Residues

RESIDUES ON BUILDINGS, AND DE-RHAM COHOMOLOGY
OF p-ADIC SYMMETRIC DOMAINS
EHUD DE SHALIT
Table of Contents
Part I : Local systems on buildings attached to hyperplane arrangements
1. The Bruhat-Tits building
2. The local system of algebras
3. Interpretation of the local system by harmonic cochains
4. Acyclicity (d ≤ 2)
5. Acyclicity (in general)
Part II : Residues on the Bruhat-Tits building, and the cohomology of X
6. The structure of X, and the definition of the residue
7. The cohomology of X(τ )
8. The cohomology in the large
9. The cohomology of Γ\X
10. Appendix : Rigid de-Rham cohomology
0.1. Background. Let K be a finite extension of the field Qp . Drinfel’d’s p-adic

symmetric domain of dimension d over K, is the complement X in Pd of the union
of all the K-rational hyperplanes. It is endowed with a structure of a rigid analytic
space (and even with a finer structure of a formal scheme, which we shall not need).
The group G = P GLd+1 (K) acts on X. Together with a certain family of étale
coverings, also due to Drinfel’d, X plays a central role in the study of Shimura
varieties, and in the representation theory of G (“Carayol’s program”).
Unlike the real symmetric domains, X is not simply-connected in the étale topol-
ogy, and its cohomologies are quite intricate. They are natural infinite dimensional
representation spaces for G. In their paper [S-S] Schneider and Stuhler computed
the cohomology of X for any cohomology theory satisfying certain reasonable ax-
ioms. Their computations apply to rigid de-Rham cohomology, and produce, for
k
each 0 ≤ k ≤ d, an isomorphism between HdR (X) and a certain space C k of k-
cochains on (part of) the Bruhat-Tits building T of G. See [S-S], corollary 17 (the
notation C k is ours - see section 8.3 below). Denote this isomorphism by
(0.1) [η] 7→ cSS
η .
The approach taken by Schneider and Stuhler left some open questions. Let us
list a few of them.
Date: June 1, 1999.

1
2 EHUD DE SHALIT
• For any two K-rational hyperplanes with equations a = 0 and b = 0, the log-
1
arithmic form d log(a/b) represents a class in HdR (X). The subring generated
by these in the cohomology we call the subring of logarithmic classes. Is it
dense ? What are all the relations between logarithmic classes ? Can one give
a formula for the cochain cSS
η for logarithmic η ?
• There is a well-known reduction map r : X → |T |, which we review in section
6.1 below. Is the map (0.1) local in the sense that the value of cSS η on a
k-simplex σ depends only on the restriction of [η] to the pre-image of σ under
r?
• Schneider and Stuhler make the remark that the space C k should be viewed
as a space of “harmonic cochains” on the building, but they don’t make
this notion precise. In fact, for 0 < k < d, their k-cochains are defined on
some, but not all, of the k-cells, and on which k-cells their cochains take
values depend on the choice of coordinates in projective space. In retrospect,
this prevents them from defining analogues of the Hodge conditions “d =
0, d∗ = 0”. Can their cochains be extended to all the simplices of T , and
can the harmonicity conditions be given a coordinate-free description which
is analogous to the notion of harmonicity on classical symmetric domains ?
• It has been known for some time that in the case d = 1 the cochain attached
to a 1-form may be defined via its residues along annuli in X. In the first
half of [S-T] Schneider and Teitelbaum developed a theory of residues for
top-dimensional (k = d) classes for arbitrary d, and proved that the residue
cochain of a d-form is harmonic. However, even in the case of top-dimensional
forms, they did not quite prove that the residue map is an isomorphism onto
the space of all harmonic cochains. Can the notion of a residue be defined for
an arbitrary k-form ? Is the residue cochain map harmonic ? If so, does it
k k
give an isomorphism of HdR (X) onto the space Char of all harmonic cochains
?
The primary purpose of this work is to develop a theory of residues in any

dimension, and as a result give a new interpretation of the cohomology of X, which
is independent of the methods of Schneider and Stuhler, but recovers their main
theorem. We also answer in the affirmative all the afore-mentioned questions.
Our approach centers on a detailed combinatorial study of the ring of logarithmic
classes. Only at the very end (in the appendix) do we show that these span the
entire cohomology in the appropriate sense. This is in remarkable contrast with the
approach of [S-S], where the logarithmic forms do not show up. The “combinatorics
of logarithmic classes” reveals interesting connections with the theory of hyperplane
arrangements. A secondary goal of this work is to explain these relations. Since
we believe that there is sufficient interest in this second aspect, we gathered all the
combinatorics and group theory in Part I, from which the geometry is absent. In
Part II we apply these results to the problem of computing the cohomology of X.
0.2. Survey of the main results (Part I). Let K be a field of characteristic
0, and VK a finite dimensional vector space over K of dimension d + 1, d ≥ 0. As
usual we identify lines through the origin in VK with linear hyperplanes in the dual
space VK∗ . Let A ⊆ P(VK ) be a family of lines in VK . For a ∈ A denote by Ha the
hyperplane in the projective space P(VK∗ ), given by the equation a = 0. When A is
RESIDUES AND COHOMOLOGY 3
finite, a great deal of work has been done on the cohomology of

[
(0.2) MA = P(V ∗ ) − Ha ,
a∈A
the complement of the hyperplane arrangement A. In particular Orlik and Solomon

attached to A a certain finite dimensional, graded, alternating K-algebra A, and
established (following work of Arnold and Brieskorn) a canonical isomorphism of
graded algebras
(0.3) A ' HdR (MA )
with the de-Rham ring of MA . A nice property of A is that it depends only on
the intersection pattern of the Ha ’s, so geometric questions are reduced to combi-
natorics, though quite non-trivial. See [O-S], and [O-T], Chapters 3 and 5.
Now assume that K is p-adic (a finite extension of Qp ). Let T be the Bruhat-Tits

building of G = P GL(VK ). In Section 2 we attach to each r-simplex (0 ≤ r ≤ d) τ
in T a certain finite-dimensional, graded, alternating algebra A(τ ), and study some
important properties of the collection {A(τ )}. Our construction is inspired by that
of Orlik and Solomon, but the set A need not be finite anymore. In fact, our main
interest lies in the case where A consists of all the K-rational hyperplanes in VK∗ .
This case enjoys extra symmetries, coming from the fact that A is G-invariant.
If v is a vertex of T , then v determines (up to a homothety) a lattice L in VK ,

hence an integral structure. This allows us to reduce the hyperplanes in A, obtaining
a finite arrangement of hyperplanes A(v) in the projective space P(L∗ /πL∗ ) over
Fq . This arrangement depends on the integral structure, namely on v. Let MA(v) be
the complement of this hyperplane arrangement, which is a smooth affine variety
over Fq . Then one can prove that
¡ ¢
(0.4) A(v) ' HM W MA(v)
(Monsky-Washnitzer cohomology [M-W]; de-Rham cohomology is not well-behaved

in characteristic p). When τ is not a vertex A(τ ) still admits an interpretation as
the rigid de-Rham cohomology of a certain rigid-analytic space, but this time not
a space with good reduction. Neither of these interpretations is needed in Part I.
Of the properties of the collection {A(τ )} that we study, those pertaining to the
individual algebras A(τ ) may be regarded as straightforward generalizations of the
classical results. The connection with the building T , however, brings in at least
two new and deep phenomena.
In Section 3 we define the notion of harmonic k-cochains on T (0 ≤ k ≤ d), as
maps from the (pointed) k-cells of T to K, satisfying certain harmonicity conditions.
We then show that Ak (τ ) can be identified with the space which is obtained by
restricting harmonic k-cochains to T̂k (τ ), the collection of k-cells contiguous to τ
(two cells σ and τ in T are contiguous if they are both faces of another cell of
k
T ). This space is contained in Char (τ ), which is the space of all maps T̂k (τ ) → K
satisfying the harmonicity conditions “when applicable”. For τ = v, a vertex, they
even coincide.
4 EHUD DE SHALIT
Another new feature of our construction is that for every face inclusion τ ≤ σ,
we get a canonical homomorphism
(0.5) rστ : A(τ ) → A(σ),
which makes the collection of algebras A(τ ) into a (cohomological) local system A
on T . When A = P(VK ), it is a G-equivariant local system. The main question
in this context is whether this local system is acyclic. In Section 4 (which for
A = P(VK ) is superseded by Section 5, so can be omitted) we give a direct proof of
the acyclicity for d ≤ 2. The case d = 0 is empty (the building is a point), d = 1 is
very easy (the building is a tree), but d = 2 is already quite involved. The methods
which work for d ≤ 2 do not generalize easily. Luckily, for A = P(VK ) we were
able to use a deep result of Schneider and Stuhler [S-S1], which supplies canonical
resolutions for certain smooth representations of G, to prove acyclicity in general.
This is the contents of Section 5.
k
0.3. Survey of the main results (Part II). In Section 6, to every [η] ∈ HdR (X)
we attach its residues resσ η ∈ K along pointed k-cells σ ∈ T̂k . We then show that
the cochain cη given by cη (σ) = resσ η is harmonic (in the sense of Section 3). The
residue of a logarithmic class is given by a simple combinatorial formula.
Our main theorem says that the residue cochain map c sending η to cη is an
isomorphism of G-modules
k k
(0.6) c : HdR (X) ' Char ,
onto the space of all harmonic k-cochains on T .
For the proof we rely heavily on the combinatorics of the logarithmic classes.
We must therefore know in advance that these classes generate the cohomology of
(at least) some nice covering of X, indexed by cells τ of T , which we denote by
{X(τ )}. This fact is proved in the appendix (Section 10) using a Gysin long exact
sequence in rigid de-Rham cohomology.
The sets X(τ ) are defined as follows (see Section 6). For every vertex v of T ,
X(v) is the subdomain of X, which is the inverse image of the star St(v) of v, under
the reduction map r : X → |T |. These X(v) form a covering
T of X by Stein domains,
whose nerve is T . If τ is a cell of T , we let X(τ ) = v∈τ X(v).
Fundamental to our proof is a triangle of maps

A· (τ ) → ·
HdR (X(τ ))
(0.7) & . .
·
Char (τ )
The arrow on the top takes a generator ea −eb of the algebra A· (τ ) to the logarithmic
differential form d log(a/b) (here a and b are non-vanishing linear forms on X). Of
the two diagonal arrows, the one on the left is the map from part I, Section 3.
Recall that it is injective, and an isomorphism if τ = v is a vertex. The arrow on
the right is the residue cochain map, constructed in Section 6, localized to X(τ ).
The triangle commutes and its top arrow is surjective (by the result proven in the
appendix). We conclude that
(0.8) A· (τ ) ' HdR
·
(X(τ )),
· ·
and for a vertex v also HdR (X(v)) ' Char (v).
k k
In order to globalize the isomorphism HdR (X(v)) ' Char (v), we use a Mayer-
Vietoris spectral sequence, relative to the covering of X by the X(v)’s. This is where
knowledge of the cohomology of all the X(τ )’s is needed. The spectral sequence
degenerates, thanks to the acyclicity of the local system A, proven in Section 5.
This degeneration leads immedaitely to the main theorem (theorem 8.2).
In section 8.3 we determine the relation of our map c to cSS , the isomorphism
of Schneider and Stuhler. Theorem 8.3 is based on a recent preprint of Iovita
and Spiess [I-S], written after this work has been completed, in which the authors
compute cSS on the logarithmic forms. As a result of their computation, one can
show that the space C k of Schneider and Stuhler is¡ obtained
¢ by restricting our
harmonic cochains to cells of minimal type (there are kd types of pointed k-cells in
general). Since a harmonic cochain is determined by its values on cells of minimal
type, C k and Char
k
are isomorphic, only the harmonicity conditions, when phrased
in terms of cells of minimal type, become less natural. Furthermore, identifying
the two spaces in this way, the isomorphisms of [S-S] and of this paper coincide:
cη (σ) = cSS k
η (σ). It should be remarked that the fact that the spaces C and Char
k
are the same, is a purely combinatorial problem, which was solved directly and
independently by Gil Alon. In theorem 8.3 we circumvent it, and get the stronger
k
result that not only the spaces coincide, but the two isomorphisms of HdR (X) with
them agree as well.
In the last section we investigate applications to the cohomology of Γ\X, where Γ

is discrete and cocompact in G. The main question treated there, the conjecture of
Schneider on a Hodge-like decomposition of H d (Γ\X), was also settled affirmatively
in [I-S].
I would like to thank E. Dror-Farjoun, G. Kalai, R. Livné and S. Mozes for

many discussions related to this work, and P. Schneider for his general interest,
and for suggesting to look at [S-S1].
Part 1. Local systems on buildings attached to hyperplane

arrangements
1. The Bruhat-Tits building
1.1. The Bruhat-Tits building. Let K be a finite extension of Qp , π a uni-
formizer of K, and q the cardinality of OK /(π). We normalize the absolute value on
b̄ so that |π| = q −1 . Let us review
the algebraic closure K̄ (and on its completion K)
some definitions and standard results on the Bruhat-Tits building of P GLd+1 (K)
([B-T], [Br], [Mus]).
Fix an integer d ≥ 0 and a d + 1 dimensional vector space VK over K. Write VK∗
for the dual space and V , V ∗ for the associated affine schemes. By a basis of VK
(or VK∗ ) we always mean an ordered basis. A basis of VK determines a dual basis
of VK∗ and vice versa. A lattice in a finite dimensional vector space over K is a
finitely generated OK -module spanning the vector space over K. Two lattices L
and L0 lie in the same dilation class if there exists a λ ∈ K × such that L0 = λL.
Let G = P GL(VK ). The Bruhat-Tits building of G, T , is a simplicial com-
plex which may be described as follows. The vertices T0 of T consist of dilation
6 EHUD DE SHALIT
classes [L] of lattices L ⊂ VK . The k-cells Tk (0 ≤ k ≤ d, we use cell and simplex

synonymously) consist of k + 1-tuples {[L0 ], [L1 ], ..., [Lk ]} where
(1.1) L0 ⊃ L1 ⊃ · · · ⊃ Lk ⊃ πL0
(strict inclusion). We call such a sequence a lattice flag, or in short a flag, in L0 .
It is confined to lie between L0 and πL0 . Notice also that there is a natural cyclic
ordering of the vertices of a k-cell. We write τ ≤ σ to denote that τ is a face of σ.
A pointed k-cell of T consists of a k-cell together with a distinguished vertex
v0 = [L0 ]. Equivalently, a pointed k-cell is given by an ordered k + 1-tuple σ =
([L0 ], ..., [Lk ]) where the Li are as above. We shall write
(1.2) σ = (L0 ⊃ L1 ⊃ · · · ⊃ Lk ⊃ πL0 )
if the Li are chosen as in (1.1) and [L0 ] is the distinguished vertex. They are
then unique up to a dilation by a common factor. Let T ck be the set of pointed
k-cells. Any pointed k-cell σ can be assigned a type, which is the sequence t(σ) =
(e0 , e1 , ..., ek ), defined by ei = dim Li /Li+1 (here and below we put Lk+1 = πL0 ,
Pk
and the dimension is over OK /(π) ' Fq ). Clearly each ei is positive and i=0 ei =
¡d¢
d + 1. There are k types of pointed k-cells.
As a rule, we shall not distinguish in our notation between a pointed k-cell σ
and the underlying non-pointed cell, and it should be clear from the context which
properties depend on the distinguished vertex and which do not.
The σ ∈ T ck with distinguished vertex v0 are in one-to-one correspondence with
flags in L0 . We shall occasionally write Vi = Li /πL0 , and denote dim Vi by di , so
that d0 = d + 1 and ei = di − di+1 . If a ∈ L0 we shall denote by ā its image in
V0 = L0 /πL0 .
A basis of VK is called adapted to (the pointed k-cell) σ if it induces an
d+1
isomorphism of OK with a lattice L0 representing the distinguished vertex, and
if under the induced isomorphism of Fd+1
q with V0 we have Vi = {ā ∈ V0 ; āj = 0
for j ≥ di }.
1.2. The combinatorial metric ρ. There is a metric ρ on T0 defined as follows.

If u = [L] and v = [M ] and if, for suitable representatives M ⊂ L
(1.3) L/M = OK /(π m0 ) ⊕ · · · ⊕ OK /(π md )
with 0 ≤ m0 ≤ · · · ≤ md , then
(1.4) ρ(u, v) = md − m0 .
If σ ∈ Tk and τ ∈ Tr we put
(1.5) ρ(τ, σ) = max ρ(u, v).
u∈τ, v∈σ
Then ρ(τ, σ) ≤ ρ(τ, ω)+ρ(ω, σ). We caution the reader that there is another notion
of distance between chambers (top dimensional cells) of T [Br], which is related to
the length of elements in the Weyl group, and does not coincide with this ρ. Each
notion has its own pathologies. In particular, for our ρ, ρ(σ, σ) = 1, unless σ is a
vertex. In Section 1.5 below we shall define yet another, third, metric d on T0 .
A set of k + 1 vertices forms a k-cell precisely when any two of them are at
distance 1 from each other. Two cells τ and σ in T are called contiguous when
ρ(τ, σ) ≤ 1. Equivalently, τ ∪ σ is a cell. The two cells may or may not share
vertices in common. We shall denote by Tk (τ ) the set of k-cells contiguous to an

arbitrary cell τ, and by Tbk (τ ) the corresponding pointed k-cells. For example, if
v0 is a vertex, Tk (v0 ) consists of k-cells that either contain v0 , or form a k + 1-cell
with it.
Another way to say that σ ∈ Tk and τ ∈ Tr are contiguous is that we can find
lattice flags
(1.6) πL0 ⊂ Lk ⊂ ... ⊂ L1 ⊂ L0 , πM0 ⊂ Mr ⊂ · · · ⊂ M1 ⊂ M0
representing them which can be interlaced - i.e., whose union is also a lattice flag.
If τ is pointed and the distinguished vertex is [M0 ], then σ is automatically given a
distinguished vertex [L0 ], determined by the condition that M0 ⊇ L0 ⊃ Lk ⊃ πM0 .
We can write the interlaced flag as blocks of L0i s and blocks of Mj0 s alternating, the
only non-uniqueness occuring where some Li is also an Mj . The type of σ with
respect to τ, written t(σ, τ ) is the combinatorial data consisting of the relative
positions of the lattices, and the dimensions of the successive quotients. One can
think of it as the type t(τ ∪ σ) defined before, relative to the distinguished vertex
[M0 ], with the extra information saying which are the vertices of each of the two
cells.
1.3. The apartments of T . Fix a basis α = (α0 , . . . , αd ) of VK . The apartment

Aα determined by α is the simplicial subcomplex of T supported on the vertices of
the form
(1.7) vα (m0 , . . . , md ) = [hπ m0 α0 , . . . , π md αd i]
for mi ∈ Z. Here h·i denotes the span over OK . Every apartment is a triangulation
of a d-dimensional Euclidean space. The vertices with mi − mj = n, for some fixed
n, span a wall W in the apartment. A basic fact (see [Mus]) is that every two
simplices σ and τ belong to a common apartment (and in fact to infinitely many
apartments).
1.4. The action of G. The group G acts on the left on T . The action is transitive
on the set of pointed k-cells of a given type. It is also transitive on pairs of pointed
cells (σ, τ ) of the same relative type t(σ, τ ). It acts as isometries on T0 . Upon
introduction of coordinates adapted to some vertex v0 = [L0 ] we may identify G
with P GLd+1 (K), and the stabilizer of v0 with P GLd+1 (OK ). The stabilizer Bσ of
a pointed k-cell σ = (v0 , ..., vk ) is then conjugate in P GLd+1 (OK ) to the standard
parahoric subgroup of type t(σ) (and is precisely equal to it if the basis is
adapted to σ as well). This is the subgroup which has along the diagonal k + 1
blocks of sizes ek , ..., e1 , e0 with arbitrary OK entries, above them arbitrary entries
too, and below them entries divisible by π.
If v is a vertex, the stabilizer Bv of v is isomorphic to P GLd+1 (OK ), so posesses
a distinguished sequence Uvn of normal pro-p subgroups (n ≥ 1), namely the prin-
cipal congruence subgroups of level π n . The group Uvn is in fact independent of
the choice of coordinates, because it can be defined intrinsically as the (pointwise)
stabilizer of the ball {u ∈ T0 ; ρ(u, v) ≤ n}.
The choice of a basis α for VK determines a maximal torus Tα in G, which,
under the isomorphism of G with P GLd+1 (K) induced by α, corresponds to the
diagonal matrices. For every wall W in the apartment Aα there is a (non-unique)
8 EHUD DE SHALIT
involution sW ∈ G which normalizes Tα and on Aα induces a reflection in the wall

W.
1.5. The topological realization of T and the Euclidean metric d. We let

|T | be the topological simplicial complex associated to T . If σ is a k-cell as above,
Pk
then |σ|Pis the open simplex consisting (if k > 0) of all the points t = i=0 ti vi ,
where ti = 1, 0 < ti < 1. It is known that |T | is contractible (see [Mus], theorem
1.1(1)). We shall denote by St(v0 ) the star of a vertex v0 . By definition, this is the
union of the open simplices |σ| for v0 ∈ σ. The closure St(v0 ) is then the compact
set which is the union of all |σ| for σ contiguous to v0 . Finally we put, for any cell
τ
[ \
(1.8) St(τ ) = |σ| = St(v)
τ ≤σ v∈τ
[ \
(1.9) St(τ ) = |σ| = St(v).
ρ(τ,σ)≤1 v∈τ
If Aα is an apartment as in Section 1.3, determined by the basis α, we can map

|Aα | onto Rd+1 /R · (1, 1, . . . , 1) by mapping vα (m0 , . . . , md ) to (m0 , . . . , md )modR ·
(1, 1, . . . , 1). This identification with Euclidean space induces a metric dα on |Aα |,
and in fact a canonical G-invariant Euclidean metric d(x, y) on |T |, because any
two points belong to a common apartment Aα , and the distance between them is
independent of α ([Br], Ch. VI.3). This metric on |T | should not be confused with
the combinatorial metric ρ on T0 , although on the vertices they are equivalent. (For
example, when d ≥ 3, not all the vertices of a simplex are at equal d-distance √ from
each other. When d = 2 they are, but ρ(u, v) = 2 implies d(u, v) = 2 or 3, etc.)
The geodesic connecting two vertices u and v in the metric d is the straight
line connecting them in the realization of some (any) apartment containing both,
endowed with its Euclidean structure. It lies in the intersection of all the apartments
containing u and v, and is independent of the apartment used to define it.
1.6. Pictures. There are very nice pictures of T (d = 1 and d = 2) in the new
book by Paul Garrett [Ga].
2. The local system of algebras on T

2.1. A certain anti-commutative graded algebra. The construction described
below is inspired by that of the Orlik-Solomon algebra for central hyperplane ar-
rangements (see [O-T], chapter 3, and [O-S]).
Let A be a non-empty subset of P(VK ). We view A as a set of lines in VK , or
hyperplanes in VK∗ . It may be finite or infinite, and in particular may consist of all
of P(VK ). We write ea for the line represented by a ∈ VK − {0}, so that eλa = ea
for any scalar λ 6= 0. By abuse of language we shall sometimes say a ∈ A to mean
that ea ∈ A.
Let Ee be the free exterior algebra over K, on the set A. It is graded and
alternating, and its multiplication is written as ∧. Let Ẽ k be the k th graded piece
of Ẽ, and Ẽ[1] the graded (left) Ẽ -module defined by a shift in the grading :
Ẽ[1]k+1 = Ẽ k .
There is a unique derivation δ of Ẽ, homogeneous of degree -1, mapping each

ea to 1. It satisfies the following properties.
k
X
(2.1) δ(ea0 ∧ · · · ∧ eak ) = (−1)i ea0 ∧ · · · ∧ ec
ai ∧ · · · ∧ eak
i=0
(the symbol eba means that ea is omitted from the product)

(2.2) δ2 = 0
(2.3) δ(ea0 ∧ · · · ∧ eak ) = (ea1 − ea0 ) ∧ · · · ∧ (eak − ea0 )

(follows from (2.1)),
(2.4) δ(ea0 ∧ · · · ∧ eak ) = δ(ea0 ∧ · · · ∧ eam ) ∧ δ(eam ∧ · · · ∧ eak )
(follows from (2.3) after a rearrangement of the indices), and
(2.5) δ(ea ∧ x) = x − ea ∧ δx.
It follows that
(2.6) E = Im(δ) = Ker(δ)
is the subalgebra generated by ea − eb , and that there is an exact sequence
δ
(2.7) 0 → E → Ẽ → E[1] → 0.
Any ea in A supplies a splitting Ẽ ← E[1], ea ∧ x ¾ x.
Definition 2.1. Let τ ∈ Tbr , and let M0 ⊃ · · · ⊃ Mr ⊃ πM0 be a lattice flag

representing τ. The index ιτ (a) of a ∈ A ∩ (M0 − πM0 ) is the unique i, 0 ≤ i ≤ r,
such that a ∈ Mi − Mi+1 . For general a ∈ A the index of a with respect to τ is
defined to be the index of λa, for a suitable scalar λ such that λa ∈ M0 − πM0 .
The index is well defined, because a representative a for ea can always be found
in M0 − πM0 , and it is then uniquely determined up to multiplication by a unit.
Here and below we agree that Mr+1 = πM0 .
Definition 2.2. Let τ be as above, and assume that the representatives of A lie
in M0 − πM0 . Let I(τ ) be the ideal in Ẽ generated by δ(ea0 ∧ · · · ∧ eam ) for any
set {a0 , . . . , am } ⊂ A ∩ (Mi − Mi+1 ) which is linearly dependent modulo Mi+1
(0 ≤ i ≤ r).
We call I(τ ) the ideal of relations determined by τ, or in short the ideal of
τ -relations. It is independent of the distinguished vertex of τ.
Definition 2.3. Let
(2.8) Ã(τ ) = Ẽ/I(τ ), and A(τ ) = (E + I(τ ))/I(τ ) ' E/E ∩ I(τ ).
Since I(τ ) is homogeneous and δ-invariant, these are graded algebras, and there
is an exact sequence
δ
(2.9) 0 → A(τ ) → Ã(τ ) → A(τ )[1] → 0.
At the risk of a slight confusion, we shall keep the notation ea also for the image
of ea in Ã(τ ). Any ea supplies a splitting Ã(τ ) ← A(τ )[1], ea ∧ x ¾ x.
10 EHUD DE SHALIT
Proposition 2.1. (i) Ã(τ )k = 0 for k > d + 1 and A(τ )k = 0 for k > d.
(ii) A(τ ) and Ã(τ ) are generated by the elements in degree 1.
(iii) A(τ ) and Ã(τ ) are finite dimensional over K.
Proof. Part (ii) is clear because it holds in Ẽ and E. As for parts (i) and (iii) we
shall soon see that Ã(τ ) (and similarly A(τ )) is a quotient of Ã(v) for any vertex
v of τ, so it is enough to prove the claim for τ = v = [M0 ]. If k > d + 1 take any
a0 , . . . , ak in A, and normalize them to lie in M0 − πM0 . Since {a1 , . . . , ak } are
linearly dependent in VK , hence clearly modulo πM0 , x = ea1 ∧ · · · ∧ eak satisfies
that δ(x) and δ(ea0 ∧ x) ∈ I(v). It follows that x = δ(ea0 ∧ x) + ea0 ∧ δ(x) ∈ I(v)
as well. Thus I(v) contains Ẽ k . This proves (i).
To prove (iii) observe that it is enough to prove that Ã(v)1 is finite dimensional.
We may assume that A ⊂ M0 − πM0 . Let a range over representatives of A in
P(M0 /πM0 ). Then for every b ∈ A we can find a scalar λ and an a such that
b ≡ λamodπM0 , hence ea − eb ∈ I(v). It follows that the ea span Ã(v)1 , so
q d+1 − 1
(2.10) dim Ã(v)1 ≤
q−1
is finite.
2.2. The restriction map. If τ ≤ σ, then I(τ ) ⊆ I(σ), so we get a restriction

map
(2.11) rστ : Ã(τ ) → Ã(σ)
and similarly for A. Moreover, the following holds.
Proposition 2.2. If τ and σ are contiguous, so that τ ∪ σ is a cell, then
(2.12) I(τ ∪ σ) = I(τ ) + I(σ).
P
Proof. Equivalently, we should prove that I(τ ) = I(vi ) if τ = (v0 , . . . , vr ). Sup-
pose that M0 ⊃ · · · ⊃ Mr ⊃ πM0 ([Mi ] = vi ) is a flag representing τ, and
{a0 , . . . , am } ⊂ Mi − Mi+1 are linearly dependent modulo Mi+1 (0 ≤ i ≤ r). Then
{a0 , . . . , am }P⊂ π −1 Mi+1 − Mi+1 , hence δ(ea0 ∧ · · · ∧ eam ) ∈ I(vi+1 ). This proves
that I(τ ) ⊆ I(vi ).
Conversely, pick a typical generator δ(ea0 ∧ · · · ∧ eam ) of I(v0 ) (say), where
{a0 , . . . , am } ⊂ M0 − πM0 are linearly dependent modulo πM0 . We want to show
that it belongs to I(τ ). By (2.4) we may assume that {a0 , . . . , am } is a minimal such
set - no proper subset of it is dependent modulo πM0 . Let i be the largest index
such that {a0 , . . . , am } are not contained in Mi+1 . After rearranging the ai we may
assume that {a0 , . . . , an } are contained in Mi − Mi+1 and the rest lie in Mi+1 . We
then must have a non-trivial linear dependence between {a0 , . . . , an } modulo Mi+1 ,
hence δ(ea0 ∧ · · · ∧ ean ) lies in I(τ ). By (2.4) again, so does δ(ea0 ∧ · · · ∧ eam ).
The algebras Ã(τ ) and A(τ ) form local systems on T (see Section 4.1). They
are graded, and the restriction maps are surjective and preserve the grading.
Corollary 2.3. The map
(2.13) A(τ ∩ σ) → A(τ ) ×A(τ ∪σ) A(σ)
is surjective.
Proof. Given ατ and ασ in Ẽ which are congruent modulo I(τ ∪ σ) = I(τ ) + I(σ),
we can find ετ ∈ I(τ ) and εσ ∈ I(σ) such that α = ατ − ετ = ασ − εσ . Then
αmodI(τ ∩ σ) maps to (ατ modI(τ ), ασ modI(σ)), and this proves the proposition
for the Ã. A similar argument proves it for A.
It is not true that the map in the corollary is an isomorphism, for it is not true
in general that I(τ ∩ σ) = I(τ ) ∩ I(σ). Consider for example the situation where
M0 ⊃ M1 ⊃ M2 ⊃ πM0 , and a, b are elements of M0 − M1 , which are congruent
modulo M2 but not modulo πM0 . Then letting τ = (M0 ⊃ M1 ⊃ πM0 ) and
σ = (M0 ⊃ M2 ⊃ πM0 ) we have v = τ ∩ σ = [M0 ]. Now ea − eb ∈ / I(v) but
ea − eb ∈ I(τ ) ∩ I(σ).
2.3. The linear functionals (σ, −). Let 0 ≤ k ≤ d. To simplify the notation
write S for the ordered k + 1 -tuple (a0 , . . . , ak ), and eS = ea0 ∧ · · · ∧ eak . For any
σ ∈ Tbk consider
(2.14) π = (ισ (a0 ), . . . , ισ (ak )).
Define (σ̃, eS ) = sgn(π) if π is a permutation of (0, 1, . . . , k). Define (σ̃, eS ) = 0
otherwise, and extend the definition by linearity to Ẽ k+1 .
Lemma 2.4. Let τ ∈ Tr be an r-cell contiguous to σ. Then the linear functional
(σ̃, −) annihilates E k+1 + I k+1 (τ ), and therefore factors through Ak (τ ).
P
Proof. Since I(τ ) = v∈τ I(v), it is enough to consider the case where τ = v =
[M0 ] is a vertex contiguous to σ. Let σ = (L0 ⊃ · · · ⊃ Lk ⊃ πL0 ). We may
assume that M0 ⊇ L0 ⊃ Lk ⊃ πM0 ⊇ πL0 , for it is clear that a change in the
distinguished vertex of σ results only in a change of sign in (σ̃, −). For any set
{a0 , . . . , am } ⊂ A ∩ (M0 − πM0 ), linearly dependent modulo πM0 (m ≤ k + 1), and
arbitrary {am+1 , . . . , ak+1 }, we have to show that
m
X
(2.15) (σ̃, (−1)i ea0 ∧ · · · ∧ ebai ∧ · · · ∧ eak+1 ) = 0.
i=0
Observe that the ισ (ai ), 0 ≤ i ≤ m can not be distinct, for if they were distinct, the
ai could not be dependent modulo πM0 . Assume therefore that ισ (a0 ) = ισ (a1 ).
Then the first two summands in the expression above cancel each other, and the
rest vanish. This proves that I k+1 (τ ) is annihilated by (σ̃, −).
Now let {a0 , . . . , ak+1 } be arbitrary. We claim that
k+1
X
(2.16) (σ̃, (−1)i ea0 ∧ · · · ∧ ebai ∧ · · · ∧ eak+1 ) = 0.
i=0
By the pigeonhole principle we may assume, rearranging the ai , that ισ (a0 ) =

ισ (a1 ). As before, the first two terms will cancel each other, and the rest will
contribute null. This proves that E k+1 is annihilated.
Finally, δ induces an isomorphism Ẽ k+1 /(E k+1 + I k+1 (σ)) ' E k /(E k ∩ I k (σ)) =
k
A (σ).
Definition 2.4. Let σ ∈ Tbk . For e ∈ E k , choose ẽ ∈ Ẽ k+1 with δ(ẽ) = e (e.g.
ẽ = ea ∧ e for some a). Define
(2.17) (σ, e) = (σ̃, ẽ).
12 EHUD DE SHALIT
The lemma guarantees that (σ, −) is well-defined, and that if σ ∈ T̂k (τ ), it defines
a linear functional on Ak (τ ), for which we still use the same notation.
Definition 2.5. Let e ∈ E k . Define
(2.18) ce : Tbk → K
to be the map ce (σ) = (σ, e). We call ce the global cochain associated with e.
Similarly, if we restrict ce to T̂k (τ ), it depends only on the image of e in Ak (τ ),
and we call it the local cochain associated with e at τ.
2.4. The broken circuit theorem. The broken circuit theorem for complex hy-
perplane arrangements [O-T, theorem 3.55] is a theorem which gives an explicit
basis for the Orlik-Solomon algebra. We shall now prove a similar theorem for the
algebras Ã(τ ) and A(τ ). For that we have to assume first that A is finite. The case
of infinite A, and in particular the case where A consists of all the hyperplanes in
VK∗ , can be deduced from it by a simple limit argument.
Fix τ , and assume that
(2.19) τ = (M0 ⊃ M1 ⊃ · · · ⊃ Mr ⊃ πM0 ).
We may assume that the lines in A are represented by a ∈ M0 − πM0 . Let
(2.20) Al = Al (τ ) = {a ∈ A; ιτ (a) = l} = A∩(Ml − Ml+1 ),
(0 ≤ l ≤ r, Mr+1 = πM0 ), and fix a linear ordering ≺ of A such that max Al+1 ≺
min Al . We call such an ordering compatible with τ . Notice that if τ ≤ σ and
they both share the same distingushed vertex [M0 ], the partition of A determined
by σ is a refinement of the partition determined by τ, and therefore a linear ordering
which is compatible with σ will automatically be compatible with τ .
If we ignore the distinguished vertex of τ, we still get a cyclic ordering of the
Al , and a linear ordering within each of them.
If
(2.21) S = (a0 Â a1 Â · · · Â ak )
is a decreasing sequence of elements of A, we shall call S τ -special if when we
divide S into blocks Sl = S ∩ Al , and write
(2.22) Sl = (ail Â · · · Â ail+1 −1 )
(meaning that Sl = ∅ if il = il+1 , i0 = 0 and ir+1 = k + 1), then for every
il ≤ j < il+1 ,
(2.23) aj = max(A ∩ haj , . . . , ail+1 −1 , Ml+1 i).
Here h·i denotes the span over OK . Notice that if a appears in a τ -special sequence
S, and ιτ (a) = l, then in particular a is the maximal element of A ∩ ha, Ml+1 i, and
that if S is τ -special then Sl is linearly independent modulo Ml+1 . In particular
the length of Sl is at most el = dim Ml /Ml+1 , and k ≤ d.
If τ 0 is obtained from τ by a cyclic permutation of the vertices, there is a new
ordering of A compatible with τ 0 , which induces the same cyclic ordering as before.
Simply permute the blocks Al cyclically, and maintain the linear ordering within
each block. In particular S is τ -special if and only if S 0 , the same set in the new
ordering, is τ 0 -special. Thus the notion of τ -special is, in this sense, independent of
the distinguished vertex. Let S(τ ) be the collection of τ -special sequences.
Theorem 2.5. The set {eS ; S ∈ S(τ )} is a basis of Ã(τ ).
Proof. Fix 0 ≤ k ≤ d, and we shall show that the eS for S ∈ Sk+1 (τ ), the τ -
special sequences of length k + 1, form a basis of Ãk+1 (τ ). Introduce a lexicographic
ordering Â on the decreasing sequences of length k + 1 : If S is as above and
T = (b0 Â · · · Â bk ) say that S Â T if a0 = b0 , . . . , aj−1 = bj−1 , but aj Â bj .
Clearly Ãk+1 (τ ) is spanned by the eS for decreasing S of length k + 1. Let S be
a decreasing k + 1-tuple, and assume that it is not τ -special. Then for some l, and
some il ≤ j < il+1 ,
(2.24) a0j = max(A ∩ haj , . . . , ail+1 −1 , Ml+1 i) Â aj .
Thus {a0j , aj , . . . , ail+1 −1 } ⊂ Ml − Ml+1 , and are linearly dependent modulo Ml+1 .
This implies that
(2.25) δ(ea0j ∧ eaj ∧ · · · ∧ eail+1 −1 ) ∈ I(τ ).
In Ãk+1 (τ ) we may therefore replace the string eaj ∧ · · · ∧ eail+1 −1 by a linear
combination of similar strings beginning with ea0j . This means that we have replaced
eS by a linear combination of eT with T Â S. Repeating the process we see that
the τ -special eS already span Ãk+1 (τ ).
Let µ be the maximal element in A. Then we have a splitting

(2.26) Ãk+1 (τ ) = Ak+1 (τ ) ⊕ eµ ∧ Ak (τ ).
Since eµ ∧ x = eµ ∧ δ(eµ ∧ x) we see that eµ ∧ Ak (τ ) = eµ ∧ Ãk (τ ), and is therefore
spanned by the eµ ∧ eS , S ∈ Sk (τ ). But eµ ∧ eS = e(µ,S) and either µ is the maximal
element of S, in which case eµ ∧ eS = 0, or (µ, S) ∈ Sk+1 (τ ). Write Sk+1 (τ )0 for
the τ -special S with µ > max(S), and Sk+1 (τ )00 for those with µ = max(S), and
note that
Sk+1 (τ ) = Sk+1 (τ )0 ∪ Sk+1 (τ )00 (disjoint union), and
(2.27) Sk+1 (τ )00 = {eµ ∧ eS ; S ∈ Sk (τ )0 } .

We shall prove that Sk+1 (τ )00 is a basis of eµ ∧Ak (τ ). This will show that #Sk (τ )0 =
dim Ak (τ ). Using this with k + 1 instead of k we shall finally get
(2.28) dim Ãk+1 (τ ) = #Sk+1 (τ )0 + #Sk+1 (τ )00 = #Sk+1 (τ ).
Thus Sk+1 (τ ) is a basis.
To prove that Sk+1 (τ )00 is a basis of eµ ∧ Ak (τ ) it is enough to prove that
they are independent. We need a certain construction. We associate with each
T = (µ = b0 Â · · · Â bk ) ∈ Sk+1 (τ )00 a k + 1-cell σT ∈ Tbk (τ ) as follows. If
Tl = T ∩ Al = (bil Â · · · Â bil+1 −1 ) 6= ∅, and il ≤ j < il+1 , let
(2.29) Lj = hbj , . . . , bil+1 −1 , Ml+1 i,
except for j = 0, where we put L0 = M0 . The fact that T is τ -special guarantees
that the Lj are in fact distinct and decreasing, and
(2.30) σT = (L0 ⊃ · · · ⊃ Lk ⊃ πL0 )
is interlaceable with τ , because Ml ⊇ Lj ⊃ Ml+1 . Thus σT and τ are contiguous.
The linear independence of Sk+1 (τ )00 will be a consequence of the following lemma.
Lemma 2.6. If S Â T are in Sk+1 (τ )00 , then (σ̃T , eS ) = 0, (σ̃S , eS ) = 1.

14 EHUD DE SHALIT
Proof. Suppose S = (a0 Â · · · Â ak ) and T = (b0 Â · · · Â bk ), a0 = b0 = µ, . . . ,

aj−1 = bj−1 , but aj Â bj (j ≥ 1). Suppose also bj ∈ Ml − Ml+1 (0 ≤ l ≤ r). For
0≤i≤j
®
(2.31) ai Â bj = max(A ∩ bj , . . . , bil+1 −1 , Ml+1 ),
so in particular ai ∈ / Lj . On the other hand ai ∈ M0 = L0 . Thus 0 ≤ ισT (ai ) < j
and there must be two a0i s with the same σT -index. This proves that (σ̃T , eS ) = 0,
and (σ̃S , eS ) = 1 is clear from the definition.
Corollary 2.7. The collection

(2.32) {(σT , −); T ∈ Sk+1 (τ )00 }
is a basis of the dual space of Ak (τ ).
Proof. This follows from lemma 2.6 and the proof of theorem 2.5.
2.5. The case A = P(VK ). In this case A is G-invariant. Every g ∈ G induces a

map g : Ẽ → Ẽ, g(ea ) = ega , and g(I(τ )) = I(gτ ). We therefore have maps
(2.33) gτ : A(τ ) → A(gτ )
(and similarly on Ã ) which commute with the restriction maps. They satisfy the
usual formula
0
(2.34) ggτ ◦ gτ = (g 0 g)τ .
The linear functionals λσ = (σ, −) satisfy
(2.35) (gσ, ge) = (σ, e)
for every σ ∈ T̂k and e ∈ E k , which can be phrased also as

(2.36) g(λσ ) = λgσ .
2.6. Open questions. There are several interesting questions regarding the ideals
I(σ). For example, does the distributive law
(2.37) (I(σ) + I(ρ)) ∩ I(τ ) = I(σ) ∩ I(τ ) + I(ρ) ∩ I(τ )
hold whenever σ, τ, and ρ are contiguous ? What is the meaning of I(σ) + I(τ )
when σ and τ are not contiguous ? When are I(σ) and I(τ ) co-maximal ? What
is the intersection of all the I(σ) ?
3. Ak (τ ) as a space of local harmonic cochains

In this section we explore averaging properties of the cochains ce , and derive
an alternative description of Ak (τ ), as a space of certain harmonic cochains c :
Tbk (τ ) → K. For τ a vertex and A = P(VK ), we shall see that the image consists of
all the local harmonic cochains.
3.1. Harmonic cochains. A map c : Tbk → K will be called harmonic if it

satisfies the following 4 conditions :
• (Property A). If σ = (v0 , . . . , vk ) ∈ Tbk and σ 0 = (v1 , . . . , vk , v0 ), then
(3.1) c(σ) = (−1)k c(σ 0 ).
• (Property B). Fix σ ∈ Tbk−1 , fix a type t of pointed k-cells, and consider
B = B(σ, t) = {σ 0 ∈ T̂k ; σ ≤ σ 0 , σ and σ 0 share the same distinguished vertex,
and t(σ 0 ) = t}. Then
X
(3.2) c(σ 0 ) = 0.
σ 0 ∈B
• (Property C). Let k ≥ 1. Fix σ ∈ T ck and an index 0 ≤ j ≤ k. Let C = C(σ, j)

be the collection of all σ 0 ∈ Tck for which L0 = Li if i 6= j, L0 ⊆ Lj and
i j
0
dim Lj /Lj+1 = 1. Then
X
(3.3) c(σ) = c(σ 0 ).
σ 0 ∈C
• (Property D). Let σ = (v0 , v1 , . . . , vk+1 ) ∈ T[

k+1 . Let σj = (v0 , . . . , vbj , . . . , vk+1 ).
Then
k+1
X
(3.4) (−1)j c(σj ) = 0.
j=0
Remarks. (i) When k = d properties C and D are void, and in Property B only
one type exists. When k = 0, only D is a non-empty condition. When d = 1 the
story is much older, see [Se].
(ii) Harmonic analysis on Γ\T , for a discrete group Γ with a finite quotient
Γ\T , is developed in [G], following work of Hodge and Eckman. Using Garland’s
notation, condition D is “d = 0”, condition B is “δ = 0, refined by type”, C is a
condition that allows one to switch types, and A is the orientation condition.
(iii) For each lattice L in VK let L∗ be the dual lattice in VK∗ . This establishes
an order-reversing bijection between lattices in VK∗ and VK . It leads to a bijection
σ ←→ σ ∗ between the k-cells of T , and the k-cells of the Bruhat-Tits building
T ∗ of VK∗ , which is an isomorphism of buildings (σ ≤ τ if and only if σ ∗ ≤ τ ∗ ).
However, the type t(σ ∗ ) is equal to t(σ) read backwards. Properties A, B and D are
invariant under σ 7→ σ ∗ , but property C is not. One may call a harmonic cochain
on T ∗ , when transported to T , co-harmonic (the distinction between harmonic
and co-harmonic has no parallel in the classical theory). In general harmonicity
does not imply co-harmonicity. This shows in particular that condition C does not
follow from A, B and D. Since it is intuitively clear that A, B and D are logically
independent of each other, it seems that all four conditions are in fact needed to
characterize harmonicity.
(iv) The origin for the refinement “by type” of the usual “d = 0, δ = 0” condition,
is that the action of G on T is not transitive in degrees 0 < k < d. There are reasons
to view this refinement as the refinement of the usual “d = 0, d∗ = 0” conditions
in Riemannian geometry by the “∂ = 0, ∂¯ = 0, ∂ ∗ = 0, ∂¯∗ = 0” conditions in
Kähler geometry. The ultimate justification for the definition is that it leads to
nice results. But there may be other, equivalent formulations.
16 EHUD DE SHALIT
k
Definition 3.1. Denote by Char the space of harmonic cochains on T̂k . Denote
k
by Char (τ ) the space of local harmonic cochains at τ. These are the maps c :
T̂k (τ ) → K which satisfy properties A − D “when applicable”, i.e. when, in (B)
and (C) we have to add the requirement that the collections B and C are contained
in T̂k (τ ).
k k k
Remark. The good notion is that of Char , and the map Char → Char (τ ) need
not be surjective. There might be relations involving only T̂k (τ ), forced on a global
harmonic cochain by the rather simple conditions A − D, but which are not formal
consequences of the same conditions restricted to collections of cells lying entirely
within T̂k (τ ).
It therefore comes as a surprise that to this remark there is an exception. When
k k k
τ = v is a vertex, Char → Char (v) is surjective, and the definition of Char (v) indeed
captures all the combinatorial restrictions imposed on a global cochain, when we
localize it (see corollary 3.4 below).
3.2. Harmonicity of the cochains ce .

Theorem 3.1. The cochains ce are harmonic.
Proof. Fix ẽ ∈ Ẽ k+1 with δ(ẽ) = e. Property A is clear from the definition. We
proceed by checking the remaining three properties.
Property B : It is enough to verify the following special case of B, from which

all the other cases follow in combination with property C. Fix 1 ≤ j ≤ k, assume
(3.5) σ = (L0 ⊃ · · · ⊃ Lj−1 ⊃ Lj+1 · ·· ⊃ Lk ⊃ Lk+1 = πL0 )
ck with
and dim Lj−1 /Lj+1 ≥ 2. Let B = B(σ, j) be the collection of σ 0 ∈ T
(3.6) σ 0 = (L0 ⊃ · · · ⊃ Lj−1 ⊃ L0j ⊃ Lj+1 · ·· ⊃ Lk ⊃ Lk+1 = πL0 )
and dim L0j /Lj+1 = 1. Then

X
(3.7) (σ̃ 0 , ẽ) = 0.
σ 0 ∈B
We may assume that ẽ = ea0 ∧ ea1 ∧ · · · ∧ eak . Write ισ (ai ) = m(i). (Here by m(i)
we understand the last l such that ai ∈ Ll . Thus m(i) 6= j.) If m(i) 6= j − 1,
then for every σ 0 ∈ B, ισ0 (ai ) = m(i) as well. On the other hand, if m(i) = j − 1,
then ισ0 (ai ) = j − 1 or j. If for some l, 0 ≤ l ≤ j − 2 or j + 1 ≤ l ≤ k, either
l = m(i) for two distinct i0 s, or for no i at all, then the same holds for every σ 0 ∈ B,
and every (σ̃ 0 , ẽ) = 0. We may therefore assume that the m(i) cover the indices
{0, . . . , j − 1, j + 1, . . . , k} with only j − 1 being repeated twice, and all the rest
once. Rearranging the ai we may assume that m(i) = i for i 6= j, and m(j) = j − 1.
Now there are two possibilities. If aj−1 and aj represent the same line modulo
Lj+1 , then for every σ 0 ∈ B they will together lie in, or out of, L0j . Thus every
(σ̃ 0 , ẽ) = 0. If, on the other hand, aj and aj−1 are independent modulo Lj+1 ,
then there will be precisely two σ 0 with (σ̃ 0 , ẽ) 6= 0, namely those corresponding to
L0j = hLj+1 , aj−1 i and to L0j = hLj+1 , aj i . Their contribution to the sum will be
−1 and +1 respectively, so will cancel each other.
Property C : Fix σ ∈ Tck (k ≥ 1), and an index 0 ≤ j ≤ k. Let C = C(σ, j) be the

c
collection of all σ ∈ Tk for which L0i = Li if i 6= j, L0j ⊆ Lj and dim L0j /Lj+1 = 1.
0
We have to show that for ẽ = ea0 ∧ ea1 ∧ · · · ∧ eak

X
(3.8) (σ̃, ẽ) = (σ̃ 0 , ẽ).
σ 0 ∈C
Note first that if ισ (ai ) = l 6= j, then for every σ 0 ∈ C also ισ0 (ai ) = l, while
if ισ (ai ) = j, then ισ0 (ai ) = j or j − 1modk + 1. (Note that since k ≥ 1, j and
j − 1modk + 1 are distinct. The arguments below fail for k = 0). Distinguish three
cases.
(a) If (σ̃, ẽ) = 1, we may assume, after renaming the ai , that ισ (ai ) = i for each
i. We then have ισ0 (ai ) = i for i 6= j, and any σ 0 . We also have ισ0 (aj ) = j for the
unique σ 0 ∈ C where L0j = hLj+1 , aj i, hence (σ̃ 0 , ẽ) = 1 for that σ 0 . For any other
σ 0 ∈ C we have ισ0 (aj ) = j − 1mod k + 1, hence (σ̃ 0 , ẽ) = 0.
(b) If (σ̃, ẽ) = 0 and (b1) there is an l 6= j with more than one ai with ισ (ai ) = l,
or (b2) there is an l 6= j − 1mod k + 1 with no ai with ισ (ai ) = l, then the same
holds true for every σ 0 ∈ C, so each (σ̃ 0 , ẽ) = 0.
(c) If (σ̃, ẽ) = 0 and we are not in case (b), then after rearranging the ai we may
assume that ισ (ai ) = i for i 6= j − 1mod k + 1 while ισ (aj−1modk+1 ) = j. As in
the previous proposition, we have two possibilities. If aj−1modk+1 and aj span the
same line in Lj /Lj+1 , then they will lie together in, or out of, any L0j , and each
(σ̃ 0 , ẽ) = 0. If they are independent modulo Lj+1 , then precisely two σ 0 will pair
non-trivially with ẽ, but their contributions will cancel each other.
Property D : Consider ẽ = ea0 ∧ · · · ∧ eak ,

(3.9) σ = (L0 ⊃ L1 ⊃ · · · ⊃ Lk+1 ⊃ πL0 ),
and we shall show that
k+1
X
(−1)j (σ̃j , ẽ) = 0.
j=0
Choosing the ai to lie in L0 − πL0 we write mi = ισ (ai ) for their indices with
respect to σ. First note that if ισ (a) = ισ (b), then for every σj also ισj (a) = ισj (b).
(The only case where some care has to be taken is j = 0, where if a, hence b, lie in
L0 − L1 , they have to be replaced by πa and πb to get representatives in L1 − πL1 .)
Thus if the mi are not all distinct, the alternating sum of the (σ̃j , ẽ) vanishes.
If the mi are distinct we may assume, because of property (A), and renaming
the a0i s if necessary, that mi = i + 1 for every 0 ≤ i ≤ k. If j ≥ 2 then we will have
(σ̃j , ẽ) = 0. For j = 0 and j = 1 we shall have (σ̃j , ẽ) = 1, and we are done.
3.3. Injectivity of the local cochain homomorphism. Now fix τ = (M0 ⊃

· · · ⊃ Mr ⊃ Mr+1 = πM0 ) (0 ≤ r). Since the global cochains ce are harmonic, so
are their restrictions to T̂k (τ ), and we get a homomorphism
(3.10) c : Ak (τ ) → Char
k
(τ )
sending e to ce .
Theorem 3.2. The local cochain homomorphism c is injective.
18 EHUD DE SHALIT
Proof. Given the results of Section 2, this is now an easy consequence of corollary
00
2.7, because the σT constructed there (for T ∈ Sk+1 (τ )) all belong to T̂k (τ ), and
k
the (σT , −) span the dual space of A (τ ).
3.4. The isomorphism Ak (v) ' Chark

(v), when A = P(VK ). To get surjectivity
statements about c, we must assume that the collection A ⊂ P(VK ), with which we
started, is large.
Theorem 3.3. Let v0 = [M0 ] be a vertex of T , and assume that A contains a
representative for each line in P(M0 /πM0 ) (for example, take A = P(VK )). Then
c induces an isomorphism Ak (v0 ) ' Char
k
(v0 ).
Proof. We may replace the set A, when dealing with Ak (v0 ), by a finite subset
containing precisely one representative for every line in P(M0 /πM0 ). Fix such an
A, and order it linearly once and for all as in Section 2.4. Let µ = max A.
We have to establish the surjectivity of Ak (v0 ) → Char k
(v0 ). We view cells
k
from T̂k (v0 ) as linear functionals on Char (v0 ) in a natural way. They clearly span
k
Char (v0 )∗ . We shall show that {σT ; T ∈ Sk+100
(v0 )} is still is a spanning set, and
therefore, by corollary 2.7, the map Char (v0 )∗ → Ak (v0 )∗ is injective, which is the
k
desired result.
k
Let us use the phrase equivalent to mean equal, as linear functionals on Char (v0 ).
00
Recall that Sk+1 (v0 ) consists of S = (a0 Â a1 Â · · · Â ak ) where a0 = µ and
aj = max(A ∩ hπM0 , ak , . . . , aj i). Let σ ∈ Tbk (v0 ). If v0 ∈ / σ then σ ∪ {v0 } is a
k + 1-cell, and property (D) shows that σ is equivalent to a linear combination
of σ 0 s containing v0 . Property (A) allows us to assume that σ = (v0 , . . . , vk ), and
repeated application of property (C) allows us then to assume that the type of σ is
(d + 1 − k, 1, 1, . . . , 1), in other words that dim Li /πL0 = k + 1 − i if i ≥ 1. Call
such a type minimal. We shall now show, that by repeated use of property (B),
every such σ of minimal type is equivalent to a linear combination of σT (note that
the σT are of minimal type as well).
If σ = (L0 ⊃ · · · ⊃ Lk ⊃ πL0 ), L0 = M0 , recall that we wrote Vi = Li /πL0 , and
A is a system of representatives in L0 − πL0 for P(V0 ). To simplify the notation
write max(Vi ) for max(A ∩ Li ). We shall prove, by decreasing induction on j, that
every σ of minimal type with
(3.11) max(Vk ) ≺ · · · ≺ max(Vj+1 ) ¹ max(Vj ) ¹ · · · ¹ max(V0 )
is equivalent to a linear combination of σ 0 of minimal type with

(3.12) max(Vk ) ≺ · · · ≺ max(Vj+1 ) ≺ max(Vj ) ¹ · · · ¹ max(V0 )
(k ≥ j ≥ 0), and that for the indices below j, wherever we had a strict inequality in
the first string of inequalities, we can maintain it in the second. First let us remark
that the last condition, although important for the induction procedure, will hold
automatically, because we shall not alter the spaces Vi for i ≤ j in the j th step of
the induction.
If k = j there is nothing to prove. Assume that k > j ≥ 0, σ is a k-cell of minimal
type as above, and max(Vj+1 ) = max(Vj ). Consider σ0 = (L0 ⊃ · · · ⊃ Lj ⊃ Lj+2 ⊃
· · · ⊃ Lk+1 = πL0 ). We shall apply the condition (B) with B(σ0 , j +1) (see the proof
of theorem 3.1 for the notation). Since dim Lj /Lj+2 = 2 and max Vj = max Vj+1 Â
max Vj+2 , when we consider all the possibilities for Lj ⊃ L0j+1 ⊃ Lj+2 , every other
L0j+1 6= Lj+1 will produce a σ 0 with
0
(3.13) max(Vk ) ≺ · · · ≺ max(Vj+2 ) ¹ max(Vj+1 ) ≺ max(Vj ) ¹ · · · ¹ max(V0 ).
Our assumption implies that σ is equivalent to the negative of the sum of all the
other σ 0 ∈ B(σ0 , j + 1). The induction hypothesis allows us now to replace each of
these σ 0 by pointed k-cells of minimal type with strict inequalities all the way down
to index j + 1. Furthermore, we may do so without spoiling the strict inequality
between indices j + 1 and j, as desired.
We therefore could have assumed, to begin with, that σ = (L0 ⊃ ··· ⊃ Lk ⊃ πL0 ),
L0 = M0 , dim Li /πL0 = k + 1 − i for i ≥ 1, and when we put bi = max(A ∩ Li ), we
00
have b0 = µ Â b1 Â · · · Â bk . Calling this sequence T, we find that T ∈ Sk+1 (v0 ),
and σ = σT .
k k
Corollary 3.4. The map Char → Char (v0 ) is surjective.
Proof. Every element of Ak (v0 ) lifts to an element of E k .
As we shall see later, the image of Ak (τ ) in Char
k
(τ ), for a general simplex τ,
consists of the elements which admit an extension to a global harmonic cochain in
k
Char .
4. Acyclicity (d ≤ 2)
4.1. Cohomology of local systems. Quite generally, let T be a simplicial com-
plex, which is assumed for simplicity to be finite-dimensional and locally finite. We
write Tk for the k-simplices of T , and Tbk for the ordered k + 1-tuples (v0 , . . . , vk )
of distinct vertices such that {v0 , . . . , vk } ∈ Tk .
A (cohomological) local system A of abelian groups (or algebras etc.) on T
is the assignment of an abelian group A(τ ) to every simplex τ of T , and homomor-
phisms (restriction maps) rστ : A(τ ) → A(σ) to every face inclusion τ ≤ σ, such
that rρσ ◦ rστ = rρτ whenever τ ≤ σ ≤ ρ, and rττ is the identity.
Given a local system A, define the group of alternating p-cochains (0 ≤ p ≤
dim T ) C p (T , A) to consist of the maps c assigning to each τ ∈ Tbp an element
c(τ ) ∈ A(τ ) such that
(4.1) c(τ 0 ) = sgn(π)c(τ )
if τ 0 is obtained from τ by applying the permutation π to the vertices. Define
(4.2) ∂ : C p (T , A) → C p+1 (T , A)
by the rule
p+1
X
(4.3) ∂c(τ ) = (−1)i rττi (c(τi ))
i=0
if τ = (v0 , . . . , vp+1 ) and τi = (v0 , . . . , vbi , . . . , vp+1 ). Then (C · (T , A), ∂) is a complex

(∂ 2 = 0), and its cohomology groups are denoted H p (T , A). We call A acyclic if
H p (T , A) = 0 for p > 0.
It is well known that if A is locally constant (all the restriction maps are iso-
morphisms), then if T is contractible (as is the case with the Bruhat-Tits build-
ing), A is acyclic. In general, even if T is contractible, A need not be acyclic. If
20 EHUD DE SHALIT
the restriction maps are not assumed to be surjective, one has non-acyclic local
systems already on the 1-simplex. Even if we require the restriction maps to be
surjective, one can take for example T as the triangle with vertices u, v, w, and
A(u) = A(v) = A(w) = A(uv) = A(uw) = A(vw) = A, but A(uvw) = A/a for
some nontrivial ideal a. The 1-cochain c given by c(uv) = c(vw) = 0, c(uw) = a ∈ a
is a 1-cocycle, but not a coboundary if a 6= 0. One may argue that what goes wrong
here is that A(u) → A(uv) ×A(uvw) A(uw) is not surjective, but counterexamples
can be found under this assumption too, and so on.
We are concerned with the question whether H p (T , A) = 0 for p > 0 for the local
system which we have constructed on the Bruhat-Tits building T . This breaks down
to the statements
(4.4) H p (T , Aq ) = 0
for 0 ≤ q ≤ d. When q = 0 the local system A0 is constant (and equal to K), so
the claim is true by the contractibility of T . Note that the short exact sequence
q
(4.5) 0 → Aq → Ã → Aq−1 → 0
is split, and it is enough to prove the acyclicity of Ãq .
4.2. The case d = 1. Here T is the q + 1 regular tree. Fix a vertex v0 , and a
1-cochain c (every 1-cochain is a cocycle). We shall modify it by coboundaries
step-by-step. At the nth step we shall modify c by a coboundary of a 0-cochain
supported on the vertices at distance n from v0 , and the resulting cochain will be
0 on all the edges up to these vertices. This will show that c is a coboundary. Now
suppose we reached step n. Every vertex v at distance n has a unique vertex u at
distance n − 1 from v0 contiguous to it. Since A(v) → A(uv) is surjective we can
perform the nth step and continue inductively.
4.3. The case d = 2. Here we already have to take care of H 1 and H 2 . Fix a vertex
v0 as before. Let Γ0 (n) denote the set of vertices at distance n (in the metric ρ
introduced in Section 1.2) from v0 . The edges at distance n, denoted Γ1 (n), are of
two kinds. If one end is at distance n − 1 and the second at distance n we say
that the edge points to v0 , and we denote the set of these edges by Γ1 (n, 1). If both
ends u and v are at distance n we say that (u, v) is parallel, and we denote the
set of edges of this type by Γ1 (n, 2). Similarly the triangles (u, v, w) at distance n,
denoted Γ2 (n), will either have one vertex at distance n, and two at distance n − 1,
in which case we say that it points outwards, and call it of type Γ2 (n, 1); or one
vertex will be at distance n − 1 and two at distance n, and we say that the triangle
points inwards, or that it is of type Γ2 (n, 2).
4.4. Vanishing of H 1 . Let c be a 1-cocycle with values in Ã. We shall modify c

step-by-step as we did in the 1-dimensional case. In the nth step we shall modify
c by the coboundary of a 0-cochain supported on Γ0 (n), arriving at a 1-cocycle
vanishing on Γ1 (m) for all m ≤ n. The nth step is divided into two substeps: in
step (n, 1) we take care of the edges of type Γ1 (n, 1), and in step (n, 2) we take care
of those of type Γ1 (n, 2).
Assume therefore that c already vanishes on Γ1 (m) for all m ≤ n − 1.
Step (n, 1).

Lemma 4.1. Every v ∈ Γ0 (n) has 1 or 2 contiguous vertices u at distance n − 1

from v0 .
Proof. We may choose a basis {e0 , e1 , e2 } and lattices M and M0 representing v and
v0 such that M0 = he0 , e1 , e2 i, M = he0 , π m e1 , π n e2 i and 0 ≤ m ≤ n. The question
is that of classifying the submodules W of W0 = M0 /M = OK /(π m ) × OK /(π n )
killed by π, and such that W0 /W is killed by π n−1 . If m = n there is a unique such
W, namely (π n−1 )/(π n ) × (π n−1 )/(π n ). If m < n then we must have
(4.6) (0) × (π n−1 )/(π n ) ⊆ W ⊆ (π m−1 )/(π m ) × (π n−1 )/(π n )
so there are precisely two such W 0 s.
If u is unique, by the surjectivity of Ã(v) → Ã(vu) we can assign v a value such
that the modified c will vanish on (vu). If there are two such u0 s call them u1 and u2
and observe (this follows from the proof of the lemma) that - perhaps after switching
u1 and u2 - (v, u1 , u2 ) ∈ Tb2 . Since c(u1 , u2 ) = 0 by the induction hypothesis,
and since ∂c(v, u1 , u2 ) = 0, c(v, u1 ) and c(v, u2 ) restrict to the same element of
Ã(v, u1 , u2 ). According to corollary 2.3 there is an element of Ã(v) restricting to
them both under the respective restriction maps, so again we can assign v a value
such that the modified c will vanish on (v, u1 ) and on (v, u2 ).
This concludes step (n, 1), so we assume that c vanishes now on Γ1 (m) for all m
≤ n − 1, and also on Γ1 (n, 1).
Step (n, 2).

Lemma 4.2. Let (v1 , v2 ) ∈ Γ1 (n, 2) (a parallel edge). Then there is a unique
u ∈ Γ0 (n − 1) such that - possibly after switching the vi - (u, v1 , v2 ) ∈ Tb2 .
Proof. We observed in the previous lemma that there are at most two such u0 s.
Furthermore, if there are two such u0 s, say u1 and u2 , they form triangles with
both v1 and v2 , so the four vertices will be contiguous, contradicting d = 2.
Note that v1 and v2 do not play a symmetric role now. We label them so that
u = [L], vi = [Mi ] and L ⊃ M1 ⊃ M2 ⊃ πL. Consider the collection of all the v1
that appear. Each parallel edge will ”belong” to precisely one of them. Also, v1
determines u. This is clear if there is only one vertex from Γ0 (n − 1) contiguous
to v1 , and in the other case, if u 6= u0 ∈ Γ0 (n − 1) is also contiguous to v1 , then
(u, v1 , u0 ) is a triangle, and the cyclic ordering of the vertices is well-defined, so
there can’t be a triangle of the shape (u0 , v1 , v2 ). Thus with each v1 comes a unique
u and we can consider the collection V = V(v1 ) of all the v such that (u, v1 , v) is a
triangle. Say v = [M ] where
(4.7) L ⊃ M1 ⊃ M ⊃ πL
and each inclusion signifies a drop of 1 in dimension. Either none or precisely one
of the v ∈ V lies in Γ0 (n − 1). All the rest lie in Γ0 (n) and make up a list of the
possible v20 s which form a parallel edge (v1 , v2 ). At any rate, no v is another ”v1 ”.
In Step (n, 2) we shall modify c by a coboundary of a 0-cochain supported only
on the v10 s. Note that for every v ∈ V(v1 ), c(v1 , v) may be assumed to be represented
by an element of I(u), since from ∂c(u, v1 , v) = 0, and from the vanishing of c on
Γ1 (n, 1), we get that a representative of c(v1 , v) lies in I(u) + I(v1 ) + I(v). Choose
therefore representatives bv ∈ I(u) of c(v1 , v).
22 EHUD DE SHALIT
Lemma 4.3. The map

Y
(4.8) I(u) → I(u)/(I(u) ∩ I(v1 , v))
v∈V(v1 )
is surjective.
By the lemma we can find an element of Ã(v1 ), represented by an element b of
I(u), which maps to bv modI(v1 , v) for every v ∈ V(v1 ). In particular, modifying c
by the coboundary of the 0-cochain so produced will not damage the vanishing of
c on Γ1 (n, 1) (on (v1 , u) because b ∈ I(u), and if there is a u0 ∈ Γ0 (n − 1) ∩ V(v1 ),
also on (v1 , u0 ) because bu0 ≡ 0modI(v1 , u0 )). It will of course make c vanish also
on the edges in Γ1 (n, 2).
Proof. Let x = δ(ea0 ∧··· ∧eam ) be a typical generator of I(u) where {a0 , . . . , am } =
S ⊆ L−πL and is a minimally dependent set modulo πL. We claim that it lies in all
I(v1 , v) except possibly for one v. In fact, if it does not lie in I(v1 , v), then S must
be contained in M , otherwise a subset of S will be contained in π −1 M − M and
will be dependent modulo M, so our element x will lie in I(v). However, if S ⊂ M
then M is spanned over πL by S (because dim M/πL = 1), M is determined by S,
and so is v.
It follows that the image of x is 0 in all but one factor on the right, and since
these x0 s generate any of the factors, the lemma is proved.
4.5. Vanishing of H 2 . This is easier, and done as before step-by-step. The nth
step is again divided into two: in step (n, 1) we take care of the triangles of type
Γ2 (n, 1), and in step (n, 2) we take care of those of type Γ2 (n, 2).
Let c be a 2-cocycle in Ã. Assume that it already vanishes on Γ2 (m) for all
m < n. Every σ ∈ Γ2 (n, 1) is of the form (v, u1 , u2 ) for some v ∈ Γ0 (n) (and
ui ∈ Γ0 (n − 1)). Furthermore an edge (v, u1 ) ∈ Γ1 (n, 1) belongs to at most one
σ ∈ Γ2 (n, 1) by lemma 4.1. By the surjectivity of Ã(vu1 ) → Ã(vu1 u2 ) we may
assign each (v, u1 ) which appears a value which will make the modified c vanish
on all Γ2 (n, 1). Note that we have modified c by the coboundary of a 1-cochain
supported only on the edges in Γ1 (n, 1) of type (v, u1 ). The edges of type (v, u2 ) or
of type (v, u) when there is a unique u ∈ Γ0 (n − 1) contiguous to v, get the value 0.
In step (n, 2) we assume that c already vanishes on Γ2 (n, 1). The σ ∈ Γ2 (n, 2)
are in one-to-one correspondence with the parallel edges at distance n, i.e. with
Γ1 (n, 2), by lemma 4.2. We may therefore find (by the surjectivity of the restriction
maps) a 1-cochain supported on Γ1 (n, 2) whose coboundary will match c on Γ2 (n, 2),
and will of course change nothing on Γ2 (m) for m < n or on Γ2 (n, 1). Modifying c
by it concludes step (n, 2). We summarize.
Theorem 4.4. Let A be an arbitrary hyperplane arrangement in dimension d ≤ 2.
Then
(4.9) H p (T , Aq ) = 0
for p > 0 and every q.
If A = P(VK ), and without any restriction on d, we always have
(4.10) H 0 (T , Aq ) ∼ q
= Char .
Proof. We have checked the first assertion. As for the second, an element of
H 0 (T , Aq ) is by definition a collection e(v) ∈ Aq (v), for every vertex v, such that
e(v) and e(u) restrict to the same element of Aq (uv) whenever u and v are con-
tiguous. By theorems 3.2 and 3.3 this is the same as a collection of local harmonic
q
cochains c(v) ∈ Char (v) which glue together to give a global cochain c. But such a c
will clearly be harmonic, because each time we have to check one of the properties
A − D we can do so within one T̂k (v). Conversely, each global harmonic cochain
gives rise to a collection of c(v) which agree in the intersections.
In the next section we remove the restriction d ≤ 2, provided A = P(VK ).
5. Acyclicity (in general, A = P(VK ))

5.1. Homology of local systems. There is a dual notion to that of Section 4.1.
Let T once again be an arbitrary finite-dimensional and locally-finite simplicial
complex.
A (homological) local system B of abelian groups (or algebras etc.) on T is
the assignment of an abelian group B(τ ) to every simplex τ of T , and homomor-
phisms rτσ : B(σ) → B(τ ) for every face inclusion τ ≤ σ, such that rτσ ◦ rσρ = rτρ
whenever τ ≤ σ ≤ ρ, and rττ is the identity. The dual of a homological local system
is a cohomological one, and vice versa.
The group of alternating p-chains Cp (T , B) consists of all the finitely supported
maps γ which assign to each τ ∈ T̂p an element γ(τ ) ∈ B(τ ) such that
(5.1) γ(τ 0 ) = sgn(π)γ(τ )
if τ 0 is obtained from τ by applying the permutation π to the vertices. Define
(5.2) ∂ : Cp+1 (T , B) → Cp (T , B)
by the rule
X
(5.3) ∂γ(τ ) = [τ 0 : τ ]γ(τ 0 )
τ0
where τ runs over the collection of p + 1-simplices in Tp+1 such that τ ≤ τ 0 , and
0
in the sum each τ 0 is given an arbitrary orientation, and the incidence number
[τ 0 : τ ] is determined according to the rule
(5.4) [(v0 , . . . , vp+1 ) : (v0 , . . . , v̂i , . . . , vp+1 )] = (−1)i .
Lemma 5.1. Let B be a homological local system of finite dimensional vector
spaces, and B ∗ the dual cohomological local system. Then
(5.5) C p (T , B ∗ ) = Cp (T , B)∗
and the map ∂ in cohomology is the dual of the map ∂ in homology.
Proof. Clear. Note that the dual of a direct sum of vector spaces is the direct
product of their duals, but not vice versa.
Corollary 5.2. Suppose that the complex (C· (T , B), ∂) is acyclic at p > 0. Then
(C · (T , B ∗ ), ∂) is acyclic at p too.
We now recall the all-important acyclicity theorem of Schneider and Stuhler.

Recall that a linear representation Λ of G is called smooth if the stabilizer of
every vector is open, and admissible if, in addition, the invariant subspace of
every open subgroup is finite dimensional. Although smooth representations are
24 EHUD DE SHALIT
usually complex, we shall consider vector spaces over K. The terminology goes
through without any modifications.
If v is a vertex of T we denoted by Bv its stabilizer in G, which is isomorphic to
P GLd+1 (OK ). For n ≥ 1, denote by Uvn ⊂ Bv the principal congruence subgroup
of level π n (this is independent of theSchoice of coordinates). For τ ∈ Tr denote by
Uτn the subgroup of G generated by {Uvn ; v is a vertex of τ }. It is a pro-p group.
When v, u are contiguous vertices (ρ(u, v) ≤ 1), Uv1 ⊂ Bu , so Uv1 ⊂ Bτ if v is a
vertex of τ. Since each Uv1 is clearly normal in Bτ (because it is normal in Bv ) it
lies in the unique(!) maximal normal pro-p subgroup of Bτ , and so does Uτ1 . In fact
Uτ1 is equal to the unique maximal normal pro-p subgroup of Bτ ([S-S], Section 6,
n
Lemma 2). We also have Ugτ = gUτn g −1 for every g ∈ G.
Theorem 5.3 (S-S1). Suppose that Λ is a smooth representation of G which is
generated, as a G-module, by its Uvn -fixed vectors for some vertex v and some
n ≥ 1. Let Λ be the local system
n
(5.6) Λ(τ ) = ΛUτ
with inclusion as restriction maps. Then
(5.7) C· (T , Λ) → Λ → 0
L n
is a resolution of Λ (the augmentation map is the natural sum map v ΛUv → Λ).
In particular, the complex (C· (T , Λ), ∂) is acyclic at p > 0.
The local system Λ figuring in the theorem has the property that it is G-
equivariant : there are maps
(5.8) gτ : Λ(τ ) → Λ(gτ ),
commuting with the restriction maps, for every g ∈ G, satisfying 1τ = identity and
0
ggτ ◦ gτ = (g 0 g)τ . The complex C· (T , Λ) carries a G-action too, and the resolution
is a projective resolution in the category of G-modules. However we shall only need
its exactness. To be able to apply the theorem in our situation we have to assume
from now on that we are in the G-equivariant case, namely A = P(VK ) (see Section
2.5).
5.2. The definition of Λ. We now apply the above to our situation. Let Bk (τ ) =
Ak (τ )∗ . We want to define a smooth representation Λk as in the theorem such that
the local systems Λk and B k are naturally isomorphic. Recall that for each σ ∈ T̂k
we defined a linear functional (σ, −) on E k in Section 2.3, and we have
(5.9) (gσ, ge) = (σ, e)
for every g ∈ G. Denote the linear functional (σ, −) by λσ and let Λk be the linear
span of all the λσ in Hom(E k , K). Of course, the λσ are far from independent,
as they satisfy the harmonicity conditions. The action of G is compatible with its
action on T̂k : g(λσ ) = λgσ . It is also clear that Λk is a smooth representation.
Furthermore, if v is any vertex of σ, and Uv = Uv1 , then Uv fixes all the vertices
at distance 1 from v, hence it fixes σ, and also λσ . It follows that Λk satisfies the
condition of the theorem with n = 1. For every τ write Uτ = Uτ1 . The next theorem
shows that we have a natural isomorphism
(5.10) ΛU k ∗
k ' A (τ ) .
τ
Theorem 5.4. A linear functional λ ∈ Λk factors through Ak (τ ) if and only if it

is fixed by Uτ .
Proof. One direction follows easily from Corollary 2.7 : every linear functional
which annihilates I(τ ) ∩ E k and therefore factors through Ak (τ ), can be written
as a linear combination of λσ with σ for which ρ(σ, τ ) ≤ 1. Such σ’s are fixed by
Uτ , because each of their vertices is fixed by Uτ . For the converse direction we may
assume that τ is a vertex u. Indeed, if τ = {u0 , . . . , ur } and λ is fixed by Uτ , then
it is fixed by each Uui , and if we know the theorem in the case of a vertex, it will
follow that λ annihilates E k ∩ I(ui ) for every i, hence also E k ∩ I(τ ), by proposition
2.2 and by the fact that E k ∩ I(τ ) = δ(I k+1 (τ )).
Let us therefore assume that λ is fixed by Uu , u ∈ T0 . We shall use the har-

monicity conditions to show that λ can be written as a linear combination ofPλσ for
σ which are contiguous to u. We shall use repeatedly the fact that if λ = cσ λσ
is fixed by Uu , and every σ is fixed by Uun (this will always be the case for n large
enough) we may average over the coset space Uu /Uun , not changing λ, and may
therefore assume that for every g ∈ Uu , the coefficients cσ and cgσ are equal. Fur-
thermore, every σ in the linear combination expressing λ will have k vertices, each
at a certain distance from u. In the averaging process we shall, at worst, replace
certain σ by gσ for g’s fixing u, so this set of distances (and the order in which
they occur) will be the same for σ and gσ. The harmonicity conditions will allow
us to replace a sum over k-cells which are farther away from u by a sum over closer
σ’s (to define farther and closer precisely we shall use the Euclidean metric d, see
Section 1.5). Then we will average again, smoothening the sum, but not altering
the distances, and apply the harmonicity conditions inductively, until we reach σ’s
which are contiguous to u.
We begin with two lemmas on the building.
Lemma 5.5. Let σ = (v0 , . . . , vk ) and assume that ρ(σ, u) > 1. Fix an apartment
A centered at u and containing σ. Then there exists a wall W in A with the property
that precisely k vertices of σ lie on W, and the remaining vi and u lie (strictly) on
opposite sides of W.
Proof. Consider, for every vi , the collection of all the walls W in A passing
through {v0 , . . . , v̂i , . . . , vk } but not through vi . Each of them divides A into two
closed half-spaces A+ (W ) and A− (W ), the former containing vi , the latter not.
If the conclusion of the lemma doesn’t hold then u ∈ A+ (W ) for every such W ,
for every vi (this allows also for the case where u lies on W ). But it is clear that
T +
W A (W ) is the star of σ in A - the union of all the (closed) chambers (top
dimensional simplices) of A containing σ - and since ρ(σ, u) > 1, u can not belong
to it.
Lemma 5.6. Let σ = (v0 , . . . , vd ) be a chamber (top dimensional simplex) in

the apartment A as in the previous lemma, ρ(σ, u) > 1, and assume that the
wall W contains all its vertices except vi , and that u and vi lie in opposite sides
of W. Let sW ∈ G be an involution inducing on A a reflection in W. Then if
we let vi0 run over the q + 1 vertices of T which form a chamber together with
{v0 , . . . , vi−1 , vi+1 , . . . , vd }, precisely two of them, vi and sW (vi ) will lie in A. Ev-
ery vi0 6= sW (vi ) will be of the form g(vi ) for g ∈ Uu fixing the wall W pointwise (g
folds A along W ).
26 EHUD DE SHALIT
Proof. The vertices vi and sW (vi ) belong to A, and they are clearly the only ones
among the vi0 , because A is a Euclidean space. Since sW (vi ) is closer to u than vi , in
the metric d, it can’t be of the form g(vi ) for a g fixing u. That the other q vi0 are of
the form g(vi ) for g ∈ Uu fixing W pointwise can be seen by introducing coordinates,
and expressing the vi0 in terms of the basis used to describe the apartment.
Consider our λ, fixed by Uu , and let E be the collection of all the finite expressions
X
(5.11) ε= cσ,σ̃ (σ, σ̃)
where the pairs (σ, σ̃)

Prun over σ ∈ T̂k and σ̃ ∈ Td such that σ ≤ σ̃, the coefficients
cσ,σ̃ ∈ K, and λε = cσ,σ̃ λσ is equal to λ (we say that ε represents λ). We shall
use the Euclidean metric d on |T | to measure the “closeness” of ε to u. Put
(5.12) ||ε|| = max sup d(u, x),
(σ,σ̃) x∈|σ̃|
where the max is over all the pairs with cσ,σ̃ 6= 0, and the sup is then the distance
of the farthest point in the closure of the chamber σ̃, from u. It is clear that the
set of possible “norms” ||ε|| is discrete. It is also clear that we may smoothen ε,
replacing it by
X X
(5.13) εsm = [Uu : Uun ]−1 cσ,σ̃ (gσ, gσ̃)
g∈Uu /Uun
for some large enough n, and εsm ∈ E is of the same norm : ||εsm || = ||ε||. The
advantage of εsm over ε is that it is invariant under Uu .
The proof of the theorem will be completed by showing that if ε is not supported
only on (σ, σ̃) such that u ∈ σ̃ (and therefore σ is contiguous to u), then there exists
an ε0 ∈ E with ||ε0 || < ||ε||. We shall then be able to reduce ||ε||, and since this
process can not go on indefinitely, we shall reach an ε representing λ which is
supported on σ’s contiguous to u, as desired.
First, we may assume that ε is smooth, ε = εsm . Consider a pair (σ, σ̃) in the
support of ε where ||ε|| is attained. Fix an apartment A containing σ̃ and u, and
a wall W as in the first lemma, applied to σ̃ (note that since u ∈ / σ̃, ρ(u, σ̃) > 1).
Now distinguish cases.
(i) If σ ⊂ W (which happens when the vertex of σ̃ which lies in A+ (W ) is not
a vertex of σ), replace (σ, σ̃) by (σ, sW (σ̃)), and observe that sW (σ̃) is closer to u
than σ̃.
(ii) Suppose that σ̃ = (v0 , . . . , vd ) = (L0 ⊃ · · · ⊃ Ld ⊃ πL0 ) and the vertex
off W is vi . Since we have dealt with case (i), we may assume that vi ∈ σ. As-
sume now that vi+1 ∈ σ, but vi−1 ∈ / σ. Consider the q + 1 possible vi0 such that
0
(v0 , . . . , vi−1 , vi , vi+1 , . . . , vd ) is a chamber. Let σ̃j (1 ≤ j ≤ q) be the q among these
chambers where vi0 = g(vi ) for g ∈ Uu fixing W, and let σj be the corresponding
k-cells g(σ). Since ε is smooth, the pairs (σj , σ̃j ) appear in ε with the same coeffi-
cient. Let σ̃0 = sW (σ̃) be the remaining chamber, and σ0 = sW (σ). By property
C of the harmonicity conditions, we may replace the part of the sum in ε involving
(σj , σ̃j ) by a combination of (σ0 , σ̃0 ) and (σ 0 , σ̃0 ) where σ 0 is obtained from σ by
replacing vi by vi−1 (which was assumed not to be a vertex of σ). Note that σ 0 lies
on W, hence is contained in sW (σ̃) = σ̃0 . As before, σ̃0 is closer to u than σ̃.
(iii) Suppose that σ̃ is as in (ii), but both vi+1 and vi−1 belong to σ (as well as
vi of course). Once again consider the q + 1 possible vi0 , and use the same notation
for the σj , 0 ≤ j ≤ q. Property B this time implies that the sum of all the λσj
vanishes, hence the part of ε involving (σj , σ̃j ) for 1 ≤ j ≤ q (which appear with
the same coefficients) can be replaced by a multiple of (σ0 , σ̃0 ).
(iv) Finally assume that σ̃ is as in (ii), vi ∈ σ, but vi+1 ∈ / σ. Let σ 0 be the
0
k + 1-cell σ = σ ∪ {vi+1 }. Let σj (0 ≤ j ≤ k + 1) be the k-cells obtained from it by
removing any one of the vertices. One of them is σ, another one - when we remove
vi - falls under case (i), and the others have both vi and vi+1 in them so fall under
cases (ii) or (iii). Since by property D (the last one we haven’t used !) the sum of
the λσj vanishes, we may replace (σ, σ̃) by a combination of terms that fall under
cases (i)-(iii).
Treating in this way all the pairs (σ, σ̃) where ||ε|| is attained, we arrive at
another ε0 ∈ E of a smaller norm. This concludes the proof of the theorem.
5.3. The acyclicity theorem. We summarize our main theorem in this section.
Theorem 5.7. Let A = P(VK ). Then
(5.14) H p (T , Aq ) = 0
for p > 0 and every q, and
(5.15) H 0 (T , Aq ) ∼ q
= Char .
Proof. The acyclicity follows from the last theorem and from the discussion pre-
ceding it. The second assertion is proved as in theorem 4.4.
k
Corollary 5.8. Char is the algebraic dual of the representation Λk .
Proof. We have Char k

= Ker(C 0 (T , Ak ) → C 1 (T , Ak )), which is dual to the coker-
nel of C1 (T , Λk ) → C0 (T , Λk ). But by the theorem of Schneider and Stuhler that
cokernel is Λk .
Proposition 5.9. The representation Λk is admissible and irreducible.
Proof. (Sketch) We have seen that it is smooth. In a way completely analogous to

Un
the proof of theorem 5.4 one may show that for any fixed n any λ ∈ Λk u can be
written as a linear combination of λσ for σ in a bounded ball around u. It follows
Un
that Λk u is finite-dimensional, and since the Uun form a fundamental system of
neighborhoods at 1 in G, Λk is admissible.
The irreducibility should follow as in [Bu], Proposition 4.2.3, by considerations
involving the Hecke algebra.
Part 2. Residues on the Bruhat-Tits building and the cohomology of X

In the second part of this work we shall apply our knowledge of the local system
A to the problem of computing the de-Rham cohomology of Drinfeld’s p-adic sym-
metric domain X. We begin with a few observations on a certain tiling of X, and
define the notion of residue which is central to our approach.
28 EHUD DE SHALIT
6. The structure of X, and the definition of the residue

6.1. Drinfel’d’s p-adic symmetric space and the reduction map. Let X
be Drinfel’d’s d-dimensional space [Dr] (we depart from the more traditional
notation Ω(d) because we reserve the character Ω to denote differential forms). This
is a rigid analytic space [B-G-R] over K, defined as the complement in P(V ∗ ) of all
the K-rational hyperplanes. It is stable under the (contragredient) action of G. We
let X̃ be its pre-image in V ∗ . We shall likewise denote by X̃ the pre-image in V ∗
of any set X in P(V ∗ ).
We refer to Section 1 for the definition and basic notation concerning the Bruhat-
Tits building T of G. There is an important “reduction map”
(6.1) r : X(K̄) → |T |
which we now recall. First identify |T | with the space of all dilation classes of real
norms on VK . To achieve this write, for every lattice L in VK , and a ∈ VK ,
(6.2) αL (a) = min {|λ|; λ−1 a ∈ L}.
If πL0 ⊂ Lk ⊂ ... ⊂ L1 ⊂ L0 is a lattice flag, so that vi = [Li ] form a pointed k–cell

Pk
P= (v0 , ..., vk ) ∈ T̂k , then points in |σ| can be written uniquely as t = i=0 ti vi ,
σ
ti = 1, 0 < ti . Noting that αL0 ≤ αL1 ≤ ... ≤ αLk ≤ qαL0 , we associate with
such a t the dilation class [αt ] of the norm
(6.3) αt = max(αL0 , q −t0 αL1 , q −t0 −t1 αL2 , ..., q −t0 −t1 ··· −tk−1 αLk ).
This is independent of the distinguished vertex of σ because if we perform a cyclic

permutation of the vertices we multiply αt by q t0 . Likewise, if we change the lattices
by a homothety, all have to change by the same factor, so the dilation class of αt is
unchanged. We conclude that t 7→ [αt ] is a well defined map from |T | to the space
of dilation classes of real norms on VK . A theorem of Goldman and Iwahori states
that this map is bijective.
Once we have identified |T | with the space of dilation classes of real norms on
VK , we proceed to define the reduction map. Pick x ∈ VK̄∗ . Then |a|x = |x(a)| is a
semi-norm on VK , whose dilation class depends only on [x] ∈ P(VK̄∗ ). Furthermore,
if [x] ∈ X(K̄) this is a norm. We let
(6.4) r([x]) = the dilation class of | · |x .
This reduction map is compatible with the action of G on X and T .

If S is a finite set of cells of T and
[
|S| = |σ|,
σ∈S
then r−1 (|S|) is an open rigid analytic subspace of X. If |S| is connected (in the
real topology), then r−1 (|S|) is connected (in the rigid analytic topology) and we
call it a simplicial subdomain. If |S| is compact (which happens precisely when
every face of a cell in S is also a cell in S), r−1 (|S|) is an affinoid.
6.2. The pre-image of a vertex under the reduction map. Let v0 = [L0 ] ∈
T0 . Then
(6.5) Xv0 = r−1 (v0 ) = {[x]; ∀a ∈ L0 − πL0 , |x(a)| = 1}.
d+1
This is isomorphic (upon introduction of coordinates in which L0 = OK ) to
(6.6) {[x0 , x1 , ..., xd ]; |xi | = 1, x̄i are linearly independent over Fq },
which is the affinoid obtained by removing from Pd (K̄) the q d +···+q+1 open tubular
neighborhoods of radius 1 around the K -rational hyperplanes. For future reference
we introduce some notation concerning these tubular neighborhoods. Suppose a ∈
L0 − πL0 . Then we let Ha be the hyperplane in P(V ∗ ) defined by x(a) = 0, and
for any 0 < r < 1 we let
Ha (r− ) = {[x]; max |x(b)| = 1, |x(a)| < r}
b∈L0
(6.7) Ha (r) = {[x]; max |x(b)| = 1, |x(a)| ≤ r}.
b∈L0
(In coordinates as above the condition maxb∈L0 |x(b)| = 1 is equivalent to max |xi | =
1). We call them the open (closed) tubular neighborhoods of Ha of radius r de-
termined by L0 . Like Ha , they only depend on [a] ∈ P(VK ). For |π| ≤ r < 1
(resp. |π| < r < 1) Ha (r) (resp. Ha (r− )) depends only on the image ā of a in
V0 = L0 /πL0 , and in fact only on its class [ā] ∈ P(V0 ).
6.3. The pre-images of simplices and stars. Fix σ = (L0 ⊃ L1 ⊃ · · · ⊃ Lk ⊃

πL0 ) ∈ Tbk . Let
(6.8) Xσ = {[x]; ∃ |π|R0 < Rk < ... < R0 s.t. ∀ a ∈ Li − Li+1 , |x(a)| = Ri } .
We claim that Xσ = r−1 (|σ|). Pick [x] ∈ Xσ and let ti ∈ (0, 1) be such that
(6.9) q 1−t0 −t1 ··· −ti = ri = Ri /Rk .
Pk
Then t = i=0 ti vi ∈ |σ|. If a ∈ Li − Li+1 then
(6.10) αL0 (a) = · · · = αLi (a) = 1 < q = αLi+1 (a) = · · · = αLk (a)
so αt (a) = ri . This proves that, up to a scalar, αt (a) = |x(a)| for every a, hence
[x] ∈ r−1 (t) ⊂ r−1 (|σ|). Conversely, starting with an arbitrary [x] ∈ r−1 (|σ|) let
t = r([x]) and define the ri accordingly. Then [x] ∈ Xσ .
It follows from the characterization of the Xσ that
(6.11) X(v0 ) = r−1 (St(v0 )) = {[x]; ∀ a ∈ L0 − πL0 , |π| < |x(a)| ≤ 1}.
Equivalently, fix a set A of representatives of P(V0 ) in L0 − πL0 . Then
[
(6.12) r−1 (St(v0 )) = P(V ∗ ) − Ha (|π|).
a∈A
−1
More generally, writing X(σ) = r (St(σ)), we claim that
(6.13) X(σ) = {[x]; ∃ |π|R0 < Rk < ... < R0 s.t. a ∈ Li ⇔ |x(a)| ≤ Ri } .
Indeed, if [x] ∈ X(σ) then [x] ∈ Xτ for some σ ≤ τ. If we put Ri = maxa∈Li |x(a)|
then Li is characterized as the set of a0 s for which |x(a)| ≤ Ri . Conversely, suppose
such a sequence of Ri exists for a fixed x. Let rm < · · · < r0 be the set of values
obtained as |x(a)| for a ∈ L0 − πL0 , and let Mi = {a; |x(a)| ≤ ri }. Clearly [x] ∈ Xτ
where τ = ([M0 ], . . . .[Mm ]). But r0 ≤ R0 , so we get L0 ⊆ M0 ⊆ L0 , hence M0 = L0 .
30 EHUD DE SHALIT
Also, if 0 < i ≤ k then for some j ≥ 1 we must have rj ≤ Ri < rj−1 , so we get
Mj ⊆ Li ⊆ Mj , hence Mj = Li . It follows that σ ≤ τ, and [x] ∈ X(σ).
6.4. Coordinates on Xσ and X̃σ . Let t(σ) = (e0 , e1 , ..., ek ) be the type of σ, and
write di = dim Vi = ei + · · · + ek (here Vi = Li /πL0 ). First choose a basis of VK
adapted to σ, and let x0 , . . . , xd be the corresponding coordinate functions on V ∗ .
Let
xdi −1
ti = (1 ≤ i ≤ k)
xd
xj
(6.14) yj = (0 ≤ i ≤ k, di+1 ≤ j < di − 1).
xdi −1
These are well-defined coordinates on Xσ . The “compact coordinates” yj come in
blocksPof sizes e0 − 1, e1 − 1, ..., ek − 1 (some or all of these blocks may be empty).
Since ei = d+1, we have d coordinates altogether, and there are k “non-compact”
coordinates (the ti ).
m
For any integer
Pm m let Cm be the affinoid subdomain of A defined by the con-
dition |c0 + i=1 ci xi | = 1 for every vector of ci ∈ OK with max |ci | = 1. Then the
coordinates (ydi+1 , . . . , ydi −2 ) map Xσ onto Cei −1 and altogether the ti and the yj
induce an isomorphism
(6.15) Xσ ∼
= Ce0 −1 × · · · × Cek −1 × Ak ,
where Ak is the “multiannulus”
(6.16) |π| < |tk | < · · · < |t1 | < 1.
In a similar fashion we define zi = xdi −1 (0 ≤ i ≤ k). Then the zi and the yj

induce an isomorphism
(6.17) X̃σ ∼
= Ce0 −1 × ... × Cek −1 × Ãk+1 ,
where Ãk+1 is the multiannulus

(6.18) |πz0 | < |zk | < · · · < |z1 | < |z0 |.
Observe that Ak is the successive fibration of annuli k times (the cyclic order in
which this is done is a conformal invariant), and Ãk+1 = Ak × Gm .
6.5. Residues of closed k + 1-forms on pointed k-cells. Let σ be a pointed

k-cell, and η a rigid analytic k + 1-form on X̃σ . Introduce coordinates as above.
Then writing C = Ce0 −1 × ... × Cek −1 we may write
k+1
X XX
(6.19) η= γi,I,J ∧ z J dzI .
i=0 |I|=i J
Here we used the following abbreviations and conventions. I runs over all the
sequences 0 ≤ ι1 < ι2 < ... < ιi ≤ k, and dzI = dzι1 ∧ ... ∧ dzιi . J runs over all
Qk
the sequences (j0 , ..., jk ) ∈ Zk+1 and z J = l=0 zljl . For each I and J, γi,I,J is a
k + 1 − i form on the affinoid C. Using the obvious norms on such forms we must
have |γi,I,J |C RJ → 0 whenever |π|R0 < Rk < ... < R1 < R0 . Now suppose that η
is closed. Then
k+1
XXX
0 = dη = (dγi,I,J ∧ z J dzI +
i=0 |I|=i J
k
X
k+1−i
(6.20) (−1) γi,I,J ∧ jι z J−ει dzι ∧ dzI ),
ι=0
where the notation is self-explanatory. Collecting terms, for each I of length i,
i ≤ k + 1, and each J, the coefficient of z J dzI in dη, which is a k + 2 − i form on
C, should vanish, and we get
i
X
(6.21) 0 = dγi,I,J + (−1)k−i γi−1,I−{ιl },J+ειl (−1)l−1 (jιl + 1).
l=1
In particular, if ι ∈ I ⇒ jι = −1, then γi,I,J is a closed form on C, and if i = k + 1,
it is a constant function. Note that the latter case occurs only once : when J =
(−1, −1, ..., −1). Although it may seem, at this stage, that we are not making full
use of the assumption that η is closed, we make the following definition.
ck (0 ≤ k ≤ d). Let η be a closed k + 1-form
Definition 6.1. Let σ = (v0 , ..., vk ) ∈ T
on X̃σ . Choose coordinates as above and define the residue of η along σ to be
(6.22) resσ η = γk+1,I,J
where I = {0 < 1 < · · · < k} and J = (−1, −1, ..., −1).
Lemma 6.1. The residue is well-defined, i.e. it is independent of the coordinates
chosen and depends only on σ and η. If η is exact its residue vanishes.
Proof. It is clear that the residue of an exact differential vanishes. We have to show
that it is independent of the choice of the coordinates. This will be established later
as a corollary to a formula which will give the residue of certain differential forms
k
which represent the classes in HdR (X̃σ ). That formula will be purely combinatorial,
and will have no reference to coordinates.
Lemma 6.2. Let σ = (v0 , v1 , . . . , vk ) and σ 0 = (v1 , . . . , vk , v0 ). Let η be a closed
k + 1 -form on X̃σ = X̃σ0 . Then
(6.23) resσ η = (−1)k resσ0 η.
Thus for even k the residue is independent of the distinguished vertex, while for odd
k it changes sign when we permute the vertices cyclically.
Proof. Let x0 , . . . , xd be coordinates in VK∗ adapted to σ. Then
(6.24) {x00 , . . . , x0d } = {π −1 xd1 , . . . , π −1 xd , x0 , . . . , xd1 −1 }
are coordinates which are adapted to σ 0 . Defining zi0 and yj0 as above from the x0i ,
we see that the yj0 are nothing but the old yj , with the indices permuted, while
(6.25) z00 = z1 , . . . , zk−1
0
= zk , zk0 = π −1 z0 .
Thus
dz00 dz 0 dz0 dzk
(6.26) 0 ∧ · · · ∧ 0k = (−1)k ∧···∧
z0 zk z0 zk
32 EHUD DE SHALIT
and this proves the lemma.
7. The cohomology of X(τ )

Fix τ ∈ Tr . Our purpose in this chapter is to compute the de-Rham cohomology
of the pre-image, under the reduction map, of St(τ ), using the results of Part I.
When τ is a vertex v0 , we saw that X(v0 ) is an “open tubular neighborhood” of
Xv0 , and Xv0 is an affinoid whose reduction is the complement Z in P(V0∗ ) of all the
Fq -rational hyperplanes. The cohomology of X(v0 ) is the same as the cohomology
of Xv0 , and this in turn is (by definition) the Monsky-Washnitzer cohomology of
Z. Enough is known about Monsky-Washnitzer cohomology (in particular, the
Gysin exact sequence is valid, see [Mo]) to immitate the approach of Orlik-Solomon
and compute it for Z. The answer is given by the algebra A(v0 ), which is a direct
analogue, in characteristic p, of the Orlik-Solomon algebra.
However, for later purposes, when we come to gluing the local computations
using a Mayer-Vietoris spectral sequence, it is not enough to know the cohomology
of the X(v)0 s, but also of their intersections
\
(7.1) X(τ ) = X(v).
v∈τ
This is how the full array of algebras A(τ ) shows up.

We shall have two descriptions of the cohomology. One, as in the theory of
complex hyperplane arrangements, through generators and relations, namely via the
algebras A(τ ). The other, which is peculiar to the p-adic setting, as certain spaces
of harmonic cochains on the Bruhat-Tits building. It is this second description that
glues well together in X, and should allow us later to recover the main theorem,
describing the cohomology of X.
Our approach will be as follows. We shall soon construct a surjective homomor-
phism
(7.2) ω : Ak (τ ) → HdR
k
(X(τ )), e 7→ ω(e).
The residue map defined above will produce another homomorphism
k k
(7.3) c : HdR (X(τ )) → Char (τ ), η 7→ cη ,
k
where Char (τ ) is the space of local harmonic cochains on Tbk (τ ), defined in Section
3. The composition c ◦ ω will be shown to be the cochain homomorphism e 7→ ce
(whence the ambiguity in notation : cω(e) = ce ). We proved that it is injective, and
an isomorphism if τ = {v0 } is a vertex. It will follow that (i) ω is an isomorphism
and (ii) c is injective (for every τ ), and an isomorphism if τ = {v0 }. 1
We shall begin by working with the X̃(τ ), and later on descend to their image
in projective space, X(τ ).
1 The reason why c is not always an isomorphism is that the harmonicity conditions can not
be applied to characterize the image of c in general : they involve replacing a σ ∈ Tbk (τ ) by the
sum over certain neighboring σ 0 , and some of these may not lie anymore in Tbk (τ ). However, the
harmonicity conditions do suffice if τ is a vertex.
7.1. The homomorphism ω from Ã(τ ) to HdR (X̃(τ )). We refer to the Appen-
dix for a discussion of the rigid de-Rham cohomology of a rigid analytic space X
defined over K. We denote it by HdR (X). It is a graded, anti-commutative K
-algebra.
We refer to Section 2 for definitions and notation concerning the algebras A(τ )
and Ã(τ ).
Fix τ = (M0 ⊃ M1 ⊃ · · · ⊃ Mr ⊃ πM0 ) ∈ Tr (the choice of the distinguished
vertex is still at our disposal), and consider the map of graded K -algebras Ẽ →
HdR (X̃(τ )) defined by ea 7→ [ωa ], where ωa = da/a. Here we view a as a nowhere
vanishing function on X̃(τ ) ⊂ V ∗ .
Proposition 7.1. This map annihilates the ideal I(τ ), and therefore defines a ho-
momorphism
(7.4) ω : Ã(τ ) → HdR (X̃(τ )).
The homomorphism ω is surjective. The canonical projection X̃(τ ) → X(τ ) induces

a commutative diagram with exact rows and surjective vertical maps
0 → A(τ ) → Ã(τ ) → A(τ )[1] → 0
(7.5) ↓ ↓ ↓ .
0 → HdR (X(τ )) → HdR (X̃(τ )) → HdR (X(τ ))[1] → 0
Proof. Suppose first that {a0 , . . . , ak } are linearly dependent. We want to show
that ω ◦ δ(ea0 ∧ · · · ∧ eak ) = 0. By (2.4) we may assume that the ai are a minimal
set of dependent elements, and since eλa = ea for any λ 6= 0, we may assume that
Pk
the linear dependence among them is i=0 ai = 0. We then compute, using the
Pk−1
fact that dak = − i=0 dai ,
Xk
1 ci ∧ · · · ∧ dak
ω ◦ δ(ea0 ∧ · · · ∧ eak ) = (−1)i ai da0 ∧ · · · ∧ da
a0 · · · ak i=0
k
da0 ∧ · · · ∧ dak−1 X
(7.6) = (−1)k ai = 0.
a0 · · · ak i=0
Next suppose that a and b lie in Mi − Mi+1 but are congruent modulo Mi+1 .
Then from a = b(1 + (a − b)/b) we get
(7.7) ωa = ωb + ω1+(a−b)/b .
But on X̃(τ ) we have |(a − b)/b| < 1, so log(1 + (a − b)/b) is rigid analytic there,
and [ωa ] = [ωb ].
Now if a0 , . . . , ak are in Mi − Mi+1 , but are linearly dependent modulo Mi+1 ,
then again we may assume they are a minimal set of this type, and dividing them by
Pk−1
appropriate units from K we may assume that ak −( i=0 ai ) ∈ Mi+1 . By what was
P k−1
said above we may replace ak by a0k = i=0 ai without affecting ω◦δ(ea0 ∧···∧eak ).
But then the ai are linearly dependent, so ω ◦ δ(ea0 ∧ · · · ∧ eak ) = 0. This proves
the first assertion, namely that ω(I(τ )) = 0.
The surjectivity of ω is proved in the Appendix.
The choice of any non-zero a ∈ VK gives us an isomorphism X̃(τ ) ' X(τ ) × Gm
(in fact, X̃ ' X × Gm ) through x 7→ ([x], x(a)). The Künneth formula yields the
34 EHUD DE SHALIT
factorization
M
k+1 k+1 k
(7.8) HdR (X̃(τ )) = HdR (X(τ )) [ωa ] ∧ HdR (X(τ )).
1
Since A(τ ) is generated by ea − eb , and [ωa − ωb ] ∈ HdR (X(τ )), we see that
k+1 k+1 k k
ω(A (τ )) ⊆ HdR (X(τ )), and ω(ea ∧ A (τ )) ⊆ [ωa ] ∧ HdR (X(τ )). The splitting
M
(7.9) Ãk+1 (τ ) = Ak+1 (τ ) ea ∧ Ak (τ )
shows that each of the two direct summands in Ãk+1 (τ ) gets mapped surjectively
k+1
onto the corresponding direct summand in HdR (X̃(τ )). Finally, the induced map
between short exact sequences is independent of the a used to split them.
k+1 k
We shall denote the canonical map HdR (X̃(τ )) → HdR (X(τ )) also by δ. Thus
k
for any a and [η] ∈ HdR (X(τ )),
(7.10) δ([ωa ] ∧ [η]) = [η],
and
(7.11) δ ◦ ω = ω ◦ δ.
This δ on cohomology can be defined goemetrically, without recourse to the splitting
supplied by the Künneth formula, as a residue map.
7.2. Computation of residues. We now compute the residue of ωa0 ∧···∧ωak on
a σ ∈ Tbk (0 ≤ k ≤ d). To simplify the notation write S for the ordered k + 1-tuple
(a0 , . . . , ak ), and ωS for ωa0 ∧ · · · ∧ ωak .
Let σ = (L0 ⊃ L1 ⊃ · · · ⊃ Lk ⊃ πL0 ). We refer to Section 2.3 for the definition
of the linear functionals (σ̃, −) and (σ, −).
Proposition 7.2. Let ẽ ∈ Ẽ k+1 . Then
(7.12) resσ (ω(ẽ)) = (σ̃, ẽ).
Proof. We may assume that ẽ = eS where S = (a0 , . . . , ak ), and each ai ∈ L0 −πL0 .
Pd
Fix coordinates x0 , . . . , xd adapted to σ, and write ai = j=0 aij xj (0 ≤ i ≤ k).
Using the zl and the yj as variables on X̃σ (see above)
k
X
(7.13) ai = cil (y)zl
l=0
where the cil are non-homogeneous linear forms in the yj which belong to the lth
block. Furthermore on X̃σ we have
(7.14) |πz0 | < |zk | < · · · < |z1 | < |z0 |
(7.15) |cil (y)| ≤ 1, all l
(7.16) |cil (y)| = 1 for l = ισ (ai )

(see Definition 2.1 for the definition of the index of a with respect to σ),
(7.17) |cil (y)| ≤ |π| for 0 ≤ l < ισ (ai ).
Writing m = m(i) = ισ (ai ) for simplicity we deduce that
(7.18) ai = cim (y)zm (1 + A)
where
X cil (y) zl
(7.19) A=
cim (y) zm
l6=m
satisfies |A| < 1 throughout X̃σ . It follows that log(1 + A) is rigid analytic in X̃σ
and
dai dcim dzm
≡ + mod (exact differentials).
ai cim zm
Since the residue of an exact differential vanishes,
(7.20)
da0 dak dzm(0) dzm(k)
resσ ( ∧···∧ ) = resσ ( ∧···∧ ) = (σ̃, ea0 ∧ · · · ∧ eak )
a0 ak zm(0) zm(k)
as required.
Corollary 7.3. The residue is well defined, independently of the choice of coordi-
nates.
Proof. Any closed differential form on X̃σ is, up to exact forms, an ω(e). But the
expression derived above is independent of the choice of coordinates.
k+1
Corollary 7.4. For any [η] ∈ HdR (X(σ)), resσ (η) = 0, and therefore resσ factors
k
through HdR (X(σ)).
Proof. This was proved in Lemma 2.4 for (σ̃, −).
7.3. Defining the local residue cochains cη . Let τ ∈ Tr , and recall that Tbk (τ )
is the collection of pointed k-cells σ contiguous to τ. We want to attach to any
k+1
[η] ∈ HdR (X̃(τ )) its residue along σ, denoted resσ (η) for every σ ∈ Tbk (τ ). If τ ≤ σ
then X̃σ ⊆ X̃(τ ), and the residue is defined. In general, if r > k say, there may
be no k-cells σ with τ ≤ σ, so we need to make sense of this extended notion of a
residue. First a lemma.
Lemma 7.5. Let σ 0 ∈ Tbm be a pointed m-cell, and η a closed k + 1-form on X̃,
where m ≥ k. Then if η|X̃σ0 is exact,
(7.21) resσ (η) = 0
for every σ ∈ Tbk which is a face of σ 0 .

Proof. Let σ ≤ σ 0 , and put Ỹ = X̃σ0 ∪ X̃σ . Then Ỹ is the cone over a simplicial
subdomain of X. Let
(7.22) σ 0 = (L0 ⊃ L1 ⊃ · · · ⊃ Lm ⊃ Lm+1 = πL0 )
and
(7.23) σ = (L0 ⊃ Li1 ⊃ · · · ⊃ Lik ⊃ Lm+1 = πL0 ).
Here i0 = 0 and ik+1 = m + 1, and we have assumed without loss of generality that
[L0 ] is a common vertex to σ and σ 0 . Introduce coordinates adapted to σ 0 , as in
Section 6. Then these coordinates are adapted to σ too, and we may consider the
ensuing coordinates zil (0 ≤ l ≤ k) and yj (0 ≤ l ≤ k, dil+1 ≤ j < dil − 1) given by
36 EHUD DE SHALIT
(6.14), but with respect to the filtration defined by σ. These coordinates induce an
isomorphism similar to (6.17)
(7.24) Ỹ ∼
= D0 × · · · × Dk × Ãk+1 ,
where Ãk+1 is the multiannulus
(7.25) |πz0 | < |zik | < · · · < |zi1 | < |z0 |.
The projection of X̃σ to Dl is an affinoid of the type Ce−1 , where e = dim Lil /Lil+1 .
However, the projection of X̃σ0 to Dl is itself of the form
(7.26) Ce0 −1 × · · · × Cet −1 × At ,
where 1 + t = il − il+1 is the “number of steps in σ 0 refining the lth step in σ” and
the ej are the dimension jumps in these steps.
At any rate, from the Künneth decomposition applied to Ỹ we deduce a decom-
position of η as in (6.19), where the γi,I,J are k+1−i forms on D = D0 ×···×Dk and
the z 0 s are taken from {z0 , zi1 , . . . , zik }. Now resσ (η) is the coefficient γk+1,I,J |X̃σ
where I = (0, i1 , . . . , ik ) and J = (−1, . . . , −1), which is constant since η is closed.
But if η|X̃σ0 is exact, γk+1,I,J |X̃σ0 vanishes. Since Ỹ is connected
(7.27) γk+1,I,J |X̃σ = γk+1,I,J |X̃σ0
and the lemma follows.
Suppose now that τ ∈ Tr and σ ∈ Tbk (τ ). Let σ 0 = σ ∪ τ (with the distinguished
k+1
vertex of σ). Let [η] ∈ HdR (X̃(τ )), and assume that η is defined on all of X̃
(see the Appendix). If η is exact on X̃(τ ), it is exact on X̃σ0 ⊆ X̃(τ ), and by the
lemma its residue on σ vanishes. Thus resσ (η) depends only on the class of η in
k+1
HdR (X̃(τ )), and this enables us to define
k+1
(7.28) resσ : HdR (X̃(τ )) → K, whenever ρ(σ, τ ) ≤ 1.
k k+1
Definition 7.1. Let [η] ∈ HdR (X(τ )). Consider any [η̃] ∈ HdR (X̃(τ )) such that
δ([η̃]) = [η] (e.g. η̃ = ωa ∧ η). Let cη : Tbk (τ ) → K be defined by
(7.29) cη (σ) = resσ (η̃).
Since η̃ is well-defined up to exact forms and up to forms in the image of δ, and
since the residue vanishes on both such forms, cη is well-defined. We call cη the
local residue cochain (at τ ) attached to η.
Corollary 7.6. We have the following formula, using the notation of Definition
2.5 :
(7.30) cω(e) = ce
for every e ∈ Ak (τ ).
Proof. Pick ẽ with δ(ẽ) = e. Then ω(e) = δ(ω(ẽ)). Writing η = ω(e) and η̃ = ω(ẽ)
we may assume that they are globally defined forms, and so we have
(7.31) cη (σ) = resσ (η̃) = (σ̃, ẽ) = (σ, e) = ce (σ)
for every σ ∈ T̂k (τ ).
Theorem 7.7. The residue cochain cη is harmonic. The local residue cochain is a
homomorphism
k k
(7.32) c : HdR (X(τ )) → Char (τ ).
This homomorphism is injective, and bijective if τ = v0 is a vertex.
The map ω is an isomorphism
(7.33) ω : Ak (τ ) ' HdR
k
(X(τ ))
for every τ.
Proof. The thorem follows from the discussion above and the corresponding theo-
rems in Section 3.
8. The cohomology in the large

Consider the tiling of X by the sets X(v), v ∈ T0 . The k th -cohomology of these
simplicial subdomains (0 ≤ k ≤ d) was determined in the previous section, both
by means of generators and relations (the Ak (v)), and as spaces of local harmonic
k
cochains (the Char (v)). It is natural to try a Mayer-Vietoris argument to glue
the local results, in order to obtain a characterization of the cohomology of X via
residues.
8.1. The Mayer-Vietoris spectral sequence. The considerations in this sub-

section will be of a general nature. Let X be a smooth connected rigid analytic
space, and
(8.1) U = {X(v) → X}
an admissible cover by open subdomains. For each σ = {v0 , v1 , . . . , vk } let X(σ) =
Tk q
i=0 X(vi ). Let H be the presheaf
q
(8.2) Hq (Y ) = HdR (Y /K).
Proposition 8.1. There exists a first-quadrant spectral sequence
(8.3) E2pq = Ȟ p (U, Hq ) =⇒ HdR
n
(X/K).
Proof. Consider a quasi-isomorphism of complexes of sheaves in the rigid analytic
(Grothendieck) topology Ω· → I · , where I · are injective sheaves, and I n = 0 for
n < 0. Consider the double complex of abelian groups
2 C 2 (U , I 0 ) → C 2 (U , I 1 ) → C 2 (U, I 2 ) → ···
δ↑ ↑ ↑
1 C 1 (U , I 0 ) → C 1 (U , I 1 ) → C 1 (U, I 2 ) → ···
(8.4) δ↑ ↑ ↑
∂ ∂
q=0 C 0 (U , I 0 ) → C 0 (U , I 1 ) → C 0 (U, I 2 ) → ···
p=0 1 2
where C q (U , I p ) is the space of (alternating) q-cochains with values in I p , δ is the
Čech derivation and ∂ theLderivation of the complex I · , modified by (−1)q . We have
δ∂ + ∂δ = 0. Let (C n = p+q=n C q (U, I p ), d = ∂ + δ) be the total complex of the
38 EHUD DE SHALIT
double complex. Then there are two spectral sequences abutting to the homology
groups H n (C · ). The first is
(8.5)
½
0 0 q>0 since I p is injective, hence flasque
E1pq = Ȟδq (U, I p ) =
Γ(X, I p ) q = 0
0
and dp,0 p
1 is induced from ∂ : I → I
p+1
. We therefore have
½
0 0 q>0
(8.6) E2pq = p
H p (Γ(X, I · )) = Hp (X, Ω· ) = HdR (X/K) q = 0
and H n (C · ) = HdR
n
(X/K).
The second spectral sequence is
00
(8.7) E1pq = C p (U, H∂q )
00
where H∂q is the presheaf whose value on U is H q (Γ(U, I · )) = HdR
q
(U/K), and dpq
1
is induced from δ. We therefore have
00
(8.8) E2pq = Ȟ p (U, Hq ) =⇒ H n (C · ) = HdR
n
(X/K)
as desired.
8.2. The main theorem. We now go back to the framework of this paper. The
algebras Ak (τ ) form a local system A on T . In Section 5, we proved that the local
system A is acyclic : H p (T , Aq ) = 0 for p > 0 and every q.
Theorem 8.2. The residue cochain map induces an isomorphism
k k
(8.9) HdR (X) ' Char .
Proof. Let us apply the spectral sequence to our situation. We then have by theo-
rem 7.7 Hq (X(τ )) = Aq (τ ), and the acyclicity assumption amounts to
(8.10) Ȟ p (U, Hq ) = 0,
(p > 0). The spectral sequence therefore degenerates at the E2 term, and we have
k
HdR (X) = Ȟ 0 (U, Hk ), which, by theorem 7.7 again, and by theorem 5.7, is equal
k
to Char (X).
8.3. The main theorem, an alternative approach. We would like to discuss
the relationship between the residue cochain homomorphism
k k
(8.11) c : HdR (X) → Char , [η] 7→ cη
and the isomorphism
(8.12) cSS : HdR
k
(X) ' C k , [η] 7→ cSS
η
of [S-S], corollary 17. Here (as in the introduction) C k is a short-hand for the space
(8.13) HomZ (Cc∞ (G/BI , Z)/RI , K)
where we have borrowed the notation of [S-S] :
• I = {1, 2, . . . , d − k}
• G = P GLd+1 (K)
• For any set J ⊂ {1, . . . , d}, BJ = the subgroup of G generated by the standard
Iwahori subgroup B = B∅ , and the transpositions si (i ∈ J), si = (i, i + 1),
viewed as permutation matrices
• Cc∞ = locally constant, compactly supported functions
• RI = the G-submodule generated by the functions

χByj BI − χBI , 0 ≤ j ≤ d
(8.14) χBI∪{i} , d − k < i ≤ d
j
z }| { z d+1−j }| {
where yj = diag(1, . . . , 1, π, . . . , π) (χT is the characteristic function of the
set T ).
k
We first have to relate Char and C k . Recall that G was defined in a coordinate-
free fashion, as P GL(VK ). We write h·, ·i for the pairing VK ×VK∗ → K, and identify
∗ ∗
G −1
also with
® P GL(VK ), so that if x ∈ VK , y ∈ VK ∗and g ∈ G,∗ we have hx, gyi =
g x, y . If L is a lattice in VK the dual lattice L = {y ∈ VK ; hL, yi ⊆ OK }.
Fix once and for all an ordered basis of VK , hence a dual basis for VK∗ . In these
bases we think of vectors in VK = t K d+1 (resp. VK∗ = K d+1 ) as row (resp. column)
vectors. We write x (resp. y) for the row (resp. column) vector representing
∗
−1 y ∈®VK ). Identify G with G = P GLd+1 (K) using these bases. Thus
x ∈ VK (resp.
hx, gyi = g x, y = xgy if g is the matrix of g ∈ G.
Now BI is the stabilizer of the lattice flag
(8.15) L∗0 ⊂ L∗1 ⊂ · · · ⊂ L∗k ⊂ π −1 L∗0
d+1
in VK∗ = K d+1 , where L∗0 = OK and for 1 ≤ i ≤ k, L∗i = yd+i−k π −1 L∗0 is the
lattice of column vectors whose top d + i − k entries lie in π −1 OK , and the rest lie
in OK . Dualizing, BI is the stabilizer in VK of
(8.16) σ0 = (L0 ⊃ L1 ⊃ · · · ⊃ Lk ⊃ πL0 ),
−1 d+1
which is a lattice flag of type tmin = (d+1−k, 1, . . . , 1). Here Li = t (πyd+i−k OK )
is the dual lattice of L∗i . We call a lattice flag (or a pointed k-cell) of this type of
minimal type. Observe that by property (C) defining harmonic cochains, every
k
c ∈ Char is determined by its values on the k-cells of minimal type. Thus G/BI
may be identified with the collection T̂k,min of all the pointed k-cells of type tmin ,
and in this identification the coset BI /BI is mapped to σ0 . We write σ0 also for
the characteristic function of this coset, as an element of Cc∞ (G/BI , Z). We may
therefore evaluate cSSη at σ0 .
In the next theorem we show how a comparison of our residue cochain map
c with the Schneider-Stuhler map cSS leads to theorem 8.2 through an approach
independent of the acyclicity theorem and the Mayer-Vietoris argument.
Theorem 8.3. Suppose that for every η of the form η = δ(ωa0 ∧ · · · ∧ ωak )
(8.17) cSS
η (σ0 ) = (σ0 , η).
Then for every η, cSS

η is the restriction of cη to T̂k,min , and η 7→ cη is an isomor-
k k k
phism HdR (X) ' Char . Furthrtmore the map Char → C k , c 7→ c|T̂k,min , where we
identify K-valued functions on T̂k,min = G/BI with HomZ (Cc∞ (G/BI , Z), K), is
an isomorphism of G-modules. In fact, it is the isomorphism (cSS · )
−1
◦ c· .
Proof. The assumption, and proposition 7.2, imply that for every η, cη and cSS η
agree on σ0 . Since cgη (gσ0 ) = cη (σ0 ) (resp. cSS
gη (gσ0 ) = cSS
η (σ0 )) for every g ∈ G,
SS
and G acts transitively on T̂k,min , cη and cη agree on all of T̂k,min . This proves the
first part of the theorem. The injectivity of the map c is now obvious, since cSS is
40 EHUD DE SHALIT
an isomorphism. It remains to prove that c is surjective, or that the harmonicity

conditions suffice to characterize the image of c. At this point it is perhaps natural
to show directly that the relations in RI are formal consequences of the harmonicity
conditions. However this involves some non-trivial combinatorics, and we prefer to
argue as follows.
Let Mk be the space of finitely-supported measures on T̂k,min which are formal
consequences of the harmonicity conditions. By this we mean the following. Each
harmonicity condition is a finite linear combination of k-cells (viewed as linear
functionals on the space of all k-cochains) and is thus a finitely-supported measure
on T̂k . Consider the KG-module spanned by these measures, and let Mk denote
those measures in it which are supported on T̂k,min . Similarly let Nk be the K-span
of the ZG-module denoted RI above. Thus C k is the space of all cochains on T̂k,min
which annihilate Nk . By weak duality2 Nk is the precise annihilator of C k in the
space of finitely-supported measures on T̂k,min . If µ ∈ Mk and c ∈ C k then c = cSS η
for some η, so c is the restriction to T̂k,min of cη . Since µ is supported on T̂k,min , and
is orthogonal to cη , µ is orthogonal to c. Since this holds for every c ∈ C k , µ ∈ Nk .
We therefore know that Mk ⊆ Nk , and we have to prove equality.
Now Nk has the pleasant property that it is generated, as a KG-module, by
measures whose support is contained in a ball of radius 1. By this we mean measures
µ for which there exists a vertex v such that µ is supported on T̂k (v). This is verified
with little effort for the measures used to define RI . Suppose Mk 6= Nk . Then there
exists such a measure µ in Nk but not in Mk . Clearly µ does not belong to the
G-module of harmonicity conditions. We may therefore find a harmonic cochain
k
c ∈ Char not annihilated by µ. However, since µ is supported on some T̂k (v) the
restriction of c to T̂k (v) will not be annihilated by µ. This restriction, by theorem
2.7, is also the restriction of a certain cη , so µ will not annihilate cη . But as µ is
supported on T̂k,min , it will not annihilate cSS η either, contradicting the fact that
µ ∈ Nk .
After this paper was written we received the preprint [I-S] by Iovita and Spiess. It
can be checked that their main theorem ([I-S], theorem 4.5) implies the hypothesis
of theorem 8.3, namely (8.17) is valid. It follows that the map
k
(8.18) Char → C k , c 7→ c|T̂k,min
is a well-defined G-isomorphism which intertwines the isomorphism produced by
Schneider and Stuhler and the one constructed by us.
9. The cohomology of Γ\X

In this section we study the two spectral sequences abutting to the de-Rham
cohomology of a variety of the form Γ\X, for Γ cocompact and discrete in G. One
is the Hodge-to-de-Rham spectral sequence, and the other the covering spectral
sequence. Our treatment is similar to [S], except that by tying the exposition to
our previous results we can exhibit them as the two spectral sequences associated to
a certain double complex. We address a conjecture of Schneider that the filtrations
induced on the cohomology by the two spectral sequences are opposite to each
2 Weak
duality is the following elementary fact from linear algebra. Let V be a vector space,
V its linear dual, W a subspace of V, W ⊥ its annihilator in V ∗ , and ⊥ (W ⊥ ) the annihilator of
∗
W ⊥ in V. Then ⊥ (W ⊥ ) = W. Weak refers to the fact that we do not need reflexivity.

other. This would lead to a “Hodge decomposition” of the cohomology of Γ\X.

For simplicity we do not introduce a module of coefficients (a representation of Γ),
although the more general case could probably be handled by our methods as well.
9.1. Preliminaries. Let Γ be a discrete, cocompact, torsion-free subgroup of G.

Then for every vertex v, the stabilizer of v in Γ, Γ ∩ Bv = {1}. Just to be on the
safe side we make the stronger assumption that if 1 6= γ ∈ Γ, then ρ(v, γv) ≥ 2.
This can be achieved by passing to a subgroup of finite index, and implies that the
subdomains X(v) and X(γv) are disjoint.
Let XΓ = Γ\X. By a fundamental theorem of Mumford (d = 1) and Mustafin
(in general), this rigid analytic space is the space associated to a smooth projective
algebraic variety defined over K, which we denote by XΓ,K [Mus]. We let πΓ : X →
XΓ be the canonical projection. For any cell τ we let XΓ (τ ) = πΓ (X(τ )). According
to our assumption on Γ, πΓ induces an isomorphism of X(τ ) onto XΓ (τ ): the former
is a fundamental domain for the latter. We shall make use of the covering U of
X introduced in Section 8, consisting of the X(v), and the covering UΓ of XΓ ,
consisting of the XΓ (v). Both are admissible open coverings in the rigid analytic
topology. Their nerves are the simplicial complexes T and TΓ = Γ\T .
The algebraic variety XΓ,K is the generic fiber of a scheme XΓ over OK , canoni-
cally associated with Γ, which is flat, projective, and nonsingular ([Mus], theorems
3.1 and 4.1). The special fiber XΓ,Fq is reduced, and is the union of transversally
crossing smooth divisors. The simplicial degeneration complex of XΓ,Fq (see [Mus]
for the definition) is canonically isomorphic to TΓ . The reduction map r induces
on XΓ a reduction map rΓ : XΓ → |TΓ |, and upon the identification of XΓ with
the K̄-valued points of XΓ , rΓ−1 (St(v)) = πΓ (X(v)) = XΓ (v) consists of the points
specializing to the component of XΓ,Fq indexed by v. Inside it, the part specializing
to smooth points is precisely rΓ−1 (v) = πΓ (Xv ).
9.2. A double complex. Let Ωs be the presheaf of rigid-analytic s-forms. Re-

stricted to the rigid analytic topology on XΓ it is a (rigid analytic) coherent sheaf.
The covering UΓ consists of the open sets XΓ (v), whose intersections are the XΓ (τ ),
and these are Stein spaces in the sense of Kiehl. By Kiehl’s version of Cartan’s the-
orem B, for any rigid analytic coherent sheaf G, H r (XΓ (τ ), G) = 0 for r > 0. It
follows that the rigid analytic sheaf cohomology H r (XΓ , G), which coincides with
the rigid analytic Čech cohomology Ȟ r (XΓ , G), can be computed using UΓ :
(9.1) H r (XΓ , G) = Ȟ r (UΓ , G),
without the need to pass to finer and finer coverings.
Consider the double complex (in non-negative degrees only)
(9.2) K r,s = C r (UΓ , Ωs ) = C r (U , Ωs )Γ
of alternating Čech cochains, with coboundary maps d0 = δ, the Čech differentia-

tion, and d00 = (−1)r d, the differentiation of forms modified by (−1)r :
K r,s+1
(9.3) ↑ (−1)r d .
K r,s δ
−
→ K r+1,s
L
The total complex (K · , dK ) is given as usual by K n = r+s=n K r,s , dK = d0 + d00 .
By the reasoning of the previous paragraph, the homology of the total complex K ·
42 EHUD DE SHALIT
is nothing but the de-Rham cohomology of XΓ :

(9.4) hn (K · , dK ) = HdR
n
(XΓ ),
and by GAGA rigid de-Rham cohomology and algebraic de-Rham cohomology co-
incide.
n
As usual there are two spectral sequence abutting to HdR (XΓ ), and two associ-
n
ated filtrations on HdR (XΓ ). Let us compute them, and start with the second one.
It is
(9.5) 00
E1r,s = Ȟ s (UΓ , Ωr ) = H s (XΓ , Ωr )
and 00 dr,s
1 is induced from d00 . This is of course the Hodge-to-de-Rham (Hodge-
Fröhlicher) spectral sequence, and the associated filtration is the Hodge filtration
· n
FdR HdR (XΓ ). It is well-known that this spectral sequence degenerates at E1 , and
consequently
00 r,s
E∞ = H s (XΓ , Ωr ) and
X
n
(9.6) bn = dim HdR (XΓ ) = hr,s
r+s=n
r,s s r
where h = dim H (XΓ , Ω ) are the Hodge numbers. According to [S], Section 2,
Proposition 1, hr,s = 0 unless 0 ≤ r, s ≤ d, and in that square we have
(9.7) hr,s = µ(Γ)δr+s,d + δr,s ,
where µ(Γ) = dim H d (Γ, K). (Note that by results of Kazhdan, Garland, Casselman
and Borel-Wallach H i (Γ, K) = 0 for 0 < i < d.) In particular the only interesting
d
cohomology is the middle one, HdR (XΓ ).
The first spectral sequence is 0 E1r,s = C r (UΓ , H sdR ), because for any of the XΓ (τ )
we know, since it is Stein, that its de-Rham cohomology is the cohomology of the
complex of global rigid analytic forms on XΓ (τ ). But this is also
(9.8) 0
E1r,s = C r (U, H sdR )Γ = C r (T , As )Γ .
The last isomorphism stems from the identification As (τ ) = HdR s
(X(τ )) proved in
theorem 7.7, and from the fact that T is the nerve of the covering U. The differential
0 r,s
d1 is induced from d0 = δ. Recall, from Section 5, that (C · (T , As ), δ) is a resolu-
s
tion of Char , the space of harmonic s-cochains on T . It is dual to the Schneider-
Stuhler resolution of the admissible representation Λs by the C· (T , (As )∗ ). Now
Cr (T , (As )∗ ) are clearly free of finite rank as K[Γ]-modules. Therefore C r (T , As )
are Γ-injectives and
(9.9) 0
E2r,s = hr (C · (T , As )Γ ) = H r (Γ, Char
s
) = H r (Γ, HdR
s
(X)).
We see that this is the covering spectral sequence ([S-S] Section 5, Proposition 2,
n
and [S], Section 1). The associated filtration on HdR (XΓ ) is the covering filtration
·
FΓ .
According to [S], Section 1, Theorem 2 the covering spectral sequence degenerates
at E2 , and consequently
0 r,s
(9.10) E∞ = H r (Γ, HdR
s
(X)).
The terms H r (Γ, HdR

s
(X)) were computed by Schneider, using the work of Borel
and Wallach. They turn out to be 0 outside the square 0 ≤ r, s ≤ d, and in that
square
(9.11) dim H r (Γ, HdR
s
(X)) = µ(Γ)δr+s,d + δr,s = hr,s .
9.3. The conjecture on a Hodge decomposition. One says that the filtrations
·
FdR and FΓ· are opposite if for every 0 ≤ p ≤ n + 1
p
M n+1−p
(9.12) H n = FdR Hn FΓ H n.
In view of the results of the previous paragraphs on the dimensions of the graded
pieces of the two filtrations, this is trivial for n 6= d, and for n = d it would follow
if
p
(9.13) FdR H d ∩ FΓd+1−p H d = 0
for every p. Setting, for p + q = d,
p
(9.14) H p,q = FdR H d ∩ FΓq H d
we get that if the two filtrations are opposite then
M
d
(9.15) HdR (XΓ ) = H p,q ,
p+q=d
which we regard as a p-adic analogue of the classical Hodge decomposition for

the variety XΓ .
The following conjecture was made by Schneider in [S].
·
Conjecture 9.1. The filtrations FdR and FΓ· are opposite to each other.
Let us give now a general criterion for the two filtrations coming from a double
complex (K r,s , d0 , d00 ) in non-negative degrees, to be opposite. Let (K · , d) be the
total complex, so that H n = hn (K · , d). First recall the two filtrations. Let
½ r,s ¾
0 p r,s K if p ≤ r
(9.16) F K =
0 otherwise
0 p n
L r,n−r
and F K = p≤r K the associated total complex. Then
0
(9.17) F p H n = Im {hn (0 F p K · , d) → hn (K · , d)} .
One similarly defines the second filtration 00 F p reversing the roles of the two indices.
In our case 0 F · = FΓ· and 00 F · = FdR
·
.
r,s
Lemma 9.2. Let ZII = Ker(d00 : K r,s → K r,s+1 ). Then
(i) If for every r and s such that r + s = n the map
·,s 0
(9.18) hr (ZII , d ) → hr (K ·,s , d0 )
is surjective then for every p we have H n = 0 F p + 00 F n+1−p .
(ii) If for every r and s such that r + s = n the above map is injective then for
every p we have 0 F p ∩ 00 F n+1−p = 0.
The proof of the lemma is by simple diagram chasing, and we omit it. A similar
criterion holds by reversing the roles of the indices, namely letting ZIr,s = Ker(d0 :
K r,s → K r+1,s ) and looking at the map
(9.19) hs (ZIr,· , d00 ) → hs (K r,· , d00 ).
44 EHUD DE SHALIT
One should note that the conditions are sufficient, but not necessary, so nothing can
be deduced from their failure. In our case we derive in fact four criteria, because
in each of the two ways the lemma can be applied, it is enough to prove either (i)
or (ii), as explained above.
·,s 0
Now applying the lemma in our situation we get that the map hr (ZII ,d ) →
hr (K ·,s , d0 ) is
(9.20) Ȟ r (UΓ , Ωscl ) → Ȟ r (UΓ , Ωs ),
and we retrieve the following criterion (compare [S], Section 2, Lemma 2).
Proposition 9.3. Suppose that Ȟ r (UΓ , Ωscl ) → Ȟ r (UΓ , Ωs ) is surjective for all r
and s such that r + s = d. Then Conjecture 4.1 is true.
Of course, we could stipulate instead that this map is injective. For example,
when s = 0, both are known to be true: since T is contractible Ȟ d (UΓ , Ω0cl ) =
Ȟ d (UΓ , K) = H d (Γ, K), while Ȟ d (UΓ , Ω0 ) = H d (XΓ , OX ) = H d (Γ, OX ), and
(9.21) H d (Γ, K) → H d (Γ, OX )
is an isomorphism.
It is instructive to compare the situation with the classical one. Let for the
moment X be a compact Kähler manifold, and let Ωscl be the sheaf of closed s-
forms, which by the Poincaré lemma is also the sheaf of exact s-forms (in the
classical complex topology). Using the exact sequence
s−1
(9.22) 0 → Ωcl → Ωs−1 → Ωscl → 0,
and arguing inductively on s, we see that the map H r (X, Ωscl ) → H r (X, Ωs ) is
surjective, and in fact
M 0 0
(9.23) H r (X, Ωscl ) = FdR
s r+s
HdR (X) = H r+s−s (X, Ωs ).
s0 ≥s
We hope to return to this question in the future.
Remark. Conjecture 4.1 was also settled in the affirmative in [I-S]. From the
point of view of our paper their method consists in showing that the criterion of
Proposition 9.3 holds. In fact, they show that the complex of logarithmic bounded
differential forms (with maps d = 0) injects densely into the complex Ω· and this
injection induces isomorphism on the level of Čech cohomology. In particular, this
injection factors through the complex of closed differential forms, so the hypothesis
of proposition 9.3 is valid.
10. Appendix : rigid de-Rham cohomology

In the appendix we collect some basic definitions concerning rigid de-Rham co-
homology, and prove the following.
Theorem 10.1. For every τ ∈ Tr , the cohomology ring of X̃(τ ) is generated by
the classes of the logarithmic forms ωa = d log a.
Corollary 10.2. The cohomology of X(τ ) is generated by the classes of ωa − ωb .
k
Corollary 10.3. The logarithmic k-classes span a dense subspace in HdR (X).
The theorem was used in Proposition 7.1, and the proof, which is technical, was
deferred to the appendix. The first corollary is immediate. The second corollary
is not a corollary of the theorem, but of all that we did in this paper. Indeed, we
k k
obtained the isomorphism HdR (X) ' Char ' Λ∗k (theorem 8.2, and corollary 5.8).
By definition λ ∈ Λk vanishes if and only if
(10.1) λ(e) = hλ, [ωe ]i = 0
for every e ∈ E k . In other words, every λ ∈ Λk which is orthogonal to all the log-
arithmic forms ωe , vanishes identically. This proves that the classes of logarithmic
forms are dense in the weak ∗ -topology on HdRk
(X).
It is probably true that the logarithmic classes are dense in the cohomology of
any simplicial subdomain of X, but except for the X(τ ) and X itself, we do not
prove it.
10.1. On the definition of de-Rham cohomology. For any smooth rigid an-
alytic space X over K, the complex Ω·X/K of rigid analytic forms is a complex of
coherent sheaves, and de-Rham cohomology is its hypercohomology
k
(10.2) HdR (X) = Hk (X, Ω· ).
This notion is not a good one in general. Even in the simplest case of the affinoid
1-disk the first de-Rham cohomology, while expected to vanish, turns out to be
infinite dimensional. Several ways to correct the annomalies have been suggested.
Following ideas of Monsky-Washnitzer and, 30 years later, of Berthelot (both do not
attach cohomology groups to rigid analytic spaces, but to schemes in characteristic
p), Elmar Große-Klönne [G-K] circumvented the problem by replacing the category
of rigid analytic spaces with a category of dagger-spaces. These spaces are modeled
on (quotients of) the algebra of over-convergent power-series rather than the Tate
algebra. Every dagger space X † gives rise, functorially, to a rigid analytic space X =
(X † )rig , but starting with a rigid analytic space X, the existence of a dagger model
X † is only guaranteed locally, and when it exists it is not unique, in general. The de-
Rham cohomology of smooth dagger-spaces is well-behaved. If X † is of finite type,
its cohomology is finite dimensional, and there are comparison theorems between
de-Rham cohomology of dagger spaces and Monsky-Washnitzer or (Berthelot’s)
rigid cohomologies. For example, if A is a flat w.c.f.g. algebra over OK (see [M-
W], Section 2) with smooth reduction A0 = A/(π), and A† = A ⊗OK K is the
k
associated dagger-algebra, then there is a canonical isomorphism HdR (Sp(A† )) '
k
HM W (Spec(A0 )).
Although to develop the general machinery one has to go through a great deal
of trouble, the rigid analytic spaces X(τ ) that we consider in this work are of a
very simple nature. Each X(τ ) is Stein : it is the increasing union of affinoid
subdomains, each contained in the “interior” of the next one ([G-K], Section 2.2,
or [Ki], definition 2.3). In this case there is no distinction between over-convergent
forms and rigid-analytic forms on X(τ ). More precisely, there is a unique “dagger-
space” structure X(τ )† on X(τ ) compatible with its given rigid-analytic structure,
and the category of dagger-coherent sheaves on it is equivalent to the category
of rigid-coherent sheaves (see [G-K], Satz 2.17). Thus HdR (X(τ )) = HdR (X(τ )† )
46 EHUD DE SHALIT
([G-K], Korollar 4.6). Moreover, by Kiehl’s “Theorem B” coherent cohomology on

X(τ ) vanishes in positive degrees, hence
k
(10.3) HdR (X(τ )) = Hk (X(τ ), Ω· ) = hk (Γ(X(τ ), Ω· )).
The same applies to X̃(τ ) and to X. These remarks explain why we chose to work
with rigid de-Rham cohomology despite its well-known annomalies : for the spaces
under consideration it gives just the right answer.
10.2. Mayer-Vietoris revisited. Next we single out a special case of the Mayer-
Vietoris spectral sequence. Suppose that X 0 = X ∪ Y is an admissible cover of
a smooth rigid analytic space X 0 by open subdomains, and Z = X ∩ Y. Then
proposition 8.1 gives the Mayer-Vietoris long exact sequence
k−1 k
(10.4) HdR (Z) → HdR (X 0 ) → HdR
k k
(X) ⊕ HdR k
(Y ) → HdR (Z) → · · ·
We shall apply it to deduce, in the special case of X(τ ), an exact sequence of
Gysin type. Similar exact sequences appear in [Mo], [Ber] and [G-K], but we found
no easy way to relate our set-up to theirs, so we shall prove what we need from first
principles. The proof for X(v), v a vertex, will be given first, because it is simpler.
Note that we do not prove that the complement of a tubular neighborhood of a gen-
eral hyperplane arrangement has logarithmic cohomology (cohomology represented
by logarithmic forms), although we believe it is true.
10.3. Continuity of the cohomology in the domain. We shall need another

general fact on de-Rham cohomology.
Proposition 10.4. Let U ⊂ P(V ∗ ) be an admissible open subset, and {Un } (n ≥ 1)
an admissible open cover of U satisfying Un ⊂ Un+1 . Suppose that the cohomologies
of Un and U are finite-dimensional vector spaces. Then
k k
(10.5) HdR (U ) = limHdR (Un ).
←
k k
Proof. A proof that HdR (P, U ) = limHdR (P, Un ) can be found in [S-S], section 2,
←
corollary 5. The statement as we need it can be derived from this using the long
exact sequence for pairs, and the exactness of lim on the category of inverse systems
←
of finite-dimensional vector-spaces.
10.4. Cohomology of disk bundles. Recall that if D is the open unit disk
{z| |z| < 1}, then for any smooth rigid analytic space X projection from X × D
k k
onto X yields an isomorphism HdR (X) ' HdR (X × D). This is called the homo-
topy invariance of cohomology. See [S-S], p.57. We generalize it as follows. An
(open) disk bundle over a rigid analytic space X is a map p : Y → X, for
which there exists an admissible open cover U = {Xα } of X and isomorphisms
Yα = p−1 (Xα ) ≈ Xα × D compatible with p.
Let U = {Xα } be an admissible cover of X. Suppose we are given for each α a
nowhere-vanishing rigid analytic function gα on Xα such that if x ∈ Xα ∩ Xβ then
|gα (x)| = |gβ (x)|. Let
(10.6) Y = {(x, z)| x ∈ Xα , |z| < |gα (x)| for some α}.
Then Y is an open disk bundle obtained by pasting the subspaces Yα which project
onto Xα . Each Yα is isomorphic to Xα × D under the map sending (x, z) to
(x, gα (x)−1 z). We call such a Y an embedded disk bundle, because it is embedded
in X × A1 .
For any disk bundle p : Y → X and any trivializing cover {Xα } as above, we
have by the homotopy invariance
q q
(10.7) HdR (Xα ) ' HdR (Yα ).
Proposition 10.5. (compare [S-S], section 2, lemma 1) The projection p : Y → X
of a disk bundle induces
k k
(10.8) HdR (X) ' HdR (Y ).
Proof. The map p induces a homomorphism of spectral sequences p∗ from the
Mayer-Vietoris spectral sequence on X w.r.t. the cover {Xα } to the Mayer-Vietoris
spectral sequence on Y w.r.t. the cover {Yα }. Since the E2 terms coincide, so do
their abutments.
10.5. Relative annuli. Relative (closed, or affinoid) annuli over an affinoid base
are discussed in [BGR], Section 9.7.1. Here we prefer to work with open annuli.
Let X = Sp(B) be an affinoid, and λ ∈ B a rigid analytic function on X with
0 < |λ(x)| < 1 throughout X. The annulus AX (λ, 1) is the rigid analytic subspace
of X × A1 defined by
(10.9) AX (λ, 1) = {(x, z)| |λ(x)| < z < 1} .
The function x 7→ |λ(x)| is uniquely determined by the annulus (hence λ is de-
termined up to a unimodular function), and is called the conformal invariant of
AX (λ, 1). If X is reduced (so we can take the sup norm | · | on B as our Banach-
norm defining its topology, see [BGR] section 6.2.4), then rigid analytic functions
on AX (λ, 1) are power series
X
(10.10) bn z n , bn ∈ B
n∈Z
such that for every constant r < 1 we have

(10.11) lim |bn |rn = 0 and lim |bn λn |r−n = 0.
n→∞ n→−∞
Assume that X is smooth. Fix any Banach norm on the space of k-forms on X
(they are all equivalent). A k-form on AX (λ, 1) can be written uniquely as
X X
(10.12) ω= ωn z n + ηn z n dz,
n∈Z n∈Z
where ωn (resp. ηn ) are k-forms (resp. k − 1-forms) on X satisfying the same

growth conditions as the coefficients bn above. The form ω is closed if and only if
(10.13) dωn = 0, (−1)k ωn n + dηn−1 = 0
for every n. In particular η−1 should be closed and we call
k−1
(10.14) ρ(ω) = [η−1 ] ∈ HdR (X)
48 EHUD DE SHALIT
the residue of ω. The residue of an exact k-form vanishes. Conversely, suppose

that ρ(ω) = 0. Then η−1 = dξ and
 
X
(10.15) ω = ω0 + d (−1)k−1 n−1 ηn−1 z n + ξz −1 dz 
n6=0
converges (here we use the fact that the annulus is open), so ω is exact if and only
if ω0 is exact on X. This proves that the projection p : AX (λ, 1) → X induces an
isomorphism (the Künneth formula)
k k dz k−1
(10.16) HdR (AX (λ, 1)) ' HdR (X) ⊕ [ ] ∧ HdR (X).
z
The cohomology class [dz/z] depends only on the orientation of the annulus. If z 0
is another parameter on the annulus inducing the same orientation in each fiber,
then log(z/z 0 ) is rigid analytic in AX (λ, 1), so dz/z − dz 0 /z 0 is exact.
10.6. The cohomology of annulus bundles. An (open) annulus bundle is a

map p : Y → X which, locally in the rigid analytic topology, looks like AX (λ, 1) →
X. We call the annulus bundle orientable if we can choose an admissible open
covering {Xα } by affinoids, and isomorphisms
(10.17) Yα = p−1 (Xα ) ' AXα (λα , 1)
which over the intersections Xα ∩ Xβ induce orientation-preserving isomorphisms
of the fibers. We then deduce from the Künneth formula the short exact sequence
k p∗k ρ
k−1
(10.18) 0 → HdR (Xα ) → HdR (Yα ) → HdR (Xα ) → 0,
where ρ is the “residue along the annulus”. Fixing an orientation on the fiber fixes
ρ (otherwise it is determined up to a sign), and the sequence is canonically split
then. Applying the Mayer-Vietoris spectral sequence as in the case of a disk bundle,
we get the following proposition.
Proposition 10.6. Let p : Y → X be an oriented annulus bundle. Then there
exists a split short exact sequence
k p∗
k ρ
k−1
(10.19) 0 → HdR (X) → HdR (Y ) → HdR (X) → 0.
Proof. Fix a trivializing admissible open cover U = {Xα } by affinoids. For each α
let ωα be the class [dz/z] for a parameter z on the annulus over Xα . Let
q
(10.20) Hq (Yα ) = HdR (Yα )
and
q q−1
(10.21) H̃q (Yα ) = HdR (Xα ) ⊕ HdR (Xα ).
Then for each α the projection p induces an isomorphism H̃q (Yα ) ' Hq (Yα ), which
sends (ξ, η) to p∗ ξ + ωα ∧ p∗ η. The same holds on the intersections of several Yα0 s,
and the isomorphisms are compatible with the restriction maps. They therefore
induce isomorphisms of the two spectral sequences whose E2 terms are Ȟ p (U, Hq )
and Ȟ p (U, H̃q ). But by proposition 8.1 the first abuts to HdR
k
(Y ) and the second
k k−1
to HdR (X) ⊕ HdR (X).
10.7. The cohomology of X(v) (v a vertex). If L is a lattice in VK , a ∈ L − πL

and 0 < r < 1, we write, as in Section 6.2,
(10.22) H̃a (L, r)
(resp. H̃a (L, r )) for the set of z ∈ V ∗ such that |z(a)| ≤ r|z|L (resp. |z(a)| <
−
r|z|L ). Here |z|L = maxb∈L |z(b)|.

d+1
If we introduce coordinates in VK so that L = OK and a = (1, 0, . . . , 0), then
H̃a (L, r) is characterized by |z0 | ≤ r max0≤i≤d |zi |. We write Ha (L, r) for the image
of H̃a (L, r) in P(V ∗ ). If |π| ≤ r < 1 (resp. |π| < r < 1) then these sets (resp. the
same sets with r replaced by r− ) depend only on the image of a in L/πL.
If v = [L] then (see (6.12))

[
(10.23) X(v) = P(V ∗ ) − Ha (L, |π|)
a
with a ranging over representatives of L/πL − {0} in L.

More generally, consider a0 , . . . , am ∈ L − πL, 0 < r0 ≤ · · · ≤ rm < 1 and
m
[
∗
(10.24) X = X(a0 , . . . , am ; r0 , . . . , rm ) = P(V ) − Hai (L, ri ).
i=0
0
Let X = X(a1 , . . . , am ; r1 , . . . , rm ), L̄ = L/ha0 i, āi = ai |Ha0 ∈ L̄, and put
m
[
(10.25) X0 = Ha0 ∩ X 0 = Ha0 − Hāi (L̄, ri ).
i=1
Suppose that r0 < r1 and choose s such that r0 < s ≤ r1 . Let Y = Ha0 (L, s− ) ∩ X 0 .
Note that X 0 = X ∪ Y.
Lemma 10.7. Y is a disk bundle over X0 .
Pd
Proof. Choose coordinates so that z(a0 ) = z0 and z(ai ) = j=0 aij zj for 1 ≤ i ≤
m, where maxj |aij | = 1 for every i, because ai ∈ L − πL. Now z ∈ Ha0 (L, s− )
if and only if |z0 | < s maxj |zj |, and in this P case max |zj | = max1≤j |zj |. But
z ∈ Hai (L, ri ) (1 ≤ i ≤ m) if and only if | j aij zj | ≤ ri maxj |zj |. Therefore for
z ∈ Ha0 (L, s− ) and 1 ≤ i ≤ m we have z = (z0 : z1 : · · · : zd ) ∈ Hai (L, ri ) if and
only if z 0 = (0 : z1 : · · · : zd ) ∈ Hai (L, ri ).
Consider now the covering of X0 by X0,i = {z 0 ∈ X0 | max |zj | = |zi |}. Then for
0
z ∈ X0,i the point z belongs to Y if and only if |z0 | < s|zi |. The part of Y lying
over X0,i is therefore isomorphic to X0,i × D, and the lemma follows.
Similarly, X ∩ Y is an oriented annulus bundle over X0 . The Mayer-Vietoris long

exact sequence of section 10.2 for the pair {X, Y }, together with propositions 10.5
and 10.6, yield a long exact sequence
k−2 k−1
··· → HdR (X0 ) ⊕ HdR (X0 )
k k−1
(10.26) → HdR (X 0 ) → HdR
k k
(X) ⊕ HdR (X0 ) → HdR k
(X0 ) ⊕ HdR (X0 ) → · · ·.
k
Cancelling the terms HdR (X0 ), which map isomorphically onto each other, gives us
the following Gysin long exact sequence :
k−2 k ε k−1 ρ
(10.27) · · · → HdR (X0 ) → HdR (X 0 ) → HdR
k
(X) → HdR (X0 ) → · · ·
50 EHUD DE SHALIT
with ε induced by the inclusion X ⊂ X 0 and ρ induced by restricting a k-form to

X ∩ Y, then taking its residue along the annulus.
The same applies mutatis mutandis to the spaces X̃, X̃ 0 and X̃0 in V ∗ .
Proposition 10.8. (i) For every choice of ai and ri as above there is a short exact
sequence
k ε k−1 ρ
(10.28) 0 → HdR (X̃ 0 ) → HdR
k
(X̃) → HdR (X̃0 ) → 0.
The maps ε and ρ commute with the restriction maps induced by the inclusion
X(a0 , . . . , am ; r0 , . . . , rm ) ⊃ X(a0 , . . . , am ; R0 , . . . , Rm )
whenever ri ≤ Ri (and 0 < R0 ≤ · · · ≤ Rm < 1).
(ii) Restriction is an isomorphism
k k
(10.29) HdR (X̃(a0 , . . . , am ; r0 , . . . , rm )) ' HdR (X̃(a0 , . . . , am ; R0 , . . . , Rm )).
k
(iii) Every class in HdR (X̃) is represented by a logarithmic form (i.e. a form in
the subring generated by d log(ai ), 0 ≤ i ≤ m).
k k
Corollary 10.9. Every class in HdR (X(v)) = HdR (Xv† ) is represented by a loga-
rithmic form.
Proof. We prove the proposition by induction on m and the dimension d. We may
therefore assume the validity of (ii) and (iii) for X̃ 0 and X̃0 , so we know that every
k−1
class in HdR (X̃0 ) is represented by a logarithmic form η. We shall also assume at
the beginning that
(10.30) 0 < r0 < r1 ≤ · · · ≤ rm < 1
so that we shall be able to use the Gysin long exact exact sequence. Once we
establish (i) in this setting, (ii) will follow from (i) and the validity of (ii) for X̃ 0
and X̃0 . We shall then be able to relax the restriction r0 < r1 by continuity.
We shall show that ρ is surjective. The surjectivity of ρ (with k replaced by
k−1
k − 1) will imply then that ε is injective. Assume therefore that [η] ∈ HdR (X̃0 ).
By induction we may assume that η is logarithmic. Let η be a lifting of η to X̃ 0 .
0
Picking d log(ai ) as a lifting of d log(āi ) we may assume that η 0 is logarithmic too,

and in particular closed. Let ω = d log(a0 ) ∧ η 0 . Then ω is a closed (logarithmic)
form on X, and ρ([ω]) = [η].
k
This argument proves also that every class in HdR (X̃) may be modified by a
logarithmic class to land in ker(ρ) = Im(ε). But by induction we know that Im(ε)
consists of logarithmic classes, hence we obtain (iii).
It remains to relax the assumption r0 < r1 . Assume that r0 = r1 , choose ε > 0
small, and ri,n = ri + ε/n for i ≥ 1, while r0,n = r0 . Let
(10.31) Un = X̃(a0 , . . . , am ; r0,n , . . . , rm,n ).
Then we know our result for Un , and in particular we know that the inclusion
Un ⊂ Un+1 induces an isomorphism in cohomology. Proposition 10.4 then concludes
the proof.
The corollary is a consequence of (10.23),
S and of the fact that as r → 1, the limit
of the de-Rham cohomology of P(V ∗ ) − a Ha (L, r), is the dagger cohomology of
Xv† .
10.8. The cohomology of X (τ ). Let us now check how the proof of the propo-
sition has to be modified to deal with X(τ ), for a general simplex
(10.32) τ = (M0 ⊃ · · · ⊃ Mr ⊃ πM0 ).
Note that X(τ ), being the intersection of the X(vi ), vi = [Mi ], is of the form
r
[ [
(10.33) X(τ ) = P(V ∗ ) − Ha (Mi , |π|).
i=0 a∈A0i
In this expression A0i is a set of representatives of Mi − πMi modulo πMi . However,

if a ∈ A0i and a ∈ / πMi−1 then a (or a representative equivalent to it modulo
πMi−1 ) will appear also in A0i−1 , and Ha (Mi−1 , |π|) ⊃ Ha (Mi , |π|), since for every
z, |z|Mi ≤ |z|Mi−1 . We may therefore replace A0i in the above expression by Ai ,
a set of representatives of πMi−1 − πMi modulo πMi (when i = 0 understand
πM−1 = Mr ).
More generally, for the purpose of using induction, consider a sequence of lattices
(10.34) π −1 L0 ⊃ Lr ⊃ · · · ⊃ L1 ⊃ L0 ,
a collection {a0 , . . . , am } such that ali , . . . , ali+1 −1 ∈ Li+1 −Li (l0 = 0, lr+1 = m+1,
Lr+1 = π −1 L0 , some of the blocks of the a0j s may be empty), and a sequence of
real numbers
(10.35) 0 < r0 ≤ · · · ≤ rm ≤ |π|.
We assume that l1 > 0 (otherwise renumber the lattices). X(τ ) is of this shape
with all ri = |π|. Unlike the case of X(v), we must insist that all ri ≤ |π|. With
this data we associate
r li+1
[ [−1
(10.36) X = X({Li }, {ai }, {ri }) = P(V ∗ ) − Haj (π −1 Li , rj )
i=0 j=li
0
and we write X for the X associated with the same lattices, {a1 , . . . , am } and
{r1 , . . . , rm }. We also write X0 = X 0 ∩ Ha0 . It is again of the same form, with Li
replaced by L̄i = Li / ha0 i , and aj replaced by āj = aj |Ha0 (1 ≤ j).
To conclude the proof of theorem 10.1 we must show that the cohomology of X
is generated by logarithmic forms. We may assume, by induction, that this is the
case with X 0 and X0 .
Choose ε > 0 small, and let r0,ε = r0 , but ri,ε = ri + ε for 1 ≤ i ≤ m, so

that r0,ε < r1,ε . Choose r0,ε < s ≤ r1,ε . We shall work with the domains Xε , Xε0
and X0,ε defined by the perturbed radii ri,ε , and then take the limit as ε → 0,
using proposition 10.4. Since ri,ε might be bigger than |π|, we can not assume the
induction hypothesis for Xε0 and X0,ε . Let Yε = Ha0 (π −1 L0 , s− ) ∩ Xε0 and note that
Xε0 = Xε ∪ Yε .
Choose a basis {αi } of VK adapted to (10.34), so that π −1 L0 = hα0 , . . . , αd i and
(10.37) Li = hα0 , . . . .αdi −1 , παdi , . . . , παd i
where di = dim Li /L0 . Since a0 ∈ L1 − L0 , we may also assume that a0 = α0 .
Now z ∈ Ha0 (π −1 L0 , s− ) if and only if |z0 | < s max |zk |. Suppose this is the case.
Pick some 1 ≤ j ≤ m and suppose that li ≤ j < li+1 , so that aj ∈ Li+1 − Li . If
52 EHUD DE SHALIT
i = 0 we are in the situation of lemma 10.7 and we saw there that for z such that
|z0 | < s max |zk |,
(10.38)
z = (z0 : · · · : zd ) ∈ Haj (π −1 L0 , rj,ε ) ⇔ z 0 = (0 : z1 : · · · : zd ) ∈ Haj (π −1 L0 , rj,ε ).
If j ≥ l1 (i ≥ 1) then z = (z0 : z1 : · · · : zd ) ∈ Haj (π −1 Li , rj,ε ) if and only if
(10.39) |z(aj )| ≤ rj,ε max{|π −1 z0 |, . . . , |π −1 zdi −1 |, |zdi |, . . . , |zd |}.
On the other hand z 0 = (0 : z1 : · · · : zd ) ∈ Haj (π −1 Li , rj,ε ) if and only if
(10.40) |z 0 (aj )| ≤ rj,ε max{|π −1 z1 |, . . . , |π −1 zdi −1 |, |zdi |, . . . , |zd |}.
Since |z0 | < s max |zk | implies
(10.41) |z(aj ) − z 0 (aj )| < rj,ε max{|π −1 z1 |, . . . , |π −1 zdi −1 |, |zdi |, . . . , |zd |}
we see that z ∈ Yε ⇒ z 0 ∈ X0,ε .
Suppose that z 0 ∈ X0,ε . Then the above also shows that z ∈ Yε if and only if for
every j ≥ l1
(10.42) rj,ε |π −1 z0 | < |z 0 (aj )|.
This proves the following lemma.
Lemma 10.10. Yε is a disk bundle over X0,ε . The intersection Xε ∩ Yε is an
oriented annulus bundle over X0,ε .
Proof. The case of Yε is clear, because a point z = (z0 : z 0 ) ∈ Yε if and only if
z 0 ∈ X0,ε and z0 satisfies
−1
(10.43) |z0 | < s max |zk | and |z0 | < min rj,ε |π||z 0 (aj )|}.
1≤k l1 ≤j
For Xε ∩ Yε note that since r0,ε ≤ |π|, for every z 0 ∈ X0,ε

−1
(10.44) r0,ε max |zk | < min rj,ε |π||z 0 (aj )|.
1≤k l1 ≤j
Thus the fiber of Xε ∩ Yε above z 0 is a non-empty annulus.
From (10.4), propositions 10.5 and 10.6 we get a long exact sequence
k
··· → HdR (Xε0 )
k k k−1 k k+1
(10.45) → HdR (Xε ) ⊕ HdR (X0,ε ) → HdR (X0,ε ) ⊕ HdR (X0,ε ) → HdR (Xε0 ).
k
As before, cancelling the terms HdR (X0,ε ) which map isomorphically onto each
other, we get the Gysin long exact sequence
k−2 k k−1 ρ
(10.46) · · · → HdR (X0,ε ) → HdR (Xε0 ) → HdR
k
(Xε ) → HdR (X0,ε ) → · · ·.
Using the exactness of lim on the category of inverse systems of finite-dimensional
←
vector spaces, and proposition 10.4, we take the limit as ε → 0 and obtain the exact
sequence
k−2 k k−1 ρ
(10.47) · · · → HdR (X0 ) → HdR (X 0 ) → HdR
k
(X) → HdR (X0 ) → · · ·
k
in which all the radii ri ≤ |π|. From here we deduce that HdR (X) is generated by
logarithmic forms precisely as in the proof of proposition 10.8.
References
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Inv. Math. 128, 329-377 (1997)
[B-G-R] Bosch, S., Güntzer, U., Remmert, R. : Non-archimedean analysis,
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E-mail address: deshalit@math.huji.ac.il

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RESIDUES ON BUILDINGS, AND DE-RHAM COHOMOLOGY

OF p-ADIC SYMMETRIC DOMAINS

0.1. Background. Let K be a finite extension of the field Qp . Drinfel’d’s p-adic

Date: June 1, 1999.

The primary purpose of this work is to develop a theory of residues in any

finite, a great deal of work has been done on the cohomology of

the complement of the hyperplane arrangement A. In particular Orlik and Solomon

Now assume that K is p-adic (a finite extension of Qp ). Let T be the Bruhat-Tits

If v is a vertex of T , then v determines (up to a homothety) a lattice L in VK ,

(Monsky-Washnitzer cohomology [M-W]; de-Rham cohomology is not well-behaved

Fundamental to our proof is a triangle of maps

In the last section we investigate applications to the cohomology of Γ\X, where Γ

I would like to thank E. Dror-Farjoun, G. Kalai, R. Livné and S. Mozes for

Part 1. Local systems on buildings attached to hyperplane

classes [L] of lattices L ⊂ VK . The k-cells Tk (0 ≤ k ≤ d, we use cell and simplex

1.2. The combinatorial metric ρ. There is a metric ρ on T0 defined as follows.

vertices in common. We shall denote by Tk (τ ) the set of k-cells contiguous to an

1.3. The apartments of T . Fix a basis α = (α0 , . . . , αd ) of VK . The apartment

involution sW ∈ G which normalizes Tα and on Aα induces a reflection in the wall

1.5. The topological realization of T and the Euclidean metric d. We let

If Aα is an apartment as in Section 1.3, determined by the basis α, we can map

2. The local system of algebras on T

There is a unique derivation δ of Ẽ, homogeneous of degree -1, mapping each

(the symbol eba means that ea is omitted from the product)

(2.3) δ(ea0 ∧ · · · ∧ eak ) = (ea1 − ea0 ) ∧ · · · ∧ (eak − ea0 )

Definition 2.1. Let τ ∈ Tbr , and let M0 ⊃ · · · ⊃ Mr ⊃ πM0 be a lattice flag

2.2. The restriction map. If τ ≤ σ, then I(τ ) ⊆ I(σ), so we get a restriction

By the pigeonhole principle we may assume, rearranging the ai , that ισ (a0 ) =

Let µ be the maximal element in A. Then we have a splitting

(2.27) Sk+1 (τ )00 = {eµ ∧ eS ; S ∈ Sk (τ )0 } .

Lemma 2.6. If S Â T are in Sk+1 (τ )00 , then (σ̃T , eS ) = 0, (σ̃S , eS ) = 1.

Proof. Suppose S = (a0 Â · · · Â ak ) and T = (b0 Â · · · Â bk ), a0 = b0 = µ, . . . ,

Corollary 2.7. The collection

2.5. The case A = P(VK ). In this case A is G-invariant. Every g ∈ G induces a

for every σ ∈ T̂k and e ∈ E k , which can be phrased also as

3. Ak (τ ) as a space of local harmonic cochains

3.1. Harmonic cochains. A map c : Tbk → K will be called harmonic if it

• (Property C). Let k ≥ 1. Fix σ ∈ T ck and an index 0 ≤ j ≤ k. Let C = C(σ, j)

• (Property D). Let σ = (v0 , v1 , . . . , vk+1 ) ∈ T[

3.2. Harmonicity of the cochains ce .

Property B : It is enough to verify the following special case of B, from which

and dim L0j /Lj+1 = 1. Then

Property C : Fix σ ∈ Tck (k ≥ 1), and an index 0 ≤ j ≤ k. Let C = C(σ, j) be the

We have to show that for ẽ = ea0 ∧ ea1 ∧ · · · ∧ eak

Property D : Consider ẽ = ea0 ∧ · · · ∧ eak ,

3.3. Injectivity of the local cochain homomorphism. Now fix τ = (M0 ⊃

3.4. The isomorphism Ak (v) ' Chark

is equivalent to a linear combination of σ 0 of minimal type with

if τ = (v0 , . . . , vp+1 ) and τi = (v0 , . . . , vbi , . . . , vp+1 ). Then (C · (T , A), ∂) is a complex

4.4. Vanishing of H 1 . Let c be a 1-cocycle with values in Ã. We shall modify c

Step (n, 1).

Lemma 4.1. Every v ∈ Γ0 (n) has 1 or 2 contiguous vertices u at distance n − 1

Step (n, 2).

Lemma 4.3. The map

5. Acyclicity (in general, A = P(VK ))

We now recall the all-important acyclicity theorem of Schneider and Stuhler.

Theorem 5.4. A linear functional λ ∈ Λk factors through Ak (τ ) if and only if it

Let us therefore assume that λ is fixed by Uu , u ∈ T0 . We shall use the har-

Lemma 5.6. Let σ = (v0 , . . . , vd ) be a chamber (top dimensional simplex) in