ R of a commutative ring with unit R we introduce a finite chain complex of left H-modules (C_{•}, ∂_{•}) which reflects many properties of the Coxeter complex of (W, S), i.e., it is acyclic if (W, S) is non-spherical (cf. Thm. A), and H is of type FP under suitable conditions on the distinguished element q ∈ R (cf. Prop. B). There exists a canonical trace function μ̃: H → R (cf. Prop. 5.1). This trace function μ̃ evaluated on the Hattori-Stallings rank of (C_{•}, ∂_{•}) can be considered as the Euler characteristic χ_{H} of H. It will be shown that for generic values of q the Euler characteristic coincides with the reciprocal of the Poincaré series of (W, S) evaluated in q (cf. Thm. C).

1. Introduction

For any commutative ring R with unit, and any distinguished element q ∈ R one may define an R-Hecke algebra H = H_{q}(W, S) associated to any Coxeter group (W, S). This algebra can be seen as a deformation of the R-group algebra of the Coxeter group (W, S). It particular, it comes equipped with an antipodal map ♮: H → H^{op}, an augmentation ε_{q}: H → R, and an R-basis B = { T_{w} | w ∈ W }. Moreover, ε_{q}(T_{w}) = q^{ℓ(w)} for w ∈ W, where ℓ: W → ℕ_{0} denotes the length function on (W, S). The Poincaré series of (W, S) is given by

(1.1)  p_{(W,S)}(t) = ∑_{w∈W} t^{ℓ(w)} ∈ ℤ⟦t⟧.

It is well known (cf. [4, Chap. IV, §1, Ex. 25 and 26]) that p_{(W,S)}(t) is a rational function in t. Moreover, if (W, S) is spherical then p_{(W,S)}(t) ∈ ℤ[t] is just a polynomial with integer coefficients. The left H-module R_{q}, which is as R-module isomorphic to R and which action is given by h.r = ε_{q}(h)r for h ∈ H, r ∈ R_{q}, can be seen as the trivial H-module.

The main purpose of this paper is to introduce a chain complex of left H-modules C = (C_{•}, ∂_{•}) concentrated in degrees 0 to |S| − 1, which can be seen as the module theoretic analogue of the Coxeter complex associated to (W, S) (cf. [1, Chap. 3]). It is canonical up to the choice of a total ordering of the finite set S. The most significant properties of the chain complex C can be summarized as follows (cf. §3).

Theorem A. Let (W, S) be a Coxeter group with 2 ≤ |S| < ∞, and let C be the Coxeter complex of the R-Hecke algebra H = H_{q}(W, S).

Date: October 22, 2011.

2010 Mathematics Subject Classification. Primary 20C08, secondary 16S80, 20F55.

(a) If (W, S) is spherical, then H_{k}(C) = 0 unless k = 0 or k = |S| − 1. Moreover, H_{0}(C) ≃ R_{q} and H_{|S|−1}(C) ≃ R_{−1}.

(b) If (W, S) is non-spherical then C is acyclic with H_{0}(C) ≃ R_{q}.

A left A-module M of an associative R-algebra A is called to be of type FP, if it has a finite, projective resolution (P_{•}, ∂_{•}^{P} , ε_{M}). From Theorem A one may deduce a sufficient criterion on the distinguished element q ∈ R ensuring the FP-property of the trivial H-module R_{q} (cf. Prop. 5.4).

Proposition B. Suppose that for the distinguished element q ∈ R one has

(1.2)  p_{(W_{I},I)}(q) ∈ R^{×}

for every spherical parabolic subgroup (W_{I}, I) of (W, S). Then R_{q} is of type FP.

Here R^{×} ⊂ R denotes the group of invertible elements in R. If q ∈ R satisfies (1.2), one may define the Hattori–Stallings rank r_{R_{q}} of the trivial left H-module R_{q} ∈ H/[H, H] by a standard procedure (cf. [5, Chap. IX]). Moreover, every R-Hecke algebra H has a canonical trace function μ̃: H → R (cf. Prop. 5.1). We will define the Euler characteristic of H by χ_{H} = μ(r_{R_{q}}), where μ is the induced map on H/[H, H]. For generic values of q the Euler characteristic of H can be computed explicitly (cf. §5).

Theorem C. If q ∈ R satisfies (1.2), then p_{(W,S)}(q) ∈ R^{×} and

(1.3)  χ_{H} = p_{(W,S)}(q)^{−1}.

It might look surprising that the Poincaré series of a Coxeter group can be recovered from the representation theory of the associated Hecke algebra. On the other hand the alternating minus signs in the formula which is usually used to calculate the series explicitly (cf. [8, §5.12]) suggest that its reciprocal value might be an Euler characteristic of something. In the case that R = R_{∘}⟦q⟧ for some commutative ring with unit R_{∘} there is another interesting phenomenon. Obviously, q ∈ R satisfies (1.2), and p_{(W,S)}(q) can be rewritten as

(1.4)  p_{H} = p_{(W,S)}(q) = ∑_{T_{w}∈B} ε_{q}(T_{w}) ∈ R

interpreting p_{H} as a series associated to (H, ε_{q}, B). Then one has the identity p_{H} · χ_{H} = 1. In fact, a similar identity is known for a Koszul algebra A_{•} defined over a field F, i.e., in this case one has

(1.5)  h_{A_{•}}(t) · h_{H^{•,•}(A_{•},F)}(−t) = 1

where h_{A_{•}}(t) (resp. h_{H^{•,•}(A_{•},F)}(t)) denotes the Hilbert series of the graded F - algebra A_{•} (resp. H^{•,•}(A_{•}, F)) (cf. [9, §2, p. 22, Cor. 2.2]). It would be interesting to know whether there exist other types of generic R_{∘}⟦q⟧-algebras (A, ^{♮}, ε, B) satisfying the identity p_{A} · χ_{A} = 1.

Acknowledgement: The authors would like to thank F. Brenti for some very helpful discussions.

2. Coxeter groups and Hecke algebras

2.1. Coxeter groups. A Coxeter graph Γ is a finite combinatorial graph^{1} with nonoriented edges 𝔢 labelled by positive integers m(𝔢) ≥ 3 or infinity. The Coxeter group (W, S) associated to Γ consists of the group W generated by the set of involutions S = { s_{v} | v ∈ 𝔙(Γ) } subject to the relations (s_{v}s_{w})^{m(𝔢)} = 1, where 𝔢 = {v, w} ∈ 𝔈(Γ) is an edge of label m(𝔢) < ∞, and the commutation relations s_{v}s_{w} = s_{w}s_{v} whenever {v, w} ∉ 𝔈(Γ). The length function on W with respect to S will be denoted by ℓ: W → ℕ_{0}. Since S = S^{−1} is a set of involutions, ℓ(w) = ℓ(w^{−1}), and it is well known that a longest element w_{0} ∈ W exists if, and only if, W is finite. In this case it is unique and has the property that ℓ(w_{0}x) = ℓ(w_{0})−ℓ(x) for all x ∈ W. A Coxeter group which is finite is called spherical, and non-spherical otherwise.

For a subset I ⊆ S let W_{I} be the corresponding parabolic subgroup, i.e., W_{I} is the subgroup of W generated by I. It is isomorphic to the Coxeter group associated to the Coxeter subgraph Γ′ based on the vertices { v ∈ 𝔙(Γ) | s_{v} ∈ I }. The length function of W restricted to W_{I} coincides with the intrinsic length function of the Coxeter group (W_{I}, I). Put

(2.1)  W^{I} = { w ∈ W | ℓ(ws) > ℓ(w) for all s ∈ I },

and let ^{I}W = (W^{I})^{−1}, i.e.,

(2.2)  ^{I}W = { w ∈ W | ℓ(sw) > ℓ(w) for all s ∈ I }.

For w ∈ W the right ascent set is given by

(2.3)  A^{ρ}(w) = { s ∈ S | ℓ(ws) > ℓ(w) }.

One has the following properties (cf. [8, §5.12]).

Proposition 2.1. Let (W, S) be a Coxeter group, let w ∈ W and let I ⊆ S.

(a) There exist a unique element w_{I} ∈ W_{I} and a unique element w^{I} ∈ W^{I} such that w = w^{I}w_{I}. Moreover, ℓ(w) = ℓ(w^{I}) + ℓ(w_{I}).

(b) There exist a unique element _{I}w ∈ W_{I} and a unique element ^{I}w ∈ ^{I}W such that w = _{I}w^{I}w. Moreover, ℓ(w) = ℓ(_{I}w) + ℓ(^{I}w).

(c) W^{I} and ^{I}W are sets of coset representatives, distinguished in the sense that the decomposition is length-additive.

(d) The element w^{I} ∈ W^{I} is the unique shortest element in wW_{I}.

(e) Let y ∈ W^{I} and u ∈ W_{I}. Then (yu)^{I} = y, (yu)_{I} = u, and ℓ(yu) = ℓ(y) + ℓ(u).

(f) For s ∈ S one has W = ^{{s}}W ⊔ s(^{{s}}W), where ⊔ denotes disjoint union.

(g) Let I ⊆ J ⊆ S. Then W^{J} ⊆ W^{I}. In particular, W^{S} = {1} and W^{∅} = W.

(h) A^{ρ}(w) = ⋃_{w∈W^{I}} I = max{ I | w ∈ W^{I} }.

(i) The element w is contained in W^{I} if, and only if, I ⊆ A^{ρ}(w). In particular, ℓ(w^{I}) ⪇ ℓ(w) if, and only if, I ⊈ A^{ρ}(w).

2.2. Hecke algebras. Let R be a commutative ring with unit and with a distinguished element q ∈ R.^{2} The R-Hecke algebra H = H_{q}(W, S) associated to (W, S) and q is the unique associative R-algebra which is a free R-module with basis { T_{w} | w ∈ W } subject to the relations

1In this context the graph ∅ with empty vertex set is also considered as a Coxeter graph.

2For certain types it is also possible to consider multiple parameter Hecke algebras. This will be discussed in [11].

(2.4)  T_{s}T_{w} = \begin{cases}T_{sw} & if ℓ(sw) > ℓ(w) \\ (q − 1)T_{w} + qT_{sw} & if ℓ(sw) < ℓ(w), \end{cases}

for s ∈ S, w ∈ W. In particular, one has a canonical isomorphism H_{1} ≃ R[W], where R[W] denotes the R-group algebra of W. The R-algebra H comes equipped with an antipodal map ^{♮}: H → H^{op}, T_{w}^{♮} = T_{w^{−1}}, i.e., ^{♮} is an isomorphism satisfying ^{♮♮} = id_{H} (cf. [8, Chap. 7.3, Ex. 1]).

For I ⊆ S we denote by H_{I} the corresponding parabolic subalgebra, i.e., the H-subalgebra of H generated by { T_{s} | s ∈ I } which coincides with the R-module spanned by { T_{w} | w ∈ W_{I} }. For further details see [8, Chap. 7].

2.3. H-modules. Any R-algebra homomorphism λ ∈ Hom_{R-alg}(H, R) defines a 1-dimensional left H-module R_{λ}, i.e., for T_{w} ∈ H, w ∈ W, and r ∈ R_{λ} one has T_{w}.r = λ(T_{w})r. Note that the relations (2.4) force λ(T_{s}) ∈ {−1, q} for all s ∈ S. Moreover, for s, s′ ∈ S and m(s, s′) odd, one has λ(T_{s}) = λ(T_{s′}). There are two particular R-algebra homomorphisms ε_{q}, ε_{−1} ∈ Hom_{R-alg}(H, R), given by ε_{q}(T_{s}) = q, ε_{−1}(T_{s}) = −1, s ∈ S. One may consider ε_{q} as the augmentation and ε_{−1} as the sign-character. Note that ε_{q}(T_{w}) = q^{ℓ(w)} and ε_{−1}(T_{w}) = (−1)^{ℓ(w)}, and therefore ε_{q}(T_{w}) = ε_{q}(T_{w}^{♮}) and ε_{−1}(T_{w}) = ε_{−1}(T_{w}^{♮}) for all w ∈ W. For short we put R_{q} = R_{ε_{q}}, R_{−1} = R_{ε_{−1}}, and use also the same notation for the restriction of these modules to any parabolic subalgebra.

For I ⊆ S let H^{I} = span_{R}{ T_{w} | w ∈ W^{I} } ⊆ H. Multiplication in H induces a canonical map of right H_{I}-modules H^{I} ⊗_{R} H_{I} ⟶ H. Let y ∈ W^{I} and u ∈ W_{I}. As ℓ(yu) = ℓ(y) + ℓ(u) (cf. Prop. 2.1(e)), one has T_{y}T_{u} = T_{yu}. This shows that this map is an isomorphism. In particular, H is a projective and thus a flat right H_{I}-module. This implies that

(2.5)  ind_{I}^{S} = ind_{H_{I}}^{H} = H ⊗_{H_{I}} : H_{I}-mod ⟶ H-mod

is an exact functor mapping projectives to projectives. Moreover, one has the following.

Fact 2.2. The canonical map c_{I}: H^{I} → ind_{I}^{S} R_{q} given by c_{I}(T_{w}) = T_{w}η_{I}, where η_{I} = T_{1} ⊗ 1 ∈ ind_{I}^{S} R_{q} and w ∈ W^{I}, is an isomorphism of R-modules. Moreover, for w ∈ W, one has T_{w}η_{I} = ε_{q}(T_{w_{I}})T_{w^{I}} η_{I}.

Proposition 2.3. Let I be a subset of S such that W_{I} is finite. Put τ_{I} = ∑_{w∈W_{I}} T_{w}. Then one has the following:

(a) τ_{I}^{2} = p_{(W_{I},I)}(q)τ_{I}. Moreover if p_{(W_{I},I)}(q) ∈ R^{×} is invertible in R, let e_{I} = (p_{(W_{I},I)}(q))^{−1}τ_{I}. Then:

(b) The element e_{I} is a central idempotent in H_{I} satisfying e_{I}^{♮} = e_{I}.

(c) The left ideal He_{I} is a finitely generated, projective, left H-module isomorphic to ind_{I}^{S} R_{q}.

(d) T_{w}e_{I} = ε_{q}(T_{w_{I}})T_{w^{I}} e_{I}.

Proof. For s ∈ I put X_{s} = ∑_{w∈^{{s}}(W_{I})} T_{w}. Then τ_{I} = (T_{1}+T_{s})X_{s} (cf. Prop. 2.1(f)) and therefore

T_{s}τ_{I} = T_{s}(T_{1} + T_{s})X_{s} = [T_{s} + qT_{1} + (q − 1)T_{s}] X_{s} = q(T_{1} + T_{s})X_{s} = ε_{q}(T_{s})τ_{I}.

This shows (a). Part (b) is an immediate consequence of (a), and the first part of (c) follows from the decomposition of the regular module H = He_{I} ⊕ H(T_{1} − e_{I}). The canonical map π: H → ind_{I}^{S} R_{q}, π(T_{w}) = T_{w}η_{I}, is a surjective morphism of H-modules with ker(π) = H(T_{1} − e_{I}). This yields the second part of (c). Part (d) follows from part (b) and Proposition 2.1(a). □

Proposition 2.4. Let W be finite with longest element w_{0}. Assume further that p_{(W,S)}(q) ∈ R^{×} and let

z = (p_{(W,S)}(q))^{−1} ∑_{w∈W} ε_{−1}(T_{w})ε_{q}(T_{w_{0}w})T_{w} ∈ H.

Then one has the following.

(a) For w ∈ W one has T_{w}z = ε_{−1}(T_{w})z, i.e., Hz is isomorphic to R_{−1} as H-module.

(b) The element z ∈ H is a central idempotent satisfying z^{♮} = z.

(c) The left ideal Hz is a finitely generated, projective, left H-module.

Proof. Let α(w) = (p_{(W,S)}(q))^{−1}ε_{−1}(T_{w})ε_{q}(T_{w_{0}w}). Then

\begin{matrix}T_{s}z = & ∑_{w∈W} α(w)T_{s}T_{w} = ∑_{w∈^{\{s\}}W}α(w)T_{sw} + ∑_{w∉^{\{s\}}W}α(w)(qT_{sw} + (q − 1)T_{w}) \\ = & \underbrace{∑_{w∈^{\{s\}}W} α(w)T_{sw}}_{A} + \underbrace{∑_{v∈^{\{s\}}W} −α(v)T_{v}}_{B} + \underbrace{∑_{w∉^{\{s\}}W} α(w)(q − 1)T_{w}}_{C} . \end{matrix}

Here we used the fact that for w ∉ ^{{s}}W it follows that ℓ(sw) < ℓ(w). Hence for v = sw, one has ℓ(v) = ℓ(w) − 1 and therefore qα(w) = −α(v).

For w ∈ ^{{s}}W and y = sw ∉ ^{{s}}W one has ℓ(y) = ℓ(w)+1. Hence α(w) = −qα(y) and A can be rewritten as ∑_{v∉^{{s}}W} −qα(v)T_{v}. Then

A + C = ∑_{x∉^{{s}}W}α(x) [−q + (q − 1)] T_{x} = − ∑_{x∉^{{s}}W}α(x)T_{x}.

This yields (a).

It is easy to check that z^{♮} = z. Thus, by (a), z ∈ Z(H). Moreover,

z^{2} = (p_{(W,S)}(q))^{−1} ∑_{w∈W} ε_{−1}(T_{w})ε_{q}(T_{w_{0}w})T_{w}.z = (p_{(W,S)}(q))^{−1} ∑_{w∈W} ε_{q}(T_{w_{0}w})z = z.

This shows (b), and (c) is a direct consequence of (b). □

3. The Coxeter complex of a Hecke algebra

3.1. The sign map. Let “<” be a total order (which is supposed to be fixed throughout) on the finite set S. Then one has a sign-map

(3.1)  sgn: S × P(S) ⟶ {±1}, sgn(s, I) = (−1)^{|{ t∈S\I | t<s }|},

where P(S) denotes the set of subsets of S. One has:

(3.2)  sgn(s, I ⊔ {t}) = sgn(s, I)  for t ≥ s,
(3.3)  sgn(s, I ⊔ {t}) = − sgn(s, I)  for t < s,
(3.4)  sgn(s, I \ {t}) = sgn(s, I)  for t ≥ s,
(3.5)  sgn(s, I \ {t}) = − sgn(s, I)  for t < s.

Moreover, the following holds.

Fact 3.1. If I ⊆ S and s, t ∉ I, s ≠ t, then

(3.6)  sgn(t, I) sgn(s, I ⊔ {t}) + sgn(s, I) sgn(t, I ⊔ {s}) = 0.

Proof. Note that either s < t or t < s. By (3.2) and (3.3), the left-hand side of (3.6) reduces in the first case to

sgn(t, I) sgn(s, I) + sgn(s, I)(− sgn(t, I)) = 0;

while in the second case one has

sgn(t, I)(− sgn(s, I)) + sgn(s, I) sgn(t, I) = 0.  □

3.2. Induction. Let I and J be subsets of S such that I ⊆ J ⊆ S. As induction is the left adjoint to restriction one has a natural isomorphism

(3.7)  ϕ: Hom_{H_{I}} (R_{q}, R_{q}) ⟶ Hom_{H_{J}} (ind_{I}^{J}(R_{q}), R_{q})

given by ϕ(α)(h ⊗ r) = h.α(r), h ∈ ind_{I}^{J}(R_{q}), r ∈ R_{q}. Put b_{I}^{J} = ϕ(id_{R_{q}}), and let

(3.8)

d_{I}^{J} = ind_{J}^{S}(b_{I}^{J}): ind_{I}^{S}(R_{q}) ⟶ ind_{J}^{S}(R_{q}),
d_{I}^{J}(T_{w} ⊗_{H_{I}} r) = T_{w} ⊗_{H_{J}} r.

3.3. The Coxeter complex. For a subset I ⊆ S put deg(I) = |S| − |I| − 1, thus deg(I) ∈ { −1, …, |S| − 1 }. For a non-negative integer k let C_{k} be the left H-module

(3.9)  C_{k} = ∐_{\begin{matrix}I⊆S \\ deg(I)=k \end{matrix}} ind_{I}^{S} R_{q}.

The differential ∂_{k}: C_{k} → C_{k−1} is defined to be the map

(3.10)  ∂_{k} = ∑_{\begin{matrix}I,J⊆S \\ deg(I)=k, \\ deg(J)=k−1 \end{matrix}} ∂_{I,J},

where

(3.11)  ∂_{I,J} = \begin{cases}sgn(s, I)d_{I}^{J} & if J = I ⊔ \{s\} \\ 0 & if J ⊉ I, \end{cases}

and d_{I}^{J} is given as in (3.8). Obviously, ∂_{k}: C_{k} → C_{k−1} are mappings of left Hmodules, and C_{k} = 0 for k > |S| − 1.

Remark 3.2. Let C = (C_{•}, ∂_{•}) be defined as in (3.9), (3.10) and (3.11).

(a) One may apply the definition also in degree −1, i.e., C_{−1} = ind_{S}^{S} R_{q} ≃ R_{q} and one has also a map ε = ∂_{0} : C_{0} → R_{q}, where ∂_{0} = ∑_{s∈S} sgn(s, S \ {s})d_{S\{s}}^{S}.

(b) In degree |S| − 1, C_{|S|−1} = ind_{∅}^{S} R_{q} ≃ ^{reg}H coincides with the regular left H-module.

(c) By Fact 2.2, every element of C_{k} can be written uniquely as a finite R-linear combination of monomials T_{w}η_{I}, where I ⊆ S, deg(I) = k and w ∈ W^{I}.

(d) The set { T_{w}η_{I} | I ⊆ S, deg(I) = k, w ∈ W^{I} } is the standard R-basis of C_{k}.

Proposition 3.3. For all k one has ∂_{k−1} ∘ ∂_{k} = 0. In particular, (C_{•}, ∂_{•}) is a chain complex of left H-modules.

Proof. If I ⊆ S and deg I = k then

∂_{k−1}∂_{k}(η_{I}) = ∂_{k−1} ( ∑_{s∈S\I} sgn(s, I)η_{I⊔{s}})

= ∑_{s∈S\I} sgn(s, I) ∑_{t∈S\(I⊔{s})} sgn(t, I ⊔ {s})η_{I⊔{s,t}}

= ∑_{\begin{matrix}s,t∈S\I \\ t≠s \end{matrix}} sgn(s, I) sgn(t, I ⊔ \{s\})η_{I⊔\{s,t\}}

= ∑_{\begin{matrix}s,t∈S\I \\ t<s \end{matrix}} [sgn(s, I) sgn(t, I ⊔ \{s\}) + sgn(t, I) sgn(s, I ⊔ \{t\})] η_{I⊔\{s,t\}},

which vanishes by Fact 3.1. □

From now on C = (C_{•}, ∂_{•}) will be called the Coxeter complex of H. The following property will turn out to be useful for our purpose.

Proposition 3.4. Let h = ∑_{deg(I)=k} ∑_{w∈W^{I}} α(w, I)T_{w}η_{I} ∈ C_{k}, k ≥ 0, and

∂_{k}(h) = ∑_{deg(J)=k−1} ∑_{v∈W^{J}} β(v, J) T_{w}η_{J}.

Then one has, for J ⊆ S, deg(J) = k − 1 and v ∈ W^{J},

(3.12)  β(v, J) = ∑_{t∈J} ∑_{x∈W_{J}^{J\{t}}} sgn(t, J \ {t}) α(vx, J \ {t}) ε_{q}(T_{x}).

In particular, if (w̄, Ī) is such that w̄ ∈ W Ī, α(w̄, Ī) ≠ 0 and α(w, I) = 0 for all w ∈ W with ℓ(w) > ℓ(w̄) and deg(I) = k, then

(3.13)  β(w̄, J) = ∑_{t∈J} sgn(t, J \ {t}) α(w̄, J \ {t}).

Proof. For I ⊂ J ⊆ S one has W_{J} = W_{J}^{I}W_{I}. As W^{J} ⊂ W^{I}, one concludes that W^{I} = W^{J}W_{J}^{I} (cf. Prop. 2.1). Hence

∂_{k}(h) = ∑_{deg(I)=k} ∑_{w∈W^{I}} α(w, I) ∑_{t∈S\I} sgn(t, I) T_{w}η_{I⊔{t}}

= ∑_{deg(J)=k−1} ∑_{t∈J} ∑_{w∈W^{J\{t}}} sgn(t, J \ {t}) α(w, J \ {t})T_{w}η_{J}

and thus by the previous remark

= ∑_{deg(J)=k−1} ∑_{v∈W^{J}} ∑_{t∈J} sgn(t, J \ {t}) ∑_{x∈W_{J}^{J\{t}}} α(vx, J \ {t}) T_{vx}η_{J}

Thus by Fact 2.2 one concludes that

= ∑_{deg(J)=k−1} ∑_{v∈W^{J}} ∑_{t∈J} sgn(t, J \ {t}) ∑_{x∈W_{J}^{J\{t}}} α(vx, J \ {t}) ε_{q}(T_{x}) T_{v}η_{J}

This yields (3.12), and (3.13) is a direct consequence of (3.12). □

Proposition 3.5. If W is infinite, ∂_{|S|−1} is injective; while for W finite and p_{(W,S)}(q) ∈ R^{×} one has ker(∂_{|S|−1}) = Hz ≃ R_{−1} (cf. Prop. 2.4).

Proof. Put ∂ = ∂_{|S|−1}. Let ζ = ∑_{w∈W} β(w)T_{w}η_{∅} ∈ ker ∂ ⊆ C_{|S|−1}. Proposition 2.1(f) yields

0 = ∂(ζ) = ∑_{w∈W} β(w) ∑_{s∈S} sgn(s, ∅)T_{w}η_{{s}}

= ∑_{s∈S} sgn(s, ∅) ( ∑_{w∈W^{{s}}} β(w)T_{w}η_{{s}} + ∑_{v∈W^{{s}}} β(vs)T_{v}T_{s}η_{{s}})

= ∑_{s∈S} sgn(s, ∅) ∑_{x∈W^{{s}}}(β(x) + β(xs)q)T_{x}η_{{s}}.

Hence one must have

(3.14)  β(x) + qβ(xs) = 0 for all s ∈ S and x ∈ W^{{s}}.

Suppose W is infinite and that there exists x_{0} ∈ W such that β(x_{0}) ≠ 0. Then —because W is infinite— there exists a sequence of elements (x_{k})_{k∈ℕ}, x_{k} ∈ W such that x_{k+1} = x_{k}s_{k}, s_{k} ∈ W and ℓ(x_{k+1}) = ℓ(x_{k}) + 1. In particular x_{k} ∈ W^{{s_{k}}}. By induction and (3.14), one concludes that β(x_{k}) ≠ 0 for all k ∈ ℕ, a contradiction, and this shows that ∂_{|S|−1} is injective in this case.

Let W be finite with longest element w_{0}. Then by (3.14) and induction, one concludes that β(x) = (−q)^{ℓ(w_{0}x)}β(w_{0}) for all x ∈ W. In particular, for b = ε_{−1}(T_{w_{0}})p_{(W,S)}(q)β(w_{0}) ∈ R one verifies easily that ζ = bzη_{∅}. This yields the claim. □

3.4. Acyclicity of the Coxeter complex. Throughout this subsection we will assume that |S| ≥ 2. Let “⪯” be the lexicographic order on ℕ_{0}×ℕ_{0}, i.e., (ℕ_{0}×ℕ_{0}, ⪯) is a well-ordered set. For k ∈ {−1, …, |S| − 2} and h ∈ C_{k} \ {0} put

h = ∑_{deg(I)=k} α_{I}η_{I}, α_{I} = ∑_{w∈W^{I}} α(w, I)T_{w} ∈ H^{I},

where α(w, I) ∈ R (cf. Fact 2.2). Then the following are well-defined:

supp(h) = { (w, I) | I ⊆ S, deg(I) = k, w ∈ W^{I}, α(w, I) ≠ 0},
λ(h) = max{ ℓ(w) | (w, I) ∈ supp(h) } ∈ ℕ_{0},
ν(h) = |{ (w, I) ∈ supp(h) | ℓ(w) = λ(h) }| ∈ ℕ_{0}.

Obviously, for h, h′ ∈ C_{k}, h ≠ h′, and r ∈ R with rh ≠ 0, one has

(3.15) (λ, ν)(h − h′) ⪯ max{(λ, ν)(h), (λ, ν)(h′)} and (λ, ν)(rh) ⪯ (λ, ν)(h).

For short we put Ω_{k} = ker ∂_{k} \ im ∂_{k+1}, and define

Δ_{k} = (λ, ν)|_{Ω_{k}}: Ω_{k} → ℕ_{0} × ℕ_{0}.

Proof of Theorem A. Obviously, ∂_{0} : C_{0} → C_{−1} is surjective. Suppose that for k ∈ {0, …, |S|−2} the set Ω = Ω_{k} is non-empty, and put Δ = Δ_{k}. As (ℕ_{0} ×ℕ_{0}, ⪯) is well-ordered, there exists a unique minimal element min Δ ∈ im(Δ) ⊆ ℕ_{0} × ℕ_{0}. Let h ∈ Ω be such that Δ(h) = min Δ. As Ω does not contain zero, h ≠ 0. Hence there exists a pair (w̄, Ī) ∈ supp(h) such that ℓ(w̄) = λ(h). Let A = A^{ρ}(w̄) (cf. (2.3)). By Proposition 2.1(h), one has to distinguish two cases.

Case 1: Ī = A. By the hypothesis on k, Ī ≠ ∅. Choose any element s̄ ∈ Ī, and let J̄ = Ī\ {s̄}. Then one has deg(J̄) = k + 1, and by Proposition 2.1(g), w̄ ∈ W J̄. Hence Tw̄ηJ̄ is an element of the standard basis of C_{k+1}, and

∂_{k+1}(Tw̄ηJ̄) = ∑_{\begin{matrix}deg(I)=k \\ I=J̄⊔\{s\} \end{matrix}} sgn(s, J̄)ε_{q}(Tw̄_{I})T_{}_{w̄^{I}} η_{I}.

Since Ī = J̄ ⊔ {s̄} and w̄Ī = w̄, one has

sgn(s̄, J̄)∂_{k+1}(Tw̄ηJ̄) = Tw̄ηĪ + \overbrace{∑_{s∈S\Ī}sgn(s̄, J̄) sgn(s, J̄)ε_{q}(Tw̄_{J̄⊔\{s\}})T_{}_{w̄^{J̄⊔\{s\}}}η_{J̄⊔\{s\}}}^{X} .

For s ∈ S \ Ī, one has J̄ ⊔ {s} ⊈ Ī = A. Thus the elements w̄^{J̄⊔{s}} are of shorter length than w̄ (cf. Prop. 2.1(i)). Hence, if X ≠ 0, then λ(X) ⪇ ℓ(w̄) = λ(h). Put

(3.16)  h′ = h − α(w̄, Ī) sgn(s̄, J̄)∂_{k+1}(Tw̄ηJ̄) ∈ ker(∂_{k}).

As h ∉ im(∂_{k+1}), one has also h′ ∉ im(∂_{k+1}). Hence h′ ∈ Ω. Moreover, by (3.15), Δ(h′) ⪯ Δ(h). Thus the minimality of Δ(h) implies that Δ(h′) = Δ(h). In particular, λ(h′) = λ(h). However, by construction,

{ (w, I) ∈ supp(h′) | ℓ(w) = λ(h′) } = { (w, I) ∈ supp(h) | ℓ(w) = λ(h) } \ { (w̄, Ī) },

and thus ν(h′) < ν(h), a contradiction, showing that Case 1 is impossible. Case 2: Ī ⊊ A. For the chosen (w̄, Ī) define the disjoint sets

A = { (w̄, I) | I ⊆ A, deg(I) = k },
B = { (w, I) | ℓ(w) = λ(h), w ≠ w̄, I ⊆ A^{ρ}(w), deg(I) = k },
C = { (w, I) | ℓ(w) ⪇ λ(h), I ⊆ A^{ρ}(w). deg(I) = k }.

Then supp(h) ⊆ A ⊔ B ⊔ C. Let h = h_{A} + h_{B} + h_{C} be the corresponding additive decomposition of h (cf. Fact 2.2). Then h_{A} ≠ 0, λ(h_{A}) = λ(h), λ(h_{B}) ≤ λ(h), and Δ(h_{C}) ≺ Δ(h).

If Ī ⊊ J ⊆ A with deg(J) = k − 1, the element Tw̄η_{J} is an element of the standard R-basis of C_{k−1}. By hypothesis, the coefficient of ∂_{k}(h) on Tw̄η_{J} equals 0. Thus by the maximality of ℓ(w̄) and Proposition 3.4, one has

(3.17)  ∑_{t∈J} sgn(t, J \ {t}) α(w̄, J \ {t}) = 0.

Let

(3.18)  ϕ = ∑_{\begin{matrix}I⊆A \\ deg(I)=k \end{matrix}} α(w̄, I)η_{I},

i.e., Tw̄ϕ = h_{A}. Define D_{k}, k ≥ −1, to be the R-submodule

D_{k} = span_{R}({η_{I} | I ⊆ A, deg(I) = k}) ⊆ C_{k},

and let d_{k}: D_{k} → D_{k−1}, k ≥ 0, be the R-linear map given by

d_{k}(η_{I}) = ∑_{t∈A\I} sgn(t, I)η_{I⊔{t}};

one easily sees that d_{k}d_{k+1} = 0 for all k (cf. Fact 3.1). Hence (D_{•}, d_{•}) is a chain complex. Moreover, for I ⊆ A, deg(I) = k, one has

(3.19)  (∂_{k} − d_{k})(η_{I}) = ∑_{t∈S\A} sgn(t, I) η_{I⊔{t}}.

The chain complex of R-modules (D_{•}, d_{•}) concentrated in degrees k ≥ −1 is contractible (as (D_{k}, d_{k})_{k≥0} coincides with the singular chain complex of an (|A| − 1)- dimensional simplex with coefficients in R). Thus there exist homomorphisms of R-modules σ_{k}: D_{k} → D_{k+1}, k ≥ −1, satisfying d_{k+1}σ_{k} + σ_{k−1}d_{k} = id_{D_{k}}. Hence for ψ ∈ ker d_{k}, one has d_{k+1}(σ_{k}(ψ)) = ψ. Moreover, by (3.17)

(3.20)

d_{k}(ϕ) = ∑_{t∈A\I} sgn(t, I) α(w̄, I) η_{I⊔{t}}

= ∑_{\begin{matrix}J⊆A \\ deg(J)=k−1 \end{matrix}} ∑_{t∈J} sgn(t, J \ \{t\}) α(w̄, J \ \{t\}) η_{J} = 0

Claim 3.5.1. For all (w, I) ∈ supp(h_{A} − Tw̄∂_{k+1}(σ_{k}(ϕ))) one has ℓ(w) < ℓ(w̄).

Proof of Claim 3.5.1. Note that h_{A} = Tw̄ϕ. Since d_{k}(ϕ) = 0 (cf. (3.20)), one has d_{k+1}(σ_{k}(ϕ)) = ϕ. Thus by the previous remark

h_{A} − Tw̄(∂_{k+1}(σ_{k}(ϕ))) = Tw̄(ϕ − ∂_{k+1}(σ_{k}(ϕ))) = Tw̄(d_{k+1} − ∂_{k+1})(σ_{k}(ϕ)).

By (3.19), (d_{k+1} − ∂_{k+1})(σ_{k}(ϕ)) is an R-linear combination of elements η_{I} with I ⊈ A, deg(I) = k. As I ⊈ A, one has w̄_{I} ≠ 1 (cf. Prop. 2.1(i)), and therefore, ℓ(w̄^{I}) < ℓ(w̄). This yields the claim. □

Note that h_{0} = h − Tw̄∂_{k+1}(σ_{k}(ϕ)) ∈ Ω. As

(3.21)  A ⊔ B ⊔ C = { (w, I) | ℓ(w) ≤ ℓ(w̄), deg(I) = k, w ∈ W^{I} },

one concludes from Claim 3.5.1 that

(3.22)  h_{1} = h_{A} − Tw̄(∂_{k+1}(σ_{k}(ϕ))) ∈ span_{R}{ T_{w}η_{I} | (w, I) ∈ C }.

In particular, λ(h_{0}) ≤ λ(h). Thus by the minimality of min Δ one must have λ(h_{0}) = λ(h). But in this case one has by construction that ν(h_{0}) < ν(h), a contradiction, showing that Case 2 is impossible. From this one concludes that Ω = ∅; in particular, ker ∂_{k} = im ∂_{k+1}, for all k ∈ { −1, …, |S| − 1 }. Hence Proposition 3.5 completes the proof of the theorem. □

4. Traces and Euler characteristics

Throughout this section R will denote a commutative ring with unit. Without further mentioning we will always assume that an associative R-algebra A contains a unit 1 ∈ A. An R-linear isomorphism ^{♮}: A → A^{op} will be called an antipode, if ♮♮ = id_{A}. If (A, ^{♮}) is an R-algebra with antipode then ε ∈ Hom_{R-alg}(A, R) will be called an augmentation if ε(a) = ε(a^{♮}) for all a ∈ A.

4.1. Trace functions. Let A be an associative R-algebra with unit. A homomorphism of R-modules τ: A → R satisfying τ(ab) = τ(ba) for all a, b ∈ A is called a trace function on A. Let [A, A] = span_{R}(\{ ab − ba | a, b ∈ A \}), and let \underline{A} denote the R-module A/[A, A]^{3}. Then \underline{A}^{∗} = Hom_{R}(\underline{A}, R) is the R-module of all trace functions of A. The following elementary property will be useful for our purpose.

Proposition 4.1. Let (A, ^{♮}, ε) be an augmented, associative R-algebra with antipode, and let B ⊂ A be a free generating system of the R-module A with the following properties:

(a) 1 ∈ B;

(b) B^{♮} = B;

(c) the symmetric R-bilinear form

(4.1)  ⟨ , ⟩: A × A ⟶ R,  ⟨a, b⟩ = δ_{a,b}ε(a),  a, b ∈ B,

where δ_{.,.} denotes Kronecker’s δ-function, satisfies

(4.2)  ⟨ab, c⟩ = ⟨b, a^{♮}c⟩  for all a, b, c ∈ A.

Then μ̃ ∈ Hom_{R}(A, R) given by μ̃(a) = ⟨1, a⟩, a ∈ A, is a trace function.

Proof. By definition, one has for all a, b ∈ A that ⟨a^{♮}, b^{♮}⟩ = ⟨a, b⟩. Hence

(4.3)  μ̃(ab − ba) = ⟨1, ab⟩ − ⟨1, ba⟩ = ⟨a^{♮}, b⟩ − ⟨b^{♮}, a⟩ = 0.

for all a, b ∈ A. This yields the claim. □

Remark 4.2. Let (A, ^{♮}, ε, B) be an augmented, associative R-algebra with antipode containing an R-basis B ⊂ A satisfying the hypothesis of Proposition 4.1. Then the induced map μ ∈ Hom_{R}(\underline{A}, R) can be seen as the canonical trace function associated to (A, ^{♮}, ε, B).

4.2. Hattori-Stallings trace maps. For a finitely generated, projective, left Amodule P let P^{∗} = Hom_{A}(P, A). Then P^{∗} carries canonically the structure of a right A-module, and it is also finitely generated and projective. One has a canonical isomorphism γ_{P} : P^{∗} ⊗_{A} P ⟶ End_{A}(P) given by γ_{P} (p^{∗} ⊗ p)(q) = p^{∗}(q)p, p^{∗} ∈ P^{∗}, p, q ∈ P (cf. [5, Chap. I, Prop. 8.3]). The evaluation map ev_{P} : P^{∗} ⊗_{A} P → \underline{A} is given by ev_{P} (p^{∗} ⊗ p) = p^{∗}(p) + [A, A]. The map

(4.4)  tr_{P} = ev_{P} ∘γ_{P}^{−1} : End_{A}(P) ⟶ \underline{A}

is called the Hattori–Stallings trace map on P and r_{P} = tr_{P} (id_{P}) ∈ \underline{A} the Hattori– Stallings rank of P (cf. [10], [5, Chap. IX.2]). In particular, tr_{P} is R-linear, and for f, g ∈ End_{A}(P) one has

(4.5)  tr_{P} (f ∘ g) = tr_{P} (g ∘ f).

From the elementary properties of the evaluation map one concludes that if P_{1} and P_{2} are two finitely generated projective left A-modules, one has

(4.6)  r_{P_{1}⊕P_{2}} = r_{P_{1}} + r_{P_{2}}.

Let e ∈ A, e = e^{2}, be an idempotent in the R-algebra A. Then Ae is a finitely generated, projective, left A-module, and

(4.7)  r_{Ae} = e + [A, A].

3In the standard literature (cf. [3], [2], [5]) this R-module is denoted by T (A).

4.3. Finite, projective chain complexes. A chain complex P = (P_{•}, ∂_{•}^{P}) of left A-modules is called finite if { k ∈ ℤ | P_{k} ≠ 0 } is finite and P_{k} is finitely generated for all k ∈ ℤ. Moreover, P will be called projective, if P_{k} is projective for all k.

For P = (P_{•}, ∂_{•}^{P}) and Q = (Q_{•}, ∂_{•}^{Q}) finite, projective chain complexes of left A-modules we denote by (\underline{Hom}_{A}(P, Q)_{•}, d_{•}) the chain complex of right A-modules

(4.8)  \underline{Hom}_{A}(P, Q)_{k} = ∏_{j=i+k} Hom_{A}(P_{i}, Q_{j}),

with differential given by

(4.9)  (d_{k}(f_{k}))_{i,j−1} = ∂_{j}^{Q} ∘ f_{i,j} − (−1)^{k}f_{i−1,j−1} ∘ ∂_{i}^{P} ,

for f_{k} = ∑_{j=i+k} f_{i,j}. In particular, f_{0} = ∑_{i∈ℤ} f_{i,i} ∈ \underline{Hom}_{A}(P, Q)_{0} is a chain map of degree 0 if, and only if, f_{0} ∈ ker(d_{0}), and f_{0} is homotopy equivalent to the 0-map if, and only if, f_{0} ∈ im(d_{1}) (cf. [5, Chap. I]). Put \underline{Ext}_{0}^{A}(P, Q) = H_{0}(\underline{Hom}_{A}(P, Q)).

Let B = (B_{•}, ∂_{•}^{B}) be a finite, projective chain complex of right A-modules. Then (B \underline{⊗}_{A} P, ∂\underline{⊗} ; •) denotes the complex

(4.10)  \begin{matrix}(B \underline{⊗}_{A} P)_{k} = ∐_{i+j=k} B_{i} ⊗_{A} P_{j}, \\ ∂\underline{⊗} ; i+j(b_{i} ⊗ p_{j}) = ∂_{i}^{B}(b_{i}) ⊗ p_{j} + (−1)^{i}b_{i} ⊗ ∂_{j}^{P} (p_{j}). \end{matrix}

Let A⟦0⟧ denote the chain complex of left A-modules concentrated in degree 0 with A⟦0⟧_{0} = A, and let \underline{A}⟦0⟧ denote the chain complex of R-modules concentrated in degree 0 with \underline{A}⟦0⟧_{0} = \underline{A}. Then P^{⊛} = (P_{•}^{⊛}, ∂_{•}^{P^{⊛}}) = (\underline{Hom}_{A}(P, A⟦0⟧)_{•}, d_{•}),

(4.11)  \begin{matrix}P_{k}^{⊛} = Hom_{A}(P_{−k}, A), \\ ∂_{k}^{P^{⊛}}(p_{k}^{∗})(p_{1−k}) = (−1)^{k+1}p_{k}^{∗}(∂_{1−k}^{P}(p_{1−k})). \end{matrix}

is a finite, projective complex of right A-modules. Note that the differential of the complex is chosen in such a way that the standard evaluation mapping

(4.12)  \begin{matrix}ev_{P} : P^{⊛} \underline{⊗}_{A} P ⟶ \underline{A}⟦0⟧, \\ ev_{s,t}(p_{s}^{∗} ⊗ p_{t}) = δ_{s+t,0} p_{s}^{∗}(p_{t}), \end{matrix}

is a mapping of chain complexes. However, the natural isomorphism

(4.13)

γ: \underline{Hom}_{A}( _{1}, A⟦0⟧) \underline{⊗}_{A} _{2} ⟶ \underline{Hom}_{A}( _{1}, _{2})
γ_{s,t}(p_{s}^{∗} ⊗_{A} q_{t})(x_{−s}) = (−1)^{st} p_{s}^{∗}(x_{−s})q_{t}

comes equipped with a non-trivial sign (cf. [5, Chap. I, Prop. 8.3(b) and Chap. VI, §6, Ex. 1]). In this context the Hattori–Stallings trace map is given by

(4.14)  tr_{P} = H_{0}(ev_{P} ∘γ_{P,P}^{−1} ): \underline{Ext}_{0}^{A}(P, P) ⟶ \underline{A}.

It has the following properties:

Proposition 4.3. Let P = (P_{•}, ∂_{•}^{P}) be a finite, projective complex of left Amodules, and let [f], [g] ∈ \underline{Ext}_{0}^{A}(P, P), f = ∑_{k∈ℤ} f_{k}, be homotopy classes of chain maps of degree 0. Then

(a) tr_{P} ([f]) = ∑_{k∈ℤ}(−1)^{k} tr_{P_{k}}(f_{k});

(b) tr_{P} ([f] ∘ [g]) = tr_{P} ([g] ∘ [f]).

(c) Let Q = (Q_{•}, ∂_{•}^{Q}) be another finite, projective complex of left A-modules which is homotopy equivalent to P, i.e., there exist chain maps ϕ: P → Q, ψ: Q → P, which composites are homotopy equivalent to the respective identity maps. Let [h] ∈ \underline{Ext}_{0}^{A}(Q, Q) such that [ϕ] ∘ [f] = [h] ∘ [ϕ]. Then tr_{P} ([f]) = tr_{Q}([h]).

Proof. Part (a) is a direct consequence of (4.13), and (b) follows from (a) and (4.5). The left hand side quadrangle in the diagram

(4.15)  [figure: \underline{Hom}_{A}(P, P); ϕ∘ ∘ψ; P^{⊛} \underline{⊗}_{A} P; γ; ^{ev^{P}}; ψ^{⊛}⊗ϕ; \underline{A}⟦0⟧; \underline{Hom}_{A}(Q, Q); Q^{⊛} \underline{⊗}_{A} Q; γ; _{ev_{Q}}; \underline{A}⟦0⟧]

commutes, and the right hand side quadrangle commutes up to homotopy equivalence. This yields claim (c). □

Let P = (P_{•}, ∂_{•}^{P}) be a finite, projective complex of left A-modules. Then one defines the Hattori–Stallings rank of P by

(4.16)  r_{P} = tr_{P} ([id_{P}]) = ∑_{k∈ℤ}(−1)^{k}r_{P_{k}}. ∈ \underline{A}

Proposition 4.3 implies that if Q = (Q_{•}, ∂_{•}^{Q}) is another finite, projective, complex of left A-modules which is homotopy equivalent to P then r_{P} = r_{Q}.

Let K(A) denote the additive category the objects of which are finite, projective chain complexes of left A-modules. Morphisms Hom_{K(A)}(P, Q) = \underline{Ext}_{0}^{A}(P, Q) are given by the homotopy classes of chain maps of degree 0. In particular, K(A) is a triangulated category and distinguished triangles are triangles isomorphic to the cylinder/cone triangles (cf. [7], [12, Chap. 10]). Thus, if

(4.17)  \xymatrix{A \ar[r] & B \ar[r] & C \ar[r] & A[1]}

is a distinguished triangle in K(A), one has r_{B} = r_{A} + r_{C}.

4.4. Modules of type FP. A left A-module M is called of type FP, if it has a resolution (P_{•}, ∂_{•}^{P} , ε_{M}), such that P = (P_{•}, ∂_{•}^{P}) is a finite, projective complex of left A-modules. For such an A-module one defines the Hattori–Stallings rank by r_{M} = r_{P} ∈ \underline{A}. The comparison theorem in homological algebra implies that this element is well defined.

An augmented R-algebra (A, ε) is called of type FP, if the left A-module R_{ε} = R, a.r = ε(a)r, a ∈ A, r ∈ R, is of type FP. Let (A, ^{♮}, ε) be an augmented, associative R-algebra with antipode, and let B ⊂ A be a free basis of A as R-module such that

(a) A is of type FP, and

(b) B satisfies the hypothesis of Proposition 4.1. Then one defines the Euler characteristic of (A, ^{♮}, ε, B) by

(4.18)  χ_{A} = χ_{(A, ^{♮},ε,B)} = μ(r_{R_{ε}}) ∈ R,

where μ: \underline{A} → R denotes the canonical trace function (cf. Remark 4.2). The following property will be useful for our purpose.

Proposition 4.4. Let C = (C_{•}, ∂_{•}^{C}) be a chain complex of left A-modules concentrated in non-negative degrees with the following properties:

(a) C is acyclic, i.e., H_{k}(C) = 0 for k ∈ ℤ, k ≠ 0; (b) C is finitely supported, i.e., C_{k} = 0 for almost all k ∈ ℤ;

(c) C_{k} is of type FP for all k ∈ ℤ. Then H_{0}(C) is of type FP, and one has

(4.19)  r_{H_{0}(C)} = ∑_{k≥0}(−1)^{k}r_{C_{k}} ∈ \underline{A}.

Proof. Let ℓ(C) = min{ n ≥ 0 | C_{n+j} = 0 for all j ≥ 0 } denote the length of C. We proceed by induction on ℓ(C). For ℓ(C) = 1, there is nothing to prove. Suppose the claim holds for chain complexes D, ℓ(D) ≤ ℓ − 1, satisfying the hypothesis (a)–(c), and let C be a complex satisfying (a)–(c) with ℓ(C) = ℓ. Let C^{∧} be the chain complex coinciding with C in all degrees k ∈ ℤ \ {0} and satisfying C_{0}^{∧} = 0. Then C^{∧}[−1] satisfies (a)–(c) and ℓ(C^{∧}[−1]) ≤ ℓ−1. Then by induction, M = H_{1}(C^{∧}) = H_{0}(C^{∧}[−1]) is of type FP, and r_{M} = ∑_{k≥1}(−1)^{k+1}r_{C_{k}}. By construction, one has a short exact sequence of left A-modules 0 → M →^{α} C_{0} → H_{0}(C) → 0. Let (P_{•}, ∂_{•}^{P} , ε_{M}) be a finite, projective resolution of M, and let (Q_{•}, ∂_{•}^{Q}, ε_{C_{0}}) be a finite, projective resolution of C_{0}. By the comparison theorem in homological algebra, there exists a chain map α_{•}: P_{•} → Q_{•} inducing α. Let Cone(α_{•}) denote the mapping cone of α_{•}. Then (Cone(α_{•}), ˜∂_{•}, ε_{∗}) is a finite, projective resolution of H_{0}(C), i.e., H_{0}(C) is of type FP. Moreover, by the remark following (4.17) one has

(4.20)  r_{H_{0}(C)} = r_{Cone(α_{•})} = r_{Q} − r_{P} = r_{C_{0}} − r_{M}.

This yields the claim. □

4.5. Induction. Let B ⊆ A be an R-subalgebra of A. The canonical injection j: B → A induces a canonical map

(4.21)  tr_{B/A} : \underline{B} → \underline{A}.

Induction ind_{B}^{A} = A ⊗_{B} is a covariant additive right-exact functor mapping finitely generated projective left B-modules to finitely generated projective left Amodules. Moreover, if A is a flat right B-module, then ind_{B}^{A} is exact. Let P be a finitely generated left B-module, and let Q = ind_{B}^{A}(P). Then one has a canonical map ι: P → Q, ι(p) = 1 ⊗ p, which is a homomorphism of left B-modules. As induction is left adjoint to restriction, every map f ∈ End_{B}(P) induces a map ι_{∘}(f) = (ι ∘ f)_{∗} ∈ End_{A}(Q).

Let P^{∗} = Hom_{B}(P, B) and Q^{∗} = Hom_{A}(Q, A). Then for f ∈ Hom_{B}(P, B) one has an induced map ι_{∗}(f) = (j ∘ f)_{∗} ∈ Q^{∗} making the diagram

(4.22)  \xymatrix{End_{B}(P) \ar[d]_{ι_{∘}} & P^{∗} ⊗_{B} P \ar[l]_{γ_{P}} \ar[d]^{ι_{∗}⊗ι} \ar[r]^{^{ev^{P}}} & \underline{B} \ar[d]^{_{tr_{B/A}}} \\ End_{A}(Q) & Q^{∗} ⊗_{A} Q \ar[l]_{γ_{Q}} \ar[r]^{_{ev_{Q}}} & \underline{A}}

commute. This shows the following.

Proposition 4.5. Let B ⊆ A be an R-subalgebra of A such that A is a flat right B-module, and let M be a left B-module of type FP. Then ind_{B}^{A}(M) is of type FP, and one has

(4.23)  r_{ind_{A}^{B}(M)} = tr_{B/A}(r_{M}).

Let (A, ^{♮}, ε, B) be an augmented, associative, R-algebra with antipode and a distinguished R-basis B satisfying the hypothesis of Proposition 4.1. Let B ⊆ A be an R-subalgebra of A such that

(i) A is a flat right B-module; (ii) B^{♮} = B; (iii) C = B ∩ B is an R-basis of B.

Let μ_{A} : \underline{A} → R and μ_{B} : \underline{B} → R denote the associated canonical traces. Then one has a commutative diagram

(4.24)  \xymatrix{\underline{B} \ar[rr]^{^{tr^{B/A}}} \ar[dr]_{μ_{B}} &  & \underline{A} \ar[dl]^{μ_{A}} \\  & R}

From this one concludes the following direct consequence of Proposition 4.5.

Corollary 4.6. Let (A, ^{♮}, ε, B) be an augmented, associative, R-algebra with antipode and a distinguished R-basis B satisfying the hypothesis of Proposition 4.1, and let B ⊆ A be an R-subalgebra satisfying (i)-(iii). Let M be a left B-module of type FP. Then μ_{B}(r_{M}) = μ_{A}(r_{ind_{B}^{A}(M)}).

5. The Euler characteristic of a Hecke algebra

5.1. The canonical trace of a Hecke algebra. Let H = H_{q}(W, S) be the R-Hecke algebra associated to the finitely generated Coxeter group (W, S), and let B = { T_{w} | w ∈ W }. Then ^{♮}: H → H^{op}, T_{w}^{♮} = T_{w^{−1}}, is an anti-automorphism of H satisfying ^{♮♮} = id_{H} (cf. [8, Chap. 7.3, Ex. 1]) and ε_{q}(a^{♮}) = ε_{q}(a) for all a ∈ H. One has the following property.

Proposition 5.1. Let H be the Hecke algebra associated to the finitely generated Coxeter group (W, S). Then the R-bilinear map ⟨ , ⟩: H × H → R associated to (H, B, ^{♮}, ε) satisfies (4.2). In particular, μ̃_{B} = ⟨ T_{1}, ⟩ is a trace function.

Proof. By Proposition 4.1, one has to show that

(5.1)  ⟨ T_{u}T_{v}, T_{w} ⟩ = ⟨ T_{v}, T_{u^{−1}}T_{w} ⟩  for all u, v, w ∈ W.

Using induction one easily concludes that it suffices to show (5.1) in the case that u = s ∈ S. In this case one has:

λ = ⟨ T_{s}T_{v}, T_{w} ⟩ = \begin{cases}δ_{sv,w}ε_{q}(T_{sv}) & if ℓ(sv) > ℓ(v) \\ (q − 1)δ_{v,w}ε_{q}(T_{v}) + qδ_{sv,w}ε_{q}(T_{sv}) & if ℓ(sv) < ℓ(v) \end{cases} (5.2)

and

ρ = ⟨ T_{v}, T_{s}T_{w} ⟩ = \begin{cases}δ_{v,sw}ε_{q}(T_{v}) & if ℓ(sw) > ℓ(w) \\ (q − 1)δ_{v,w}ε_{q}(T_{v}) + qδ_{v,sw}ε_{q}(T_{v}) & if ℓ(sw) < ℓ(w) \end{cases} (5.3)

We proceed by a case-by-case analysis.

Case 1: ℓ(sv) > ℓ(v) and ℓ(sw) > ℓ(w). Suppose that λ ≠ 0. Then sv = w, but ℓ(w) = ℓ(sv) > ℓ(v) = ℓ(sw), a contradiction. Hence λ = 0. Reversing the rôles of v and w yields λ = ρ = 0 and thus the claim.

Case 2: ℓ(sv) > ℓ(v) and ℓ(sw) < ℓ(w). Then, v ≠ w. If λ ≠ 0, then sv = w. Hence ℓ(w) = ℓ(sv) = ℓ(v)+1, and λ = ε_{q}(T_{w}) = ε_{q}(T_{s})ε_{q}(T_{v}). On the other hand, ρ = (q − 1)δ_{v,w}ε_{q}(T_{v}) + qδ_{v,sw}ε_{q}(T_{v}) = qε_{q}(T_{v}) = λ. If λ = 0, then sv ≠ w. Hence ρ = (q − 1)δ_{v,w}ε_{q}(T_{v}) + qδ_{v,sw}ε_{q}(T_{v}) = 0 = λ.

Case 3: ℓ(sv) < ℓ(v) and ℓ(sw) > ℓ(w). Reversing the rôles of v and w one can transfer the proof for Case 2 verbatim.

Case 4: ℓ(sv) < ℓ(v) and ℓ(sw) < ℓ(w). Suppose that sv = w, or, equivalently, v = sw. Then ℓ(sv) < ℓ(v) = ℓ(sw) < ℓ(w), a contradiction. Hence sv ≠ w and v ≠ sw. Thus λ = ρ. This completes the proof. □

Remark 5.2. The trace function μ̃: H → R can be seen as the canonical trace function on H. It is straight forward to verify that for Hecke algebras of type A_{n}, B_{n} or D_{n} this trace function coincides with the Jones–Ocneanu trace evaluated in 0 (cf. [6]).

5.2. Properties of the Coxeter complex. Let (W, S) be a finite Coxeter group, and let q ∈ R be such that p_{(W,S)}(q) ∈ R^{×}. Then R_{q} ≃ He_{S} (cf. Prop. 2.3); in particular, R_{q} is a projective left H-module. This shows that for any Coxeter group (W, S) and I ⊆ S such that W_{I} is finite, ind_{H_{I}}^{H} (R_{q}) is a finitely generated, projective, left H-module. As a consequence one has the following (cf. [8, §6.8]):

Proposition 5.3. Let (W, S) be a finitely generated Coxeter group, which is either affine, or compact hyperbolic and let q ∈ R be such that p_{(W_{I},I)}(q) ∈ R^{×} for any proper parabolic subgroup (W_{I}, I). Then the Coxeter complex (C_{•}, ∂_{•}, ε) is a finite projective resolution of R_{q}.

In the general case one has the following:

Proposition 5.4. Let (W, S) be a finitely generated Coxeter group, and let q ∈ R be such that p_{(W_{I},I)}(q) ∈ R^{×} for any finite parabolic subgroup (W_{I}, I). Then (C_{•}, ∂_{•}) is a chain complex of left H-modules of type FP; in particular, R_{q} is a left H-module of type FP.

Proof. By hypothesis and the previously mentioned remark, ind_{H_{I}}^{H} (R_{q}) is a finitely generated projective H-module for any finite parabolic subgroup (W_{I}, I). We proceed by induction on d = |S|. For |S| ≤ 2, there is nothing to prove. Assume that the claim holds for all Coxeter groups (W_{J}, J) with |J| < d, and that |S| = d. By induction, for K ⊊ S, R_{q} is a left H_{K}-module of type FP. Hence ind_{H_{K}}^{H} (R_{q}) is a left H-module of type FP. Thus C_{k} is a left H-module of type FP for 0 ≤ k ≤ d−1. If (W, S) is spherical, then R_{q} is a finitely generated, projective, left H-module by the first remark. If (W, S) is non-spherical, (C_{•}, ∂_{•}) is acyclic. Hence R_{q} is a left H-module of type FP by Proposition 4.4. This completes the proof. □

5.3. The Euler characteristic of a Hecke algebra. Proposition 5.4 has the following consequence.

Proposition 5.5. Let (W, S) be a finitely generated, non-spherical Coxeter group, and let q ∈ R be such that p_{(W_{I},I)}(q) ∈ R^{×} for any finite parabolic subgroup (W_{I}, I). Then

(5.4)  r_{R_{q}} = ∑_{I⊊S}(−1)^{|S\I|−1}r_{ind_{I}^{S}(R_{q})}.

Proof. By (4.19) and (4.23), one has

(5.5)  r_{R_{q}} = ∑_{0≤k<|S|}(−1)^{k}r_{C_{k}} = ∑_{I⊊S}(−1)^{|S\I|−1}r_{ind_{I}^{S}(R_{q})}.

This yields the claim. □

Proof of Theorem C. If (W, S) is spherical, R_{q} ≃ He_{S} where e_{S} is given as in Proposition 2.3. Hence, as r_{R_{q}} = e_{S} + [H, H] (cf. (4.7)), one has χ_{H} = μ(r_{R_{q}}) = p_{(W,S)}(q)^{−1}.

If (W, S) is non-spherical, we proceed by induction on |S|. Proposition 4.5, Corollary 4.6 and Proposition 5.5 imply that

χ_{H} = μ_{H}(r_{R_{q}}) = ∑_{I⊊S}(−1)^{|S\I|−1}μ_{H}(r_{ind_{I}^{S}(R_{q})})

= ∑_{I⊊S}(−1)^{|S\I|−1}μ_{H_{I}} (R_{q})

and thus by induction

= ∑_{I⊊S}(−1)^{|S\I|−1}p_{(W_{I},I)}(q)^{−1}. (5.6)

It is well-known that the alternating sum (5.6) is equal to p_{(W,S)}(q)^{−1} (cf. [8, Prop. 5.12]). □

References

1. P. Abramenko and K. S. Brown, Buildings. theory and applications, Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008.

2. H. Bass, Euler characteristics and characters of discrete groups, Invent. Math. 35 (1976), 155–196.

3. , Traces and Euler characteristics, Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge, 1979, pp. 1–26.

4. N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 4 à 6., Springer, Berlin, 2007 (French), Réimpression inchangée de l’édition originale de 1968.

5. K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1982.

6. M. Geck, Trace functions on Iwahori-Hecke algebras, Knot theory (Warsaw, 1995), Banach Center Publ., vol. 42, Polish Acad. Sci., Warsaw, 1998, pp. 87–109.

7. S. I. Gelfand and Yu. I. Manin, Homological algebra, Springer-Verlag, Berlin, 1999.

8. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990.

9. A. Polishchuk and L. Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005.

10. J. Stallings, Centerless groups—An algebraic formulation of Gottlieb’s theorem, Topology 4 (1965), 129–134.

11. T. Terragni, Hecke algebras associated to Coxeter groups, Ph.D. thesis, Università di Milano– Bicocca, Italy, 2012, (in preparation).

12. C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.

T. Terragni, T. Weigel, Università di Milano-Bicocca, Ed. U5, Via R.Cozzi, 53, 20125 Milano, Italy

E-mail address: t.terragni@campus.unimib.it
E-mail address: thomas.weigel@unimib.it