arXiv:1703.02586v2 [math.AT] 29 Sep 2017

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS
GIOVANNI PAOLINI AND MARIO SALVETTI
Abstract. In this paper we expand the theory of weighted sheaves over posets,
and use it to study the local homology of Artin groups. First, we use such
theory to relate the homology of classical braid groups with the homology of
certain independence complexes of graphs. Then, in the context of discrete
Morse theory on weighted sheaves, we introduce a particular class of acyclic
matchings. Explicit formulas for the homology of the corresponding Morse
complexes are given, in terms of the ranks of the associated incidence matrices.
We use such method to perform explicit computations for the new affine case
C̃n , as well as for the cases An , Bn and Ãn (which were already done before
by different methods).

Contents
1. Introduction
2. Preliminaries
2.1. Weighted sheaves over posets
2.2. Decomposition and filtration
2.3. Discrete Morse theory for weighted complexes
2.4. Weighted sheaves for Artin groups
3. Homology of braid groups and independence complexes
4. Precise matchings
5. Precise matchings for Artin groups
5.1. Case An
5.2. Case Bn
5.3. Case Ãn
5.4. Case C̃n
References

1
4
4
5
6
8
9
14
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20
25
32
36
42

1. Introduction
The topological theory of Artin groups started in the 60’s with classical braid
groups [FN62, Arn69, Arn70], and was broadened to general Artin groups in the
70’s [Fuc70, BS72, Del72, Bri73, Coh76, Wei78, Gor78]. It has received big attention in connection with problems in singularity theory, hyperplane arrangements,
and combinatorics (see [Par14] for some references), until recent applications of
some special cases to conjectures about hyperbolic groups [AGM13]. Therefore any
This work was partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca,
and by University of Pisa, Project no. PRA 67.
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GIOVANNI PAOLINI AND MARIO SALVETTI

method to calculate topological invariants of Artin groups (e.g. their homology and
cohomology) should be considered interesting and subject to possible applications.
Thanks to some good topological model for the K(π, 1) space of Artin groups
[Sal87, Sal94], a combinatorial free resolution for these groups was found [DCS96],
and several (co)homology computations were carried out [DCPSS99, DCPS01, CS04,
Cal05, Cal06, CMS08a, CMS08b, CMS10] (an equivalent resolution for Artin groups
of finite type, based on purely algebraic methods, appeared in [Squ94]). The computational methods used in these papers are essentially based on filtering the algebraic
complex and using the associated spectral sequence.
In [SV13] a more combinatorial method of calculation was introduced, based
on the application of discrete Morse theory to a particular class of sheaves over
posets (called weighted sheaves). It is interesting to notice that similar ideas were
considered later in [CGN16], even if there the authors were mainly interested in
computational aspects.
In this paper we expand the theory of weighted sheaves. First, we observe a surprising relationship between some twisted (co)homology of the classical braid groups
and the (co)homology of certain independence complexes, which in their simplest
form have already been studied in a combinatorial context (see for example[Eng09]).
This relationship is obtained in an indirect way, after localization. The exact formula we are going to prove is the following (it was announced in [Sal15] without
proof).
Theorem 3.4.


R
∼
.
H∗ (Brn+1 ; R)ϕd = H̃∗−d+1 Indd−2 (An−d );
(ϕd )
Here on the left we have the homology with local coefficients of the classical braid
group and on the right we have the homology of some independence complex with
trivial coefficients (we indicate by ϕd the d-th cyclotomic polynomial). The homology of these independence complexes can be computed separately, so our results can
be used in both directions: either in order to compute the homology of these complexes from that of Artin groups, or to compute the homology of these complexes
to deduce that of Artin groups.
As said before, our method is based on the notions of weighted sheaves over
posets and of weighted matchings, which are interesting objects by themselves. In
our situation, the d-weight vd (σ) of a simplex σ is the maximal power of the dth cyclotomic polynomial which divides the Poincaré polynomial of the (finite)
parabolic subgroup generated by the vertices of the simplex. We allow matchings
between simplices of the same weight. Applying discrete Morse theory, one obtains
a Morse complex which still computes the local homology.
Then, by applying these methods to other Coxeter graphs (in particular to finite
and affine type cases), we come up with the notion of precise matching. In all
the cases we consider, it is possible to produce acyclic weighted matchings such
that the Morse complex turns out to have a very nice property: the boundary of a
critical p-simplex σ has a non-zero coefficient along a critical (p − 1)-simplex τ if
and only if vd (σ) = vd (τ ) + 1. Then the homology of the Morse complex reduces
to the computation of the ranks of the incidence matrices among critical simplices
(Theorems 4.2 and 4.4). We show that the existence of precise matchings can also
be interpreted in terms of second-page collapsing of a spectral sequence which is
naturally associated to the weighted sheaf (Proposition 4.3).

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

3

As we discuss at the beginning of Section 5, the existence of precise matchings
for a certain Artin group has strong implications for its homology. In particular we
want to highlight the following non-trivial result.
Theorem 5.2. Let GW be an Artin group (corresponding to some finitely generated Coxeter group W) that admits a ϕ-precise matching for all cyclotomic polynomials ϕ = ϕd (with d ≥ 2). Then the homology H∗ (XW ; R) does not have
ϕk -torsion for k ≥ 2.
Here XW (introduced in Section 2.4) is a finite CW-complex which is a deformation
retract of the orbit space associated to W, and π1 (XW ) = GW . When W is finite
and in a few other cases XW it is known to be a K(GW , 1), whereas this property
is only conjectured in general [Del72, Bri73, Oko79, VdL83, Hen85, Sal87, Sal94,
CD95, CMS10, Par14].
Theorem 5.2 can be seen as a result about the homology of the infinite cyclic
covering of XW and its monodromy. It is known that the conclusion of such corollary holds for all Artin groups of finite type, due to geometric reasons (see Remark
5.3). Our construction provides a new combinatorial condition that applies to a
wider class of Artin groups (including affine groups of type Ãn and C̃n , and also
others).
The methods we develop allow explicit and complete computations, in general in
a more direct way with respect to other known methods. We use them to compute
the twisted homology in the case of affine Artin groups of type C̃n , obtaining the
following result.
Theorem 5.25. Let GW be an Artin group of type C̃n . Then the ϕd -primary
component of H∗ (GW ; R) is trivial for d odd, and for d even is as follows (let
n = k d2 + r):
Hm (GW ; R)ϕd ∼
=

(

(R/(ϕd ))
0

⊕m+k−n+1

if n − k ≤ m ≤ n − 1,
otherwise.

We also carry out complete computations in the cases An , Bn and Ãn : these
cases were already done before by different methods [Fre88, DCPSS99, DCPS01,
CMS08a], but we need them to perform computations of more complicated cases.
In a recent preprint [Pao17], the first author succeeded to construct precise
matchings for all remaining Artin groups of finite and affine type. Therefore precise
matchings seem to be a suitable and effective tool for studying the homology of
Artin groups.
The paper is structured as follows. In Section 2 we recall the most important
definitions and results of [SV13]. In particular we introduce the general framework
of weighted sheaves over posets, and then explain how it can be used to compute the
homology of Artin groups. In Section 3 we study the case of braid groups (i.e. Artin
groups of type An ), and relate their homology with that of certain independence
complexes. In Section 4 we introduce precise matchings and derive a general formula
for the homology of the Morse complex. In Section 5 we apply the theory developed
in Section 4 to the case of Artin groups. We construct precise matchings for Artin
groups of type An , Bn , Ãn and C̃n , and use such matchings to explicitly compute
the homology.

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GIOVANNI PAOLINI AND MARIO SALVETTI

2. Preliminaries
In this section we are going to recall some definitions and constructions from
[SV13] (see also [MSV12]).
2.1. Weighted sheaves over posets. Let (P, ) be a finite poset.
Definition 2.1. Define a sheaf of rings over P (or a diagram of rings over P ) as
a collection
{Ax | x ∈ P }
of commutative rings together with a collection of ring homomorphisms
{ρx,y : Ay → Ax | x  y}
satisfying
ρx,x = idAx ;
x  y  z ⇒ ρx,z = ρx,y ρy,z .
In other words, a sheaf of rings over P is a functor from P op to the category of
commutative rings.
Fix a PID R; usually we take R = Q[q ±1 ]. The divisibility relation
p1 | p2 ⇔ (p1 ) ⊇ (p2 )
gives R the structure of a small category (and to R/R∗ the structure of a poset,
where R∗ is the group of units in R, with minimum element the class of the units
and maximum element 0). Any functor w : (P, ) → (R, |), which maps every x ∈ P
to some w (x) ∈ R \ {0}, defines a sheaf over P by the collection
{R/(w (x)) | x ∈ P }
and
{ix,y : R/(w (y)) → R/(w (x)) | x  y}
where ix,y is induced by the identity of R.
Definition 2.2. Given a poset P , a PID R and a morphism w : (P, ) → (R, |) as
above, we call the triple (P, R, w ) a weighted sheaf over P and the coefficients w (x)
the weights of the sheaf.
Remark 2.3. Consider the poset topology over P , where a basis for the open sets
is given by the upper intervals B = {P>p , p ∈ P }. Let (P, R, w ) be a weighted
sheaf. Then one can see w as a functor from (B, ⊇) to (R, |) and thus a weighted
sheaf defines a sheaf in the usual sense.
From now on our poset will be a simplicial complex K defined over a finite set
S, with the partial ordering
σ  τ ⇔ σ ⊆ τ.
We adopt the convention that K contains the empty simplex ∅. A weighted sheaf
over K is given by assigning to each simplex σ ∈ K a weight w (σ) ∈ R, with
σ  τ ⇒ w (σ) | w (τ ).
C∗0 (K; R)

Let
be the 1-shifted standard algebraic complex computing the simplicial
homology of K, i.e.
M
Ck0 (K; R) =
R e0σ
σ∈K
|σ|=k

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

5

where e0σ is a generator associated to a given orientation of σ (C00 (K; R) = R e0∅ ).
The boundary is given by
X
[σ : τ ] e0τ ,
∂ 0 (e0σ ) =
|τ |=k−1

where [σ : τ ] is the incidence number (which is equal to ±1 if τ ≺ σ, and otherwise
is equal to 0).
Definition 2.4. The weighted complex associated to the weighted sheaf (K, R, w )
is the algebraic complex L∗ = L∗ (K) defined by
M
R
Lk =
ēσ ,
(w(σ))
|σ|=k

with boundary ∂ : Lk → Lk−1 induced by ∂ 0 :
X
∂(aσ ēσ ) =
[σ : τ ] iτ,σ (aσ ) ēτ .
τ ≺σ

C∗0

There is a natural projection π : C∗0 (K; R) → L∗ which sends a generator eσ in
to the generator ēσ in L∗ .

Remark 2.5. The diagram {R/(w(σ))}, for σ ∈ K, also defines a sheaf over the
poset K in the sense of Remark 2.3. The sheaf cohomology associated to the open
covering given by the upper segments coincides with the homology of the weighted
sheaf.
2.2. Decomposition and filtration. Let S = (K, R, w ) be a weighted sheaf. For
any irreducible ϕ ∈ R, we define the ϕ-primary component Sϕ = (K, R, wϕ ) of the
weighted sheaf S by setting
wϕ (σ) = ϕvϕ (σ) ,
where
vϕ (σ) = maximal r such that ϕr divides w (σ).
Since R is a PID and w(σ) 6= 0, such a maximal value exists. Notice that Sϕ
is a weighted sheaf. The weighted complex (L∗ )ϕ associated to Sϕ is called the
ϕ-primary component of the weighted complex L∗ . In degree k one has:
M
R
(Lk )ϕ =
ēσ .
v
(ϕ ϕ (σ) )
|σ|=k

The complex (L∗ )ϕ has a natural increasing filtration by the following subcomplexes:
M
R
ēσ .
F s (L∗ )ϕ =
v
(ϕ ϕ (σ) )
vϕ (σ)≤s

Such filtration is associated to an increasing filtration of the simplicial complex K:
Kϕ,s = {σ ∈ K | vϕ (σ) ≤ s}.

Then F s (L∗ )ϕ is the weighted complex associated to the weighted sheaf
(Kϕ,s , R, wϕ |Kϕ,s ).

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GIOVANNI PAOLINI AND MARIO SALVETTI

Theorem 2.6 ([SV13]). Let (K, R, w ) be a weighted sheaf, with associated weighted
complex L∗ . For any irreducible ϕ ∈ R, there exists a spectral sequence
r
Ep,q
⇒ H∗ ((L∗ )ϕ )

that abuts to the homology of the ϕ-primary component of the associated algebraic
complex L∗ . Moreover the E 1 -term
E 1 = Hp+q (F p /F p−1 ) ∼
= Hp+q (Kϕ,p , Kϕ,p−1 ; R/(ϕp ))
p,q

is isomorphic to the relative homology with trivial coefficients of the simplicial pair
(Kϕ,p , Kϕ,p−1 ).
2.3. Discrete Morse theory for weighted complexes. Given x, y in a poset
(P, ≺), the notation x ⊳ y means that y covers x, i.e. x ≺ y and there is no other
z ∈ P such that x ≺ z ≺ y.
Recall the basic facts of discrete Morse theory (see for example [For98, Koz07]).
A matching in a poset P is a set M ⊆ P × P such that (x, y) ∈ M ⇒ x ⊳ y and
each x ∈ P belongs to at most one pair of M. An alternating path is a sequence
y0 ⊲ x1 ⊳ y1 ⊲ x2 ⊳ y2 ⊲ · · · ⊲ xm ⊳ ym (⊲xm+1 )
such that each pair xi ⊳ yi belongs to M and no pair xi ⊳ yi−1 belongs to M. A
cycle is a closed alternating path with y0 = ym . An acyclic matching over P is
a matching with no cycles. We are going to describe a variant of discrete Morse
theory which is suitable for our situation.
Definition 2.7. A weighted acyclic matching on a weighted sheaf (P, R, w ) over P
is an acyclic matching M on P such that
(x, y) ∈ M ⇒ (w (x)) = (w (y))
(in [SV13] we required w (x) = w (y), but the above generalization does not change
the results).
The standard theory generalizes as follows. Let S = (K, R, w ) be a weighted
sheaf over a finite simplicial complex K, and let M an acyclic weighted matching.
A critical simplex of K is a simplex σ which does not belong to any pair of M. The
following definition is equivalent to [SV13, Definition 3.3].
Definition 2.8. The Morse complex of S with respect to M is defined as the
torsion complex
M
R
LM
ēσ
∗ =
(w (σ))
σ critical

with boundary

∂ M (ēσ ) =

X

[σ : τ ]M ēτ ,

τ critical
|τ | = |σ|−1

where [σ : τ ]M ∈ Z is given by the sum over all alternating paths
σ ⊲ τ1 ⊳ σ1 ⊲ τ2 ⊳ σ2 ⊲ · · · ⊲ τm ⊳ σm ⊲ τ
from σ to τ of the quantity
(−1)m [σ : τ1 ][σ1 : τ1 ][σ1 : τ2 ][σ2 : τ2 ] · · · [σm : τm ][σm : τ ].
The boundary map ∂ M is extended by R-linearity, where some care should be taken
since each ēτ lives in a component with possibly different torsion (∂ M is the same
as the boundary map of [SV13]).

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

7

Theorem 2.9 ([SV13]). Let S = (K, R, w ) be a weighted sheaf over a simplicial
complex K. Let M be an acyclic matching for S. Then there is an isomorphism
H∗ (L∗ , ∂) ∼
= H∗ (LM , ∂ M )
∗

between the homology of the weighted complex L∗ associated to S and the homology
of the Morse complex LM
∗ of S.
If we forget about the weights, the matching M is in particular an acyclic matching for the simplicial complex K. Therefore the algebraic complex C∗0 (K; A), which
computes the (shifted and reduced) simplicial homology of K with coefficients in
some ring A, admits a “classical” algebraic Morse complex
M
C∗0 (K; A)M =
A ēσ
σ critical

with boundary map

δ M (ēσ ) =

X

[σ : τ ]M ēτ .

τ critical
|τ | = |σ|−1

Remark 2.10. Set C∗0 = C∗0 (K; R) and let π : C∗0 → L∗ be the natural projection.
The composition with the projection L∗ → (L∗ )ϕ , for some fixed irreducible element
ϕ ∈ R, yields a natural projection πϕ : C∗0 → (L∗ )ϕ . Similarly at the level of Morse
complexes we have the projection π̄ϕ : (C∗0 )M → (L∗ )M
ϕ which sends a generator ēσ
0 M
M
in (C∗ ) to the generator ēσ in (L∗ )ϕ . Then, by construction, the maps induced
in homology by the projections πϕ and π̄ϕ commute with the Morse isomorphisms
of [Koz07] and [SV13]:
H∗ (C∗0 )

(πϕ )∗

H∗ ((L∗ )ϕ )

∼
=

∼
=

H∗ ((C∗0 )M )

(π̄ϕ )∗

H∗ ((L∗ )M
ϕ ).

Let ϕ ∈ R be an irreducible element, and let M be a weighted acyclic matching
on Sϕ . Then the filtration on the weighted complex induces a filtration on the
Morse complex:
M
R
ēσ .
F p (L∗ )M
ϕ =
v
(ϕ ϕ (σ) )
σ critical
vϕ (σ)≤p

Consider also the following quotient complex:
p
M
p−1
∼
F p (L∗ )M
(L∗ )M
ϕ = F (L∗ )ϕ / F
ϕ =

M

σ critical
vϕ (σ)=p

R
ēσ .
(ϕp )

Theorem 2.11 ([SV13]). Let S and ϕ be as above and let M be an acyclic matching for Sϕ . Then the E 1 -page of the spectral sequence of Theorem 2.6 is identified
with
1 ∼
Ep,q
= Hp+q (F p (L∗ )M
ϕ ),
where (L∗ )M
ϕ is the Morse complex of Sϕ . The differential
1
1
d1p,q : Ep,q
→ Ep−1,q

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GIOVANNI PAOLINI AND MARIO SALVETTI

is induced by the boundary of the Morse complex, and thus it is also computed by
using alternating paths.
2.4. Weighted sheaves for Artin groups. We use the constructions given above
in the context of Artin groups. Let (W, S) be a Coxeter system, with |S| = n finite,
and let Γ be the corresponding Coxeter graph (with S as vertex set). Recall that W
can always be realized as a group of reflections in some Rn (for example through
the Tits representation, see [Bou68, Vin71], see also [Par14]), so that it has the
following naturally associated objects:
(i) a hyperplane arrangement in Rn
A = {H | H is the fixed point set of some reflection in W};
(ii) configuration spaces
Y = (int(U ) + iRn ) \

[

HC ,

H∈A

YW = Y/W

where U = W · C0 is the Tits cone (here C0 is a fixed chamber of the arrangement, and C0 is its topological closure);
(iii) a simplicial complex (defined over the finite set S)
K = KW = {σ ⊆ S | the parabolic subgroup Wσ generated by σ is finite}.
One can define the Artin group GW of type W as the fundamental group
π1 (YW ). It has a presentation
GW = h gs , s ∈ S | gs gs′ gs gs′ . . . = gs′ gs gs′ gs . . . i
(both the products have m(s, s′ ) factors). Recall also the following results [Sal94,
Theorems 1.4 and 1.8].
(i) The orbit space YW deformation retracts onto a finite CW -complex XW
given by a union
[
Q =
Qσ
σ∈KW

of convex polyhedra with explicit identifications of their faces.
(ii) Consider the action of the Artin group GW on the ring R = Q[q ±1 ] given by
gs 7→ [multiplication by − q] ∀s ∈ S.
Then the homology H∗ (XW ; R) is computed by the algebraic complex
M
Ck =
R eσ
σ∈KW
|σ|=k

with boundary
∂(eσ ) =

X

τ ⊳σ

where Wσ (q) =

X

w∈Wσ

q

l(w)

[σ : τ ]

Wσ (q)
eτ ,
Wτ (q)

is the Poincaré polynomial of Wσ .

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

9

Remark 2.12. The R-module structure on H∗ (XW ; R) is given by the transformation µq induced by q-multiplication. If the order of µq is n, then the homology
groups decompose into cyclic factors which are either free, of the form Rk , or torsion, of the form R/(ϕd ), where ϕd is the d-th cyclotomic polynomial with d | n.
Therefore we are interested in localizing to cyclotomic polynomials.
Theorem 2.13 ([SV13]). To a Coxeter system (W, S) we can associate a weighted
sheaf (K, R, w ) over the simplicial complex K = KW , by setting w (σ) = Wσ (q).
Also, for any cyclotomic polynomial ϕ = ϕd , the map
wϕ (σ) = maximal power of ϕ which divides Wσ (q)
defines a weighted sheaf (K, R, wϕ ) over K = KW .
The homology of the associated weighted complex is strictly related to the homology of XW . Specifically, set C∗0 = C∗0 (K; R) and consider the diagonal map
∆ : C∗ → C∗0 ,

eσ 7→ Wσ (q) e0σ .

By the formula for the boundary map it follows that ∆ is an injective chain-complex
homomorphism, so there is an exact sequence of complexes:
∆

π

0 −→ C∗ −→ C∗0 −→ L∗ −→ 0,
where
Lk =

M

σ∈KW
|σ|=k

R
ēσ
(Wσ (q))

is the quotient complex. Passing to the associated long exact sequence we get:
(1)

∆

π

∆

π

∗
∗
∗
∗
...
Hk (L∗ ) → Hk−1 (C∗ ) −−→
Hk (C∗0 ) −→
Hk+1 (L∗ ) → Hk (C∗ ) −−→
. . . −→

Then the homology of L∗ can be used to compute the homology of C∗ .
The orbit space YW (and thus the CW-complex XW ≃ YW ) is conjectured
to be a classifying space for the Artin group GW [Del72, Bri73, VdL83, Par14].
Such conjecture was proved for Artin groups of finite type [Del72], for affine Artin
groups of type Ãn , B̃n and C̃n [Oko79, CMS10], and for some other families of
Artin groups [Hen85, CD95]. Whenever the conjecture holds (in particular this is
true for all the cases we consider in this paper), the homology H∗ (XW ; R) coincides
with the twisted homology H∗ (GW ; R) of the Artin group GW .
3. Homology of braid groups and independence complexes
In this section we are going to show how the twisted homology of braid groups
(i.e. Artin groups of type An ) is related to the homology of suitable independence
complexes. Recall that a Coxeter graph of type An is a linear graph with n vertices,
usually labeled 1, 2, . . . , n (see Figure 1), and that the corresponding Artin group
is the braid group on n + 1 strands (which we denote by Brn+1 ). With a slight
abuse of notation, by An we will sometimes indicate the graph itself.
1

2

3

4

...

n−1

Figure 1. A Coxeter graph of type An .

n

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GIOVANNI PAOLINI AND MARIO SALVETTI

The homology of Brn+1 , with coefficients in the representation R = Q[q ±1 ]
described in the previous section, has been computed in [Fre88, DCPS01]. See also
[Cal06] for coefficients in Z[q ±1 ] (but we will not address such case in this paper).
Theorem 3.1 ([Fre88, DCPS01]). The ϕd -primary component of the twisted homology of a braid group is given by
(
R/(ϕd ) if n ≡ 0 or −1 (mod d),
H∗ (Brn+1 ; R)ϕd =
0
otherwise,
where the non vanishing term is in degree (d − 2)k if n = dk or n = dk − 1.
Recall that, if G is a graph with vertex set V G, an independent set of G is a
subset of V G consisting of pairwise non-adjacent vertices. Also, the independence
complex Ind(G) of G is the abstract simplicial complex with V G as set of vertices
and whose simplices are all the non-empty independent sets of G. In contrast with
the simplicial complex K introduced in Section 2, the simplicial complex Ind(G)
does not contain the empty simplex, and the dimension of a simplex σ ∈ Ind(G) is
given by |σ| − 1. The homotopy type of Ind(An ) has been computed in [Koz07] by
means of discrete Morse theory, and the result is the following.
Proposition 3.2 ([Koz07]).
Ind(An ) ≃

(

S k−1
{pt}

if n = 3k or n = 3k − 1,
if n = 3k + 1.

By comparing this with Theorem 3.1 we obtain the following relation between
the homology of the independence complex of An and the ϕ3 -primary component
of the twisted homology of Brn+1 .
Corollary 3.3.


R
∼
,
H∗ (Brn+1 ; R)ϕ3 = H̃∗−2 Ind(An−3 );
(ϕ3 )
where on the left we have local coefficients and on the right we have trivial coefficients.

In general, following [Sal15], define the r-independence complex of a graph G as
Indr (G) = {full subgraphs G′ ⊆ G such that each connected component
of G′ has at most r vertices} .
So Indr (G) is an abstract simplicial complex on the set of vertices V G of the graph
G, which coincides with Ind(G) for r = 1. The case r = 0 also makes sense:
Ind0 (G) = ∅ for any graph G. On page 12 we are going to prove the following
generalization of Corollary 3.3.
Theorem 3.4.


R
H∗ (Brn+1 ; R)ϕd ∼
,
= H̃∗−d+1 Indd−2 (An−d );
(ϕd )
where on the left we have local coefficients and on the right we have trivial coefficients.
From the expression of the ϕd -primary component of the local homology of the
braid group (Theorem 3.1) we obtain the following consequence.

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

11

Corollary 3.5.
H̃∗ (Indd−2 (An )) ∼
=

(

H̃∗ (S dk−2k−1 ) for n = dk or n = dk − 1,
0
otherwise.

Therefore the knowledge of the twisted homology of the braid group gives the
homology (with trivial coefficients) of Indd (An ). Conversely the knowledge of the
homology of Indd (An ) gives the twisted homology of the braid group.
The Poincaré polynomial of a Coxeter group of type Ak is given by
Y
qk − 1
WAk (q) = [k + 1]q ! where [k]q =
=
ϕd
q−1
d|k
d≥2

(see for example [BB06]). Consider now a simplex σ ⊆ S ∼
= {1, . . . , n}. Denote by
Γ(σ) the subgraph of An induced by σ. Denote by Γ1 (σ), Γ2 (σ), . . . , Γm (σ) the
connected components of Γ(σ), and by n1 , n2 , . . . nm their cardinalities (see Figure
2). The i-th connected component is a Coxeter graph of type Ani , so the entire
Coxeter graph induced by σ has Poincaré polynomial
Wσ (q) = [n1 + 1]q ! · [n2 + 1]q ! · · · [nm + 1]q !
=
=

m Y
Y

⌊

ϕd

ni +1
⌋
d

i=1 d≥2
Y Pm

ϕd

i=1 ⌊

ni +1
d ⌋

.

d≥2

Then we are interested in the homology of the weighted complex (L∗ )ϕ associated
to the weighted sheaf (K, R, wϕ ), where ϕ = ϕd is a cyclotomic polynomial (with
d ≥ 2) and
m
X
 ni +1 
.
wϕ (σ) = ϕvϕ (σ) , vϕ (σ) =
d
i=1

Notice that only the connected components with at least d−1 vertices contribute to
the ϕ-weight. Therefore, the weighted complex (L∗ )ϕ is generated by the subgraphs
having at least one component with ≥ d − 1 vertices.
1

2

3
Γ1 (σ)

4

5

6

7

Γ2 (σ)

8

9
Γ3 (σ)

Figure 2. An example with n = 9 and σ = {2, 3, 5, 6, 7, 9} ∈
Ind3 (A9 ). In this case |Γ1 (σ)| = 2, |Γ2 (σ)| = 3, |Γ3 (σ)| = 1.
Theorem 3.6. There is a weighted acyclic matching M on K such that the set of
critical simplices is given by
Cr(M) = {σ | Γ(σ) = Γ1 ⊔ · · · ⊔ Γm−1 ⊔ Ad−1 , |Γi | ≤ d − 2} ∪ Indd−2 (An )
= {τ ⊔ Ad−1 | τ ∈ Indd−2 (An−d )} ∪ Indd−2 (An ),
where Ad−1 is the linear graph on the vertices n − d + 2, . . . , n.

12

GIOVANNI PAOLINI AND MARIO SALVETTI

Proof. First notice that removing the d-th vertex from an Ak component leaves the
ϕ-weight unchanged:




= 1 + k+1−d
= vϕ (Ad−1 ⊔ Ak−d )
vϕ (Ak ) = k+1
d
d

(see Figure 3).

...

...
k vertices

...
d − 1 vertices

...
k − d vertices

Figure 3. The ϕ-weight of an Ak component remains the same if we
remove the d-th vertex, splitting Ak into Ad−1 ⊔ Ak−d .
Let K0 = Indd−2 (An ) ⊆ K (this is the set of the simplices σ ∈ K such that all
the connected components of the induced subgraph Γ(σ) have cardinality < d − 1).
Let us define on K ′ = K \ K0 the following matching M. For σ ∈ K ′ , with
Γ(σ) = Γ1 ⊔ · · · ⊔ Γm , set
i(σ) = min{i | |Γi | ≥ d − 1}.
Then match σ with the simplex τ obtained by adding or removing from σ the
(d − 1)-th vertex of the component Γi(σ) . By the remark at the beginning of the
proof, M is a weighted matching. We prove that it is acyclic. In fact, suppose that
an alternating path contains some subpath
τ ⊳ σ ⊲ τ ′ ⊳ σ′
with τ 6= τ ′ . Then either i(τ ) < i(τ ′ ) or (set j = i(τ ) = i(τ ′ ))
Γj (τ ′ ) = {a, a + 1, . . . , a + d − 2},

Γj (τ ′ ) = {a + 1, . . . , a + d − 1}

for some a ≤ n − d + 1. In this case the first vertex of Γj (τ ′ ) is greater than the
first vertex of Γj (τ ). Therefore an alternating path in K ′ cannot be closed.
The set of critical elements of M in K ′ is given by
Cr(M) = {σ | Γ(σ) = Γ1 ⊔ · · · ⊔ Γm−1 ⊔ Ad−1 , |Γi | ≤ d − 2},
where Ad−1 is as in the statement of the theorem.



Proof of Theorem 3.4. Consider the matching M of Theorem 3.6. Since K0 =
Indd−2 (An ) does not contribute to the weighted complex (L∗ )ϕ , we concentrate on
the poset K ′ = K \ K0 . Notice that there are no non-trivial alternating paths
between critical elements of K ′ . Therefore the boundaries between simplices in
Cr′ (M) = {τ ⊔ Ad−1 | τ ∈ Indd−2 (An−d )}
are the same as the boundaries of Indd−2 (An−d ). In the long exact sequence (1) the
algebraic complex C∗0 has vanishing homology in all degrees (because K is the full
simplicial complex on n vertices), so we have an isomorphism H∗+1 (L∗ ) ∼
= H∗ (C∗ ).
Therefore
H∗ (Brn+1 ; R)ϕ ∼
= H∗ (C∗ )ϕ
d

d

∼
= H∗+1 (L∗ )ϕd

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

13



R
∼
.
= H̃∗−d+1 Indd−2 (An−d );
(ϕd )
In the last isomorphism there is a (d − 1)-shift in degree due to the loss of d − 1
vertices when passing from Cr′ (M) to Indd−2 (An−d ); there is then a further 1shift in degree due to the fact that in L∗ a simplex σ has dimension |σ|, and in
Indd−2 (An−d ) it has dimension |σ| − 1; finally, it is necessary to pass to reduced
homology because the empty simplex is missing in Indd−2 (An−d ).

For the sake of completeness we also give an independent and purely topological
proof of Corollary 3.5, which is a straightforward generalization of the one in [Koz07]
for the case d = 3.
Proof of Corollary 3.5. We denote here for brevity K0 = Indd−2 (An ). Let n =
qd + r be the euclidean division of n by d.
Let Q = {cd > c2d > c3d > · · · > cqd > c∗ } be a linearly ordered set, where c∗
is the minimum element. Define a map f : K0 → Q as follows. If σ ∈ K0 does not
contain any multiples of d, then set f (σ) = c∗ . Otherwise, set f (σ) = cjd if jd ∈ σ
but j ′ d 6∈ σ for j ′ < j. Clearly f is a poset map (if we remove a vertex from σ, j
increases or remains the same).
In f −1 (cd ) consider the matching M1 = {(σ \ {1}, σ ∪ {1}) | σ ∈ f −1 (cd )}, which
is justified by:
σ ∈ f −1 (cd ) ⇒ {2, . . . , d − 1} 6⊆ σ.
Similarly, in f −1 (cjd ) for j ≤ q, consider the matching
o
n

Mj =
σ \ {(j − 1)d + 1}, σ ∪ {(j − 1)d + 1} σ ∈ f −1 (cjd ) ,
justified by:

σ ∈ f −1 (cjd ) ⇒ {(j − 1)d + 2, . . . , jd − 1} 6⊆ σ.
Each Mj is acyclic in f −1 (cjd ), thus
M=

q
[

Mj

j=1

is an acyclic matching on K0 by the Patchwork Theorem (see for example [Jon08,
Lemma 4.2], [Koz07, Theorem 11.10]). Since each Mj is a perfect matching on
f −1 (cjd ) for j = 1, . . . , q, and K1 = f −1 (c∗ ) is a subcomplex of K0 , it follows that
K0 deformation retracts over K1 .
Notice now that K1 is the join of q copies of Indd−2 (Ad−1 ) ∼
= S d−3 and one copy
of Indd−2 (Ar ). For r = 0 we get
K0 ∼
= S q(d−3)+q−1 = S qd−2q−1
= (S d−3 )∗q ∼
and for r = d − 1 we get
K0 ∼
= S (q+1)(d−3)+q = S (q+1)d−2(q+1)−1 ,
= (S d−3 )∗(q+1) ∼
which give the first case in the corollary. If r is not 0 or d − 1 then Indd−2 (Ar ) is a
full simplex ∆r−1 on r vertices, therefore K0 is contractible.


14

GIOVANNI PAOLINI AND MARIO SALVETTI

4. Precise matchings
In this section we are going to introduce the notion of precise matching on a
weighted sheaf. The motivation comes from the study of the twisted homology of
some families of Artin groups, as we will show in the subsequent sections. For now
we go back to the general framework of weighted sheaves over simplicial complexes
(later we will specialize in the case of Artin groups).
Assume from now on that the PID R contains some field K. Our main case of
interest is K = Q and R = Q[q ±1 ]. Let S = (K, R, w ) be a weighted sheaf over the
finite simplicial complex K, with associated weighted complex L∗ . Given a fixed
irreducible element ϕ of R, let Sϕ be the ϕ-primary component of S and let (L∗ )ϕ
be its associated weighted complex. Let M be a weighted acyclic matching for Sϕ .
Let GM be the incidence graph of the corresponding Morse complex: the vertices
of GM are the critical simplices of K, and there is an (oriented) edge σ → τ
whenever [σ : τ ]M is not 0 in R (or equivalently in K), where [σ : τ ]M ∈ Z is the
incidence number between σ and τ in the Morse complex of K. In other words,
there is an edge σ → τ if [σ : τ ]M is not multiple of char K = char R. When K = Q,
this simply means that [σ : τ ]M 6= 0.
Let I be the set of connected components of GM (computed ignoring the orientation of the edges). Recall that vϕ (σ) is the maximal k ∈ N such that ϕk divides
w (σ).
Definition 4.1. The matching M is ϕ-precise (or simply precise) if, for any edge
σ → τ of GM , we have that vϕ (σ) = vϕ (τ ) + 1.
In other words M is precise if, for any two simplices σ and τ lying in the same
connected component i ∈ I, the following relation holds:
vϕ (σ) − vϕ (τ ) = |σ| − |τ |.
Equivalently the quantity |σ| − vϕ (σ), as a function of σ, is constant within a
fixed connected component of GM . This definition is motivated by the fact that
precise matchings exist in many cases of interest (we will see this in Section 5), and
that the homology of the Morse complex is much simpler to compute (and takes a
particularly nice form) when the matching is precise. The name “precise” has been
chosen because for a generic matching one only has vϕ (σ) ≥ vϕ (τ ) (when σ → τ is
an edge of GM ), and we require vϕ (σ) to be precisely vϕ (τ ) + 1.
Assume from now on in this section that M is a ϕ-precise matching. To simplify
M
0
M M
the notation, set (A∗ , ∂) = ((L∗ )M
ϕ , ∂ ) and (V∗ , δ) = (C∗ (K, K) , δ ). Our
aim is to derive a formula for the homology of the Morse complex A∗ . Since the
differential δ vanishes between simplices in different connected components of GM ,
the complex (V∗ , δ) splits as follows:
M
(V∗ , δ) =
(V∗i , δ i ),
i∈I

where
V∗i =

M

K ēσ

σ critical
σ∈i

and the boundary map δ i : V∗i → V∗i is the restriction of δ to V∗i . The differential
∂ of A∗ is induced by δ, and thus it also vanishes between simplices in different

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

15

connected components. Therefore we have an analogous splitting for (A∗ , ∂):
M
(A∗ , ∂) =
(Ai∗ , ∂ i ),
i∈I

where

Ai∗ =

M

R
ēσ
(wϕ (σ))

σ critical
σ∈i
Ai∗ is simply

and the boundary map ∂ i : Ai∗ →
the restriction of ∂ to Ai∗ .
Fix now a connected component i ∈ I. Since M is precise, there exists some
k ∈ Z (which depends on i) such that vϕ (σ) = |σ| + k for all σ ∈ i. Therefore in
degree m we have


M
M
R
R
R
Aim =
ē = 
K ēσ  ⊗K m+k = Vmi ⊗K m+k ,
m+k ) σ
(ϕ
(ϕ
)
(ϕ
)
i
i
σ∈Crm

σ∈Crm

Crim

where
is the set of critical simplices σ such that σ ∈ i and |σ| = m. By
i
construction the boundary ∂m
: Aim → Aim−1 factors accordingly:
i
i
∂m
= δm
⊗K πm ,

where
πm :

R
(ϕm+k )

→

R
(ϕm+k−1 )

i
i
, each Vmi
is the projection induced by the identity R → R. Since im δm+1
⊆ ker δm
splits (as a vector space over K) as a direct sum of linear subspaces:
i
i
i
Vmi = Wm,1
⊕ Wm,2
⊕ Wm,3
,
i
i
i
i
i
where Wm,1
= im δm+1
and Wm,1
⊕ Wm,2
= ker δm
. Then
i
i
ker(∂m
) = ker(δm
⊗K πm )


i
i
⊗K
= Wm,1
⊕ Wm,2

i
i
im (∂m+1
) = im (δm+1
⊗K πm+1 )
R
i
= Wm,1
⊗K m+k .
(ϕ
)

R
(ϕm+k )





(ϕm+k−1 )
i
⊕ Wm,3
⊗K
;
(ϕm+k )

Therefore the homology of (Ai∗ , δ i ) is given, as an R-module, by
i
ker(∂m
)
i
im (∂m+1 )

 

R
(ϕm+k−1 )
i
i
= Wm,2 ⊗K m+k
⊕ Wm,3 ⊗K
(ϕ
)
(ϕm+k )

 

i
R
R
i i
rk δm
∼
⊕ K
⊗K
.
= Hm (V∗ , δ ) ⊗K m+k
(ϕ
)
(ϕ)

Hm (Ai∗ ) =

i
i
In the last isomorphism we used the fact that dim Wm,3
= dim Vmi − dim ker δm
=
i
rk δm .
Recall that the previous formula holds for a fixed connected component i ∈ I,
and k depends on i. Since we now need to take the direct sum over the connected
components, let ki be the value of k for the component i.

16

GIOVANNI PAOLINI AND MARIO SALVETTI

Theorem 4.2. The homology of (L∗ )ϕ is given, as an R-module, by
! 

M
R
R
rk δm
i
∼
⊗K
⊕ K
Hm ((L∗ )ϕ , ∂) =
.
Hm (V∗ ) ⊗K m+ki
)
(ϕ
(ϕ)
i∈I

M
Proof. By Theorem 2.9, Hm ((L∗ )ϕ , ∂) ∼
= Hm ((L∗ )M
ϕ , ∂ ). Using what we have
done in this section, we have that
M
Hm ((L∗ )M
ϕ , ∂ ) = Hm (A∗ , ∂)
M
=
Hm (Ai∗ , ∂)
i∈I

∼
=
∼
=

!

!

R
⊕
⊗K
K
Hm (V∗i ) ⊗K m+ki
)
(ϕ
(ϕ)
i∈I
i∈I
! 

M
R
R
⊕ Krk δm ⊗K
.
Hm (V∗i ) ⊗K m+ki
)
(ϕ
(ϕ)
R

M

M

i
rk δm

!


i∈I

Let us see how the existence of a precise matching can be interpreted in terms
of the spectral sequence associated to the weighted sheaf (see Theorem 2.6).
Proposition 4.3. If a ϕ-precise matching M exists, then the spectral sequence
r
Ep,q
associated to the weighted sheaf Sϕ collapses at the E 2 -page.
Proof. By Theorem 2.11, the E 1 -page can be computed through the Morse complex
of our matching M:
1 ∼
Ep,q
= Hp+q (F p (L∗ )M
ϕ ),
and the differential d1p,q is induced by the boundary of the Morse complex. The
spectral sequence then splits as a direct sum over the connected components of
GM :
M
r
r,i
Ep,q
=
Ep,q
,
i∈I

where

1,i
Ep,q

∼
= Hp+q (F p Ai∗ ). Since the matching is precise, for m 6= p − ki we have

F p Aim = F p Aim / F p−1 Aim = 0.
0,i ∼
1,i
This means that the page Ep,q
is non-trivial only in the row q = −ki , and
= Ep,q
r
the entire spectral sequence Ep,q collapses at the E 2 -page.

What we have done so far in this section assumed ϕ to be some fixed irreducible
element of the PID R. In order to recover the full homology of L∗ we need to make
ϕ vary among all equivalence classes of irreducible elements of R modulo the units.
Suppose from now on to have a ϕ-precise matching Mϕ on Sϕ , for each ϕ. The
following results follows immediately from Theorem 4.2, provided that we add a “ϕ”
subscript (or superscript) to all the quantities that depend on the matching Mϕ .
Theorem 4.4. The homology of L∗ is given, as an R-module, by




M M
ϕ
R
R
rk δm
ϕ,i
∼


.
⊕ K
⊗K
Hm (L∗ , ∂) =
Hm (V∗ ) ⊗K m+kϕ,i
(ϕ
(ϕ)
)
ϕ
i∈Iϕ



WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

17

For later applications, we finally need to study how the isomorphism of Theorem
4.4 behaves with respect to the projection π : C∗0 → L∗ of Remark 2.10. Such
projection is the direct sum over ϕ of the projections
πϕ : C∗0 → (L∗ )ϕ .
Instead of studying the induced map (πϕ )∗ : Hm (C∗0 ) → Hm ((L∗ )ϕ ), we study the
map
(π̄ϕ )∗ : Hm ((C∗0 )M ) → Hm ((L∗ )M
ϕ )
between the Morse complexes (here M = Mϕ is a precise matching which depends
on ϕ). For i ∈ Iϕ , let πi : (C∗0 )M → V∗ϕ,i ⊗K R ⊆ (C∗0 )M be the projection on the
subcomplex corresponding to the connected component i, and let (πi )∗ be the map
0 M
induced in homology. Let [c] ∈ Hm ((C∗0 )M ), for some cycle c ∈ ker δ M ⊆ (Cm
) .
Applying the map (π̄ϕ )∗ : Hm ((C∗0 )M ) → Hm ((L∗ )M
)
we
obtain
ϕ


X
(πi )∗ ([c])
(π̄ϕ )∗ ([c]) = (π̄ϕ )∗ 
i∈Iϕ

=

X

i∈Iϕ



(π̄ϕ )∗ (πi )∗ ([c]) .

Applying the isomorphism of Theorem 4.2, this element is sent to

M
X
R
(πi )∗ ([c]) ⊗K [1] ∈
Hm (V∗ϕ,i ) ⊗K m+kϕ,i .
)
(ϕ
i∈Iϕ

i∈Iϕ

We are going to use these computations to prove the following result which
describes the kernel and the cokernel of π∗ .
Proposition 4.5. The cokernel of π∗ : Hm (C∗0 ) → Hm (L∗ ) is given by
ϕ
M  R ⊕ rk δm
∼
coker π∗ =
.
(ϕ)
ϕ
In addition, the kernel of π∗ is a free R-module isomorphic to Hm (C∗0 ).
Proof. Throughout the proof, consider the following R-modules identified one with
each other, without explicitly mentioning the isomorphisms between them:
M
Hm (L∗ ) ∼
Hm ((L∗ )ϕ )
=
ϕ

∼
=

M

Hm ((L∗ )M
ϕ )

ϕ

∼
=

M
ϕ




M

i∈Iϕ

Hm (V∗ϕ,i ) ⊗K





ϕ
 ⊕ Krk δm ⊗K R .
(ϕm+kϕ,i )
(ϕ)
R

Recall that the matching M depends on ϕ, although we write M instead of Mϕ
in order to make the notations more readable. Also recall that the isomorphisms
Hm ((L∗ )ϕ ) ∼
= Hm ((L∗ )M
ϕ ) occur in the commutative diagram of Remark 2.10.

18

GIOVANNI PAOLINI AND MARIO SALVETTI

We want to show that the image of π∗ : Hm (C∗0 ) → Hm (L∗ ) is given by


M M
R

im π∗ =
Hm (V∗ϕ,i ) ⊗K m+kϕ,i  ⊆ Hm (L∗ ).
)
(ϕ
ϕ
i∈Iϕ

Let ψϕ : H∗ (C∗0 ) → H∗ ((L∗ )M
ϕ ) be the map defined as the composition
∼
=

(π̄ϕ )∗

H∗ (C∗0 ) −→ H∗ ((C∗0 )M ) −−−→ H∗ ((L∗ )M
ϕ ).
By commutativity of the diagram of Remark 2.10, the image of π∗ =
the same as the image of
M
M
ψϕ : H∗ (C∗0 ) →
H∗ ((L∗ )M
ϕ ).
ϕ

L

ϕ (πϕ )∗

is

ϕ

We have already proved that, for any [c] ∈ Hm ((C∗0 )M ),

X
M
(π̄ϕ )∗ ([c]) =
(πi )∗ ([c]) ⊗R [1] ∈
Hm (V∗ϕ,i ) ⊗K
i∈Iϕ

i∈Iϕ

R
,
(ϕm+kϕ,i )

which means in particular that
im (π̄ϕ )∗ ⊆

M

Hm (V∗ϕ,i ) ⊗K

i∈Iϕ

R
.
(ϕm+kϕ,i )

Therefore we immediately have the inclusion
im (π∗ ) ⊆

X
ϕ

im ψϕ =

X

im (π̄ϕ )∗ ⊆

ϕ

M
ϕ




M

Hm (V∗ϕ,i ) ⊗K

i∈Iϕ

R
(ϕm+kϕ,i )

To prove the opposite inclusion, we show that any element of the form
R
[c] ⊗K [1] ∈ Hm (V∗ϕ,i ) ⊗K m+kϕ,i
(ϕ
)



.

is in the image of π∗ (for any fixed ϕ and i). To do so, choose α ∈ R such that
α ≡ 1 (mod ϕm+kϕ,i ) and α ≡ 0 (mod η m+kη,j ) for any irreducible element η 6= ϕ
which divides some weight w (σ), and for any connected component j ∈ Iη (there
is only a finite number of such η up to multiplication by units, because K is finite).
0
0 M
The element c ⊗K α is a cycle in Vmϕ,i ⊗K R ⊆ Cm
(K, K)M ⊗K R ∼
) . Then,
= (Cm
∼
=
0
if [c̃] is the preimage of [c ⊗K α] under the isomorphism H∗ (C∗ ) −→ H∗ ((C∗0 )M ),
we have that:
R
ψϕ ([c̃]) = (π̄ϕ )∗ ([c ⊗K α]) = [c] ⊗K [1] ∈ Hm (V∗ϕ,i ) ⊗K m+kϕ,i ;
)
(ϕ
′
ψη ([c̃]) = (π̄η )∗ ([c ⊗K α]) = 0 for any η 6= ϕ,
where [c′ ] is the image of [c] under the isomorphism
∼
=

∼
=

H∗ ((C∗0 )Mϕ ) −→ H∗ (C∗0 ) −→ H∗ ((C∗0 )Mη ).
Therefore [c] ⊗K [1] is in the image of π∗ . We have thus proved that


M M
R

im π∗ =
Hm (V∗ϕ,i ) ⊗K m+kϕ,i  .
)
(ϕ
ϕ
i∈Iϕ

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

19

Then the cokernel of π∗ can be easily computed:
ϕ

 M
⊕ rk δm
ϕ
R
R
Hm (L∗ ) M
rk δm
K
⊗K
=
.
=
coker π∗ =
im π∗
(ϕ)
(ϕ)
ϕ
ϕ

The R-module Hm (C∗0 ) ∼
= Hm (C∗0 (K; K)) ⊗K R is free and finitely generated,
0
because Hm (C∗ (K; K)) is a finite-dimensional vector space over K (recall that K
is a finite simplicial complex). The kernel of π∗ is a submodule of Hm (C∗0 ), so it is
itself a free R-module with lower or equal rank. Let [c1 ], . . . , [ck ] be an R-base of
Hm (C∗0 ). Consider the non-zero ideal
\ \
I=
(ϕm+ki ) ⊆ R,
ϕ i∈Iϕ

where ϕ varies among the (finitely many) irreducible elements which divide some
weight w (σ) (for σ ∈ K). Fix any non-zero element α ∈ I. Then the elements α[c1 ], . . . , α[ck ] generate a free submodule of ker π∗ of rank k = rk Hm (C∗0 ).
Therefore ker π∗ and Hm (C∗0 ) have the same rank, so they are isomorphic as Rmodules.

5. Precise matchings for Artin groups
Consider now the case of Artin groups, as in Section 2.4. For a Coxeter system
(W, S) we have constructed a weighted sheaf S = (K, R, w ) with
K = {σ ⊂ S | the parabolic subgroup Wσ is finite}
and R = Q[q ±1 ]. The associated weighted complex L∗ fits into the short exact
sequence
∆

π

→ L∗ → 0,
0 → C∗ −→ C∗0 −
which gives rise to the long exact sequence (1). In order to compute H∗ (C∗ ) =
H∗ (XW ; R), we split such long exact sequence into the short exact sequences
∆

∗
ker π∗ → 0,
0 → coker π∗ → Hm (C∗ ) −−→

where on the left we have the cokernel of π∗ : Hm+1 (C∗0 ) → Hm+1 (L∗ ) and on the
right we have the kernel of π∗ : Hm (C∗0 ) → Hm (L∗ ). Since ker π∗ is a free R-module,
these short exact sequences split:
Hm (C∗ ) ∼
= coker π∗ ⊕ ker π∗ .
Recall that the only irreducible elements of R that occur in the factorization of
the weights are the cyclotomic polynomials ϕd for d ≥ 2. As in Section 4, suppose
from now on that we have constructed a ϕ-precise matching Mϕ for each cyclotomic
polynomial ϕ = ϕd (with d ≥ 2). Then we have an explicit description of coker π∗
and ker π∗ thanks to Proposition 4.5, and we obtain the following result.
Theorem 5.1. Under the above hypothesis, the homology of XW with coefficients
in the representation R = Q[q ±1 ] is given by
!
ϕ
M  R ⊕ rk δm+1
∼
⊕ Hm (C∗0 ).

Hm (XW ; R) =
(ϕ)
ϕ

20

GIOVANNI PAOLINI AND MARIO SALVETTI

In particular the term Hm (C∗0 ) gives the free part of the homology, and the
other direct summands give the torsion part. The torsion part actually takes a very
particular form, and we are going to highlight this in the following corollary.
Theorem 5.2. Let GW be an Artin group that admits a ϕ-precise matching for
all cyclotomic polynomials ϕ = ϕd (with d ≥ 2). Then the homology H∗ (XW ; R)
does not have ϕk -torsion for k ≥ 2.

We are particularly interested in Artin groups of finite and affine type. When
GW is an Artin group of finite type with n generators, K is the full simplicial complex on S ∼
= {1, . . . , n} and therefore C∗0 has trivial homology in every dimension.
Thus the formula of Theorem 5.1 reduces to
ϕ
M  R ⊕ rk δm+1
∼
.
Hm (XW ; R) =
(ϕ)
ϕ

When GW is an Artin group of affine type with n+ 1 generators, K is the simplicial
complex on S ∼
= {0, . . . , n} which misses only the full simplex. Then we have

R
for m = n,
ϕ
Hm (XW ; R) ∼
= L  R ⊕ rk δm+1

for m < n.
ϕ (ϕ)

Remark 5.3. When GW is an Artin group of finite type, the corresponding reflection arrangement of hyperplanes A is finite. In this case it is well known that there
is an R-module isomorphism between the twisted homology H∗ (XW ; R) and the
homology with constant coefficients H∗ (F ; Q) of the Milnor fiber F of A [Cal05].
The q-multiplication on the homology of XW corresponds to the action of the monodromy operator on the homology of F . If N = |A|, the square of the defining
polynomial of the arrangement is W-invariant, thus the order of the monodromy
of the Milnor fibration divides 2N . It follows that the polynomial q 2N − 1 must
annihilate the homology. Since q 2N − 1 is square-free in characteristic 0, the homology cannot have ϕk -torsion for k ≥ 2. So the conclusion of Theorem 5.2 is not
surprising in the case of Artin groups of finite type.

In the rest of this paper we are going to construct precise matchings for Artin
groups of type An , Bn , Ãn and C̃n . For each of these cases we are then going to
use Theorem 5.1 to compute the homology H∗ (XW ; R).
Let us first introduce some notations. We will have S = {1, 2, . . . , n} for the
finite cases (An and Bn ), and S = {0, 1, 2, . . . , n} for the affine cases (Ãn and C̃n ).
A simplex σ ⊆ S will be also represented as a string of bits ǫi ∈ {0, 1} (for i ∈ S),
where ǫi = 1 if i ∈ S and ǫi = 0 if i 6∈ S. For example, if S = {1, 2, 3, 4}, the string
representation of σ = {1, 2, 4} is 1101. Also, for σ ⊆ S and v ∈ S, let
(
σ ∪ {v} if v 6∈ σ,
σ⊻v =
σ \ {v} if v ∈ σ.
(it can be regarded as the bitwise xor between the string representation of σ and
the string with ǫv = 1 and ǫi = 0 for i 6= v).
5.1. Case An . Many properties of the homology H∗ (XW ; R) in the case An have
been thoroughly discussed in Section 3. Using precise matchings we are going to
obtain a new proof of Theorem 3.1.

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

21

Let S = {1, 2, . . . , n} be the vertices of the Coxeter graph of type An , as in
Section 3, Figure 1. In this case K is the full simplicial complex on S. Fix now an
integer d ≥ 2 and set ϕ = ϕd . The ϕ-weight of a simplex σ ∈ K has been computed
in Section 3 and is as follows:
vϕ (σ) =

m
X

ωϕ (Ani ),

i=1

where ni is the size of the i-th (linear) connected component of the subgraph induced
by σ, and ωϕ (Ak ) stands for the ϕ-weight of a connected component of type Ak ,
given by


k+1
.
ωϕ (Ak ) =
d
Fix also an integer f with 0 ≤ f ≤ d − 1. Let Kn,f ⊆ K be as follows:
Kn,f = {σ ∈ K | 1, 2, . . . , f ∈ σ}.
Notice that Kn,0 = K, and Kn,f is not a subcomplex of K for f ≥ 1 (but it is
still a subposet of K, so it makes sense to define a matching on it). We are going
to construct a ϕ-precise matching on Kn,f . In particular, for f = 0, we will get a
ϕ-precise matching for K. The precise matchings on Kn,f for f ≥ 1 will become
useful when treating the cases Bn , Ãn and C̃n .
For a fixed d, the matching will be constructed recursively in n and f for n ≥ 0
A
and 0 ≤ f ≤ d − 1. In the subsequent sections we will also write Kn,f
instead of
Kn,f to avoid confusion with other K’s. The matching is as follows.
(a) If {1, . . . , d − 1} ⊆ σ then match σ with σ ⊻ d (unless n = d − 1, in which case
σ is critical). Notice that for f = d − 1 this is always the case, thus in the
subsequent steps we can assume f ≤ d − 2. Here σ is matched with a simplex
which also occurs in step (a).
(b) Otherwise, if n = f then σ is critical.
(c) Otherwise, if f + 1 ∈ σ then match σ with σ \ {f + 1}. Notice that σ \ {f + 1}
occurs in step (d).
(d) Otherwise, if {f + 2, . . . , d − 1} * σ then match σ with σ ∪ {f + 1}. Notice that
σ ∪ {f + 1} occurs in step (c).
(e) We are left with the simplices σ such that {1, . . . , f, f + 2, . . . , d − 1} ⊆ σ and
f + 1 6∈ σ. If we ignore the vertices 1, . . . , f + 1 we are left with the simplices
on the vertex set {f + 2, . . . , n} which contain f + 2, . . . , d − 1; relabeling the
vertices, these are the same as the simplices on the vertex set {1, . . . , n − f − 1}
which contain 1, . . . , d − 2 − f . Then construct the matching recursively as in
Kn−f −1, d−2−f .
Example 5.4. For n = 5, d = 3 and f = 1, Kn,f contains 24 = 16 simplices
of which 14 are matched and 2 are critical. For instance, consider the simplex
σ = {1, 4, 5}. Step (e) applies because 1 ∈ σ and 2 6∈ σ; the recursion requires to
consider the new simplex σ ′ = {2, 3} ∈ K3,0 . Again step (e) applies because 1 6∈ σ ′
and 2 ∈ σ ′ ; the recursion requires to consider the new simplex σ ′′ = {1, 2} ∈ K2,1 .
Finally step (a) applies because {1, 2} ⊆ σ ′′ , and σ ′′ is critical because σ ′′ = {1, 2}.
Therefore σ = {1, 4, 5} ∈ K5,1 is also critical. See Table 1 for an explicit description
of the matching for K5,1 , d = 3.

22

GIOVANNI PAOLINI AND MARIO SALVETTI

vϕ (σ)

Step

−→

2

(a)

−→

1

(a)

−→

1

(a)

−→

1

(a)

−→

1

(e) → (a)

−→

0

(e) → (c)/(d)

−→

0

(e) → (c)/(d)

(critical)

1

(e) → (e) → (a)

(critical)

0

(e) → (e) → (e) → (b)

Simplices

Table 1. Matching in the case An with n = 5, d = 3, f = 1. The
last columns indicates the step where simplices occur. When step (e)
is reached, an arrow indicates how the recursion continues (after K5,1 ,
the recursion involves K3,0 , K2,1 and K0,0 ).

Remark 5.5. A peculiarity of this matching is that, if σ ′ → τ ′ is in the matching,
then σ ′ = τ ′ ∪ {v} with v ≡ f + 1 or v ≡ 0 (mod d). This can be easily checked by
induction.
Lemma 5.6. The matching described above is an acyclic matching on Kn,f .
Proof. The proof is by induction on n, the case n = 0 being trivial. Assume that
n > f since, for n ≤ f , in Kn,f there are either 1 or 0 simplices. Consider the map
η : Kn,f → {pa > pe > pc,d } which sends σ ∈ Kn,f to the px such that σ occurs
in step (x) (here steps (c) and (d) are united). For instance, η(σ) = pa if and only
if {1, . . . , d − 1} ⊆ σ. Notice that the map η is compatible with the matching, i.e.
two matched simplices lie in the same fiber of η.
Let us prove that η is a poset map. Given two simplices σ ≥ τ , we want to prove
that η(σ) ≥ η(τ ). If η(τ ) = pa then {1, . . . , d − 1} ⊆ τ ⊆ σ, so also η(σ) = pa . If
η(τ ) = pe then {1, . . . , f, f + 2, . . . , d − 1} ⊆ τ ⊆ σ, thus η(σ) ∈ {pa , pe }. Finally, if
η(τ ) = pc,d there is nothing to prove.
Since η is a poset map, our matching is acyclic if and only if it is acyclic on each
fiber of η. The restriction of the matching to η −1 (pa ) consists of edges σ → τ with
σ = τ ∪ {d}, so there is no alternating path of length ≥ 4 and the matching is
acyclic. The restriction to η −1 (pc,d ) is acyclic for the same reason (it is always the
vertex f + 1 which is added or removed). The restriction to η −1 (pe ) is acyclic by
induction.


WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

23

Lemma 5.7. The matching described above is a weighted matching on Kn,f .
Proof. The proof is by induction on n, the case n = 0 being trivial.
In step (a), vϕ (σ) = vϕ (σ ⊻ d) by what we have already said in Section 3 (see the
proof of Theorem 3.6). In steps (c) and (d), both the simplices σ and σ ⊻ (f + 1)
do not contain {1, . . . , d − 1}. Thus the (possibly empty) connected component
of 1 has size ≤ d − 2 and does not contribute to the weight. Therefore vϕ (σ) =
vϕ (σ ⊻ (f + 1)). Finally, suppose that σ and τ are simplices that occur in step (e).
Let σ̂, τ̂ ∈ Kn−f −1, d−2−f be the simplices obtained ignoring the vertices 1, . . . ,
f + 1, as described above. Since f ≤ d − 2 we have that
vϕ (σ) = vϕ (τ )

if and only if

vϕ (σ̂) = vϕ (τ̂ ),

so we are done by induction on n.



The critical simplices of the matching on Kn,f are quite simple to describe. If
f ≤ d − 2, n > f and n ≡ f or −1 (mod d), there are 2 critical simplices, one of
weight 1 and one of weight 0. If n = f there is 1 critical simplex (in fact there
is only one simplex in Kn,f ). In all the other cases the matching is perfect. We
are going to prove this in the following theorem. See Table 2 and Figures 4 and 5
for an explicit description of the critical simplices. Notice in particular that, when
there are 2 critical simplices, one is a face of the other in Kn,f .
Case

n>f
f ≤d−2

Simplices

n = kd + f

(Fig. 4)

(1f 01d−2−f 0)k−1 1f 01d−1
f

(1 01
f

n = kd − 1

(Fig. 5)

(1 01
f

(1 01

d−2−f

n − 2k + 1

1

n − 2k

0

k−1 d−1

n − 2k + 2

1

n − 2k + 1

0

n

1 or 0

0) 1

0)

1

k−1 f

0)

1 01

1f

n=f

vϕ (σ)

k f

d−2−f

d−2−f

|σ|

d−2−f

Table 2. Description of the critical simplices for An .

k − 1 times
...
f vertices

...
d − 2 − f vertices

... ...
f vertices

...
d − 1 vertices

k times
...
f vertices

...
d − 2 − f vertices

... ...
f vertices

Figure 4. Critical simplices for f ≤ d − 2 and n = kd + f (k ≥ 1).
Theorem 5.8. The critical simplices for the matching on Kn,f are those listed
in Table 2. In particular the matching is always precise, and is perfect if n 6≡
f, −1 (mod d) or n > f = d − 1. In addition, when there are two critical simplices
there is only one alternating path between them (the trivial one).

24

GIOVANNI PAOLINI AND MARIO SALVETTI

k − 1 times
...
f vertices

...
d − 2 − f vertices

...

...
d − 1 vertices

k − 1 times
...
f vertices

...
d − 2 − f vertices

... ...
f vertices

...
d − 2 − f vertices

Figure 5. Critical simplices for f ≤ d − 2 and n = kd − 1 (k ≥ 1).
Proof. We prove the first part by induction on n, the case n = 0 being trivial (there
is one critical cell for f = 0, namely the empty simplex ∅, and no critical cells for
f > 0). Let σ ∈ Kn,f be a simplex. Let us consider each of the five cases that can
occur in the construction of the matching.
(a) We have {1, . . . , d − 1} ⊆ σ. Then σ is critical if and only if n = d − 1. If
n = d − 1 then σ is indeed listed in Table 2, as the first of the two critical
simplices in the case n = kd − 1 (here k = 1). Conversely, the only simplex
of Table 2 which contains {1, . . . , d − 1} is the first one of the case n = kd − 1
when k = 1.
(b) We have n = f and σ = {1, . . . , f }. Then σ is critical, and it is indeed listed
as the second of the two critical simplices in the case n = kd + f (here k = 0).
(c) We have {1, . . . , f + 1} ⊆ σ. Then σ is not critical. The only listed simplex
which contains {1, . . . , f + 1} is the first one of the case n = kd − 1 for k = 1,
but it is equal to {1, . . . , d − 1} and so it must be different from σ.
(d) We have f + 1 6∈ σ and {f + 2, . . . , d − 1} * σ. Then σ is not critical. It is
easy to check that all the listed simplices τ with n > f and f + 1 6∈ τ satisfy
{f + 2, . . . , d − 1} ⊆ τ , so none of them can be equal to σ.
(e) We have {1, . . . , f, f + 2, . . . , d − 1} ⊆ σ and f + 1 6∈ σ. Then σ is critical if
and only if the simplex σ̂ ∈ Kn−f −1, d−2−f is critical, where σ̂ is constructed
from σ ignoring the first f + 1 vertices. By induction, σ̂ is critical if and only
if it is listed in Table 2. By taking the simplices of Table 2 for Kn−f −1, d−2−f
and adding 1f 0 at the beginning, one obtains exactly the simplices of Table
2 for Kn,f (but the two cases are exchanged). This concludes the induction
argument.
For fixed n and f , the quantity |σ|−vϕ (σ) is constant among the critical simplices
σ. More precisely, it is equal to n − 2k for n = kd + f (except for f = d − 1) and
to n − 2k + 1 for n = kd − 1. Therefore the matching is precise.
Finally we need to prove that the only alternating path between critical simplices
is the trivial one. Consider the case n = kd + f for k ≥ 1 (the case n = kd − 1 is
analogous). Let σ and τ be the two critical simplices (the ones of Figure 4). Notice
that σ = τ ∪ {n − f }. Suppose by contradiction to have a non-trivial alternating
path
σ ⊲ τ1 ⊳ σ1 ⊲ τ2 ⊳ σ2 ⊲ · · · ⊲ τm ⊳ σm ⊲ τ, m ≥ 1.
Since τ1 6= τ , we must have σ = τ1 ∪ {v1 } with v1 6= n − f . Notice that n − f is
the only vertex v ∈ σ which satisfies v ≡ f + 1 or v ≡ 0 (mod d). Therefore, by
Remark 5.5, the vertex v1 will never be recovered throughout the alternating path.
This is a contradiction since v1 ∈ τ .


WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

25

As a consequence we can compute the homology H∗ (GW ; R) for GW of type
An , and obtain a proof of Theorem 3.1.
Proof of Theorem 3.1. We simply need to apply the formula given by Theorem 5.1
using our precise matching on K = Kn,0 . Since f = 0, there are critical cells for
ϕ
n = kd or n = kd − 1. The boundary δm+1
is non-trivial only for
(
m = n − 2k = k(d − 2)
if n = kd,
m = n − 2k + 1 = k(d − 2) if n = kd − 1.
ϕ
In both cases for m = k(d − 2) we have rk δm+1
= 1, and all the other boundaries
are trivial. Theorem 3.1 follows.


Remark 5.9. A relation with independence complexes can be found for Kn,f also
when 1 ≤ f ≤ d − 2. Indeed, choosing a suitable weighted matching (similar to the
one of Theorem 3.6), the set of critical simplices of positive weight is
{σ ⊔ Ad−1 | σ ∈ Indd−2 (An−d ) ∩ Kn−d, f }.
Remark 5.10. Throughout this section, the condition f ≤ d − 1 is not very
important. We chose to restrict to this case in order to have simpler results and
proofs, but everything can be easily extended to the cases in which f ≥ d (ignoring
the first sd vertices, where sd is the biggest multiple of d which is ≤ f ). We will
apply this argument in Section 5.3, when constructing the matching for the case
Ãn .
5.2. Case Bn . Consider now a Coxeter graph of type Bn , as in Figure 6. In this
case K is again the full simplex on vertices {1, 2, . . . , n}. The Poincaré polynomial
of a Coxeter group of type Bk is given by
WBk (q) = [2k]q !! =

n
Y

[2i]q =

Y

ϕωϕ (Bk ) ,

ϕ

i=1

where, for a given cyclotomic polynomial ϕ = ϕd , we have
( k
if d is odd,
d
ωϕ (Bk ) = j k k
if d is even
d/2

(see for example [BB06]). Then we can compute the ϕ-weight of any simplex σ ∈ K
by looking at the subgraph Γ(σ) induced by σ. Let Γ1 (σ), . . . , Γm (σ) be the
connected components of Γ(σ), with cardinality n1 , . . . , nm respectively, where
Γ1 (σ) is the (possibly empty) component that contains the vertex 1 ∈ S. Then
vϕ (σ) = ωϕ (Bn1 ) +

m
X

ωϕ (Ani ).

i=2

1

4

2

3

4

...

n−1

Figure 6. A Coxeter graph of type Bn .

n

26

GIOVANNI PAOLINI AND MARIO SALVETTI

The situation is quite different depending on the parity of d. If d is odd, the
ϕ-weight of a Bk+1 component is equal to the ϕ-weight of an Ak component:


k+1
ωϕ (Bk+1 ) =
= ωϕ (Ak ) (d odd).
d
For this reason it is possible to construct a very simple perfect matching on K:
match each simplex σ ∈ K with σ ⊻ 1. Such matching is acyclic and perfect, but
we also want it to be a weighted matching.
Lemma 5.11. The matching constructed above for d odd is a weighted matching
on K.
Proof. Let σ ∈ K. The only difference between Γ(σ) and Γ(σ ⊻ 1) is the leftmost
connected component, which in one case is of type Bk+1 and in the other case is of
type Ak . Therefore vϕ (σ) = vϕ (σ ⊻ 1).

Suppose from now on that d is even. The simplicial complex K is partitioned as




G
|Γ1 (σ)|
K=
Kq , where Kq = σ ∈ K
=q .
d/2
q≥0

Here Γ1 (σ) is the (possibly empty) connected component of Γ(σ) which contains
the vertex 1. Notice that each Kq is a subposet of K, but not a subcomplex in
general. For a given simplex σ ∈ Kq , let |Γ1 (σ)| = q d2 + r with 0 ≤ r < d2 . The
matching on Kq is as follows.

(a) If r ≥ 1 (i.e. q d2 + 1 ∈ σ) then match σ with σ \ q d2 + 1 .

(b) If r = 0 (i.e. q d2 + 1 6∈ σ) and q d2 + 2, . . . , (q + 1) d2 * σ, then match σ with

σ ∪ q d2 + 1 (unless n = q d2 , in which case σ is critical).
(c) We are left 
with the simplices σ for which neither (a) nor (b) apply, i.e. with
r = 0 and q d2 + 2, . . . , (q + 1) d2 ⊆ σ. Ignore the first q d2 + 1 vertices and
relabel the remaining ones from 1 to n − q d2 − 1, so that we are left exactly with
A
A
the simplices of Kn−q
. Then construct the matching on Kn−q
d
d
d
d
2 −1, 2 −1
2 −1, 2 −1
as in Section 5.1.
Putting together the matchings on each Kq we obtain a matching on the full simplicial complex K.
Example 5.12. For n = 4 and d = 4, the simplicial complex K contains 24 = 16
simplices of which 12 are matched and 4 are critical. For instance, consider σ =
{1, 2} ∈ K. Then q = 1 and r = 0. Since 4 6∈ σ, step (b) occurs. Therefore σ
is matched with σ ∪ {3} = {1, 2, 3}. See Table 3 for an explicit description of the
matching in this case.
Lemma 5.13. The matching constructed above for d even is an acyclic matching
on K.
j
k
1 (σ)|
Proof. The map K → (N, ≤) which sends σ to q = |Γd/2
is a poset map compatible with the matching, and its fibers are exactly the subsets Kq for q ∈ N.
Therefore we only need to prove that the matching on each fiber Kq is acyclic.
For a fixed q ∈ N, consider the map η : Kq → {pc > pa,b } which sends σ to the
px such that σ occurs in step (x) (here steps (a) and (b) are united). Clearly η is
compatible with the matching. We want to prove that it is a poset map, and for
this we only need to show that if η(τ ) = pc and τ ≤ σ then also η(σ) = pc . We have

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

vϕ (σ)

q

Step

1

1

(a)/(b)

0

0

(a)/(b)

0

0

(a)/(b)

0

0

(a)/(b)

0

0

(a)/(b)

(critical)

2

2

(b)

(critical)

1

1

(c)

(critical)

1

0

(c)

0

0

(c)

0

0

(c)

Simplices
4
4
4
4
4

−→
−→
−→
−→
−→

4
4
4
4
4

−→

27

4
4
4
4
4

4

(critical)

Table 3. Matching in the case Bn with n = 4 and d = 4.


that 1, . . . , q d2 , q d2 + 2, . . . , (q + 1) 2d ⊆ τ ⊆ σ. The simplex σ cannot contain the
vertex q d2 + 1, because otherwise we would have σ ∈ Kq+1 . Therefore η(σ) = pc .
On the fiber η −1 (pa,b ) the matching is acyclic because the same vertex q d2 + 1 is
always added or removed. On the fiber η −1 (pc ) the matching is acyclic by Lemma
5.6. Therefore the entire matching on Kq is acyclic.

Lemma 5.14. The matching constructed above for d even is a weighted matching
on K.
Proof. Let σ ∈ Kq be a simplex which occurs
 either in step (a) or step (b). We
want to show that vϕ (σ) = vϕ σ ⊻ q d2 + 1 . Suppose without
 loss of generality
that σ occurs in step (a), i.e. r ≥ 1 and q d2 + 1 ∈ σ. Let τ = σ \ q d2 + 1 . The only
difference between Γ(σ) and Γ(τ ) is that Γ(σ) has a Bq d +r component, whereas
2
Γ(τ ) has a Bq d component and an Ar−1 component instead. Since 1 ≤ r < d2 , we
2
have that
ωϕ (Bq d +r ) = ωϕ (Bq d ) = q
2

2

and ωϕ (Ar−1 ) = 0
(the second equality holds because r − 1 ≤ d2 − 1 ≤ d − 2). Therefore vϕ (σ) = vϕ (τ ).
For simplices occurring in step (c), the matching only involves changes in the
connected components not containing the first vertex (i.e. connected components
of type Ak ). Therefore Lemma 5.7 applies.


28

GIOVANNI PAOLINI AND MARIO SALVETTI

Now we are going to describe the critical simplices. The matching is perfect
when n is not a multiple of d2 . On the other hand, if n = k d2 , we have two families
of critical simplices: σq (for 0 ≤ q ≤ k − 2) and σq′ (for 0 ≤ q ≤ k). See Table 4 and
Figure 7 for the definition of such simplices. For instance, in Example 5.12, the
critical simplices are: σ0 = {2, 3, 4}, σ2′ = {1, 2, 3, 4}, σ1′ = {1, 2, 4} and σ0′ = {1, 3}.
See also Table 5, where the critical simplices are listed by dimension.
Simplices
σq = 1
σq′

qd
2

(01

=1

d
2 −1

qd
2

)k−q−2 01d−1

(01

d
2 −1

|σ|

vϕ (σ)

n−k+q+1

q+1

0≤q ≤k−2

n−k+q

q

0≤q≤k

k−q

)

Table 4. Description of the critical simplices for Bn , where d is even
and n = k d2 .

k − q − 2 times
4

...
q d2

...

...
d
2

vertices

...

...
d − 1 vertices

− 1 vertices
k − q times

4

...
q d2

...

...
d
2

vertices

− 1 vertices

Figure 7. Critical simplices for Bn , where d is even and n = k d2 .
Above is the simplex σq (0 ≤ q ≤ k − 2) and below is the simplex σq′
(0 ≤ q ≤ k).

|σ|

vϕ (σ)

Simplices

n

k

σk′

n−1

k−1

′
σk−2 , σk−1

n−2
..
.

k−2
..
.

′
σk−3 , σk−2
..
.

n−k+1

1

σ0 , σ1′

n−k

0

σ0′

Table 5. Critical simplices for Bn by dimension, where d is even and
n = k d2 .

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

29

Theorem 5.15. The critical simplices for the matching on K are those defined
in Table 4. In particular the matching
is always precise, and is perfect if d is odd


or if d is even and n 6≡ 0 mod d2 . In addition, if d is even and n ≡ 0 mod d2 ,
the incidence numbers between the critical simplices in the Morse complex are as
follows:
d

[σq : σq−1 ]M = (−1)(q−1) 2 ,
d

[σq : σq′ ]M = (−1)(k−1)( 2 −1)+q ,
d

[σq′ : σq−2 ]M = (−1)k( 2 −1)+q ,
d

′
[σq′ : σq−1
]M = (−1)(q−1) 2 .

Proof. For d odd there is nothing to prove. Suppose from now on that d is even.
For a given simplex σ ∈ K, let us consider each of the three cases that can occur
in the construction of the matching.
(a) We have q d2 + 1 ∈ σ, and σ is not critical. Indeed, none of the simplices of
d
Table 4 contains the vertex
 qd 2 + 1.
d
(b) We have q 2 + 1 6∈ σ and q 2 + 2, . . . , (q + 1) 2d * σ. In this case σ is critical
if and only if n = q d2 . Indeed, the only simplex of this type in Table 4 is σk′
(which occurs for q = k i.e. n = q d2 ).
A
. By Theorem
(c) In the remaining case, we apply the matching on Kn−q
d
d
2 −1, 2 −1
5.8, such matching admits critical simplices if and only if
n − q d2 − 1 ≡

d
2 − 1 (mod d)

d
2 . Notice that

or n − q d2 − 1 ≡ −1 (mod d),

d
i.e. for n ≡ 0 mod
if f = d2 − 1 then
 also d − 2 − f = 2 − 1.
d
Therefore, again by Theorem 5.8, for n ≡ 0 mod 2 the critical simplices are
exactly the ones listed in Table 4.
For a fixed n = k d2 , the quantity |σ| − vϕ (σ) is constant among the critical simplices and is equal to n − k. Thus the matching is precise.
Finally we need to prove the formulas for the incidence numbers. We are going
to show that there is exactly one alternating path for each of the four pairs. Notice
that, if σ → τ is in the matching of Kq , then σ = τ ∪ {v} with v ≡ 1 mod d2
and v ≥ q d2 + 1. In particular, if at a certain point of an alternating path one
be added again. But
removes a vertex v 6≡ 1 mod d2 , then that vertex will never

path
all the critical simplices contain every vertex 6≡ 1 mod d2 , so any alternating

between critical simplices cannot ever drop a vertex 6≡ 1 mod d2 . Let us now
consider each of the four pairs.
• (σq , σq−1 ). We have that σq = σq−1 ∪ {v} where v = (q − 1) d2 + 1, so there
is the trivial alternating path σq → σq−1 which contributes to the incidence
number [σq : σq−1 ]M by
d

[σq : σq−1 ] = (−1)|{w∈σq |w<v}| = (−1)(q−1) 2 .
Suppose by contradiction that there exists some other (non-trivial) alternating path
σq ⊲ τ1 ⊳ ρ1 ⊲ τ2 ⊳ ρ2 ⊲ · · · ⊲ τm ⊳ ρm ⊲ σq−1 ,

m ≥ 1.

Let σq = τ1 ∪ {v}. By the previous considerations, we must have v ≡
1 mod d2 . If v = (k − 1) d2 + 1 then τ1 = σq′ , but this is not possible since

30

GIOVANNI PAOLINI AND MARIO SALVETTI

alternating paths stop at critical simplices. Similarly, if v = (q − 1) 2d + 1
then we would stop at τ1 = σq−1 . Therefore we must have v ≤ (q − 2) 2d + 1.
But then τ1 ∈ Kq′ with q ′ ≤ q − 2, and by induction all the subsequent
simplices in the alternating path must lie in
[
Kq′′ .
q′′ ≤q−2

In particular σq−1 ∈ Kq′′ for some q ′′ ≤ q − 2, but this is a contradiction
since σq−1 ∈ Kq−1 .
• (σq , σq′ ). This case is similar to the previous one, except for the fact that
σq = σq′ ∪ {v} for v = (k − 1) d2 + 1. So the only alternating path is the
trivial one, which contributes to the incidence number by
[σq : σq′ ] = (−1)|{w∈σq |w<v}|
d

d

= (−1)q 2 +(k−q−1)( 2 −1)
d

= (−1)(k−1)( 2 −1)+q .
′
• (σq′ , σq−1
). This case is also similar to the previous ones. Here we have
′
′
σq = σq−1
∪ {v} with v = (q − 1) d2 + 1, so the contribution to the incidence
number due to the trivial alternating path is
d

′

′
[σq′ : σq−1
] = (−1)|{w∈σq |w<v}| = (−1)(q−1) 2 .

• (σq′ , σq−2 ). This case is more complicated because the only alternating path
is non-trivial. Suppose to have an alternating path
σq′ ⊲ τ1 ⊳ ρ1 ⊲ τ2 ⊳ ρ2 ⊲ · · · ⊲ τm ⊳ ρm ⊲ σq−2 ,

m≥1

(m must be at least 1 because σq−2 is not a face of σq′ in K). Let σq′ =

′
and we
τ1 ∪ {v}, with v ≡ 1 mod d2 . If v = (q − 1) d2 + 1 then τ1 = σq−1
d
′
must stop. If v ≤ (q − 3) 2 + 1 we fall into some Kq′ with q ≤ q − 3 and it is
not possible to reach σq−2 ∈ Kq−2 . Therefore necessarily v = (q − 2) d2 + 1.
Then τ1 is matched with

ρ1 = τ1 ∪ q d2 + 1 .
Then again, τ2 = ρ1 \ {v2 } and the only possibility for v2 (in order to have
τ2 6= τ1 and not to fall into some Kq′ with q ′ ≤ q − 3) is v2 = (q − 1) d2 + 1.
Proceeding by induction we obtain that

τi = ρi−1 \ (q − 3 + i) d2 + 1 ,

ρi = τi ∪ (q − 1 + i) d2 + 1 .

The path stops at τk−q+1 = σq−2 , and its length is m = k − q. See Table 6
for an example. The contribution of this path to the incidence number is
(−1)k−q [σq′ : τ1 ][ρ1 : τ1 ] · · · [ρk−q : τk−q ][ρk−q : σq−2 ]
where
d

d

d

d

d

d

[ρi−1 : τi ] = (−1)(q−2) 2 +(i−1)( 2 −1) ,
[ρi : τi ] = (−1)(q−2) 2 +(i−1)( 2 −1)+d−1
= (−1)(q−2) 2 +(i−1)( 2 −1)+1

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

31

(these formulas also hold for ρ0 = σq′ and τk−q+1 = σq−2 ). Then
!
k−q
Y
′
M
k−q
[ρi−1 : τi ][ρi : τi ] [ρk−q : σq−2 ]
[σq : σq−2 ] = (−1)
i=1

k−q

= (−1)

d

d

(−1)k−q (−1)(q−2) 2 +(k−q)( 2 −1)

d

= (−1)k( 2 −1)+q .

Simplices
σ2′

=

τ1

=

ρ1

=

τ2

=

ρ2

=

τ3

=

ρ3

=

σ0

=

4
4
4
4
4
4
4
4



vϕ (σ)
2
1
1
1
1
1
1
1

Table 6. The only alternating path from σq′ to σq−2 for n = 10, d = 4,
k = 5 and q = 2.

Having a complete description of a precise matching on K, we can now compute
the homology H∗ (GW ; R) for GW of type Bn . We recover the result of [DCPSS99].
Theorem 5.16 ([DCPSS99]). For an Artin group GW of type Bn , we have
(
d
∼ R/(ϕd ) if d is even, n = k 2 and n − k ≤ m ≤ n − 1,
Hm (GW ; R)ϕd =
0
otherwise.
ϕ
. Assume that
Proof. For a fixed ϕ = ϕd , we need to compute the boundary δm+1
d
n = k 2 , otherwise there are no critical simplices. In top dimension (m + 1 = n) we
′
have rk δnϕ = 1, because [σk′ : σk−1
] 6= 0. For m ≤ n−k−1 the boundary δnϕ vanishes,
because there are no critical simplices in dimension ≤ n − k − 1. For m = n − k
ϕ
we have rk δm+1
= 1, because [σ1′ : σ0′ ] 6= 0. Finally, for n − k + 1 ≤ m ≤ n − 2, we
have (set q = m − n + k):


′
[σq : σq−1 ] [σq+1
: σq−1 ]
ϕ
rk δm+1 = rk
′
[σq : σq′ ]
[σq+1
: σq′ ]

32

GIOVANNI PAOLINI AND MARIO SALVETTI
d

(−1)(q−1) 2
= rk
d
(−1)(k−1)( 2 −1)+q

d

(−1)k( 2 −1)+q+1
d
(−1)q 2

!

= 1,

because
d

(−1)(q−1) 2
det
d
(−1)(k−1)( 2 −1)+q

d

(−1)k( 2 −1)+q+1
d
(−1)q 2

!

d

d

= (−1) 2 − (−1) 2 = 0.

ϕ
ϕ
To summarize, we have: rk δm+1
= 1 for n − k ≤ m ≤ n − 1, and rk δm+1
= 0
otherwise. We conclude applying Theorem 5.1.


5.3. Case Ãn . Consider now the case of affine Artin groups of type Ãn (see Figure
8 for a picture of the corresponding Coxeter graph). Recall that, for affine Artin
groups, the simplicial complex K consists of all the simplices σ ⊆ S = {0, . . . , n}
except for the full simplex σ = {0, . . . , n}. For any σ ∈ K, the induced subgraph
Γ(σ) consists only of connected components of type Ak . The weight vϕ (σ) is then
computed as in Section 5.1.
2
3

1

4

0

...

5

n

6
Figure 8. A Coxeter graph of type Ãn .
For h ∈ {0, . . . , n} set
Kh = {σ ∈ K | h 6∈ σ and {0, . . . , h − 1} ⊆ σ}.
Since every simplex σ ∈ K misses at least one vertex, the subsets Kh form a
partition of K:
n
G
K=
Kh .
h=0

Construct a matching on a fixed Kh as follows. Let h = qd + r with 0 ≤ r ≤ d − 1,
and let m = n − qd. Consider the following clockwise relabeling of the vertices:
r − 1 becomes 1, r − 2 becomes 2, . . . , 0 becomes r, n becomes r + 1, . . . , h + 2
becomes m − 1, h + 1 becomes m (the vertices r, r + 1, . . . , h are forgotten). Such
relabeling induces a poset isomorphism
∼
=

A
Kh −
→ Km,r
.
A
Then put on Kh the pull-back of the matching on Km,r
defined in Section 5.1 (for
h = n the only simplex in Kh is critical).

Lemma 5.17. The matching defined above is an acyclic matching on K.

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

33

Proof. Consider the map η : K → (N, ≤) which sends σ ∈ K to min{h ∈ N | h 6∈ σ}.
A
Then η is a poset map with fibers η −1 (h) = Kh . The matching on each Kh ∼
= Km,r
is acyclic by Lemma 5.6, therefore the whole matching on K is acyclic.

Lemma 5.18. The matching defined above is a weighted matching on K.
∼
=

A
Proof. Fix h ∈ {0, . . . , n}. The bijection Kh −
→ Km,r
is such that, if σ 7→ σ̂, then
vϕ (σ) = vϕ (σ̂) + q. Indeed, Γ(σ) is obtained from Γ(σ̂) by adding qd vertices to one
(possibly empty) connected component, and this increases the weight by q. Then,
A
since the matching on Km,r
is weighted, its pull-back also is.


Case

Simplices

n = kd + r

τq

6≡ −1 (mod d)

τq′

n = kd − 1

|σ|

vϕ (σ)

n − 2(k − q) + 1 q + 1
n − 2(k − q)

0≤q ≤k−1

q

0≤q≤k

σq,r

n − 2(k − q) + 2 q + 1 0 ≤ q ≤ k − 2, 0 ≤ r ≤ d − 2

′
σq,r

n − 2(k − q) + 1

q

n

k

n

σ̄ = 1 0

0 ≤ q ≤ k − 1, 0 ≤ r ≤ d − 2

Simplices
τq = 1qd+r 01d−1 0(1r 01d−2−r 0)k−q−1
τq′ = 1qd+r 0(1r 01d−2−r 0)k−q
σq,r = 1qd+r 01d−1 0(1d−2−r 01r 0)k−q−2 1d−2−r 0
′
σq,r
= 1qd+r 0(1d−2−r 01r 0)k−q−1 1d−2−r 0

Table 7. Description of the critical simplices for Ãn .

Theorem 5.19. The critical simplices for the matching on K are those listed in
Table 7. The only non-trivial incidence numbers between critical cells are:
[τq : τq′ ]M = ±1 (for n = kd + r with 0 ≤ r ≤ d − 2);
′
[σq,r : σq,r
]M = ±1,
′
[σ̄ : σk−1,r
]M = ±1 (for n = kd − 1).

In particular the matching is precise.
Proof. For n 6≡ −1 (mod d), the matching on Kh has critical simplices only for
A
h ≡ n (mod d). If we write n = kd + r with 0 ≤ r ≤ d − 2, then Kqd+r ∼
= Kn−qd,
r
has two critical simplices for 0 ≤ q ≤ k − 1 and one critical simplex for q = k. Such
simplices are those listed in Table 7.
Suppose now that n = kd − 1. For any h = qd + r, with 0 ≤ r ≤ d − 2, the
A
complex Kqd+r ∼
= Kn−qd,
r has two critical simplices because n − qd ≡ −1 (mod d).
A
Moreover, for h = n the complex Kn ∼
= Kd−1,
d−1 has one critical simplex. In the
remaining cases the matching on Kh is perfect. Again, the critical simplices are
those listed in Table 7.

34

GIOVANNI PAOLINI AND MARIO SALVETTI

Let us go back to the case n = kd + r, with 0 ≤ r ≤ d − 2. We want to find
the incidence numbers between critical simplices of consecutive dimensions. First,
let us look for alternating paths from τq to τq′ (for 0 ≤ q ≤ k − 1). Set h = qd + r.
Suppose to have one such path:
τq ⊲ ζ1 ⊳ ρ1 ⊲ ζ2 ⊳ ρ2 ⊲ · · · ⊲ ζm ⊳ ρm ⊲ τq′ .
If at some point a vertex v ∈ {0, ..., h − 1} is removed, then the path falls into some
Kh′ with h′ < h and can never return in Kh . Therefore the path must be entirely
A
′
contained in Kh ∼
= Kn−qd,
r , and by Theorem 5.8 it must be the trivial path τq ⊲ τq .
′
Thus [τq : τq ] = ±1.
′
The other pairs of critical simplices in consecutive dimensions are (τq+1
, τq ) for
′
0 ≤ q ≤ k − 1. There is a trivial path τq+1 ⊲ τq which consists in removing the
vertex h = qd + r, and contributes to the incidence number by
′
[τq+1
: τq ] = (−1)h .

Suppose to have some other (non-trivial) alternating path:
′
τq+1
⊲ ζ1 ⊳ ρ1 ⊲ ζ2 ⊳ ρ2 ⊲ · · · ⊲ ζm ⊳ ρm ⊲ τq ,

m ≥ 1.

′
Let ρm = τq ∪ {v}. If v = h then ρm = τq+1
, which is excluded. Then v must be
one of the other vertices that don’t belong to τq . They are of the form v = n − sd
(for 0 ≤ s ≤ k − q − 1) or v = n − (d − 1 − r) − sd (for 0 ≤ s ≤ k − q − 2). If
v = n − (d − 1 − r) − sd, then ρm is matched with ρm ∪ {n − (s + 1)d}, impossible
because ρm must be matched with some ζm ⊳ ρm . Similarly, if v = n − sd with
0 ≤ s ≤ k − q − 2, then ρm is matched with ρm ∪ {n − (d − 1 − r) − sd} and not
with some ζm ⊳ ρm . The only remaining possibility is v = n − sd with s = k − q − 1,
i.e. v = n − (k − q − 1)d = (q + 1)d + r. In this case ρm is matched with ζm =
ρm \ {qd + 2r + 1}. Going on with the same argument, we have exactly one way to
continue the alternating path (from right to left), and we eventually end up with
′
τq+1
. From left to right, the obtained alternating path is the following:
′
ζ1 = τq+1
\ {qd + 2r + 1},

ρ1 = ζ1 ∪ {n},
ζ2 = ρ1 \ {qd + r},
ρ2 = ζ2 ∪ {n − (d − 1 − r)},
ζ3 = ρ2 \ {n},
ρ3 = ζ3 ∪ {n − d},
ζ4 = ρ3 \ {n − (d − 1 − r)},
..
.
ζm = ρm−1 \ {(q + 1)d + 2r + 1},
ρm = ζm ∪ {qd + 2r + 1},
τq = ρm \ {(q + 1)d + r}.
The length of the path is m = 2(k − q). Apart from qd + r, which is the vertex in
′
τq+1
\ τq , the other vertices are added and removed exactly once during the path. If
a certain v is added in some ρi and removed in some ζj (the removal might possibly

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

35

come before the addition), then
[ρi : ζi ][ρj−1

(
1
if v = n,
: ζj ] =
−1 otherwise.

This is true because, except for v = n, between the addition and the removal of
v an odd number of vertices < v has been added/removed. For instance, between
the addition of v = n − (d − 1 − r) (which occurs for i = 2) and its removal (which
occurs for j = 4), one vertex < d has been added (namely n − d) and none has been
removed. Therefore this alternating path contributes to the incidence number by
(−1)m · (−1)m−1 · [ζ2 : ρ1 ] = (−1)qd+r+1 = (−1)h+1 .
Finally, the incidence number is given by
′
[τq+1
: τq ]M = (−1)h + (−1)h+1 = 0.

Consider now the case n = kd − 1. Consider a generic alternating path starting
from σ̄ = {0, . . . , n − 1}:
σ̄ ⊲ ζ1 ⊳ ρ1 ⊲ ζ2 ⊳ ρ2 ⊲ · · · ⊲ ζm ⊳ ρm ⊲ ζm+1 .
′
Let σ̄ = ζ1 ∪ {v}. If n − d + 1 ≤ v ≤ n − 1 then ζ1 = σk−1,r
for r = n − 1 − v.
′
Therefore we have trivial alternating paths from σ̄ to any of the σk−1,r
. If v ≤ n − d
then ζ1 ∈ Kh with h ≤ n − d = (k − 1)d − 1. None of such Kh contains critical
simplices ζm+1 with |ζm+1 | = n − 1, and the alternating path cannot return in any
Kh′ with h′ ≥ (k − 1)d. Thus there no other alternating paths from σ̄ to critical
simplices of K. Then the non-trivial incidence numbers involving σ̄ are:
′
[σ̄ : σk−1,r
]M = ±1.
′
. Suppose to have one:
Let us check the alternating paths from σq,r1 to σq,r
2
′
.
σq,r1 ⊲ ζ1 ⊳ ρ1 ⊲ ζ2 ⊳ ρ2 ⊲ · · · ⊲ ζm ⊳ ρm ⊲ σq,r
2
′
′
causes the creation of a connected compo∪ {v}. Adding v to σq,r
Let ρm = σq,r
2
2
nent Γi of size ≡ −1 (mod d). This means that ρm is matched with a simplex of
higher dimension, obtained by adding the vertex immediately after Γi (clockwise),
or is not matched at all (this happens for v = (q + 1)d − 1). Therefore the alternating path must be trivial, and it occurs only for r1 = r2 . Then the non-trivial
′
]M are:
incidence numbers of the form [σq,r1 : σq,r
2
′
[σq,r : σq,r
]M = ±1.
′
to σq,r2 . Suppose to
Finally we need to check the alternating paths from σq+1,r
1
have one:
′
⊲ ζ1 ⊳ ρ1 ⊲ ζ2 ⊳ ρ2 ⊲ · · · ⊲ ζm ⊳ ρm ⊲ σq,r2 .
σq+1,r
1
As before, we work backwards. Let ρm = σq,r2 ∪ {v}. Apart from the choices
v = qd + r and v = (q + 1)d + r, in all other cases ρm has a connected component
of size ≡ −1 (mod d) and this prevents the continuation of the alternating path.
′
, so we have a trivial alternating path from
For v = qd + r we obtain ρm = σq+1,r
2
′
σq+1,r2 to σq,r2 . For v = (q + 1)d + r, iterating the same argument, we have exactly
one way to continue the alternating path (from right to left) and we end up with
′
. From left to right, the path is as follows (set r = r2 ):
σq+1,r
2
′
ζ1 = σq+1,r
\ {(q + 1)d − 1},

ρ1 = ζ1 ∪ {n},

36

GIOVANNI PAOLINI AND MARIO SALVETTI

ζ2 = ρ1 \ {qd + r},
ρ2 = ζ2 ∪ {n − (d − 1 − r)},
ζ3 = ρ2 \ {n},
ρ3 = ζ3 ∪ {n − d},
ζ4 = ρ3 \ {n − (d − 1 − r)},
..
.
ζm = ρm−1 \ {(q + 2)d − 1},
ρm = ζm ∪ {(q + 1)d − 1},
σq,r = ρm \ {(q + 1)d + r}.
The length of the path is m = 2(k − q) − 1. Like what happened in the case
n = kd + r, the contribution of this alternating path to the incidence number is
given by
(−1)m · (−1)m−1 · [ζ2 : ρ1 ] = (−1)qd+r+1 .
′
The contribution of the trivial path σq+1,r
⊲ σq,r is given by (−1)qd+r . Therefore
′
: σq,r ]M = (−1)qd+r + (−1)qd+r+1 = 0.
[σq+1,r



We are now able to recover the result of [CMS08b] about the homology H∗ (GW ; R)
when GW is an Artin group of type Ãn .
Theorem 5.20 ([CMS08b]). For an Artin group GW of type Ãn , we have

⊕d−1

if n = kd − 1 and m = n − 2i + 1 (1 ≤ i ≤ k),
(R/(ϕd ))
Hm (GW ; R)ϕd ∼
if n = kd + r and m = n − 2i (1 ≤ i ≤ k),
= R/(ϕd )


0
otherwise.
In the second case, the condition on r is 0 ≤ r ≤ d − 2.

Proof. It is a matter of applying Theorems 5.1 and 5.19. For n = kd + r (0 ≤ r ≤
ϕ
d − 2), the boundary map δm+1
has rank 1 when m + 1 = n − 2(k − q) + 1 (for
0 ≤ q ≤ k − 1, due to τq ); it has rank 0 otherwise. For n = kd − 1, the boundary
ϕ
map δm+1
has rank d − 1 if m + 1 = n − 2(k − q) + 2 (for 0 ≤ q ≤ k − 2, due to the
simplices σq,r ) or if m + 1 = n (due to σ̄); it has rank 0 otherwise.

5.4. Case C̃n . The last case we are going to consider is that of Artin groups of
type C̃n . The corresponding Coxeter graph Γ is shown in Figure 9. For any σ ∈ K,
the subgraph Γ(σ) of Γ splits as a union of connected components of type Bk (those
containing the first or the last vertex) and of type Ak (the remaining ones).
0

4

1

2

3

...

n−1 4

n

Figure 9. A Coxeter graph of type C̃n .
B
Let Km
be the full simplicial complex on {1, . . . , m}, endowed with the weight
function of Bm (see Section 5.2). We are going to construct a precise matching for
B
the case C̃n using the precise matching on Km
.

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

37

For h ∈ {0, . . . , n} set
Kh = {σ ∈ K | h 6∈ σ and {0, . . . , h − 1} ⊆ σ}.
Like in the Ãn case, since every simplex σ ∈ K misses at least one vertex, the
subsets Kh form a partition of K:
n
G
K=
Kh .
h=0

Construct a matching on a fixed Kh as follows. Ignore the vertices 0, . . . , h and
relabel the remaining ones from right to left: n becomes 1, n − 1 becomes 2, . . . ,
h + 1 becomes n − h. This induces a poset isomorphism
∼
=

B
Kh −
→ Kn−h
.
B
Then put on Kh the pull-back of the matching on Kn−h
constructed in Section 5.2
(for h = n the only simplex in Kh is critical).

Remark 5.21. For d odd we are simply matching σ with σ ⊻ n for all σ except
σ = {0, . . . , n − 1}.
Lemma 5.22. The matching defined above is an acyclic matching on K.
Proof. As in the proof of Lemma 5.17, the subsets Kh are the fibers of a poset map
B
.

and the matching is acyclic on each Kh ∼
= Kn−h
Lemma 5.23. The matching defined above is a weighted matching on K.
∼
=

B
Proof. If a simplex σ ∈ Kh is sent to σ̂ by the isomorphism Kh −
→ Kn−h
, then
B
vϕ (σ) = vϕ (σ̂) + ωϕ (Bh ). Since the matching on Kn−h is weighted, the matching
on Kh also is.


As we have already said, for d odd there is exactly one critical simplex. Suppose
now that d is even. Let n = k d2 + r, with 0 ≤ r ≤ d2 − 1. In Kh there are critical
simplices if and only if n − h ≡ 0 mod d2 , i.e. when h = q1 d2 + r for some q1 (with
0 ≤ q1 ≤ k). By Theorem 5.15 there are two families of critical simplices: σq1 ,q2
(for 0 ≤ q1 + q2 ≤ k − 2) and σq′ 1 ,q2 (for 0 ≤ q1 + q2 ≤ k), as shown in Table 8 and
Figure 10. In a fixed dimension m = n − k + l (0 ≤ l ≤ k) there are: 2l + 1 critical
simplices if l ≤ k − 1; l critical simplices if l = k. See Table 9.
k − q1 − q2 − 2 times
4

...

...
...
d − 1 vertices

q1 d2 + r vertices

...
d
2

...

...

...

q2 d2 vertices

− 1 vertices

k − q1 − q2 times
4

...

q1 d2 + r vertices

...

...
d
2

− 1 vertices

...

4

...

4

q2 d2 vertices

Figure 10. Critical simplices for C̃n , with d even and n = k d2 + r.
Above is the simplex σq1 ,q2 and below is the simplex σq′ 1 ,q2 .

38

GIOVANNI PAOLINI AND MARIO SALVETTI

Case

Simplices

|σ|

vϕ (σ)

d odd

σ̄ = 1n 0

n

1 or 0

d even

σq1 ,q2

n − k + q1 + q2 + 1 q1 + q2 + 1 0 ≤ q1 + q2 ≤ k − 2

σq′ 1 ,q2

n − k + q1 + q2

q1 + q2

0 ≤ q1 + q2 ≤ k

Simplices
σq1 ,q2 = 1

q1 d
2 +r

d

d

01d−1 0(1 2 −1 0)k−q1 −q2 −2 1q2 2
d

d

d

σq′ 1 ,q2 = 1q1 2 +r 0(1 2 −1 0)k−q1 −q2 1q2 2

Table 8. Description of the critical simplices for C̃n . When d is even,
set n = k d2 + r.

Simplices
σl−1,0

′
σl,0

σl−2,1
..
.

′
σl−1,1
..
.

σ1,l−2

′
σ2,l−2

σ0,l−1

′
σ1,l−1
′
σ0,l

Table 9. Critical simplices for C̃n in dimension m = n − k + l (0 ≤
l ≤ k), where d is even and n = k d2 + r. For l = k only the second
column occurs.

Theorem 5.24. The critical simplices for the matching on K are those defined
in Table 8. In particular the matching is precise. Moreover the only non-trivial
incidence numbers between critical simplices in the Morse complex are as follows
(for d even and n = k d2 + r):
[σq1 ,q2 : σq1 , q2 −1 ]M = (−1)α ,
[σq1 ,q2 : σq1 −1, q2 ]M = (−1)β+1 ,
d

[σq1 ,q2 : σq′ 1 ,q2 ]M = (−1)β+ 2 +1 ,
[σq′ 1 ,q2 : σq1 , q2 −2 ]M = (−1)β+1 ,
[σq′ 1 ,q2 : σq1 −2, q2 ]M = (−1)β ,
[σq′ 1 ,q2 : σq′ 1 , q2 −1 ]M = (−1)α+1 ,
d

where α = (k − q2 )

d
2

[σq′ 1 ,q2 : σq′ 1 −1, q2 ]M = (−1)β+ 2 ,

− 1 + q1 + r + d2 and β = q1 d2 + r.

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

39

Proof. We have already proved that the critical simplices are those listed in Table
8. For d odd there is nothing more to prove, so let d be even and n = k d2 + r.
Consider a generic alternating path starting from any critical cell ρ0 and ending in
a critical cell of the form σq1 ,q2 :
ρ0 ⊲ τ1 ⊳ ρ1 ⊲ τ2 ⊳ ρ2 ⊲ · · · ⊲ τm ⊳ ρm ⊲ σq1 ,q2 .
Let ρm = σq1 ,q2 ∪ {v}. If v = q1 d2 + r then ρm = σq′ 1 +2, q2 and the alternating path
stops. If v = n − q2 d2 then the alternating path stops at ρm = σq1 , q2 +1 . Suppose
q1 + q2 ≤ k − 3, otherwise there are no more cases. If q1 d2 + r + d < v < n − q2 d2 ,
then Γ(ρm ) has at least one connected component of size d − 1 and therefore ρm is
matched with a simplex of higher dimension; thus the path stops without having
reached a critical simplex. If v = q1 d2 + r + d then ρm is matched with

d
d
d
d
τm = ρm \ q1 d2 + r + d2 = 1q1 2 +r 01 2 −1 01d−1 0(1 2 −1 0)k−q1 −q2 −3 01q2 2 .

From here we can continue in many ways. Let ρm−1 = τm ∪ {w}.
• If w = q1 d2 + r, we end up with ρm−1 = σq1 +1, q2 .
• If w = q1 d2 + r + d2 we would be going back to ρm .
• If w > q1 d2 + r + 23 d, then Γ(ρm−1 ) has at least one connected component of
size d−1 and therefore ρm−1 is matched with a simplex of higher dimension.
• If w = q1 d2 + r + 32 d, then ρm−1 is matched with

τm−1 = ρm−1 \ q1 d2 + r + d
d

d

d

d

d

= 1q1 2 +r 01 2 −1 01 2 −1 01d−1 0(1 2 −1 0)k−q1 −q2 −4 01q2 2 .

By induction, repeating the same argument as above, this path can be
continued in exactly one way and it eventually arrives at the critical simplex
σq′ 1 , q2 +2 . The path has length m = k − q1 − q2 − 2 and is as follows.

τ1 = σq′ 1 , q2 +2 \ n − q2 d2 ,

ρ1 = τ1 ∪ n − q2 d2 − d ,

τ2 = ρ1 \ n − q2 d2 − d2 ,

ρ2 = τ2 ∪ n − q2 d2 − 23 d ,

τ3 = ρ2 \ n − q2 d2 − d ,
..
.


ρm−1 = τm−1 ∪ q1 d2 + r + d ,

τm = ρm−1 \ q1 d2 + r + 23 d ,

ρm = τm ∪ q1 d2 + r + d2 ,

σq1 ,q2 = ρm \ q1 d2 + r + d .

Consider now a generic alternating path starting from any critical cell ρ0 and
ending in a critical cell of the form σq′ 1 ,q2 :
ρ0 ⊲ τ1 ⊳ ρ1 ⊲ τ2 ⊳ ρ2 ⊲ · · · ⊲ τm ⊳ ρm ⊲ σq1 ,q2 .
As usual, let ρm = σq′ 1 ,q2 ∪ {v}. For the same reasons as above, there are only three
possibilities: v = q1 d2 + r, v = q1 d2 + r + d2 , v = n − q2 d2 ; in all the other cases, ρm is
matched with a simplex of higher dimension. If v = q1 d2 + r then the path ends (to

40

GIOVANNI PAOLINI AND MARIO SALVETTI

the left) at ρm = σq′ 1 +1, q2 . If v = q1 d2 + r + d2 then the path ends at σq1 ,q2 . Finally,
if v = n − q2 d2 then the path ends at σq′ 1 , q2 +1 .
We have seven families of incidence numbers to compute, each coming from one
of the alternating paths we have found.
• From σq′ 1 +2, q2 to σq1 ,q2 . The alternating path is trivial and consists in
removing the vertex v = q1 d2 + r, so
d

[σq′ 1 +2, q2 : σq1 ,q2 ]M = (−1)|{w∈σq1 ,q2 |w<v}| = (−1)q1 2 +r ,
d

=⇒ [σq′ 1 ,q2 : σq1 −2, q2 ]M = (−1)q1 2 +r = (−1)β .
• From σq1 , q2 +1 to σq1 ,q2 . Again the path is trivial, and it consists in removing
v = n − q2 d2 . Therefore
[σq1 , q2 +1 : σq1 ,q2 ]M = (−1)|{w∈σq1 ,q2 |w<v}|
d

= (−1)n−q2 2 −(k−q1 −q2 −1)
d

=⇒

[σq1 ,q2 : σq1 , q2 −1 ]M

d

= (−1)k 2 +r−q2 2 −k+q1 +q2 +1

(k−q2 ) d
−1 +r+q1 +1
2
= (−1)
,

d
= (−1)(k−q2 +1) 2 −1 +r+q1 +1
= (−1)α .

• From σq1 +1, q2 to σq1 ,q2 . The path consists in removing q1 d2 + r, adding
q1 d2 + r + d2 , and removing q1 d2 + r + d. Therefore
d

d

d

d

[σq1 +1, q2 : σq1 ,q2 ]M = (−1)(−1)q1 2 +r (−1)q1 2 +r+ 2 −1 (−1)q1 2 +r+d−1
d

d

= (−1)q1 2 +r+ 2 +1 ,
=⇒

d

d

[σq1 ,q2 : σq1 −1, q2 ]M = (−1)(q1 −1) 2 +r+ 2 +1
= (−1)β+1 .

• From σq′ 1 , q2 +2 to σq1 ,q2 . The path is the one we explicitly wrote before.
Notice that from ρi−1 to τi one removes the vertex vi = n − (q2 − i + 1) d2 ,
and from τi to ρi one adds the vertex vi′ = n − (q2 − i − 1) d2 . Therefore
′

[ρi−1 : τi ][ρi : τi ] = (−1)|{w∈τi |vi <w<vi }| = (−1)d−1 = −1.
Then we can compute the incidence number in the Morse complex:
!
m
Y
M
m
′
[ρi−1 : τi ][ρi : τi ] [σq1 ,q2 : ρm ]
[σq1 , q2 +2 : σq1 ,q2 ] = (−1)
i=1

d

m

= (−1) (−1)m (−1)q1 2 +r+(d−1)
d

= (−1)q1 2 +r+1 ,
=⇒

d

[σq′ 1 ,q2 : σq1 , q2 −2 ]M = (−1)q1 2 +r+1
= (−1)β+1 .

WEIGHTED SHEAVES AND HOMOLOGY OF ARTIN GROUPS

41

• From σq′ 1 +1, q2 to σq′ 1 ,q2 . The path is trivial and consists in removing v =
q1 d2 + r. Then
d

[σq′ 1 +1, q2 : σq′ 1 ,q2 ]M = (−1)q1 2 +r ,
d

d

=⇒ [σq′ 1 ,q2 : σq′ 1 −1, q2 ]M = (−1)(q1 −1) 2 +r = (−1)β+ 2 .
• From σq1 ,q2 to σq′ 1 ,q2 . The path is trivial and consists in removing v =
q1 d2 + r + d2 . Then
d

d

d

[σq1 , q2 : σq′ 1 ,q2 ]M = (−1)q1 2 +r+ 2 −1 = (−1)β+ 2 +1 .
• From σq′ 1 , q2 +1 to σq′ 1 ,q2 . The path is trivial and consists in removing v =
n − q2 d2 . Then
d

[σq′ 1 , q2 +1 : σq′ 1 ,q2 ]M = (−1)n−q2 2 −(k−q1 −q2 )
d

=⇒

[σq′ 1 ,q2 : σq′ 1 , q2 −1 ]M

d

= (−1)k 2 +r−q2 2 −k+q1 +q2

d
= (−1)(k−q2 ) 2 −1 +r+q1 ,

d
= (−1)(k−q2 +1) 2 −1 +r+q1
= (−1)α+1 .



We can finally compute the homology H∗ (GW ; R) for Artin groups of type C̃n .
Theorem 5.25. Let GW be an Artin group of type C̃n . Then the ϕd -primary
component of H∗ (GW ; R) is trivial for d odd, and for d even is as follows (let
n = k d2 + r):
(
⊕m+k−n+1
if n − k ≤ m ≤ n − 1,
∼ (R/(ϕd ))
Hm (GW ; R)ϕd =
0
otherwise.
Proof. In order to apply Theorem 5.1 we need to find the rank the boundary maps
ϕ
δm+1
of the Morse complex. For d odd there is only one critical cell, thus all the
boundaries vanish. Suppose from now on that d is even, and let n = k d2 + r. In
ϕ
order to have a non-trivial boundary δm+1
we must have at least one critical simplex
both in dimension m and in dimension m + 1, thus m = n − k + l with 0 ≤ l ≤ k − 1
(see Table 9).
ϕ
We are going to prove that, for l ≤ k − 2, a basis for the image of δ = δm+1
is
given by

B = δσq1 ,q2 | q1 + q2 = l .
By Theorem 5.24 we obtain the following formula for δσq1 ,q2 (whenever a simplex
does not exist, the corresponding term simply vanishes):
d

δσq1 ,q2 = (−1)α σq1 , q2 −1 + (−1)β+1 σq1 −1, q2 + (−1)β+ 2 +1 σq′ 1 ,q2 ,

where α = (k − q2 ) d2 − 1 + q1 + r + d2 and β = q1 d2 + r. The term σq′ 1 ,q2 appears in
δσq1 ,q2 but not in any other element of B, thus B is made of linearly independent
vectors. Moreover, if q1 + q2 = l + 1, we have that
d

δσq′ 1 ,q2 = (−1)β+1 σq1 , q2 −2 + (−1)β σq1 −2, q2 + (−1)α+1 σq′ 1 , q2 −1 + (−1)β+ 2 σq′ 1 −1, q2


d
d
d
= (−1)α+β+ 2 (−1)α+ 2 +1 σq1 , q2 −2 + (−1)β+1 σq1 −1, q2 −1 + (−1)β+ 2 +1 σq′ 1 , q2 −1

42

GIOVANNI PAOLINI AND MARIO SALVETTI



d
d
+ (−1) 2 +1 (−1)α+1 σq1 −1, q2 −1 + (−1)β+ 2 +1 σq1 −2, q2 + (−1)β+1 σq′ 1 −1, q2
d

d

= (−1)α+β+ 2 δσq1 , q2 −1 + (−1) 2 +1 δσq1 −1, q2 .

ϕ
ϕ
Therefore B generates the image of δm+1
. Thus rk δm+1
= |B| = l + 1 for l ≤ k − 2.
For l = k − 1, i.e. m = n − 1, the situation is a bit different because there are no
critical simplices of the form σq1 ,q2 with q1 + q2 = l. However we can still define
d

ǫq1 ,q2 = (−1)α σq1 , q2 −1 + (−1)β+1 σq1 −1, q2 + (−1)β+ 2 +1 σq′ 1 ,q2
for q1 + q2 = l, and

B = ǫq1 ,q2 | q1 + q2 = l .

As above we have that, for q1 + q2 = l + 1,
d

d

δσq′ 1 ,q2 = (−1)α+β+ 2 ǫq1 , q2 −1 + (−1) 2 +1 ǫq1 −1, q2 .
Then, for the same reasons as above, B is a basis for the image of δnϕ .
ϕ
We have proved that, for 0 ≤ l ≤ k − 1, the rank of δm+1
is equal to l + 1 =
m + k − n + 1. Then we conclude applying Theorem 5.1.

Remark 5.26. Future works will focus on other families of Artin groups. In
particular it seems that precise matchings can be constructed in all finite and affine
cases (see [Pao17]), possibly allowing explicit homology computations.
The methods developed in this paper are particularly powerful when the coefficients are over a PID, but Artin groups with roots of different lengths (e.g. Bn , B̃n
and C̃n ) also admit natural representations over polynomial rings with more than
one variable. We believe that some of our theory can be extended to such cases,
and this can also be the aim of future works.
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Giovanni Paolini
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa (Italy)
E-mail address: giovanni.paolini@sns.it
Mario Salvetti
Department of Mathematics, University of Pisa, Largo Pontecorvo 5, 56127 Pisa (Italy)
E-mail address: salvetti@dm.unipi.it

