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 ℚ_{p}. Drinfel’d’s p-adic symmetric domain of dimension d over K, is the complement 𝔛 in ℙ^{d} 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 GL_{d+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 topology, 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 axioms. Their computations apply to rigid de-Rham cohomology, and produce, for each 0 ≤ k ≤ d, an isomorphism between H_{dR}^{k}(𝔛) and a certain space C^{k} of kcochains 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

[η] ↦ c_{η}^{SS}. (0.1)

The approach taken by Schneider and Stuhler left some open questions. Let us list a few of them.

Date: June 1, 1999.

• For any two K-rational hyperplanes with equations a = 0 and b = 0, the logarithmic form d log(a/b) represents a class in H_{dR}^{1}(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 c_{η}^{SS} 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 c_{η}^{SS} 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 give an isomorphism of H_{dR}^{k}(𝔛) onto the space C_{har}^{k} 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 V_{K} a finite dimensional vector space over K of dimension d + 1, d ≥ 0. As usual we identify lines through the origin in V_{K} with linear hyperplanes in the dual space V_{K}^{∗}. Let A ⊆ ℙ(V_{K}) be a family of lines in V_{K}. For a ∈ A denote by H_{a} the hyperplane in the projective space ℙ(V_{K}^{∗}), given by the equation a = 0. When A is finite, a great deal of work has been done on the cohomology of

M_{A} = ℙ(V^{∗}) − ⋃_{a∈A} H_{a}, (0.2)

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

A ≃ H_{dR} (M_{A}) (0.3)

with the de-Rham ring of M_{A}. A nice property of A is that it depends only on the intersection pattern of the H_{a}’s, so geometric questions are reduced to combinatorics, 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 ℚ_{p}). Let T be the Bruhat-Tits building of G = P GL(V_{K}). 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 V_{K}^{∗}. 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 V_{K}, 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 𝔽_{q}. This arrangement depends on the integral structure, namely on v. Let M_{A(v)} be the complement of this hyperplane arrangement, which is a smooth affine variety over 𝔽_{q}. Then one can prove that

A(v) ≃ H_{MW} (M_{A(v)}) (0.4)

(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 A^{k}(τ) can be identified with the space which is obtained by restricting harmonic k-cochains to T̂_{k}(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 T ). This space is contained in C_{har}^{k}(τ), which is the space of all maps T̂_{k}(τ) → K satisfying the harmonicity conditions “when applicable”. For τ = v, a vertex, they even coincide.

Another new feature of our construction is that for every face inclusion τ ≤ σ, we get a canonical homomorphism

r_{σ}^{τ} : A(τ) → A(σ), (0.5)

which makes the collection of algebras A(τ) into a (cohomological) local system \underline{A} on T . When A = ℙ(V_{K}), 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(V_{K}) 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(V_{K}) 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.

0.3. Survey of the main results (Part II). In Section 6, to every [η] ∈ H_{dR}^{k}(𝔛) we attach its residues res_{σ}η ∈ K along pointed k-cells σ ∈ T̂_{k}. We then show that the cochain c_{η} given by c_{η}(σ) = res_{σ}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 c_{η} is an isomorphism of G-modules

c : H_{dR}^{k}(𝔛) ≃ C_{har}^{k}, (0.6)

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 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

\begin{aligned}A^{·}(τ) & & → & & H_{dR}^{·}(X(τ)) \\ & ↘ & & ↙ \\ & & C_{har}^{·}(τ) \end{aligned} . (0.7)

The arrow on the top takes a generator e_{a}−e_{b} 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

A^{·}(τ) ≃ H_{dR}^{·}(X(τ)), (0.8)

and for a vertex v also H_{dR}^{·}(X(v)) ≃ C_{har}^{·}(v).

In order to globalize the isomorphism H_{dR}^{k}(X(v)) ≃ C_{har}^{k}(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 \underline{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 c^{SS}, 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 c^{SS} 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 \binom{d}{k} types of pointed k-cells in general). Since a harmonic cochain is determined by its values on cells of minimal type, C^{k} and C_{har}^{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: c_{η}(σ) = c_{η}^{SS}(σ). It should be remarked that the fact that the spaces C^{k} and C_{har}^{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 result that not only the spaces coincide, but the two isomorphisms of H_{dR}^{k}(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 ℚ_{p}, π a uniformizer of K, and q the cardinality of O_{K}/(π). We normalize the absolute value on the algebraic closure K̄ (and on its completion ˆ K̄) so that |π| = q^{−1}. Let us review some definitions and standard results on the Bruhat-Tits building of P GL_{d+1}(K) ([B-T], [Br], [Mus]).

Fix an integer d ≥ 0 and a d + 1 dimensional vector space V_{K} over K. Write V_{K}^{∗} for the dual space and V, V^{∗} for the associated affine schemes. By a basis of V_{K} (or V_{K}^{∗}) we always mean an ordered basis. A basis of V_{K} determines a dual basis of V_{K}^{∗} and vice versa. A lattice in a finite dimensional vector space over K is a finitely generated O_{K}-module spanning the vector space over K. Two lattices L and L′ lie in the same dilation class if there exists a λ ∈ K^{×} such that L′ = λL.

Let G = P GL(V_{K}). The Bruhat-Tits building of G, T , is a simplicial complex which may be described as follows. The vertices T_{0} of T consist of dilation classes [L] of lattices L ⊂ V_{K}. The k-cells T_{k} (0 ≤ k ≤ d, we use cell and simplex synonymously) consist of k + 1-tuples {[L_{0}], [L_{1}], ..., [L_{k}]} where

L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0} (1.1)

(strict inclusion). We call such a sequence a lattice flag, or in short a flag, in L_{0}. It is confined to lie between L_{0} and πL_{0}. 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 v_{0} = [L_{0}]. Equivalently, a pointed k-cell is given by an ordered k + 1-tuple σ = ([L_{0}], ..., [L_{k}]) where the L_{i} are as above. We shall write

σ = (L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0}) (1.2)

if the L_{i} are chosen as in (1.1) and [L_{0}] is the distinguished vertex. They are then unique up to a dilation by a common factor. Let T̂_{k} be the set of pointed k-cells. Any pointed k-cell σ can be assigned a type, which is the sequence 𝔱(σ) = (e_{0}, e_{1}, ..., e_{k}), defined by e_{i} = dim L_{i}/L_{i+1} (here and below we put L_{k+1} = πL_{0}, and the dimension is over O_{K}/(π) ≃ 𝔽_{q}). Clearly each e_{i} is positive and ∑_{i=0}^{k} e_{i} = d + 1. There are \binom{d}{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 ∈ T̂_{k} with distinguished vertex v_{0} are in one-to-one correspondence with flags in L_{0}. We shall occasionally write V_{i} = L_{i}/πL_{0}, and denote dim V_{i} by d_{i}, so that d_{0} = d + 1 and e_{i} = d_{i} − d_{i+1}. If a ∈ L_{0} we shall denote by ā its image in V_{0} = L_{0}/πL_{0}.

A basis of V_{K} is called adapted to (the pointed k-cell) σ if it induces an isomorphism of O_{K}^{d+1} with a lattice L_{0} representing the distinguished vertex, and if under the induced isomorphism of 𝔽_{q}^{d+1} with V_{0} we have V_{i} = {ā ∈ V_{0}; ā_{j} = 0 for j ≥ d_{i}}.

1.2. The combinatorial metric ρ. There is a metric ρ on T_{0} defined as follows. If u = [L] and v = [M] and if, for suitable representatives M ⊂ L

L/M = O_{K}/(π^{m_{0}}) ⊕ ⋯ ⊕ O_{K}/(π^{m_{d}}) (1.3)

with 0 ≤ m_{0} ≤ ⋯ ≤ m_{d}, then

ρ(u, v) = m_{d} − m_{0}. (1.4)

If σ ∈ T_{k} and τ ∈ T_{r} we put

ρ(τ, σ) = max_{u∈τ, v∈σ} ρ(u, v). (1.5)

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 T_{0}.

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 T_{k}(τ) the set of k-cells contiguous to an arbitrary cell τ, and by T̂_{k}(τ) the corresponding pointed k-cells. For example, if v_{0} is a vertex, T_{k}(v_{0}) consists of k-cells that either contain v_{0}, or form a k + 1-cell with it.

Another way to say that σ ∈ T_{k} and τ ∈ T_{r} are contiguous is that we can find lattice flags

πL_{0} ⊂ L_{k} ⊂ ... ⊂ L_{1} ⊂ L_{0}, πM_{0} ⊂ M_{r} ⊂ ⋯ ⊂ M_{1} ⊂ M_{0} (1.6)

representing them which can be interlaced - i.e., whose union is also a lattice flag. If τ is pointed and the distinguished vertex is [M_{0}], then σ is automatically given a distinguished vertex [L_{0}], determined by the condition that M_{0} ⊇ L_{0} ⊃ L_{k} ⊃ πM_{0}. We can write the interlaced flag as blocks of L′_{i}s and blocks of M′_{j}s alternating, the only non-uniqueness occuring where some L_{i} is also an M_{j}. 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 [M_{0}], 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 = (α_{0}, …, α_{d}) of V_{K}. The apartment A_{α} determined by α is the simplicial subcomplex of T supported on the vertices of the form

v_{α}(m_{0}, …, m_{d}) = [⟨π^{m_{0}}α_{0}, …, π^{m_{d}}α_{d}⟩] (1.7)

for m_{i} ∈ ℤ. Here ⟨·⟩ denotes the span over O_{K}. Every apartment is a triangulation of a d-dimensional Euclidean space. The vertices with m_{i} − m_{j} = 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 T_{0}. Upon introduction of coordinates adapted to some vertex v_{0} = [L_{0}] we may identify G with P GL_{d+1}(K), and the stabilizer of v_{0} with P GL_{d+1}(O_{K}). The stabilizer B_{σ} of a pointed k-cell σ = (v_{0}, ..., v_{k}) is then conjugate in P GL_{d+1}(O_{K}) 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 e_{k}, ..., e_{1}, e_{0} with arbitrary O_{K} entries, above them arbitrary entries too, and below them entries divisible by π.

If v is a vertex, the stabilizer B_{v} of v is isomorphic to P GL_{d+1}(O_{K}), so posesses a distinguished sequence U_{v}^{n} of normal pro-p subgroups (n ≥ 1), namely the principal congruence subgroups of level π^{n}. The group U_{v}^{n} is in fact independent of the choice of coordinates, because it can be defined intrinsically as the (pointwise) stabilizer of the ball {u ∈ T_{0}; ρ(u, v) ≤ n}.

The choice of a basis α for V_{K} determines a maximal torus T_{α} in G, which, under the isomorphism of G with P GL_{d+1}(K) induced by α, corresponds to the diagonal matrices. For every wall W in the apartment A_{α} there is a (non-unique) involution s_{W} ∈ G which normalizes T_{α} and on A_{α} 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, then |σ| is the open simplex consisting (if k > 0) of all the points t = ∑_{i=0}^{k} t_{i}v_{i}, where ∑ t_{i} = 1, 0 < t_{i} < 1. It is known that |T | is contractible (see [Mus], theorem 1.1(1)). We shall denote by St(v_{0}) the star of a vertex v_{0}. By definition, this is the union of the open simplices |σ| for v_{0} ∈ σ. The closure \overline{St}(v_{0}) is then the compact set which is the union of all |σ| for σ contiguous to v_{0}. Finally we put, for any cell τ

St(τ) = ⋃_{τ≤σ} |σ| = ⋂_{v∈τ} St(v) (1.8)

\overline{St}(τ) = ⋃_{ρ(τ,σ)≤1} |σ| = ⋂_{v∈τ} \overline{St}(v). (1.9)

If A_{α} is an apartment as in Section 1.3, determined by the basis α, we can map |A_{α}| onto ℝ^{d+1}/ℝ · (1, 1, …, 1) by mapping v_{α}(m_{0}, …, m_{d}) to (m_{0}, …, m_{d})modℝ · (1, 1, …, 1). This identification with Euclidean space induces a metric d_{α} on |A_{α}|, and in fact a canonical G-invariant Euclidean metric d(x, y) on |T |, because any two points belong to a common apartment A_{α}, 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 T_{0}, 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 \sqrt{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 arrangements (see [O-T], chapter 3, and [O-S]).

Let A be a non-empty subset of ℙ(V_{K}). We view A as a set of lines in V_{K}, or hyperplanes in V_{K}^{∗}. It may be finite or infinite, and in particular may consist of all of ℙ(V_{K}). We write e_{a} for the line represented by a ∈ V_{K} − {0}, so that e_{λa} = e_{a} for any scalar λ ≠ 0. By abuse of language we shall sometimes say a ∈ A to mean that e_{a} ∈ A.

Let Ẽ 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 e_{a} to 1. It satisfies the following properties.

δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}}) = ∑_{i=0}^{k}(−1)^{i}e_{a_{0}} ∧ ⋯ ∧ ê_{a_{i}} ∧ ⋯ ∧ e_{a_{k}} (2.1)

(the symbol ê_{a} means that e_{a} is omitted from the product)

δ^{2} = 0 (2.2)

δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}}) = (e_{a_{1}} − e_{a_{0}}) ∧ ⋯ ∧ (e_{a_{k}} − e_{a_{0}}) (2.3)

(follows from (2.1)),

δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}}) = δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{m}}) ∧ δ(e_{a_{m}} ∧ ⋯ ∧ e_{a_{k}}) (2.4)

(follows from (2.3) after a rearrangement of the indices), and

δ(e_{a} ∧ x) = x − e_{a} ∧ δx. (2.5)

It follows that

E = Im(δ) = Ker(δ) (2.6)

is the subalgebra generated by e_{a} − e_{b}, and that there is an exact sequence

0 → E → Ẽ →^{δ} E[1] → 0. (2.7)

Any e_{a} in A supplies a splitting Ẽ ← E[1], e_{a} ∧ x ↢ x.

Definition 2.1. Let τ ∈ T̂_{r}, and let M_{0} ⊃ ⋯ ⊃ M_{r} ⊃ πM_{0} be a lattice flag representing τ. The index ι_{τ} (a) of a ∈ A ∩ (M_{0} − πM_{0}) is the unique i, 0 ≤ i ≤ r, such that a ∈ M_{i} − M_{i+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 ∈ M_{0} − πM_{0}.

The index is well defined, because a representative a for e_{a} can always be found in M_{0} − πM_{0}, and it is then uniquely determined up to multiplication by a unit. Here and below we agree that M_{r+1} = πM_{0}.

Definition 2.2. Let τ be as above, and assume that the representatives of A lie in M_{0} − πM_{0}. Let I(τ) be the ideal in Ẽ generated by δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{m}}) for any set {a_{0}, …, a_{m}} ⊂ A ∩ (M_{i} − M_{i+1}) which is linearly dependent modulo M_{i+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

Ã(τ) = Ẽ/I(τ), and A(τ) = (E + I(τ))/I(τ) ≃ E/E ∩ I(τ). (2.8)

Since I(τ) is homogeneous and δ-invariant, these are graded algebras, and there is an exact sequence

0 → A(τ) → Ã(τ) →^{δ} A(τ)[1] → 0. (2.9)

At the risk of a slight confusion, we shall keep the notation e_{a} also for the image of e_{a} in Ã(τ). Any e_{a} supplies a splitting Ã(τ) ← A(τ)[1], e_{a} ∧ x ↢ x.

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 = [M_{0}]. If k > d + 1 take any a_{0}, …, a_{k} in A, and normalize them to lie in M_{0} − πM_{0}. Since {a_{1}, …, a_{k}} are linearly dependent in V_{K}, hence clearly modulo πM_{0}, x = e_{a_{1}} ∧ ⋯ ∧ e_{a_{k}} satisfies that δ(x) and δ(e_{a_{0}} ∧ x) ∈ I(v). It follows that x = δ(e_{a_{0}} ∧ x) + e_{a_{0}} ∧ 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 ⊂ M_{0} − πM_{0}. Let a range over representatives of A in ℙ(M_{0}/πM_{0}). Then for every b ∈ A we can find a scalar λ and an a such that b ≡ λamodπM_{0}, hence e_{a} − e_{b} ∈ I(v). It follows that the e_{a} span Ã(v)^{1}, so

dim Ã(v)^{1} ≤ \frac{q^{d+1} − 1}{q − 1} (2.10)

is finite.

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

r_{σ}^{τ} : Ã(τ) → Ã(σ) (2.11)

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

I(τ ∪ σ) = I(τ) + I(σ). (2.12)

Proof. Equivalently, we should prove that I(τ) = ∑ I(v_{i}) if τ = (v_{0}, …, v_{r}). Suppose that M_{0} ⊃ ⋯ ⊃ M_{r} ⊃ πM_{0} ([M_{i}] = v_{i}) is a flag representing τ, and {a_{0}, …, a_{m}} ⊂ M_{i} − M_{i+1} are linearly dependent modulo M_{i+1} (0 ≤ i ≤ r). Then {a_{0}, …, a_{m}} ⊂ π^{−1}M_{i+1} − M_{i+1}, hence δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{m}}) ∈ I(v_{i+1}). This proves that I(τ) ⊆ ∑ I(v_{i}).

Conversely, pick a typical generator δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{m}}) of I(v_{0}) (say), where {a_{0}, …, a_{m}} ⊂ M_{0} − πM_{0} are linearly dependent modulo πM_{0}. We want to show that it belongs to I(τ). By (2.4) we may assume that {a_{0}, …, a_{m}} is a minimal such set - no proper subset of it is dependent modulo πM_{0}. Let i be the largest index such that {a_{0}, …, a_{m}} are not contained in M_{i+1}. After rearranging the a_{i} we may assume that {a_{0}, …, a_{n}} are contained in M_{i} − M_{i+1} and the rest lie in M_{i+1}. We then must have a non-trivial linear dependence between {a_{0}, …, a_{n}} modulo M_{i+1}, hence δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{n}}) lies in I(τ). By (2.4) again, so does δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{m}}).

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

A(τ ∩ σ) → A(τ) ×_{A(τ∪σ)} A(σ) (2.13)

is surjective.

Proof. Given α_{τ} and α_{σ} in Ẽ which are congruent modulo I(τ ∪ σ) = I(τ) + I(σ), we can find ε_{τ} ∈ I(τ) and ε_{σ} ∈ I(σ) such that α = α_{τ} − ε_{τ} = α_{σ} − ε_{σ}. Then αmodI(τ ∩ σ) maps to (α_{τ} modI(τ), α_{σ}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 M_{0} ⊃ M_{1} ⊃ M_{2} ⊃ πM_{0}, and a, b are elements of M_{0} − M_{1}, which are congruent modulo M_{2} but not modulo πM_{0}. Then letting τ = (M_{0} ⊃ M_{1} ⊃ πM_{0}) and σ = (M_{0} ⊃ M_{2} ⊃ πM_{0}) we have v = τ ∩ σ = [M_{0}]. Now e_{a} − e_{b} ∉ I(v) but e_{a} − e_{b} ∈ 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 (a_{0}, …, a_{k}), and e_{S} = e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}}. For any σ ∈ T̂_{k} consider

π = (ι_{σ}(a_{0}), …, ι_{σ}(a_{k})). (2.14)

Define (σ̃, e_{S}) = sgn(π) if π is a permutation of (0, 1, …, k). Define (σ̃, e_{S}) = 0 otherwise, and extend the definition by linearity to Ẽ^{k+1}.

Lemma 2.4. Let τ ∈ T_{r} be an r-cell contiguous to σ. Then the linear functional (σ̃, −) annihilates E^{k+1} + I^{k+1}(τ), and therefore factors through A^{k}(τ).

Proof. Since I(τ) = ∑_{v∈τ} I(v), it is enough to consider the case where τ = v = [M_{0}] is a vertex contiguous to σ. Let σ = (L_{0} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0}). We may assume that M_{0} ⊇ L_{0} ⊃ L_{k} ⊃ πM_{0} ⊇ πL_{0}, 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 {a_{0}, …, a_{m}} ⊂ A ∩ (M_{0} − πM_{0}), linearly dependent modulo πM_{0} (m ≤ k + 1), and arbitrary {a_{m+1}, …, a_{k+1}}, we have to show that

(σ̃, ∑_{i=0}^{m}(−1)^{i}e_{a_{0}} ∧ ⋯ ∧ ê_{a_{i}} ∧ ⋯ ∧ e_{a_{k+1}}) = 0. (2.15)

Observe that the ι_{σ}(a_{i}), 0 ≤ i ≤ m can not be distinct, for if they were distinct, the a_{i} could not be dependent modulo πM_{0}. Assume therefore that ι_{σ}(a_{0}) = ι_{σ}(a_{1}). 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 {a_{0}, …, a_{k+1}} be arbitrary. We claim that

(σ̃, ∑_{i=0}^{k+1}(−1)^{i}e_{a_{0}} ∧ ⋯ ∧ ê_{a_{i}} ∧ ⋯ ∧ e_{a_{k+1}}) = 0. (2.16)

By the pigeonhole principle we may assume, rearranging the a_{i}, that ι_{σ}(a_{0}) = ι_{σ}(a_{1}). 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)) = A^{k}(σ).

Definition 2.4. Let σ ∈ T̂_{k}. For e ∈ E^{k}, choose ẽ ∈ Ẽ^{k+1} with δ(ẽ) = e (e.g. ẽ = e_{a} ∧ e for some a). Define

(σ, e) = (σ̃, ẽ). (2.17)

The lemma guarantees that (σ, −) is well-defined, and that if σ ∈ T̂_{k}(τ), it defines a linear functional on A^{k}(τ), for which we still use the same notation.

Definition 2.5. Let e ∈ E^{k}. Define

c_{e} : T̂_{k} → K (2.18)

to be the map c_{e}(σ) = (σ, e). We call c_{e} the global cochain associated with e. Similarly, if we restrict c_{e} to T̂_{k}(τ), it depends only on the image of e in A^{k}(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 hyperplane 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 V_{K}^{∗}, can be deduced from it by a simple limit argument.

Fix τ, and assume that

τ = (M_{0} ⊃ M_{1} ⊃ ⋯ ⊃ M_{r} ⊃ πM_{0}). (2.19)

We may assume that the lines in A are represented by a ∈ M_{0} − πM_{0}. Let

A_{l} = A_{l}(τ) = {a ∈ A; ι_{τ} (a) = l} = A∩(M_{l} − M_{l+1}), (2.20)

(0 ≤ l ≤ r, M_{r+1} = πM_{0}), and fix a linear ordering ≺ of A such that max A_{l+1} ≺ min A_{l}. We call such an ordering compatible with τ. Notice that if τ ≤ σ and they both share the same distingushed vertex [M_{0}], 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 A_{l}, and a linear ordering within each of them.

If

S = (a_{0} ≻ a_{1} ≻ ⋯ ≻ a_{k}) (2.21)

is a decreasing sequence of elements of A, we shall call S τ-special if when we divide S into blocks S_{l} = S ∩ A_{l}, and write

S_{l} = (a_{i_{l}} ≻ ⋯ ≻ a_{i_{l+1}−1}) (2.22)

(meaning that S_{l} = ∅ if i_{l} = i_{l+1}, i_{0} = 0 and i_{r+1} = k + 1), then for every i_{l} ≤ j < i_{l+1},

a_{j} = max(A ∩ ⟨a_{j}, …, a_{i_{l+1}−1}, M_{l+1}⟩). (2.23)

Here ⟨·⟩ denotes the span over O_{K}. Notice that if a appears in a τ-special sequence S, and ι_{τ} (a) = l, then in particular a is the maximal element of A ∩ ⟨a, M_{l+1}⟩, and that if S is τ-special then S_{l} is linearly independent modulo M_{l+1}. In particular the length of S_{l} is at most e_{l} = dim M_{l}/M_{l+1}, and k ≤ d.

If τ′ is obtained from τ by a cyclic permutation of the vertices, there is a new ordering of A compatible with τ′, which induces the same cyclic ordering as before. Simply permute the blocks A_{l} cyclically, and maintain the linear ordering within each block. In particular S is τ-special if and only if S′, the same set in the new ordering, is τ′-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 {e_{S}; S ∈ S(τ)} is a basis of Ã(τ).

Proof. Fix 0 ≤ k ≤ d, and we shall show that the e_{S} for S ∈ S_{k+1}(τ), the τspecial sequences of length k + 1, form a basis of Ã^{k+1}(τ). Introduce a lexicographic ordering ≻ on the decreasing sequences of length k + 1 : If S is as above and T = (b_{0} ≻ ⋯ ≻ b_{k}) say that S ≻ T if a_{0} = b_{0}, …, a_{j−1} = b_{j−1}, but a_{j} ≻ b_{j}.

Clearly Ã^{k+1}(τ) is spanned by the e_{S} 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 i_{l} ≤ j < i_{l+1},

a′_{j} = max(A ∩ ⟨a_{j}, …, a_{i_{l+1}−1}, M_{l+1}⟩) ≻ a_{j}. (2.24)

Thus {a′_{j}, a_{j}, …, a_{i_{l+1}−1}} ⊂ M_{l} − M_{l+1}, and are linearly dependent modulo M_{l+1}. This implies that

δ(e_{a′_{j}} ∧ e_{a_{j}} ∧ ⋯ ∧ e_{a_{i_{l+1}−1}}) ∈ I(τ). (2.25)

In Ã^{k+1}(τ) we may therefore replace the string e_{a_{j}} ∧ ⋯ ∧ e_{a_{i_{l+1}−1}} by a linear combination of similar strings beginning with e_{a′_{j}}. This means that we have replaced e_{S} by a linear combination of e_{T} with T ≻ S. Repeating the process we see that the τ-special e_{S} already span Ã^{k+1}(τ).

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

Ã^{k+1}(τ) = A^{k+1}(τ) ⊕ e_{μ} ∧ A^{k}(τ). (2.26)

Since e_{μ} ∧ x = e_{μ} ∧ δ(e_{μ} ∧ x) we see that e_{μ} ∧ A^{k}(τ) = e_{μ} ∧ Ã^{k}(τ), and is therefore spanned by the e_{μ} ∧e_{S}, S ∈ S_{k}(τ). But e_{μ} ∧e_{S} = e_{(μ,S)} and either μ is the maximal element of S, in which case e_{μ} ∧ e_{S} = 0, or (μ, S) ∈ S_{k+1}(τ). Write S_{k+1}(τ)′ for the τ-special S with μ > max(S), and S_{k+1}(τ)′′ for those with μ = max(S), and note that

S_{k+1}(τ) = S_{k+1}(τ)′ ∪ S_{k+1}(τ)′′ (disjoint union), and

S_{k+1}(τ)′′ = {e_{μ} ∧ e_{S}; S ∈ S_{k}(τ)′} . (2.27)

We shall prove that S_{k+1}(τ)′′ is a basis of e_{μ}∧A^{k}(τ). This will show that #S_{k}(τ)′ = dim A^{k}(τ). Using this with k + 1 instead of k we shall finally get

dim Ã^{k+1}(τ) = #S_{k+1}(τ)′ + #S_{k+1}(τ)′′ = #S_{k+1}(τ). (2.28)

Thus S_{k+1}(τ) is a basis.

To prove that S_{k+1}(τ)′′ is a basis of e_{μ} ∧ A^{k}(τ) it is enough to prove that they are independent. We need a certain construction. We associate with each T = (μ = b_{0} ≻ ⋯ ≻ b_{k}) ∈ S_{k+1}(τ)′′ a k + 1-cell σ_{T} ∈ T̂_{k}(τ) as follows. If T_{l} = T ∩ A_{l} = (b_{i_{l}} ≻ ⋯ ≻ b_{i_{l+1}−1}) ≠ ∅, and i_{l} ≤ j < i_{l+1}, let

L_{j} = ⟨b_{j}, …, b_{i_{l+1}−1}, M_{l+1}⟩, (2.29)

except for j = 0, where we put L_{0} = M_{0}. The fact that T is τ-special guarantees that the L_{j} are in fact distinct and decreasing, and

σ_{T} = (L_{0} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0}) (2.30)

is interlaceable with τ, because M_{l} ⊇ L_{j} ⊃ M_{l+1}. Thus σ_{T} and τ are contiguous. The linear independence of S_{k+1}(τ)′′ will be a consequence of the following lemma.

Lemma 2.6. If S ≻ T are in S_{k+1}(τ)′′, then (σ̃_{T} , e_{S}) = 0, (σ̃_{S}, e_{S}) = 1.

Proof. Suppose S = (a_{0} ≻ ⋯ ≻ a_{k}) and T = (b_{0} ≻ ⋯ ≻ b_{k}), a_{0} = b_{0} = μ, …, a_{j−1} = b_{j−1}, but a_{j} ≻ b_{j} (j ≥ 1). Suppose also b_{j} ∈ M_{l} − M_{l+1} (0 ≤ l ≤ r). For 0 ≤ i ≤ j

a_{i} ≻ b_{j} = max(A ∩ ⟨b_{j}, …, b_{i_{l+1}−1}, M_{l+1}⟩), (2.31)

so in particular a_{i} ∉ L_{j}. On the other hand a_{i} ∈ M_{0} = L_{0}. Thus 0 ≤ ι_{σ_{T}} (a_{i}) < j and there must be two a′_{i}s with the same σ_{T} -index. This proves that (σ̃_{T} , e_{S}) = 0, and (σ̃_{S}, e_{S}) = 1 is clear from the definition.

Corollary 2.7. The collection

{(σ_{T} , −); T ∈ S_{k+1}(τ)′′} (2.32)

is a basis of the dual space of A^{k}(τ).

Proof. This follows from lemma 2.6 and the proof of theorem 2.5.

2.5. The case A = ℙ(V_{K}). In this case A is G-invariant. Every g ∈ G induces a map g : Ẽ → Ẽ, g(e_{a}) = e_{ga}, and g(I(τ)) = I(gτ). We therefore have maps

g_{τ} : A(τ) → A(gτ) (2.33)

(and similarly on Ã) which commute with the restriction maps. They satisfy the usual formula

g′_{gτ} ∘ g_{τ} = (g′g)_{τ} . (2.34)

The linear functionals λ_{σ} = (σ, −) satisfy

(gσ, ge) = (σ, e) (2.35)

for every σ ∈ T̂_{k} and e ∈ E^{k}, which can be phrased also as

g(λ_{σ}) = λ_{gσ}. (2.36)

2.6. Open questions. There are several interesting questions regarding the ideals I(σ). For example, does the distributive law

(I(σ) + I(ρ)) ∩ I(τ) = I(σ) ∩ I(τ) + I(ρ) ∩ I(τ) (2.37)

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. A^{k}(τ) as a space of local harmonic cochains

In this section we explore averaging properties of the cochains c_{e}, and derive an alternative description of A^{k}(τ), as a space of certain harmonic cochains c : T̂_{k}(τ) → K. For τ a vertex and A = ℙ(V_{K}), we shall see that the image consists of all the local harmonic cochains.

3.1. Harmonic cochains. A map c : T̂_{k} → K will be called harmonic if it satisfies the following 4 conditions :

• (Property A). If σ = (v_{0}, …, v_{k}) ∈ T̂_{k} and σ′ = (v_{1}, …, v_{k}, v_{0}), then

c(σ) = (−1)^{k}c(σ′). (3.1)

• (Property B). Fix σ ∈ T̂_{k−1}, fix a type 𝔱 of pointed k-cells, and consider B = B(σ, 𝔱) = {σ′ ∈ T̂_{k}; σ ≤ σ′, σ and σ′ share the same distinguished vertex, and 𝔱(σ′) = 𝔱}. Then

∑_{σ′∈B} c(σ′) = 0. (3.2)

• (Property C). Let k ≥ 1. Fix σ ∈ T̂_{k} and an index 0 ≤ j ≤ k. Let C = C(σ, j) be the collection of all σ′ ∈ T̂_{k} for which L′_{i} = L_{i} if i ≠ j, L′_{j} ⊆ L_{j} and dim L′_{j}/L_{j+1} = 1. Then

c(σ) = ∑_{σ′∈C} c(σ′). (3.3)

• (Property D). Let σ = (v_{0}, v_{1}, …, v_{k+1}) ∈ { \atop T_{k+1}}. Let σ_{j} = (v_{0}, …, v̂_{j}, …, v_{k+1}).

Then

∑_{j=0}^{k+1}(−1)^{j}c(σ_{j}) = 0. (3.4)

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 V_{K} let L^{∗} be the dual lattice in V_{K}^{∗}. This establishes an order-reversing bijection between lattices in V_{K}^{∗} and V_{K}. It leads to a bijection σ ⟷ σ^{∗} between the k-cells of T , and the k-cells of the Bruhat-Tits building T^{∗} of V_{K}^{∗}, 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 ↦ 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.

Definition 3.1. Denote by C_{har}^{k} the space of harmonic cochains on T̂_{k}. Denote by C_{har}^{k}(τ) the space of local harmonic cochains at τ. These are the maps c : T̂_{k}(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 T̂_{k}(τ).

Remark. The good notion is that of C_{har}^{k}, and the map C_{har}^{k} → C_{har}^{k}(τ) need not be surjective. There might be relations involving only T̂_{k}(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 T̂_{k}(τ).

It therefore comes as a surprise that to this remark there is an exception. When τ = v is a vertex, C_{har}^{k} → C_{har}^{k}(v) is surjective, and the definition of C_{har}^{k}(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 c_{e}.

Theorem 3.1. The cochains c_{e} 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

σ = (L_{0} ⊃ ⋯ ⊃ L_{j−1} ⊃ L_{j+1} · ·· ⊃ L_{k} ⊃ L_{k+1} = πL_{0}) (3.5)

and dim L_{j−1}/L_{j+1} ≥ 2. Let B = B(σ, j) be the collection of σ′ ∈ T̂_{k} with

σ′ = (L_{0} ⊃ ⋯ ⊃ L_{j−1} ⊃ L′_{j} ⊃ L_{j+1} · ·· ⊃ L_{k} ⊃ L_{k+1} = πL_{0}) (3.6)

and dim L′_{j}/L_{j+1} = 1. Then

∑_{σ′∈B}(σ̃′, ẽ) = 0. (3.7)

We may assume that ẽ = e_{a_{0}} ∧ e_{a_{1}} ∧ ⋯ ∧ e_{a_{k}}. Write ι_{σ}(a_{i}) = m(i). (Here by m(i) we understand the last l such that a_{i} ∈ L_{l}. Thus m(i) ≠ j.) If m(i) ≠ j − 1, then for every σ′ ∈ B, ι_{σ′}(a_{i}) = m(i) as well. On the other hand, if m(i) = j − 1, then ι_{σ′}(a_{i}) = j − 1 or j. If for some l, 0 ≤ l ≤ j − 2 or j + 1 ≤ l ≤ k, either l = m(i) for two distinct i′s, or for no i at all, then the same holds for every σ′ ∈ B, and every (σ̃′, 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 a_{i} we may assume that m(i) = i for i ≠ j, and m(j) = j − 1.

Now there are two possibilities. If a_{j−1} and a_{j} represent the same line modulo L_{j+1}, then for every σ′ ∈ B they will together lie in, or out of, L′_{j}. Thus every (σ̃′, ẽ) = 0. If, on the other hand, a_{j} and a_{j−1} are independent modulo L_{j+1}, then there will be precisely two σ′ with (σ̃′, ẽ) ≠ 0, namely those corresponding to L′_{j} = ⟨L_{j+1}, a_{j−1}⟩ and to L′_{j} = ⟨L_{j+1}, a_{j}⟩ . Their contribution to the sum will be −1 and +1 respectively, so will cancel each other.

Property C : Fix σ ∈ T̂_{k} (k ≥ 1), and an index 0 ≤ j ≤ k. Let C = C(σ, j) be the collection of all σ′ ∈ T̂_{k} for which L′_{i} = L_{i} if i ≠ j, L′_{j} ⊆ L_{j} and dim L′_{j}/L_{j+1} = 1. We have to show that for ẽ = e_{a_{0}} ∧ e_{a_{1}} ∧ ⋯ ∧ e_{a_{k}}

(σ̃, ẽ) = ∑_{σ′∈C}(σ̃′, ẽ). (3.8)

Note first that if ι_{σ}(a_{i}) = l ≠ j, then for every σ′ ∈ C also ι_{σ′}(a_{i}) = l, while if ι_{σ}(a_{i}) = j, then ι_{σ′}(a_{i}) = 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 a_{i}, that ι_{σ}(a_{i}) = i for each i. We then have ι_{σ′}(a_{i}) = i for i ≠ j, and any σ′. We also have ι_{σ′}(a_{j}) = j for the unique σ′ ∈ C where L′_{j} = ⟨L_{j+1}, a_{j}⟩, hence (σ̃′, ẽ) = 1 for that σ′. For any other σ′ ∈ C we have ι_{σ′}(a_{j}) = j − 1mod k + 1, hence (σ̃′, ẽ) = 0.

(b) If (σ̃, ẽ) = 0 and (b1) there is an l ≠ j with more than one a_{i} with ι_{σ}(a_{i}) = l, or (b2) there is an l ≠ j − 1mod k + 1 with no a_{i} with ι_{σ}(a_{i}) = l, then the same holds true for every σ′ ∈ C, so each (σ̃′, ẽ) = 0.

(c) If (σ̃, ẽ) = 0 and we are not in case (b), then after rearranging the a_{i} we may assume that ι_{σ}(a_{i}) = i for i ≠ j − 1mod k + 1 while ι_{σ}(a_{j−1modk+1}) = j. As in the previous proposition, we have two possibilities. If a_{j−1modk+1} and a_{j} span the same line in L_{j}/L_{j+1}, then they will lie together in, or out of, any L′_{j}, and each (σ̃′, ẽ) = 0. If they are independent modulo L_{j+1}, then precisely two σ′ will pair non-trivially with ẽ, but their contributions will cancel each other.

Property D : Consider ẽ = e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}},

σ = (L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{k+1} ⊃ πL_{0}), (3.9)

and we shall show that

∑_{j=0}^{k+1}(−1)^{j}(σ̃_{j}, ẽ) = 0.

Choosing the a_{i} to lie in L_{0} − πL_{0} we write m_{i} = ι_{σ}(a_{i}) for their indices with respect to σ. First note that if ι_{σ}(a) = ι_{σ}(b), then for every σ_{j} also ι_{σ_{j}}(a) = ι_{σ_{j}}(b). (The only case where some care has to be taken is j = 0, where if a, hence b, lie in L_{0} − L_{1}, they have to be replaced by πa and πb to get representatives in L_{1} − πL_{1}.) Thus if the m_{i} are not all distinct, the alternating sum of the (σ̃_{j}, ẽ) vanishes.

If the m_{i} are distinct we may assume, because of property (A), and renaming the a′_{i}s if necessary, that m_{i} = i + 1 for every 0 ≤ i ≤ k. If j ≥ 2 then we will have (σ̃_{j}, ẽ) = 0. For j = 0 and j = 1 we shall have (σ̃_{j}, ẽ) = 1, and we are done.

3.3. Injectivity of the local cochain homomorphism. Now fix τ = (M_{0} ⊃ ⋯ ⊃ M_{r} ⊃ M_{r+1} = πM_{0}) (0 ≤ r). Since the global cochains c_{e} are harmonic, so are their restrictions to T̂_{k}(τ), and we get a homomorphism

c : A^{k}(τ) → C_{har}^{k}(τ) (3.10)

sending e to c_{e}.

Theorem 3.2. The local cochain homomorphism c is injective.

Proof. Given the results of Section 2, this is now an easy consequence of corollary 2.7, because the σ_{T} constructed there (for T ∈ S′′_{k+1}(τ)) all belong to T̂_{k}(τ), and the (σ_{T} , −) span the dual space of A^{k}(τ).

3.4. The isomorphism A^{k}(v) ≃ C_{har}^{k}(v), when A = ℙ(V_{K}). To get surjectivity statements about c, we must assume that the collection A ⊂ ℙ(V_{K}), with which we started, is large.

Theorem 3.3. Let v_{0} = [M_{0}] be a vertex of T , and assume that A contains a representative for each line in ℙ(M_{0}/πM_{0}) (for example, take A = ℙ(V_{K})). Then c induces an isomorphism A^{k}(v_{0}) ≃ C_{har}^{k}(v_{0}).

Proof. We may replace the set A, when dealing with A^{k}(v_{0}), by a finite subset containing precisely one representative for every line in ℙ(M_{0}/πM_{0}). 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 A^{k}(v_{0}) → C_{har}^{k}(v_{0}). We view cells from T̂_{k}(v_{0}) as linear functionals on C_{har}^{k}(v_{0}) in a natural way. They clearly span C_{har}^{k}(v_{0})^{∗}. We shall show that {σ_{T}; T ∈ S′′_{k+1}(v_{0})} is still is a spanning set, and therefore, by corollary 2.7, the map C_{har}^{k}(v_{0})^{∗} → A^{k}(v_{0})^{∗} is injective, which is the desired result.

Let us use the phrase equivalent to mean equal, as linear functionals on C_{har}^{k}(v_{0}). Recall that S′′_{k+1}(v_{0}) consists of S = (a_{0} ≻ a_{1} ≻ ⋯ ≻ a_{k}) where a_{0} = μ and a_{j} = max(A ∩ ⟨πM_{0}, a_{k}, …, a_{j}⟩). Let σ ∈ T̂_{k}(v_{0}). If v_{0} ∉ σ then σ ∪ {v_{0}} is a k + 1-cell, and property (D) shows that σ is equivalent to a linear combination of σ′s containing v_{0}. Property (A) allows us to assume that σ = (v_{0}, …, v_{k}), 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 L_{i}/πL_{0} = 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 σ_{T} (note that the σ_{T} are of minimal type as well).

If σ = (L_{0} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0}), L_{0} = M_{0}, recall that we wrote V_{i} = L_{i}/πL_{0}, and A is a system of representatives in L_{0} − πL_{0} for ℙ(V_{0}). To simplify the notation write max(V_{i}) for max(A ∩ L_{i}). We shall prove, by decreasing induction on j, that every σ of minimal type with

(3.11) max(V_{k}) ≺ ⋯ ≺ max(V_{j+1}) ⪯ max(V_{j}) ⪯ ⋯ ⪯ max(V_{0})

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

(3.12) max(V_{k}) ≺ ⋯ ≺ max(V_{j+1}) ≺ max(V_{j}) ⪯ ⋯ ⪯ max(V_{0})

(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 V_{i} 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(V_{j+1}) = max(V_{j}). Consider σ_{0} = (L_{0} ⊃ ⋯ ⊃ L_{j} ⊃ L_{j+2} ⊃ ⋯ ⊃ L_{k+1} = πL_{0}). We shall apply the condition (B) with B(σ_{0}, j+1) (see the proof of theorem 3.1 for the notation). Since dim L_{j}/L_{j+2} = 2 and max V_{j} = max V_{j+1} ≻ max V_{j+2}, when we consider all the possibilities for L_{j} ⊃ L′_{j+1} ⊃ L_{j+2}, every other L′_{j+1} ≠ L_{j+1} will produce a σ′ with

(3.13) max(V_{k}) ≺ ⋯ ≺ max(V_{j+2}) ⪯ max(V′_{j+1}) ≺ max(V_{j}) ⪯ ⋯ ⪯ max(V_{0}).

Our assumption implies that σ is equivalent to the negative of the sum of all the other σ′ ∈ B(σ_{0}, j + 1). The induction hypothesis allows us now to replace each of these σ′ 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 = (L_{0} ⊃ ··· ⊃ L_{k} ⊃ πL_{0}), L_{0} = M_{0}, dim L_{i}/πL_{0} = k + 1 − i for i ≥ 1, and when we put b_{i} = max(A ∩ L_{i}), we have b_{0} = μ ≻ b_{1} ≻ ⋯ ≻ b_{k}. Calling this sequence T, we find that T ∈ S′′_{k+1}(v_{0}), and σ = σ_{T} .

Corollary 3.4. The map C_{har}^{k} → C_{har}^{k}(v_{0}) is surjective.

Proof. Every element of A^{k}(v_{0}) lifts to an element of E^{k}.

As we shall see later, the image of A^{k}(τ) in C_{har}^{k}(τ), for a general simplex τ, consists of the elements which admit an extension to a global harmonic cochain in C_{har}^{k}.

4. Acyclicity (d ≤ 2)

4.1. Cohomology of local systems. Quite generally, let T be a simplicial complex, which is assumed for simplicity to be finite-dimensional and locally finite. We write T_{k} for the k-simplices of T , and T̂_{k} for the ordered k + 1-tuples (v_{0}, …, v_{k}) of distinct vertices such that {v_{0}, …, v_{k}} ∈ T_{k}.

A (cohomological) local system \underline{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 homomorphisms (restriction maps) r_{σ}^{τ} : A(τ) → A(σ) to every face inclusion τ ≤ σ, such that r_{ρ}^{σ} ∘ r_{σ}^{τ} = r_{ρ}^{τ} whenever τ ≤ σ ≤ ρ, and r_{τ}^{τ} is the identity.

Given a local system \underline{A}, define the group of alternating p-cochains (0 ≤ p ≤ dim T ) C^{p}(T , \underline{A}) to consist of the maps c assigning to each τ ∈ T̂_{p} an element c(τ) ∈ A(τ) such that

c(τ′) = sgn(π)c(τ) (4.1)

if τ′ is obtained from τ by applying the permutation π to the vertices. Define

∂ : C^{p}(T , \underline{A}) → C^{p+1}(T , \underline{A}) (4.2)

by the rule

∂c(τ) = ∑_{i=0}^{p+1}(−1)^{i}r_{τ}^{τ_{i}}(c(τ_{i})) (4.3)

if τ = (v_{0}, …, v_{p+1}) and τ_{i} = (v_{0}, …, v̂_{i}, …, v_{p+1}). Then (C^{·}(T , \underline{A}), ∂) is a complex (∂^{2} = 0), and its cohomology groups are denoted H^{p}(T , \underline{A}). We call \underline{A} acyclic if H^{p}(T , \underline{A}) = 0 for p > 0.

It is well known that if \underline{A} is locally constant (all the restriction maps are isomorphisms), then if T is contractible (as is the case with the Bruhat-Tits building), \underline{A} is acyclic. In general, even if T is contractible, \underline{A} need not be acyclic. If 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 ≠ 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 , \underline{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

H^{p}(T , \underline{A}^{q}) = 0 (4.4)

for 0 ≤ q ≤ d. When q = 0 the local system \underline{A}^{0} is constant (and equal to K), so the claim is true by the contractibility of T . Note that the short exact sequence

0 → \underline{A}^{q} → \underline{Ã}^{q} → \underline{A}^{q−1} → 0 (4.5)

is split, and it is enough to prove the acyclicity of Ã^{q}.

4.2. The case d = 1. Here T is the q + 1 regular tree. Fix a vertex v_{0}, and a 1-cochain c (every 1-cochain is a cocycle). We shall modify it by coboundaries step-by-step. At the n^{th} step we shall modify c by a coboundary of a 0-cochain supported on the vertices at distance n from v_{0}, 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 v_{0} contiguous to it. Since A(v) → A(uv) is surjective we can perform the n^{th} 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 v_{0} as before. Let Γ_{0}(n) denote the set of vertices at distance n (in the metric ρ introduced in Section 1.2) from v_{0}. The edges at distance n, denoted Γ_{1}(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 v_{0}, and we denote the set of these edges by Γ_{1}(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 Γ_{1}(n, 2). Similarly the triangles (u, v, w) at distance n, denoted Γ_{2}(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 Γ_{2}(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 Γ_{2}(n, 2).

4.4. Vanishing of H^{1}. Let c be a 1-cocycle with values in \underline{Ã}. We shall modify c step-by-step as we did in the 1-dimensional case. In the n^{th} step we shall modify c by the coboundary of a 0-cochain supported on Γ_{0}(n), arriving at a 1-cocycle vanishing on Γ_{1}(m) for all m ≤ n. The n^{th} step is divided into two substeps: in step (n, 1) we take care of the edges of type Γ_{1}(n, 1), and in step (n, 2) we take care of those of type Γ_{1}(n, 2).

Assume therefore that c already vanishes on Γ_{1}(m) for all m ≤ n − 1.

Step (n, 1).

Lemma 4.1. Every v ∈ Γ_{0}(n) has 1 or 2 contiguous vertices u at distance n − 1 from v_{0}.

Proof. We may choose a basis {e_{0}, e_{1}, e_{2}} and lattices M and M_{0} representing v and v_{0} such that M_{0} = ⟨e_{0}, e_{1}, e_{2}⟩, M = ⟨e_{0}, π^{m}e_{1}, π^{n}e_{2}⟩ and 0 ≤ m ≤ n. The question is that of classifying the submodules W of W_{0} = M_{0}/M = O_{K}/(π^{m}) × O_{K}/(π^{n}) killed by π, and such that W_{0}/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

(0) × (π^{n−1})/(π^{n}) ⊆ W ⊆ (π^{m−1})/(π^{m}) × (π^{n−1})/(π^{n}) (4.6)

so there are precisely two such W′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 u′s call them u_{1} and u_{2} and observe (this follows from the proof of the lemma) that - perhaps after switching u_{1} and u_{2} - (v, u_{1}, u_{2}) ∈ T̂_{2}. Since c(u_{1}, u_{2}) = 0 by the induction hypothesis, and since ∂c(v, u_{1}, u_{2}) = 0, c(v, u_{1}) and c(v, u_{2}) restrict to the same element of Ã(v, u_{1}, u_{2}). 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, u_{1}) and on (v, u_{2}).

This concludes step (n, 1), so we assume that c vanishes now on Γ_{1}(m) for all m ≤ n − 1, and also on Γ_{1}(n, 1).

Step (n, 2).

Lemma 4.2. Let (v_{1}, v_{2}) ∈ Γ_{1}(n, 2) (a parallel edge). Then there is a unique u ∈ Γ_{0}(n − 1) such that - possibly after switching the v_{i} - (u, v_{1}, v_{2}) ∈ T̂_{2}.

Proof. We observed in the previous lemma that there are at most two such u′s. Furthermore, if there are two such u′s, say u_{1} and u_{2}, they form triangles with both v_{1} and v_{2}, so the four vertices will be contiguous, contradicting d = 2.

Note that v_{1} and v_{2} do not play a symmetric role now. We label them so that u = [L], v_{i} = [M_{i}] and L ⊃ M_{1} ⊃ M_{2} ⊃ πL. Consider the collection of all the v_{1} that appear. Each parallel edge will ”belong” to precisely one of them. Also, v_{1} determines u. This is clear if there is only one vertex from Γ_{0}(n − 1) contiguous to v_{1}, and in the other case, if u ≠ u′ ∈ Γ_{0}(n − 1) is also contiguous to v_{1}, then (u, v_{1}, u′) is a triangle, and the cyclic ordering of the vertices is well-defined, so there can’t be a triangle of the shape (u′, v_{1}, v_{2}). Thus with each v_{1} comes a unique u and we can consider the collection V = V(v_{1}) of all the v such that (u, v_{1}, v) is a triangle. Say v = [M] where

L ⊃ M_{1} ⊃ M ⊃ πL (4.7)

and each inclusion signifies a drop of 1 in dimension. Either none or precisely one of the v ∈ V lies in Γ_{0}(n − 1). All the rest lie in Γ_{0}(n) and make up a list of the possible v′_{2}s which form a parallel edge (v_{1}, v_{2}). At any rate, no v is another ”v_{1}”.

In Step (n, 2) we shall modify c by a coboundary of a 0-cochain supported only on the v′_{1}s. Note that for every v ∈ V(v_{1}), c(v_{1}, v) may be assumed to be represented by an element of I(u), since from ∂c(u, v_{1}, v) = 0, and from the vanishing of c on Γ_{1}(n, 1), we get that a representative of c(v_{1}, v) lies in I(u) + I(v_{1}) + I(v). Choose therefore representatives b_{v} ∈ I(u) of c(v_{1}, v).

Lemma 4.3. The map

I(u) → ∏_{v∈V(v_{1})} I(u)/(I(u) ∩ I(v_{1}, v)) (4.8)

is surjective.

By the lemma we can find an element of Ã(v_{1}), represented by an element b of I(u), which maps to b_{v}modI(v_{1}, v) for every v ∈ V(v_{1}). In particular, modifying c by the coboundary of the 0-cochain so produced will not damage the vanishing of c on Γ_{1}(n, 1) (on (v_{1}, u) because b ∈ I(u), and if there is a u′ ∈ Γ_{0}(n − 1) ∩ V(v_{1}), also on (v_{1}, u′) because b_{u′} ≡ 0modI(v_{1}, u′)). It will of course make c vanish also on the edges in Γ_{1}(n, 2).

Proof. Let x = δ(e_{a_{0}} ∧··· ∧e_{a_{m}}) be a typical generator of I(u) where {a_{0}, …, a_{m}} = S ⊆ L−πL and is a minimally dependent set modulo πL. We claim that it lies in all I(v_{1}, v) except possibly for one v. In fact, if it does not lie in I(v_{1}, 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 x′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 n^{th} step is again divided into two: in step (n, 1) we take care of the triangles of type Γ_{2}(n, 1), and in step (n, 2) we take care of those of type Γ_{2}(n, 2).

Let c be a 2-cocycle in \underline{Ã}. Assume that it already vanishes on Γ_{2}(m) for all m < n. Every σ ∈ Γ_{2}(n, 1) is of the form (v, u_{1}, u_{2}) for some v ∈ Γ_{0}(n) (and u_{i} ∈ Γ_{0}(n − 1)). Furthermore an edge (v, u_{1}) ∈ Γ_{1}(n, 1) belongs to at most one σ ∈ Γ_{2}(n, 1) by lemma 4.1. By the surjectivity of Ã(vu_{1}) → Ã(vu_{1}u_{2}) we may assign each (v, u_{1}) which appears a value which will make the modified c vanish on all Γ_{2}(n, 1). Note that we have modified c by the coboundary of a 1-cochain supported only on the edges in Γ_{1}(n, 1) of type (v, u_{1}). The edges of type (v, u_{2}) or of type (v, u) when there is a unique u ∈ Γ_{0}(n − 1) contiguous to v, get the value 0.

In step (n, 2) we assume that c already vanishes on Γ_{2}(n, 1). The σ ∈ Γ_{2}(n, 2) are in one-to-one correspondence with the parallel edges at distance n, i.e. with Γ_{1}(n, 2), by lemma 4.2. We may therefore find (by the surjectivity of the restriction maps) a 1-cochain supported on Γ_{1}(n, 2) whose coboundary will match c on Γ_{2}(n, 2), and will of course change nothing on Γ_{2}(m) for m < n or on Γ_{2}(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

H^{p}(T , \underline{A}^{q}) = 0 (4.9)

for p > 0 and every q.

If A = ℙ(V_{K}), and without any restriction on d, we always have

H^{0}(T , \underline{A}^{q}) ≅ C_{har}^{q}. (4.10)

Proof. We have checked the first assertion. As for the second, an element of H^{0}(T , \underline{A}^{q}) is by definition a collection e(v) ∈ A^{q}(v), for every vertex v, such that e(v) and e(u) restrict to the same element of A^{q}(uv) whenever u and v are contiguous. By theorems 3.2 and 3.3 this is the same as a collection of local harmonic cochains c(v) ∈ C_{har}^{q}(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 T̂_{k}(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(V_{K}).

5. Acyclicity (in general, A = ℙ(V_{K}))

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 \underline{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 homomorphisms r_{τ}^{σ} : B(σ) → B(τ) for every face inclusion τ ≤ σ, such that r_{τ}^{σ} ∘ r_{σ}^{ρ} = r_{τ}^{ρ} whenever τ ≤ σ ≤ ρ, and r_{τ}^{τ} is the identity. The dual of a homological local system is a cohomological one, and vice versa.

The group of alternating p-chains C_{p}(T , \underline{B}) consists of all the finitely supported maps γ which assign to each τ ∈ T̂_{p} an element γ(τ) ∈ B(τ) such that

γ(τ′) = sgn(π)γ(τ) (5.1)

if τ′ is obtained from τ by applying the permutation π to the vertices. Define

∂ : C_{p+1}(T , \underline{B}) → C_{p}(T , \underline{B}) (5.2)

by the rule

∂γ(τ) = ∑_{τ′}[τ′ : τ]γ(τ′) (5.3)

where τ′ runs over the collection of p + 1-simplices in T_{p+1} such that τ ≤ τ′, and in the sum each τ′ is given an arbitrary orientation, and the incidence number [τ′ : τ] is determined according to the rule

[(v_{0}, …, v_{p+1}) : (v_{0}, …, v̂_{i}, …, v_{p+1})] = (−1)^{i}. (5.4)

Lemma 5.1. Let \underline{B} be a homological local system of finite dimensional vector spaces, and \underline{B}^{∗} the dual cohomological local system. Then

C^{p}(T , \underline{B}^{∗}) = C_{p}(T , \underline{B})^{∗} (5.5)

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 , \underline{B}), ∂) is acyclic at p > 0. Then (C^{·}(T , \underline{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 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 B_{v} its stabilizer in G, which is isomorphic to P GL_{d+1}(O_{K}). For n ≥ 1, denote by U_{v}^{n} ⊂ B_{v} the principal congruence subgroup of level π^{n} (this is independent of the choice of coordinates). For τ ∈ T_{r} denote by U_{τ}^{n} the subgroup of G generated by ⋃{U_{v}^{n}; v is a vertex of τ}. It is a pro-p group. When v, u are contiguous vertices (ρ(u, v) ≤ 1), U_{v}^{1} ⊂ B_{u}, so U_{v}^{1} ⊂ B_{τ} if v is a vertex of τ. Since each U_{v}^{1} is clearly normal in B_{τ} (because it is normal in B_{v}) it lies in the unique(!) maximal normal pro-p subgroup of B_{τ} , and so does U_{τ}^{1}. In fact U_{τ}^{1} is equal to the unique maximal normal pro-p subgroup of B_{τ} ([S-S], Section 6, Lemma 2). We also have U_{gτ}^{n} = gU_{τ}^{n}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 U_{v}^{n}-fixed vectors for some vertex v and some n ≥ 1. Let \underline{Λ} be the local system

Λ(τ) = Λ^{U_{τ}^{n}} (5.6)

with inclusion as restriction maps. Then

C_{·}(T , \underline{Λ}) → Λ → 0 (5.7)

is a resolution of Λ (the augmentation map is the natural sum map ⨁_{v} Λ^{U_{v}^{n}} → Λ). In particular, the complex (C_{·}(T , \underline{Λ}), ∂) is acyclic at p > 0.

The local system \underline{Λ} figuring in the theorem has the property that it is Gequivariant : there are maps

g_{τ} : Λ(τ) → Λ(gτ), (5.8)

commuting with the restriction maps, for every g ∈ G, satisfying 1_{τ} = identity and g′_{gτ} ∘ g_{τ} = (g′g)_{τ}. The complex C_{·}(T , \underline{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(V_{K}) (see Section 2.5).

5.2. The definition of Λ. We now apply the above to our situation. Let B_{k}(τ) = A^{k}(τ)^{∗}. We want to define a smooth representation Λ_{k} as in the theorem such that the local systems \underline{Λ}_{k} and \underline{B}_{k} are naturally isomorphic. Recall that for each σ ∈ T̂_{k} we defined a linear functional (σ, −) on E^{k} in Section 2.3, and we have

(gσ, ge) = (σ, e) (5.9)

for every g ∈ G. Denote the linear functional (σ, −) by λ_{σ} and let Λ_{k} be the linear span of all the λ_{σ} in Hom(E^{k}, K). Of course, the λ_{σ} are far from independent, as they satisfy the harmonicity conditions. The action of G is compatible with its action on T̂_{k} : g(λ_{σ}) = λ_{gσ}. It is also clear that Λ_{k} is a smooth representation. Furthermore, if v is any vertex of σ, and U_{v} = U_{v}^{1}, then U_{v} fixes all the vertices at distance 1 from v, hence it fixes σ, and also λ_{σ}. It follows that Λ_{k} satisfies the condition of the theorem with n = 1. For every τ write U_{τ} = U_{τ}^{1}. The next theorem shows that we have a natural isomorphism

Λ_{k}^{U_{τ}} ≃ A^{k}(τ)^{∗}. (5.10)

Theorem 5.4. A linear functional λ ∈ Λ_{k} factors through A^{k}(τ) if and only if it is fixed by U_{τ} .

Proof. One direction follows easily from Corollary 2.7 : every linear functional which annihilates I(τ) ∩ E^{k} and therefore factors through A^{k}(τ), can be written as a linear combination of λ_{σ} with σ for which ρ(σ, τ) ≤ 1. Such σ’s are fixed by U_{τ}, because each of their vertices is fixed by U_{τ} . For the converse direction we may assume that τ is a vertex u. Indeed, if τ = {u_{0}, …, u_{r}} and λ is fixed by U_{τ} , then it is fixed by each U_{u_{i}}, and if we know the theorem in the case of a vertex, it will follow that λ annihilates E^{k} ∩I(u_{i}) 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 U_{u}, u ∈ T_{0}. We shall use the harmonicity conditions to show that λ can be written as a linear combination of λ_{σ} for σ which are contiguous to u. We shall use repeatedly the fact that if λ = ∑ c_{σ}λ_{σ} is fixed by U_{u}, and every σ is fixed by U_{u}^{n} (this will always be the case for n large enough) we may average over the coset space U_{u}/U_{u}^{n}, not changing λ, and may therefore assume that for every g ∈ U_{u}, the coefficients c_{σ} and c_{gσ} are equal. Furthermore, 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 = (v_{0}, …, v_{k}) 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 v_{i} and u lie (strictly) on opposite sides of W.

Proof. Consider, for every v_{i}, the collection of all the walls W in A passing through {v_{0}, …, v̂_{i}, …, v_{k}} but not through v_{i}. Each of them divides A into two closed half-spaces A^{+}(W) and A^{−}(W), the former containing v_{i}, the latter not. If the conclusion of the lemma doesn’t hold then u ∈ A^{+}(W) for every such W, for every v_{i} (this allows also for the case where u lies on W). But it is clear that ⋂_{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 = (v_{0}, …, v_{d}) 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 v_{i}, and that u and v_{i} lie in opposite sides of W. Let s_{W} ∈ G be an involution inducing on A a reflection in W. Then if we let v′_{i} run over the q + 1 vertices of T which form a chamber together with {v_{0}, …, v_{i−1}, v_{i+1}, …, v_{d}}, precisely two of them, v_{i} and s_{W} (v_{i}) will lie in A. Every v′_{i} ≠ s_{W} (v_{i}) will be of the form g(v_{i}) for g ∈ U_{u} fixing the wall W pointwise (g folds A along W).

Proof. The vertices v_{i} and s_{W} (v_{i}) belong to A, and they are clearly the only ones among the v′_{i}, because A is a Euclidean space. Since s_{W} (v_{i}) is closer to u than v_{i}, in the metric d, it can’t be of the form g(v_{i}) for a g fixing u. That the other q v′_{i} are of the form g(v_{i}) for g ∈ U_{u} fixing W pointwise can be seen by introducing coordinates, and expressing the v′_{i} in terms of the basis used to describe the apartment.

Consider our λ, fixed by U_{u}, and let E be the collection of all the finite expressions

ε = ∑ c_{σ,σ̃}(σ, σ̃) (5.11)

where the pairs (σ, σ̃) run over σ ∈ T̂_{k} and σ̃ ∈ T_{d} such that σ ≤ σ̃, the coefficients c_{σ,σ̃} ∈ K, and λ_{ε} = ∑ c_{σ,σ̃}λ_{σ} 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

||ε|| = max_{(σ,σ̃)} sup_{x∈|σ̃|} d(u, x), (5.12)

where the max is over all the pairs with c_{σ,σ̃} ≠ 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

ε^{sm} = [U_{u} : U_{u}^{n}]^{−1} ∑_{g∈U_{u}/U_{u}^{n}} ∑ c_{σ,σ̃}(gσ, gσ̃) (5.13)

for some large enough n, and ε^{sm} ∈ E is of the same norm : ||ε^{sm}|| = ||ε||. The advantage of ε^{sm} over ε is that it is invariant under U_{u}.

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 ε′ ∈ E with ||ε′|| < ||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 = ε^{sm}. 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, s_{W} (σ̃)), and observe that s_{W} (σ̃) is closer to u than σ̃.

(ii) Suppose that σ̃ = (v_{0}, …, v_{d}) = (L_{0} ⊃ ⋯ ⊃ L_{d} ⊃ πL_{0}) and the vertex off W is v_{i}. Since we have dealt with case (i), we may assume that v_{i} ∈ σ. Assume now that v_{i+1} ∈ σ, but v_{i−1} ∉ σ. Consider the q + 1 possible v′_{i} such that (v_{0}, …, v_{i−1}, v′_{i}, v_{i+1}, …, v_{d}) is a chamber. Let σ̃_{j} (1 ≤ j ≤ q) be the q among these chambers where v′_{i} = g(v_{i}) for g ∈ U_{u} fixing W, and let σ_{j} be the corresponding k-cells g(σ). Since ε is smooth, the pairs (σ_{j}, σ̃_{j}) appear in ε with the same coefficient. Let σ̃_{0} = s_{W} (σ̃) be the remaining chamber, and σ_{0} = s_{W} (σ). By property C of the harmonicity conditions, we may replace the part of the sum in ε involving (σ_{j}, σ̃_{j}) by a combination of (σ_{0}, σ̃_{0}) and (σ′, σ̃_{0}) where σ′ is obtained from σ by replacing v_{i} by v_{i−1} (which was assumed not to be a vertex of σ). Note that σ′ lies on W, hence is contained in s_{W} (σ̃) = σ̃_{0}. As before, σ̃_{0} is closer to u than σ̃.

(iii) Suppose that σ̃ is as in (ii), but both v_{i+1} and v_{i−1} belong to σ (as well as v_{i} of course). Once again consider the q + 1 possible v′_{i}, and use the same notation for the σ_{j}, 0 ≤ j ≤ q. Property B this time implies that the sum of all the λ_{σ_{j}} vanishes, hence the part of ε involving (σ_{j}, σ̃_{j}) for 1 ≤ j ≤ q (which appear with the same coefficients) can be replaced by a multiple of (σ_{0}, σ̃_{0}).

(iv) Finally assume that σ̃ is as in (ii), v_{i} ∈ σ, but v_{i+1} ∉ σ. Let σ′ be the k + 1-cell σ′ = σ ∪ {v_{i+1}}. Let σ_{j} (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 v_{i} - falls under case (i), and the others have both v_{i} and v_{i+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 λ_{σ_{j}} 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 ε′ ∈ 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(V_{K}). Then

H^{p}(T , \underline{A}^{q}) = 0 (5.14)

for p > 0 and every q, and

H^{0}(T , \underline{A}^{q}) ≅ C_{har}^{q}. (5.15)

Proof. The acyclicity follows from the last theorem and from the discussion preceding it. The second assertion is proved as in theorem 4.4.

Corollary 5.8. C_{har}^{k} is the algebraic dual of the representation Λ_{k}.

Proof. We have C_{har}^{k} = Ker(C^{0}(T , \underline{A}^{k}) → C^{1}(T , \underline{A}^{k})), which is dual to the cokernel of C_{1}(T , \underline{Λ}_{k}) → C_{0}(T , \underline{Λ}_{k}). But by the theorem of Schneider and Stuhler that cokernel is Λ_{k}.

Proposition 5.9. The representation Λ_{k} is admissible and irreducible.

Proof. (Sketch) We have seen that it is smooth. In a way completely analogous to the proof of theorem 5.4 one may show that for any fixed n any λ ∈ Λ_{k}^{U_{u}^{n}} can be written as a linear combination of λ_{σ} for σ in a bounded ball around u. It follows that Λ_{k}^{U_{u}^{n}} is finite-dimensional, and since the U_{u}^{n} form a fundamental system of neighborhoods at 1 in G, Λ_{k} 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 \underline{A} to the problem of computing the de-Rham cohomology of Drinfeld’s p-adic symmetric 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.

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”

r : 𝔛(K̄) → |T | (6.1)

which we now recall. First identify |T | with the space of all dilation classes of real norms on V_{K}. To achieve this write, for every lattice L in V_{K}, and a ∈ V_{K},

α_{L}(a) = min {|λ|; λ^{−1}a ∈ L}. (6.2)

If πL_{0} ⊂ L_{k} ⊂ ... ⊂ L_{1} ⊂ L_{0} is a lattice flag, so that v_{i} = [L_{i}] form a pointed k–cell σ = (v_{0}, ..., v_{k}) ∈ T̂_{k}, then points in |σ| can be written uniquely as t = ∑_{i=0}^{k} t_{i}v_{i}, ∑ t_{i} = 1, 0 < t_{i}. Noting that α_{L_{0}} ≤ α_{L_{1}} ≤ ... ≤ α_{L_{k}} ≤ qα_{L_{0}}, we associate with such a t the dilation class [α_{t}] of the norm

α_{t} = max(α_{L_{0}}, q^{−t_{0}}α_{L_{1}}, q^{−t_{0}−t_{1}}α_{L_{2}}, ..., q^{−t_{0}−t_{1}··· −t_{k−1}}α_{L_{k}}). (6.3)

This is independent of the distinguished vertex of σ because if we perform a cyclic permutation of the vertices we multiply α_{t} by q^{t_{0}}. Likewise, if we change the lattices by a homothety, all have to change by the same factor, so the dilation class of α_{t} is unchanged. We conclude that t ↦ [α_{t}] is a well defined map from |T | to the space of dilation classes of real norms on V_{K}. 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 V_{K}, we proceed to define the reduction map. Pick x ∈ V ∗ ; K̄. Then |a|_{x} = |x(a)| is a semi-norm on V_{K}, whose dilation class depends only on [x] ∈ ℙ(V ∗ ; K̄). 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∈S} |σ|,

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 v_{0} = [L_{0}] ∈ T_{0}. Then

X_{v_{0}} = r^{−1}(v_{0}) = {[x]; ∀a ∈ L_{0} − πL_{0}, |x(a)| = 1}. (6.5)

This is isomorphic (upon introduction of coordinates in which L_{0} = O_{K}^{d+1}) to

(6.6) {[x_{0}, x_{1}, ..., x_{d}]; |x_{i}| = 1, x̄_{i} are linearly independent over 𝔽_{q}},

which is the affinoid obtained by removing from ℙ^{d}(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 ∈ L_{0} − πL_{0}. Then we let H_{a} be the hyperplane in ℙ(V^{∗}) defined by x(a) = 0, and for any 0 < r < 1 we let

H_{a}(r^{−}) = {[x]; max_{b∈L_{0}} |x(b)| = 1, |x(a)| < r}
(6.7)  H_{a}(r) = {[x]; max_{b∈L_{0}} |x(b)| = 1, |x(a)| ≤ r}.

(In coordinates as above the condition max_{b∈L_{0}} |x(b)| = 1 is equivalent to max |x_{i}| = 1). We call them the open (closed) tubular neighborhoods of H_{a} of radius r determined by L_{0}. Like H_{a}, they only depend on [a] ∈ ℙ(V_{K}). For |π| ≤ r < 1 (resp. |π| < r < 1) H_{a}(r) (resp. H_{a}(r^{−})) depends only on the image ā of a in V_{0} = L_{0}/πL_{0}, and in fact only on its class [ā] ∈ ℙ(V_{0}).

6.3. The pre-images of simplices and stars. Fix σ = (L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0}) ∈ T̂_{k}. Let

X_{σ} = {[x]; ∃ |π|R_{0} < R_{k} < ... < R_{0} s.t. ∀ a ∈ L_{i} − L_{i+1}, |x(a)| = R_{i}} . (6.8)

We claim that X_{σ} = r^{−1}(|σ|). Pick [x] ∈ X_{σ} and let t_{i} ∈ (0, 1) be such that

q^{1−t_{0}−t_{1}··· −t_{i}} = r_{i} = R_{i}/R_{k}. (6.9)

Then t = ∑_{i=0}^{k} t_{i}v_{i} ∈ |σ|. If a ∈ L_{i} − L_{i+1} then

α_{L_{0}}(a) = ⋯ = α_{L_{i}}(a) = 1 < q = α_{L_{i+1}}(a) = ⋯ = α_{L_{k}}(a) (6.10)

so α_{t}(a) = r_{i}. This proves that, up to a scalar, α_{t}(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 r_{i} accordingly. Then [x] ∈ X_{σ}.

It follows from the characterization of the X_{σ} that

X(v_{0}) = r^{−1}(St(v_{0})) = {[x]; ∀ a ∈ L_{0} − πL_{0}, |π| < |x(a)| ≤ 1}. (6.11)

Equivalently, fix a set A of representatives of ℙ(V_{0}) in L_{0} − πL_{0}. Then

r^{−1}(St(v_{0})) = ℙ(V^{∗}) − ⋃_{a∈A} H_{a}(|π|). (6.12)

More generally, writing X(σ) = r^{−1}(St(σ)), we claim that

X(σ) = {[x]; ∃ |π|R_{0} < R_{k} < ... < R_{0} s.t. a ∈ L_{i} ⇔ |x(a)| ≤ R_{i}} . (6.13)

Indeed, if [x] ∈ X(σ) then [x] ∈ X_{τ} for some σ ≤ τ. If we put R_{i} = max_{a∈L_{i}} |x(a)| then L_{i} is characterized as the set of a′s for which |x(a)| ≤ R_{i}. Conversely, suppose such a sequence of R_{i} exists for a fixed x. Let r_{m} < ⋯ < r_{0} be the set of values obtained as |x(a)| for a ∈ L_{0} − πL_{0}, and let M_{i} = {a; |x(a)| ≤ r_{i}}. Clearly [x] ∈ X_{τ} where τ = ([M_{0}], ….[M_{m}]). But r_{0} ≤ R_{0}, so we get L_{0} ⊆ M_{0} ⊆ L_{0}, hence M_{0} = L_{0}.

Also, if 0 < i ≤ k then for some j ≥ 1 we must have r_{j} ≤ R_{i} < r_{j−1}, so we get M_{j} ⊆ L_{i} ⊆ M_{j}, hence M_{j} = L_{i}. It follows that σ ≤ τ, and [x] ∈ X(σ).

6.4. Coordinates on X_{σ} and X̃_{σ}. Let 𝔱(σ) = (e_{0}, e_{1}, ..., e_{k}) be the type of σ, and write d_{i} = dim V_{i} = e_{i} + ⋯ + e_{k} (here V_{i} = L_{i}/πL_{0}). First choose a basis of V_{K} adapted to σ, and let x_{0}, …, x_{d} be the corresponding coordinate functions on V^{∗}. Let

t_{i} = \frac{x_{d_{i}−1}}{x_{d}} (1 ≤ i ≤ k)
(6.14)  y_{j} = \frac{x_{j}}{x_{d_{i}−1}} (0 ≤ i ≤ k, d_{i+1} ≤ j < d_{i} − 1).

These are well-defined coordinates on X_{σ}. The “compact coordinates” y_{j} come in blocks of sizes e_{0} − 1, e_{1} − 1, ..., e_{k} − 1 (some or all of these blocks may be empty). Since ∑ e_{i} = d+1, we have d coordinates altogether, and there are k “non-compact” coordinates (the t_{i}).

For any integer m let C_{m} be the affinoid subdomain of 𝔸^{m} defined by the condition |c_{0} + ∑_{i=1}^{m} c_{i}x_{i}| = 1 for every vector of c_{i} ∈ O_{K} with max |c_{i}| = 1. Then the coordinates (y_{d_{i+1}}, …, y_{d_{i}−2}) map X_{σ} onto C_{e_{i}−1} and altogether the t_{i} and the y_{j} induce an isomorphism

X_{σ} ≅ C_{e_{0}−1} × ⋯ × C_{e_{k}−1} × A_{k}, (6.15)

where A_{k} is the “multiannulus”

|π| < |t_{k}| < ⋯ < |t_{1}| < 1. (6.16)

In a similar fashion we define z_{i} = x_{d_{i}−1} (0 ≤ i ≤ k). Then the z_{i} and the y_{j} induce an isomorphism

X̃_{σ} ≅ C_{e_{0}−1} × ... × C_{e_{k}−1} × Ã_{k+1}, (6.17)

where Ã_{k+1} is the multiannulus

|πz_{0}| < |z_{k}| < ⋯ < |z_{1}| < |z_{0}|. (6.18)

Observe that A_{k} is the successive fibration of annuli k times (the cyclic order in which this is done is a conformal invariant), and Ã_{k+1} = A_{k} × 𝔾_{m}.

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 X̃_{σ}. Introduce coordinates as above. Then writing C = C_{e_{0}−1} × ... × C_{e_{k}−1} we may write

η = ∑_{i=0}^{k+1} ∑_{|I|=i} ∑_{J} γ_{i,I,J} ∧ z^{J}dz_{I}. (6.19)

Here we used the following abbreviations and conventions. I runs over all the sequences 0 ≤ ι_{1} < ι_{2} < ... < ι_{i} ≤ k, and dz_{I} = dz_{ι_{1}} ∧ ... ∧ dz_{ι_{i}}. J runs over all the sequences (j_{0}, ..., j_{k}) ∈ ℤ^{k+1} and z^{J} = ∏_{l=0}^{k} z_{l}^{j_{l}}. For each I and J, γ_{i,I,J} is a k + 1 − i form on the affinoid C. Using the obvious norms on such forms we must have |γ_{i,I,J}|_{C}R^{J} → 0 whenever |π|R_{0} < R_{k} < ... < R_{1} < R_{0}. Now suppose that η is closed. Then

0 = dη = ∑_{i=0}^{k+1} ∑_{|I|=i} ∑_{J}(dγ_{i,I,J} ∧ z^{J}dz_{I} +

(−1)^{k+1−i} ∑_{ι=0}^{k} γ_{i,I,J} ∧ j_{ι}z^{J−ε_{ι}}dz_{ι} ∧ dz_{I}), (6.20)

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}dz_{I} in dη, which is a k + 2 − i form on C, should vanish, and we get

0 = dγ_{i,I,J} + (−1)^{k−i} ∑_{l=1}^{i} γ_{i−1,I−{ι_{l}},J+ε_{ι_{l}}} (−1)^{l−1}(j_{ι_{l}} + 1). (6.21)

In particular, if ι ∈ I ⇒ j_{ι} = −1, then γ_{i,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.

Definition 6.1. Let σ = (v_{0}, ..., v_{k}) ∈ T̂_{k} (0 ≤ k ≤ d). Let η be a closed k +1-form on X̃_{σ}. Choose coordinates as above and define the residue of η along σ to be

res_{σ}η = γ_{k+1,I,J} (6.22)

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 which represent the classes in H_{dR}^{k}(X̃_{σ}). That formula will be purely combinatorial, and will have no reference to coordinates.

Lemma 6.2. Let σ = (v_{0}, v_{1}, …, v_{k}) and σ′ = (v_{1}, …, v_{k}, v_{0}). Let η be a closed k + 1 -form on X̃_{σ} = X̃_{σ′}. Then

res_{σ}η = (−1)^{k}res_{σ′}η. (6.23)

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 x_{0}, …, x_{d} be coordinates in V_{K}^{∗} adapted to σ. Then

{x′_{0}, …, x′_{d}} = {π^{−1}x_{d_{1}}, …, π^{−1}x_{d}, x_{0}, …, x_{d_{1}−1}} (6.24)

are coordinates which are adapted to σ′. Defining z′_{i} and y′_{j} as above from the x′_{i}, we see that the y′_{j} are nothing but the old y_{j}, with the indices permuted, while

z′_{0} = z_{1}, …, z′_{k−1} = z_{k}, z′_{k} = π^{−1}z_{0}. (6.25)

Thus

\frac{dz′_{0}}{z′_{0}} ∧ ⋯ ∧ \frac{dz′_{k}}{z′_{k}} = (−1)^{k} \frac{dz_{0}}{z_{0}} ∧ ⋯ ∧ \frac{dz_{k}}{z_{k}} (6.26)

and this proves the lemma.

7. The cohomology of X(τ)

Fix τ ∈ T_{r}. 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 v_{0}, we saw that X(v_{0}) is an “open tubular neighborhood” of X_{v_{0}}, and X_{v_{0}} is an affinoid whose reduction is the complement Z in ℙ(V_{0}^{∗}) of all the 𝔽_{q} -rational hyperplanes. The cohomology of X(v_{0}) is the same as the cohomology of X_{v_{0}}, 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(v_{0}), 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)′s, but also of their intersections

X(τ) = ⋂_{v∈τ} X(v). (7.1)

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 homomorphism

ω : A^{k}(τ) → H_{dR}^{k}(X(τ)), e ↦ ω(e). (7.2)

The residue map defined above will produce another homomorphism

c : H_{dR}^{k}(X(τ)) → C_{har}^{k}(τ), η ↦ c_{η}, (7.3)

where C_{har}^{k}(τ) is the space of local harmonic cochains on T̂_{k}(τ), defined in Section 3. The composition c ∘ ω will be shown to be the cochain homomorphism e ↦ c_{e} (whence the ambiguity in notation : c_{ω(e)} = c_{e}). We proved that it is injective, and an isomorphism if τ = {v_{0}} 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 = {v_{0}}. ^{1}

We shall begin by working with the X̃(τ), and later on descend to their image in projective space, X(τ).

1The 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 ∈ T̂_{k}(τ) by the sum over certain neighboring σ′, and some of these may not lie anymore in T̂_{k}(τ). However, the harmonicity conditions do suffice if τ is a vertex.

7.1. The homomorphism ω from Ã(τ) to H_{dR}(X̃(τ)). We refer to the Appendix for a discussion of the rigid de-Rham cohomology of a rigid analytic space X defined over K. We denote it by H_{dR}(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 = (M_{0} ⊃ M_{1} ⊃ ⋯ ⊃ M_{r} ⊃ πM_{0}) ∈ T_{r} (the choice of the distinguished vertex is still at our disposal), and consider the map of graded K -algebras Ẽ → H_{dR}(X̃(τ)) defined by e_{a} ↦ [ω_{a}], where ω_{a} = 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 homomorphism

ω : Ã(τ) → H_{dR}(X̃(τ)). (7.4)

The homomorphism ω is surjective. The canonical projection X̃(τ) → X(τ) induces a commutative diagram with exact rows and surjective vertical maps

\begin{aligned}0 & → & A(τ) & → & Ã(τ) & → & A(τ)[1] & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & H_{dR}(X(τ)) & → & H_{dR}(X̃(τ)) & → & H_{dR}(X(τ))[1] & → & 0 \end{aligned} . (7.5)

Proof. Suppose first that {a_{0}, …, a_{k}} are linearly dependent. We want to show that ω ∘ δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}}) = 0. By (2.4) we may assume that the a_{i} are a minimal set of dependent elements, and since e_{λa} = e_{a} for any λ ≠ 0, we may assume that the linear dependence among them is ∑_{i=0}^{k} a_{i} = 0. We then compute, using the fact that da_{k} = − ∑_{i=0}^{k−1} da_{i},

ω ∘ δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}}) = \frac{1}{a_{0} ⋯ a_{k}} ∑_{i=0}^{k}(−1)^{i}a_{i}da_{0} ∧ ⋯ ∧ d̂a_{i} ∧ ⋯ ∧ da_{k}

(7.6)  = (−1)^{k} \frac{da_{0} ∧ ⋯ ∧ da_{k−1}}{a_{0} ⋯ a_{k}} ∑_{i=0}^{k} a_{i} = 0.

Next suppose that a and b lie in M_{i} − M_{i+1} but are congruent modulo M_{i+1}. Then from a = b(1 + (a − b)/b) we get

ω_{a} = ω_{b} + ω_{1+(a−b)/b}. (7.7)

But on X̃(τ) we have |(a − b)/b| < 1, so log(1 + (a − b)/b) is rigid analytic there, and [ω_{a}] = [ω_{b}].

Now if a_{0}, …, a_{k} are in M_{i} − M_{i+1}, but are linearly dependent modulo M_{i+1}, then again we may assume they are a minimal set of this type, and dividing them by appropriate units from K we may assume that a_{k}−(∑_{i=0}^{k−1} a_{i}) ∈ M_{i+1}. By what was said above we may replace a_{k} by a′_{k} = ∑_{i=0}^{k−1} a_{i} without affecting ω∘δ(e_{a_{0}} ∧···∧e_{a_{k}}). But then the a_{i} are linearly dependent, so ω ∘ δ(e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}}) = 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 ∈ V_{K} gives us an isomorphism X̃(τ) ≃ X(τ) × 𝔾_{m} (in fact, 𝔛̃ ≃ 𝔛 × 𝔾_{m}) through x ↦ ([x], x(a)). The Künneth formula yields the

34 EHUD DE SHALIT

factorization H_{dR}^{k+1}(X̃(τ)) = H_{dR}^{k+1}(X(τ)) ⨁[ω_{a}] ∧ H_{dR}^{k}(X(τ)). (7.8) Since A(τ) is generated by e_{a} − e_{b}, and [ω_{a} − ω_{b}] ∈ H_{dR}^{1}(X(τ)), we see that ω(A^{k+1}(τ)) ⊆ H_{dR}^{k+1}(X(τ)), and ω(e_{a} ∧ A^{k}(τ)) ⊆ [ω_{a}] ∧ H_{dR}^{k}(X(τ)). The splitting Ã^{k+1}(τ) = A^{k+1}(τ) ⨁ e_{a} ∧ A^{k}(τ) (7.9) shows that each of the two direct summands in Ã^{k+1}(τ) gets mapped surjectively onto the corresponding direct summand in H_{dR}^{k+1}(X̃(τ)). Finally, the induced map between short exact sequences is independent of the a used to split them.

We shall denote the canonical map H_{dR}^{k+1}(X̃(τ)) → H_{dR}^{k}(X(τ)) also by δ. Thus for any a and [η] ∈ H_{dR}^{k}(X(τ)), δ([ω_{a}] ∧ [η]) = [η], (7.10) and δ ∘ ω = ω ∘ δ. (7.11) 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 ω_{a_{0}} ∧···∧ω_{a_{k}} on a σ ∈ T̂_{k} (0 ≤ k ≤ d). To simplify the notation write S for the ordered k + 1-tuple (a_{0}, …, a_{k}), and ω_{S} for ω_{a_{0}} ∧ ⋯ ∧ ω_{a_{k}}.

Let σ = (L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0}). 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 res_{σ}(ω(ẽ)) = (σ̃, ẽ). (7.12) Proof. We may assume that ẽ = e_{S} where S = (a_{0}, …, a_{k}), and each a_{i} ∈ L_{0}−πL_{0}. Fix coordinates x_{0}, …, x_{d} adapted to σ, and write a_{i} = ∑_{j=0}^{d} a_{ij}x_{j} (0 ≤ i ≤ k). Using the z_{l} and the y_{j} as variables on X̃_{σ} (see above)

a_{i} = ∑_{l=0}^{k} c_{il}(y)z_{l} (7.13)

where the c_{il} are non-homogeneous linear forms in the y_{j} which belong to the l^{th} block. Furthermore on X̃_{σ} we have |πz_{0}| < |z_{k}| < ⋯ < |z_{1}| < |z_{0}| (7.14)

|c_{il}(y)| ≤ 1, all l (7.15)

|c_{il}(y)| = 1 for l = ι_{σ}(a_{i}) (7.16) (see Definition 2.1 for the definition of the index of a with respect to σ), |c_{il}(y)| ≤ |π| for 0 ≤ l < ι_{σ}(a_{i}). (7.17) Writing m = m(i) = ι_{σ}(a_{i}) for simplicity we deduce that a_{i} = c_{im}(y)z_{m} (1 + A) (7.18) where

A = ∑_{l≠m} \frac{c_{il}(y)}{c_{im}(y)} \frac{z_{l}}{z_{m}} (7.19)

satisfies |A| < 1 throughout X̃_{σ}. It follows that log(1 + A) is rigid analytic in X̃_{σ} and

\frac{da_{i}}{a_{i}} ≡ \frac{dc_{im}}{c_{im}} + \frac{dz_{m}}{z_{m}} mod (exact differentials).

Since the residue of an exact differential vanishes,

res_{σ}(\frac{da_{0}}{a_{0}} ∧ ⋯ ∧ \frac{da_{k}}{a_{k}}) = res_{σ}(\frac{dz_{m(0)}}{z_{m(0)}} ∧ ⋯ ∧ \frac{dz_{m(k)}}{z_{m(k)}} ) = (σ̃, e_{a_{0}} ∧ ⋯ ∧ e_{a_{k}}) (7.20)

as required.

Corollary 7.3. The residue is well defined, independently of the choice of coordinates.

Proof. Any closed differential form on X̃_{σ} is, up to exact forms, an ω(e). But the expression derived above is independent of the choice of coordinates.

Corollary 7.4. For any [η] ∈ H_{dR}^{k+1}(X(σ)), res_{σ}(η) = 0, and therefore res_{σ} factors through H_{dR}^{k}(X(σ)).

Proof. This was proved in Lemma 2.4 for (σ̃, −).

7.3. Defining the local residue cochains c_{η}. Let τ ∈ T_{r}, and recall that T̂_{k}(τ) is the collection of pointed k-cells σ contiguous to τ. We want to attach to any [η] ∈ H_{dR}^{k+1}(X̃(τ)) its residue along σ, denoted res_{σ}(η) for every σ ∈ T̂_{k}(τ). If τ ≤ σ then X̃_{σ} ⊆ 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′ ∈ T̂_{m} be a pointed m-cell, and η a closed k + 1-form on 𝔛̃, where m ≥ k. Then if η|X̃_{σ′} is exact,

res_{σ}(η) = 0 (7.21)

for every σ ∈ T̂_{k} which is a face of σ′.

Proof. Let σ ≤ σ′, and put Ỹ = X̃_{σ′} ∪ X̃_{σ}. Then Ỹ is the cone over a simplicial subdomain of 𝔛. Let

σ′ = (L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{m} ⊃ L_{m+1} = πL_{0}) (7.22)

and

σ = (L_{0} ⊃ L_{i_{1}} ⊃ ⋯ ⊃ L_{i_{k}} ⊃ L_{m+1} = πL_{0}). (7.23)

Here i_{0} = 0 and i_{k+1} = m + 1, and we have assumed without loss of generality that [L_{0}] is a common vertex to σ and σ′. Introduce coordinates adapted to σ′, as in Section 6. Then these coordinates are adapted to σ too, and we may consider the ensuing coordinates z_{i_{l}} (0 ≤ l ≤ k) and y_{j} (0 ≤ l ≤ k, d_{i_{l+1}} ≤ j < d_{i_{l}} − 1) given by (6.14), but with respect to the filtration defined by σ. These coordinates induce an isomorphism similar to (6.17)

Ỹ ≅ D_{0} × ⋯ × D_{k} × Ã_{k+1}, (7.24)

where Ã_{k+1} is the multiannulus

|πz_{0}| < |z_{i_{k}}| < ⋯ < |z_{i_{1}}| < |z_{0}|. (7.25)

The projection of X̃_{σ} to D_{l} is an affinoid of the type C_{e−1}, where e = dim L_{i_{l}}/L_{i_{l+1}}. However, the projection of X̃_{σ′} to D_{l} is itself of the form

C_{e_{0}−1} × ⋯ × C_{e_{t}−1} × A_{t}, (7.26)

where 1 + t = i_{l} − i_{l+1} is the “number of steps in σ′ refining the l^{th} step in σ” and the e_{j} are the dimension jumps in these steps.

At any rate, from the Künneth decomposition applied to Ỹ we deduce a decomposition of η as in (6.19), where the γ_{i,I,J} are k+1−i forms on D = D_{0}×···×D_{k} and the z′s are taken from {z_{0}, z_{i_{1}}, …, z_{i_{k}}}. Now res_{σ}(η) is the coefficient γ_{k+1,I,J}|X̃_{σ} where I = (0, i_{1}, …, i_{k}) and J = (−1, …, −1), which is constant since η is closed. But if η|X̃_{σ′} is exact, γ_{k+1,I,J}|X̃_{σ′} vanishes. Since Ỹ is connected

γ_{k+1,I,J}|X̃_{σ} = γ_{k+1,I,J}|X̃_{σ′} (7.27)

and the lemma follows.

Suppose now that τ ∈ T_{r} and σ ∈ T̂_{k}(τ). Let σ′ = σ ∪ τ (with the distinguished vertex of σ). Let [η] ∈ H_{dR}^{k+1}(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 X̃_{σ′} ⊆ X̃(τ), and by the lemma its residue on σ vanishes. Thus res_{σ}(η) depends only on the class of η in H_{dR}^{k+1}(X̃(τ)), and this enables us to define

(7.28) res_{σ} : H_{dR}^{k+1}(X̃(τ)) → K, whenever ρ(σ, τ) ≤ 1.

Definition 7.1. Let [η] ∈ H_{dR}^{k}(X(τ)). Consider any [η̃] ∈ H_{dR}^{k+1}(X̃(τ)) such that δ([η̃]) = [η] (e.g. η̃ = ω_{a} ∧ η). Let c_{η} : T̂_{k}(τ) → K be defined by

c_{η}(σ) = res_{σ}(η̃). (7.29)

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, c_{η} is well-defined. We call c_{η} the local residue cochain (at τ) attached to η.

Corollary 7.6. We have the following formula, using the notation of Definition 2.5 :

c_{ω(e)} = c_{e} (7.30)

for every e ∈ A^{k}(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

c_{η}(σ) = res_{σ}(η̃) = (σ̃, ẽ) = (σ, e) = c_{e}(σ) (7.31)

for every σ ∈ T̂_{k}(τ).

Theorem 7.7. The residue cochain c_{η} is harmonic. The local residue cochain is a homomorphism

c : H_{dR}^{k}(X(τ)) → C_{har}^{k}(τ). (7.32)

This homomorphism is injective, and bijective if τ = v_{0} is a vertex.

The map ω is an isomorphism

ω : A^{k}(τ) ≃ H_{dR}^{k}(X(τ)) (7.33)

for every τ.

Proof. The thorem follows from the discussion above and the corresponding theorems in Section 3.

8. The cohomology in the large

Consider the tiling of 𝔛 by the sets X(v), v ∈ T_{0}. 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 A^{k}(v)), and as spaces of local harmonic cochains (the C_{har}^{k}(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 subsection will be of a general nature. Let X be a smooth connected rigid analytic space, and

U = {X(v) → X} (8.1)

an admissible cover by open subdomains. For each σ = {v_{0}, v_{1}, …, v_{k}} let X(σ) = ⋂_{i=0}^{k} X(v_{i}). Let H^{q} be the presheaf

H^{q}(Y) = H_{dR}^{q}(Y/K). (8.2)

Proposition 8.1. There exists a first-quadrant spectral sequence

E_{2}^{pq} = Ȟ^{p}(U, H^{q}) ⟹ H_{dR}^{n}(X/K). (8.3)

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

\begin{aligned}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}) & → & ⋯ \\ & δ ↑ & & ↑ & & ↑ \\ q = 0 & C^{0}(U, I^{0}) & →^{∂} & C^{0}(U, I^{1}) & →^{∂} & C^{0}(U, I^{2}) & → & ⋯ \\ & p = 0 & & 1 & & 2 \end{aligned} (8.4)

where C^{q}(U, I^{p}) is the space of (alternating) q-cochains with values in I^{p}, δ is the Čech derivation and ∂ the derivation 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 double complex. Then there are two spectral sequences abutting to the homology groups H^{n}(C^{·}). The first is

′E_{1}^{pq} = Ȟ_{δ}^{q}(U, I^{p}) = \{ \begin{matrix}0 & q > 0 & since I^{p} is injective, hence flasque \\ Γ(X, I^{p}) & q = 0 \end{matrix} (8.5)

and ′d_{1}^{p,0} is induced from ∂ : I^{p} → I^{p+1}. We therefore have

′E_{2}^{pq} = \{ \begin{matrix}0 & q > 0 \\ H^{p}(Γ(X, I^{·})) = ℍ^{p}(X, Ω^{·}) = H_{dR}^{p}(X/K) & q = 0 \end{matrix} (8.6)

and H^{n}(C^{·}) = H_{dR}^{n}(X/K).

The second spectral sequence is

′′E_{1}^{pq} = C^{p}(U, H_{∂}^{q}) (8.7)

where H_{∂}^{q} is the presheaf whose value on U is H^{q}(Γ(U, I^{·})) = H_{dR}^{q}(U/K), and ′′d_{1}^{pq} is induced from δ. We therefore have

′′E_{2}^{pq} = Ȟ^{p}(U, H^{q}) ⟹ H^{n}(C^{·}) = H_{dR}^{n}(X/K) (8.8)

as desired.

8.2. The main theorem. We now go back to the framework of this paper. The algebras A^{k}(τ) form a local system \underline{A} on T . In Section 5, we proved that the local system \underline{A} is acyclic : H^{p}(T , \underline{A}^{q}) = 0 for p > 0 and every q.

Theorem 8.2. The residue cochain map induces an isomorphism

H_{dR}^{k}(𝔛) ≃ C_{har}^{k}. (8.9)

Proof. Let us apply the spectral sequence to our situation. We then have by theorem 7.7 H^{q}(X(τ)) = A^{q}(τ), and the acyclicity assumption amounts to

Ȟ^{p}(U, H^{q}) = 0, (8.10)

(p > 0). The spectral sequence therefore degenerates at the E_{2} term, and we have H_{dR}^{k}(𝔛) = Ȟ^{0}(U, H^{k}), which, by theorem 7.7 again, and by theorem 5.7, is equal to C_{har}^{k}(𝔛).

8.3. The main theorem, an alternative approach. We would like to discuss the relationship between the residue cochain homomorphism

c : H_{dR}^{k}(𝔛) → C_{har}^{k}, [η] ↦ c_{η} (8.11)

and the isomorphism

c^{SS} : H_{dR}^{k}(𝔛) ≃ C^{k}, [η] ↦ c_{η}^{SS} (8.12)

of [S-S], corollary 17. Here (as in the introduction) C^{k} is a short-hand for the space

Hom_{ℤ}(C_{c}^{∞}(G/B_{I}, ℤ)/R_{I}, K) (8.13)

where we have borrowed the notation of [S-S] :

• I = {1, 2, …, d − k}

• G = P GL_{d+1}(K)

• For any set J ⊂ {1, …, d}, B_{J} = the subgroup of G generated by the standard Iwahori subgroup B = B_{∅}, and the transpositions s_{i} (i ∈ J), s_{i} = (i, i + 1), viewed as permutation matrices

• C_{c}^{∞} = locally constant, compactly supported functions • R_{I} = the G-submodule generated by the functions

χ_{By_{j}B_{I}} − χ_{B_{I}} , 0 ≤ j ≤ d
(8.14)  χ_{B_{I∪{i}}}, d − k < i ≤ d

where y_{j} = diag(\overbrace{1, …, 1}^{j}, \overbrace{π, …, π}^{d+1−j}) (χ_{T} is the characteristic function of the set T).

We first have to relate C_{har}^{k} and C^{k}. Recall that G was defined in a coordinatefree fashion, as P GL(V_{K}). We write ⟨·, ·⟩ for the pairing V_{K} ×V_{K}^{∗} → K, and identify G also with P GL(V_{K}^{∗}), so that if x ∈ V_{K}, y ∈ V_{K}^{∗} and g ∈ G, we have ⟨x, gy⟩ = ⟨g^{−1}x, y⟩ . If L is a lattice in V_{K} the dual lattice L^{∗} = {y ∈ V_{K}^{∗}; ⟨L, y⟩ ⊆ O_{K}}.

Fix once and for all an ordered basis of V_{K}, hence a dual basis for V_{K}^{∗}. In these bases we think of vectors in V_{K} = ^{t}K^{d+1} (resp. V_{K}^{∗} = K^{d+1}) as row (resp. column) vectors. We write x (resp. y) for the row (resp. column) vector representing x ∈ V_{K} (resp. y ∈ V_{K}^{∗}). Identify G with G = P GL_{d+1}(K) using these bases. Thus ⟨x, gy⟩ = ⟨g^{−1}x, y⟩ = xgy if g is the matrix of g ∈ G.

Now B_{I} is the stabilizer of the lattice flag

L_{0}^{∗} ⊂ L_{1}^{∗} ⊂ ⋯ ⊂ L_{k}^{∗} ⊂ π^{−1}L_{0}^{∗} (8.15)

in V_{K}^{∗} = K^{d+1}, where L_{0}^{∗} = O_{K}^{d+1} and for 1 ≤ i ≤ k, L_{i}^{∗} = y_{d+i−k}π^{−1}L_{0}^{∗} is the lattice of column vectors whose top d + i − k entries lie in π^{−1}O_{K}, and the rest lie in O_{K}. Dualizing, B_{I} is the stabilizer in V_{K} of

σ_{0} = (L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0}), (8.16)

which is a lattice flag of type 𝔱_{min} = (d+1−k, 1, …, 1). Here L_{i} = ^{t}(πy_{d+i−k}^{−1}O_{K}^{d+1}) 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 c ∈ C_{har}^{k} is determined by its values on the k-cells of minimal type. Thus G/B_{I} may be identified with the collection T̂_{k,min} of all the pointed k-cells of type 𝔱_{min}, and in this identification the coset B_{I}/B_{I} is mapped to σ_{0}. We write σ_{0} also for the characteristic function of this coset, as an element of C_{c}^{∞}(G/B_{I}, ℤ). We may therefore evaluate c_{η}^{SS} at σ_{0}.

In the next theorem we show how a comparison of our residue cochain map c with the Schneider-Stuhler map c^{SS} 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(ω_{a_{0}} ∧ ⋯ ∧ ω_{a_{k}})

c_{η}^{SS}(σ_{0}) = (σ_{0}, η). (8.17)

Then for every η, c_{η}^{SS} is the restriction of c_{η} to T̂_{k,min}, and η ↦ c_{η} is an isomorphism H_{dR}^{k}(𝔛) ≃ C_{har}^{k}. Furthrtmore the map C_{har}^{k} → C^{k}, c ↦ c|T̂_{k,min}, where we identify K-valued functions on T̂_{k,min} = G/B_{I} with Hom_{ℤ}(C_{c}^{∞}(G/B_{I}, ℤ), K), is an isomorphism of G-modules. In fact, it is the isomorphism (c_{·}^{SS})^{−1} ∘ c_{·}.

Proof. The assumption, and proposition 7.2, imply that for every η, c_{η} and c_{η}^{SS} agree on σ_{0}. Since c_{gη}(gσ_{0}) = c_{η}(σ_{0}) (resp. c_{gη}^{SS}(gσ_{0}) = c_{η}^{SS}(σ_{0})) for every g ∈ G, and G acts transitively on T̂_{k,min}, c_{η} and c_{η}^{SS} agree on all of T̂_{k,min}. This proves the first part of the theorem. The injectivity of the map c is now obvious, since c^{SS} is 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 R_{I} are formal consequences of the harmonicity conditions. However this involves some non-trivial combinatorics, and we prefer to argue as follows.

Let M_{k} be the space of finitely-supported measures on T̂_{k,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 T̂_{k}. Consider the KG-module spanned by these measures, and let M_{k} denote those measures in it which are supported on T̂_{k,min}. Similarly let N_{k} be the K-span of the ℤG-module denoted R_{I} above. Thus C^{k} is the space of all cochains on T̂_{k,min} which annihilate N_{k}. By weak duality^{2} N_{k} is the precise annihilator of C^{k} in the space of finitely-supported measures on T̂_{k,min}. If μ ∈ M_{k} and c ∈ C^{k} then c = c_{η}^{SS} for some η, so c is the restriction to T̂_{k,min} of c_{η}. Since μ is supported on T̂_{k,min}, and is orthogonal to c_{η}, μ is orthogonal to c. Since this holds for every c ∈ C^{k}, μ ∈ N_{k}. We therefore know that M_{k} ⊆ N_{k}, and we have to prove equality.

Now N_{k} 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 T̂_{k}(v). This is verified with little effort for the measures used to define R_{I}. Suppose M_{k} ≠ N_{k}. Then there exists such a measure μ in N_{k} but not in M_{k}. Clearly μ does not belong to the G-module of harmonicity conditions. We may therefore find a harmonic cochain c ∈ C_{har}^{k} not annihilated by μ. However, since μ is supported on some T̂_{k}(v) the restriction of c to T̂_{k}(v) will not be annihilated by μ. This restriction, by theorem 2.7, is also the restriction of a certain c_{η}, so μ will not annihilate c_{η}. But as μ is supported on T̂_{k,min}, it will not annihilate c_{η}^{SS} either, contradicting the fact that μ ∈ N_{k}.

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

C_{har}^{k} → C^{k}, c ↦ c|T̂_{k,min} (8.18)

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 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.

2Weak 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.

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 ∩ B_{v} = {1}. Just to be on the safe side we make the stronger assumption that if 1 ≠ 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 X_{Γ} = 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 X_{Γ,K} [Mus]. We let π_{Γ} : 𝔛 → X_{Γ} be the canonical projection. For any cell τ we let X_{Γ}(τ) = π_{Γ}(X(τ)). According to our assumption on Γ, π_{Γ} induces an isomorphism of X(τ) onto X_{Γ}(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 U_{Γ} of X_{Γ}, consisting of the X_{Γ}(v). Both are admissible open coverings in the rigid analytic topology. Their nerves are the simplicial complexes T and T_{Γ} = Γ\T .

The algebraic variety X_{Γ,K} is the generic fiber of a scheme X_{Γ} over O_{K}, canonically associated with Γ, which is flat, projective, and nonsingular ([Mus], theorems 3.1 and 4.1). The special fiber X_{Γ,𝔽_{q}} is reduced, and is the union of transversally crossing smooth divisors. The simplicial degeneration complex of X_{Γ,𝔽_{q}} (see [Mus] for the definition) is canonically isomorphic to T_{Γ}. The reduction map r induces on X_{Γ} a reduction map r_{Γ} : X_{Γ} → |T_{Γ}|, and upon the identification of X_{Γ} with the K̄-valued points of X_{Γ}, r_{Γ}^{−1}(St(v)) = π_{Γ}(X(v)) = X_{Γ}(v) consists of the points specializing to the component of X_{Γ,𝔽_{q}} indexed by v. Inside it, the part specializing to smooth points is precisely r_{Γ}^{−1}(v) = π_{Γ}(X_{v}).

9.2. A double complex. Let \underline{Ω}^{s} be the presheaf of rigid-analytic s-forms. Restricted to the rigid analytic topology on X_{Γ} it is a (rigid analytic) coherent sheaf. The covering U_{Γ} consists of the open sets X_{Γ}(v), whose intersections are the X_{Γ}(τ), and these are Stein spaces in the sense of Kiehl. By Kiehl’s version of Cartan’s theorem B, for any rigid analytic coherent sheaf G, H^{r}(X_{Γ}(τ), G) = 0 for r > 0. It follows that the rigid analytic sheaf cohomology H^{r}(X_{Γ}, G), which coincides with the rigid analytic Čech cohomology Ȟ^{r}(X_{Γ}, G), can be computed using U_{Γ} :

H^{r}(X_{Γ}, G) = Ȟ^{r}(U_{Γ}, G), (9.1)

without the need to pass to finer and finer coverings.

Consider the double complex (in non-negative degrees only)

K^{r,s} = C^{r}(U_{Γ}, \underline{Ω}^{s}) = C^{r}(U, \underline{Ω}^{s})^{Γ} (9.2)

of alternating Čech cochains, with coboundary maps d′ = δ, the Čech differentiation, and d′′ = (−1)^{r}d, the differentiation of forms modified by (−1)^{r} :

\begin{matrix}K^{r,s+1} \\ ↑ (−1)^{r}d \\ K^{r,s} & \underrightarrow{δ} & K^{r+1,s} \end{matrix} . (9.3)

The total complex (K^{·}, d_{K}) is given as usual by K^{n} = ⨁_{r+s=n} K^{r,s}, d_{K} = d′ + d′′. By the reasoning of the previous paragraph, the homology of the total complex K^{·} is nothing but the de-Rham cohomology of X_{Γ} :

h^{n}(K^{·}, d_{K}) = H_{dR}^{n}(X_{Γ}), (9.4)

and by GAGA rigid de-Rham cohomology and algebraic de-Rham cohomology coincide.

As usual there are two spectral sequence abutting to H_{dR}^{n}(X_{Γ}), and two associated filtrations on H_{dR}^{n}(X_{Γ}). Let us compute them, and start with the second one. It is

′′E_{1}^{r,s} = Ȟ^{s}(U_{Γ}, \underline{Ω}^{r}) = H^{s}(X_{Γ}, \underline{Ω}^{r}) (9.5)

and ′′d_{1}^{r,s} is induced from d′′. This is of course the Hodge-to-de-Rham (Hodge-Fröhlicher) spectral sequence, and the associated filtration is the Hodge filtration F_{dR}^{·}H_{dR}^{n}(X_{Γ}). It is well-known that this spectral sequence degenerates at E_{1}, and consequently

′′E_{∞}^{r,s} = H^{s}(X_{Γ}, \underline{Ω}^{r}) and

b_{n} = dim H_{dR}^{n}(X_{Γ}) = ∑_{r+s=n} h^{r,s} (9.6)

where h^{r,s} = dim H^{s}(X_{Γ}, \underline{Ω}^{r}) are the Hodge numbers. According to [S], Section 2, Proposition 1, h^{r,s} = 0 unless 0 ≤ r, s ≤ d, and in that square we have

h^{r,s} = μ(Γ)δ_{r+s,d} + δ_{r,s}, (9.7)

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 cohomology is the middle one, H_{dR}^{d}(X_{Γ}).

The first spectral sequence is ′E_{1}^{r,s} = C^{r}(U_{Γ}, \underline{H}_{dR}^{s}), because for any of the X_{Γ}(τ) we know, since it is Stein, that its de-Rham cohomology is the cohomology of the complex of global rigid analytic forms on X_{Γ}(τ). But this is also

′E_{1}^{r,s} = C^{r}(U, \underline{H}_{dR}^{s})^{Γ} = C^{r}(T , \underline{A}^{s})^{Γ}. (9.8)

The last isomorphism stems from the identification A^{s}(τ) = H_{dR}^{s}(X(τ)) proved in theorem 7.7, and from the fact that T is the nerve of the covering U. The differential ′d_{1}^{r,s} is induced from d′ = δ. Recall, from Section 5, that (C^{·}(T , \underline{A}^{s}), δ) is a resolution of C_{har}^{s}, the space of harmonic s-cochains on T . It is dual to the Schneider-Stuhler resolution of the admissible representation Λ_{s} by the C_{·}(T , (\underline{A}^{s})^{∗}). Now C_{r}(T , (\underline{A}^{s})^{∗}) are clearly free of finite rank as K[Γ]-modules. Therefore C^{r}(T , \underline{A}^{s}) are Γ-injectives and

′E_{2}^{r,s} = h^{r}(C^{·}(T , \underline{A}^{s})^{Γ}) = H^{r}(Γ, C_{har}^{s}) = H^{r}(Γ, H_{dR}^{s}(𝔛)). (9.9)

We see that this is the covering spectral sequence ([S-S] Section 5, Proposition 2, and [S], Section 1). The associated filtration on H_{dR}^{n}(X_{Γ}) is the covering filtration F_{Γ}^{·}.

According to [S], Section 1, Theorem 2 the covering spectral sequence degenerates at E_{2}, and consequently

′E_{∞}^{r,s} = H^{r}(Γ, H_{dR}^{s}(𝔛)). (9.10)

The terms H^{r}(Γ, H_{dR}^{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

dim H^{r}(Γ, H_{dR}^{s}(𝔛)) = μ(Γ)δ_{r+s,d} + δ_{r,s} = h^{r,s}. (9.11)

9.3. The conjecture on a Hodge decomposition. One says that the filtrations F_{dR}^{·} and F_{Γ}^{·} are opposite if for every 0 ≤ p ≤ n + 1

H^{n} = F_{dR}^{p}H^{n} ⨁ F_{Γ}^{n+1−p}H^{n}. (9.12)

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 ≠ d, and for n = d it would follow if

F_{dR}^{p}H^{d} ∩ F_{Γ}^{d+1−p}H^{d} = 0 (9.13)

for every p. Setting, for p + q = d,

H^{p,q} = F_{dR}^{p}H^{d} ∩ F_{Γ}^{q}H^{d} (9.14)

we get that if the two filtrations are opposite then

H_{dR}^{d}(X_{Γ}) = ⨁_{p+q=d} H^{p,q}, (9.15)

which we regard as a p-adic analogue of the classical Hodge decomposition for the variety X_{Γ}.

The following conjecture was made by Schneider in [S].

Conjecture 9.1. The filtrations F_{dR}^{·} and F_{Γ}^{·} are opposite to each other.

Let us give now a general criterion for the two filtrations coming from a double complex (K^{r,s}, d′, d′′) in non-negative degrees, to be opposite. Let (K^{·}, d) be the total complex, so that H^{n} = h^{n}(K^{·}, d). First recall the two filtrations. Let

(9.16) ′F^{p}K^{r,s} = \{ \begin{matrix}K^{r,s} if p ≤ r \\ 0 otherwise \end{matrix} \}

and ′F^{p}K^{n} = ⨁_{p≤r} K^{r,n−r} the associated total complex. Then

′F^{p}H^{n} = Im {h^{n}(′F^{p}K^{·}, d) → h^{n}(K^{·}, d)} . (9.17)

One similarly defines the second filtration′′F^{p} reversing the roles of the two indices. In our case ′F^{·} = F_{Γ}^{·} and ′′F^{·} = F_{dR}^{·}.

Lemma 9.2. Let Z_{II}^{r,s} = Ker(d′′ : K^{r,s} → K^{r,s+1}). Then (i) If for every r and s such that r + s = n the map

h^{r}(Z_{II}^{·,s}, d′) → h^{r}(K^{·,s}, d′) (9.18)

is surjective then for every p we have H^{n} = ′F^{p} + ′′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 ′F^{p} ∩ ′′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 Z_{I}^{r,s} = Ker(d′ : K^{r,s} → K^{r+1,s}) and looking at the map

h^{s}(Z_{I}^{r,·}, d′′) → h^{s}(K^{r,·}, d′′). (9.19)

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.

Now applying the lemma in our situation we get that the map h^{r}(Z_{II}^{·,s}, d′) → h^{r}(K^{·,s}, d′) is

Ȟ^{r}(U_{Γ}, \underline{Ω}_{cl}^{s}) → Ȟ^{r}(U_{Γ}, \underline{Ω}^{s}), (9.20)

and we retrieve the following criterion (compare [S], Section 2, Lemma 2).

Proposition 9.3. Suppose that Ȟ^{r}(U_{Γ}, \underline{Ω}_{cl}^{s}) → Ȟ^{r}(U_{Γ}, \underline{Ω}^{s}) 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}(U_{Γ}, \underline{Ω}_{cl}^{0}) = Ȟ^{d}(U_{Γ}, K) = H^{d}(Γ, K), while Ȟ^{d}(U_{Γ}, \underline{Ω}^{0}) = H^{d}(X_{Γ}, O_{X}) = H^{d}(Γ, O_{𝔛}), and

H^{d}(Γ, K) → H^{d}(Γ, O_{𝔛}) (9.21)

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 \underline{Ω}_{cl}^{s} be the sheaf of closed sforms, which by the Poincaré lemma is also the sheaf of exact s-forms (in the classical complex topology). Using the exact sequence

0 → \underline{Ω}_{cl}^{s−1} → \underline{Ω}^{s−1} → \underline{Ω}_{cl}^{s} → 0, (9.22)

and arguing inductively on s, we see that the map H^{r}(X, \underline{Ω}_{cl}^{s}) → H^{r}(X, \underline{Ω}^{s}) is surjective, and in fact

H^{r}(X, \underline{Ω}_{cl}^{s}) = F_{dR}^{s}H_{dR}^{r+s}(X) = ⨁_{s′≥s} H^{r+s−s′}(X, \underline{Ω}^{s′}). (9.23)

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 \underline{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 cohomology, and prove the following.

Theorem 10.1. For every τ ∈ T_{r}, the cohomology ring of X̃(τ) is generated by the classes of the logarithmic forms ω_{a} = d log a.

Corollary 10.2. The cohomology of X(τ) is generated by the classes of ω_{a} − ω_{b}.

Corollary 10.3. The logarithmic k-classes span a dense subspace in H_{dR}^{k}(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 obtained the isomorphism H_{dR}^{k}(𝔛) ≃ C_{har}^{k} ≃ Λ_{k}^{∗} (theorem 8.2, and corollary 5.8). By definition λ ∈ Λ_{k} vanishes if and only if

λ(e) = ⟨λ, [ω_{e}]⟩ = 0 (10.1)

for every e ∈ E^{k}. In other words, every λ ∈ Λ_{k} which is orthogonal to all the logarithmic forms ω_{e}, vanishes identically. This proves that the classes of logarithmic forms are dense in the weak^{∗}-topology on H_{dR}^{k}(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 analytic 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

H_{dR}^{k}(X) = ℍ^{k}(X, Ω^{·}). (10.2)

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 O_{K} (see [M-W], Section 2) with smooth reduction A_{0} = A/(π), and A^{†} = A ⊗_{O_{K}} K is the associated dagger-algebra, then there is a canonical isomorphism H_{dR}^{k}(Sp(A^{†})) ≃ H_{MW}^{k} (Spec(A_{0})).

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 “daggerspace” 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 H_{dR}(X(τ)) = H_{dR}(X(τ)^{†})

([G-K], Korollar 4.6). Moreover, by Kiehl’s “Theorem B” coherent cohomology on X(τ) vanishes in positive degrees, hence

H_{dR}^{k}(X(τ)) = ℍ^{k}(X(τ), Ω^{·}) = h^{k}(Γ(X(τ), Ω^{·})). (10.3)

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′ = X ∪ Y is an admissible cover of a smooth rigid analytic space X′ by open subdomains, and Z = X ∩ Y. Then proposition 8.1 gives the Mayer-Vietoris long exact sequence

H_{dR}^{k−1}(Z) → H_{dR}^{k}(X′) → H_{dR}^{k}(X) ⊕ H_{dR}^{k}(Y) → H_{dR}^{k}(Z) → ⋯ (10.4)

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 general 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 {U_{n}} (n ≥ 1) an admissible open cover of U satisfying U_{n} ⊂ U_{n+1}. Suppose that the cohomologies of U_{n} and U are finite-dimensional vector spaces. Then

H_{dR}^{k}(U) = lim_{←}H_{dR}^{k}(U_{n}). (10.5)

Proof. A proof that H_{dR}^{k}(ℙ, U) = lim_{←}H_{dR}^{k}(ℙ, U_{n}) 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 onto X yields an isomorphism H_{dR}^{k}(X) ≃ H_{dR}^{k}(X × D). This is called the homotopy 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 = {X_{α}} of X and isomorphisms Y_{α} = p^{−1}(X_{α}) ≈ X_{α} × D compatible with p.

Let U = {X_{α}} be an admissible cover of X. Suppose we are given for each α a nowhere-vanishing rigid analytic function g_{α} on X_{α} such that if x ∈ X_{α} ∩ X_{β} then |g_{α}(x)| = |g_{β}(x)|. Let

Y = {(x, z)| x ∈ X_{α}, |z| < |g_{α}(x)| for some α}. (10.6)

Then Y is an open disk bundle obtained by pasting the subspaces Y_{α} which project onto X_{α}. Each Y_{α} is isomorphic to X_{α} × D under the map sending (x, z) to (x, g_{α}(x)^{−1}z). We call such a Y an embedded disk bundle, because it is embedded in X × 𝔸^{1}.

For any disk bundle p : Y → X and any trivializing cover {X_{α}} as above, we have by the homotopy invariance

H_{dR}^{q}(X_{α}) ≃ H_{dR}^{q}(Y_{α}). (10.7)

Proposition 10.5. (compare [S-S], section 2, lemma 1) The projection p : Y → X of a disk bundle induces

H_{dR}^{k}(X) ≃ H_{dR}^{k}(Y). (10.8)

Proof. The map p induces a homomorphism of spectral sequences p^{∗} from the Mayer-Vietoris spectral sequence on X w.r.t. the cover {X_{α}} to the Mayer-Vietoris spectral sequence on Y w.r.t. the cover {Y_{α}}. Since the E_{2} 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 A_{X}(λ, 1) is the rigid analytic subspace of X × 𝔸^{1} defined by

A_{X}(λ, 1) = {(x, z)| |λ(x)| < z < 1} . (10.9)

The function x ↦ |λ(x)| is uniquely determined by the annulus (hence λ is determined up to a unimodular function), and is called the conformal invariant of A_{X}(λ, 1). If X is reduced (so we can take the sup norm | · | on B as our Banachnorm defining its topology, see [BGR] section 6.2.4), then rigid analytic functions on A_{X}(λ, 1) are power series

∑_{n∈ℤ} b_{n}z^{n}, b_{n} ∈ B (10.10)

such that for every constant r < 1 we have

(10.11)  lim_{n→∞} |b_{n}|r^{n} = 0 and lim_{n→−∞} |b_{n}λ^{n}|r^{−n} = 0.

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 A_{X}(λ, 1) can be written uniquely as

ω = ∑_{n∈ℤ} ω_{n}z^{n} + ∑_{n∈ℤ} η_{n}z^{n}dz, (10.12)

where ω_{n} (resp. η_{n}) are k-forms (resp. k − 1-forms) on X satisfying the same growth conditions as the coefficients b_{n} above. The form ω is closed if and only if

dω_{n} = 0, (−1)^{k}ω_{n}n + dη_{n−1} = 0 (10.13)

for every n. In particular η_{−1} should be closed and we call

ρ(ω) = [η_{−1}] ∈ H_{dR}^{k−1}(X) (10.14) the residue of ω. The residue of an exact k-form vanishes. Conversely, suppose that ρ(ω) = 0. Then η_{−1} = dξ and

ω = ω_{0} + d ((−1)^{k−1} ∑_{n≠0} n^{−1}η_{n−1}z^{n} + ξz^{−1}dz) (10.15)

converges (here we use the fact that the annulus is open), so ω is exact if and only if ω_{0} is exact on X. This proves that the projection p : A_{X}(λ, 1) → X induces an isomorphism (the Künneth formula)

H_{dR}^{k}(A_{X}(λ, 1)) ≃ H_{dR}^{k}(X) ⊕ [\frac{dz}{z}] ∧ H_{dR}^{k−1}(X). (10.16)

The cohomology class [dz/z] depends only on the orientation of the annulus. If z′ is another parameter on the annulus inducing the same orientation in each fiber, then log(z/z′) is rigid analytic in A_{X}(λ, 1), so dz/z − dz′/z′ 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 A_{X}(λ, 1) → X. We call the annulus bundle orientable if we can choose an admissible open covering {X_{α}} by affinoids, and isomorphisms

Y_{α} = p^{−1}(X_{α}) ≃ A_{X_{α}}(λ_{α}, 1) (10.17)

which over the intersections X_{α} ∩ X_{β} induce orientation-preserving isomorphisms of the fibers. We then deduce from the Künneth formula the short exact sequence

0 → H_{dR}^{k}(X_{α}) →^{p^{∗}} H_{dR}^{k}(Y_{α}) →^{ρ} H_{dR}^{k−1}(X_{α}) → 0, (10.18)

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

0 → H_{dR}^{k}(X) →^{p^{∗}} H_{dR}^{k}(Y) →^{ρ} H_{dR}^{k−1}(X) → 0. (10.19)

Proof. Fix a trivializing admissible open cover U = {X_{α}} by affinoids. For each α let ω_{α} be the class [dz/z] for a parameter z on the annulus over X_{α}. Let

H^{q}(Y_{α}) = H_{dR}^{q}(Y_{α}) (10.20)

and

H̃^{q}(Y_{α}) = H_{dR}^{q}(X_{α}) ⊕ H_{dR}^{q−1}(X_{α}). (10.21)

Then for each α the projection p induces an isomorphism H̃^{q}(Y_{α}) ≃ H^{q}(Y_{α}), which sends (ξ, η) to p^{∗}ξ + ω_{α} ∧ p^{∗}η. The same holds on the intersections of several Y′_{α}s, and the isomorphisms are compatible with the restriction maps. They therefore induce isomorphisms of the two spectral sequences whose E_{2} terms are Ȟ^{p}(U, H^{q}) and Ȟ^{p}(U, H̃^{q}). But by proposition 8.1 the first abuts to H_{dR}^{k}(Y) and the second to H_{dR}^{k}(X) ⊕ H_{dR}^{k−1}(X).

10.7. The cohomology of X(v) (v a vertex). If L is a lattice in V_{K}, a ∈ L−πL and 0 < r < 1, we write, as in Section 6.2,

H̃_{a}(L, r) (10.22)

(resp. H̃_{a}(L, r^{−})) for the set of z ∈ V^{∗} such that |z(a)| ≤ r|z|_{L} (resp. |z(a)| < r|z|_{L}). Here |z|_{L} = max_{b∈L} |z(b)|.

If we introduce coordinates in V_{K} so that L = O_{K}^{d+1} and a = (1, 0, …, 0), then H̃_{a}(L, r) is characterized by |z_{0}| ≤ r max_{0≤i≤d} |z_{i}|. We write H_{a}(L, r) for the image of H̃_{a}(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))

X(v) = ℙ(V^{∗}) − ⋃_{a} H_{a}(L, |π|) (10.23)

with a ranging over representatives of L/πL − {0} in L.

More generally, consider a_{0}, …, a_{m} ∈ L − πL, 0 < r_{0} ≤ ⋯ ≤ r_{m} < 1 and

X = X(a_{0}, …, a_{m}; r_{0}, …, r_{m}) = ℙ(V^{∗}) − ⋃_{i=0}^{m} H_{a_{i}}(L, r_{i}). (10.24)

Let X′ = X(a_{1}, …, a_{m}; r_{1}, …, r_{m}), L̄ = L/⟨a_{0}⟩, ā_{i} = a_{i}|H_{a_{0}} ∈ L̄, and put

X_{0} = H_{a_{0}} ∩ X′ = H_{a_{0}} − ⋃_{i=1}^{m} Hā_{i}(L̄, r_{i}). (10.25)

Suppose that r_{0} < r_{1} and choose s such that r_{0} < s ≤ r_{1}. Let Y = H_{a_{0}}(L, s^{−})∩X′. Note that X′ = X ∪ Y.

Lemma 10.7. Y is a disk bundle over X_{0}.

Proof. Choose coordinates so that z(a_{0}) = z_{0} and z(a_{i}) = ∑_{j=0}^{d} a_{ij}z_{j} for 1 ≤ i ≤ m, where max_{j} |a_{ij}| = 1 for every i, because a_{i} ∈ L − πL. Now z ∈ H_{a_{0}}(L, s^{−}) if and only if |z_{0}| < s max_{j} |z_{j}|, and in this case max |z_{j}| = max_{1≤j} |z_{j}|. But z ∈ H_{a_{i}}(L, r_{i}) (1 ≤ i ≤ m) if and only if | ∑_{j} a_{ij}z_{j}| ≤ r_{i} max_{j} |z_{j}|. Therefore for z ∈ H_{a_{0}}(L, s^{−}) and 1 ≤ i ≤ m we have z = (z_{0} : z_{1} : ⋯: z_{d}) ∈ H_{a_{i}}(L, r_{i}) if and only if z′ = (0 : z_{1} : ⋯: z_{d}) ∈ H_{a_{i}}(L, r_{i}).

Consider now the covering of X_{0} by X_{0,i} = {z′ ∈ X_{0}| max |z_{j}| = |z_{i}|}. Then for z′ ∈ X_{0,i} the point z belongs to Y if and only if |z_{0}| < s|z_{i}|. The part of Y lying over X_{0,i} is therefore isomorphic to X_{0,i} × D, and the lemma follows.

Similarly, X ∩Y is an oriented annulus bundle over X_{0}. 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

⋯ → H_{dR}^{k−2}(X_{0}) ⊕ H_{dR}^{k−1}(X_{0})
(10.26) → H_{dR}^{k}(X′) → H_{dR}^{k}(X) ⊕ H_{dR}^{k}(X_{0}) → H_{dR}^{k−1}(X_{0}) ⊕ H_{dR}^{k}(X_{0}) → · · ·.

Cancelling the terms H_{dR}^{k}(X_{0}), which map isomorphically onto each other, gives us the following Gysin long exact sequence :

⋯ → H_{dR}^{k−2}(X_{0}) → H_{dR}^{k}(X′) →^{ε} H_{dR}^{k}(X) →^{ρ} H_{dR}^{k−1}(X_{0}) → ⋯ (10.27)

with ε induced by the inclusion X ⊂ X′ 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′ and X̃_{0} in V^{∗}.

Proposition 10.8. (i) For every choice of a_{i} and r_{i} as above there is a short exact sequence

0 → H_{dR}^{k}(X̃′) →^{ε} H_{dR}^{k}(X̃) →^{ρ} H_{dR}^{k−1}(X̃_{0}) → 0. (10.28)

The maps ε and ρ commute with the restriction maps induced by the inclusion

X(a_{0}, …, a_{m}; r_{0}, …, r_{m}) ⊃ X(a_{0}, …, a_{m}; R_{0}, …, R_{m})

whenever r_{i} ≤ R_{i} (and 0 < R_{0} ≤ ⋯ ≤ R_{m} < 1).

(ii) Restriction is an isomorphism

H_{dR}^{k}(X̃(a_{0}, …, a_{m}; r_{0}, …, r_{m})) ≃ H_{dR}^{k}(X̃(a_{0}, …, a_{m}; R_{0}, …, R_{m})). (10.29)

(iii) Every class in H_{dR}^{k}(X̃) is represented by a logarithmic form (i.e. a form in the subring generated by d log(a_{i}), 0 ≤ i ≤ m).

Corollary 10.9. Every class in H_{dR}^{k}(X(v)) = H_{dR}^{k}(X_{v}^{†}) is represented by a logarithmic 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′ and X̃_{0}, so we know that every class in H_{dR}^{k−1}(X̃_{0}) is represented by a logarithmic form η. We shall also assume at the beginning that

0 < r_{0} < r_{1} ≤ ⋯ ≤ r_{m} < 1 (10.30)

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′ and X̃_{0}. We shall then be able to relax the restriction r_{0} < r_{1} by continuity.

We shall show that ρ is surjective. The surjectivity of ρ (with k replaced by k − 1) will imply then that ε is injective. Assume therefore that [η] ∈ H_{dR}^{k−1}(X̃_{0}). By induction we may assume that η is logarithmic. Let η′ be a lifting of η to X̃′. Picking d log(a_{i}) as a lifting of d log(ā_{i}) we may assume that η′ is logarithmic too, and in particular closed. Let ω = d log(a_{0}) ∧ η′. Then ω is a closed (logarithmic) form on X, and ρ([ω]) = [η].

This argument proves also that every class in H_{dR}^{k}(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 r_{0} < r_{1}. Assume that r_{0} = r_{1}, choose ε > 0 small, and r_{i,n} = r_{i} + ε/n for i ≥ 1, while r_{0,n} = r_{0}. Let

U_{n} = X̃(a_{0}, …, a_{m}; r_{0,n}, …, r_{m,n}). (10.31)

Then we know our result for U_{n}, and in particular we know that the inclusion U_{n} ⊂ U_{n+1} induces an isomorphism in cohomology. Proposition 10.4 then concludes the proof.

The corollary is a consequence of (10.23), and of the fact that as r → 1, the limit of the de-Rham cohomology of ℙ(V^{∗}) − ⋃_{a} H_{a}(L, r), is the dagger cohomology of X_{v}^{†}.

10.8. The cohomology of X (τ). Let us now check how the proof of the proposition has to be modified to deal with X(τ), for a general simplex

τ = (M_{0} ⊃ ⋯ ⊃ M_{r} ⊃ πM_{0}). (10.32)

Note that X(τ), being the intersection of the X(v_{i}), v_{i} = [M_{i}], is of the form

X(τ) = ℙ(V^{∗}) − ⋃_{i=0}^{r} ⋃_{a∈A′_{i}} H_{a}(M_{i}, |π|). (10.33)

In this expression A′_{i} is a set of representatives of M_{i} −πM_{i} modulo πM_{i}. However, if a ∈ A′_{i} and a ∉ πM_{i−1} then a (or a representative equivalent to it modulo πM_{i−1}) will appear also in A′_{i−1}, and H_{a}(M_{i−1}, |π|) ⊃ H_{a}(M_{i}, |π|), since for every z, |z|_{M_{i}} ≤ |z|_{M_{i−1}}. We may therefore replace A′_{i} in the above expression by A_{i}, a set of representatives of πM_{i−1} − πM_{i} modulo πM_{i} (when i = 0 understand πM_{−1} = M_{r}).

More generally, for the purpose of using induction, consider a sequence of lattices

π^{−1}L_{0} ⊃ L_{r} ⊃ ⋯ ⊃ L_{1} ⊃ L_{0}, (10.34)

a collection {a_{0}, …, a_{m}} such that a_{l_{i}}, …, a_{l_{i+1}−1} ∈ L_{i+1}−L_{i} (l_{0} = 0, l_{r+1} = m+1, L_{r+1} = π^{−1}L_{0}, some of the blocks of the a′_{j}s may be empty), and a sequence of real numbers

0 < r_{0} ≤ ⋯ ≤ r_{m} ≤ |π|. (10.35)

We assume that l_{1} > 0 (otherwise renumber the lattices). X(τ) is of this shape with all r_{i} = |π|. Unlike the case of X(v), we must insist that all r_{i} ≤ |π|. With this data we associate

X = X({L_{i}}, {a_{i}}, {r_{i}}) = ℙ(V^{∗}) − ⋃_{i=0}^{r} ⋃_{j=l_{i}}^{l_{i+1}−1} H_{a_{j}}(π^{−1}L_{i}, r_{j}) (10.36)

and we write X′ for the X associated with the same lattices, {a_{1}, …, a_{m}} and {r_{1}, …, r_{m}}. We also write X_{0} = X′ ∩ H_{a_{0}}. It is again of the same form, with L_{i} replaced by L̄_{i} = L_{i}/ ⟨a_{0}⟩ , and a_{j} replaced by ā_{j} = a_{j}|H_{a_{0}} (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′ and X_{0}.

Choose ε > 0 small, and let r_{0,ε} = r_{0}, but r_{i,ε} = r_{i} + ε for 1 ≤ i ≤ m, so that r_{0,ε} < r_{1,ε}. Choose r_{0,ε} < s ≤ r_{1,ε}. We shall work with the domains X_{ε}, X′_{ε} and X_{0,ε} defined by the perturbed radii r_{i,ε}, and then take the limit as ε → 0, using proposition 10.4. Since r_{i,ε} might be bigger than |π|, we can not assume the induction hypothesis for X′_{ε} and X_{0,ε}. Let Y_{ε} = H_{a_{0}}(π^{−1}L_{0}, s^{−})∩X′_{ε} and note that X′_{ε} = X_{ε} ∪ Y_{ε}.

Choose a basis {α_{i}} of V_{K} adapted to (10.34), so that π^{−1}L_{0} = ⟨α_{0}, …, α_{d}⟩ and

L_{i} = ⟨α_{0}, ….α_{d_{i}−1}, πα_{d_{i}}, …, πα_{d}⟩ (10.37)

where d_{i} = dim L_{i}/L_{0}. Since a_{0} ∈ L_{1} − L_{0}, we may also assume that a_{0} = α_{0}.

Now z ∈ H_{a_{0}}(π^{−1}L_{0}, s^{−}) if and only if |z_{0}| < s max |z_{k}|. Suppose this is the case. Pick some 1 ≤ j ≤ m and suppose that l_{i} ≤ j < l_{i+1}, so that a_{j} ∈ L_{i+1} − L_{i}. If i = 0 we are in the situation of lemma 10.7 and we saw there that for z such that |z_{0}| < s max |z_{k}|,

z = (z_{0} : ⋯: z_{d}) ∈ H_{a_{j}}(π^{−1}L_{0}, r_{j,ε}) ⇔ z′ = (0 : z_{1} : ⋯: z_{d}) ∈ H_{a_{j}}(π^{−1}L_{0}, r_{j,ε}). (10.38)

If j ≥ l_{1} (i ≥ 1) then z = (z_{0} : z_{1} : ⋯: z_{d}) ∈ H_{a_{j}}(π^{−1}L_{i}, r_{j,ε}) if and only if

|z(a_{j})| ≤ r_{j,ε} max{|π^{−1}z_{0}|, …, |π^{−1}z_{d_{i}−1}|, |z_{d_{i}}|, …, |z_{d}|}. (10.39)

On the other hand z′ = (0 : z_{1} : ⋯: z_{d}) ∈ H_{a_{j}}(π^{−1}L_{i}, r_{j,ε}) if and only if

|z′(a_{j})| ≤ r_{j,ε} max{|π^{−1}z_{1}|, …, |π^{−1}z_{d_{i}−1}|, |z_{d_{i}}|, …, |z_{d}|}. (10.40)

Since |z_{0}| < s max |z_{k}| implies

|z(a_{j}) − z′(a_{j})| < r_{j,ε} max{|π^{−1}z_{1}|, …, |π^{−1}z_{d_{i}−1}|, |z_{d_{i}}|, …, |z_{d}|} (10.41)

we see that z ∈ Y_{ε} ⇒ z′ ∈ X_{0,ε}.

Suppose that z′ ∈ X_{0,ε}. Then the above also shows that z ∈ Y_{ε} if and only if for every j ≥ l_{1}

r_{j,ε}|π^{−1}z_{0}| < |z′(a_{j})|. (10.42)

This proves the following lemma.

Lemma 10.10. Y_{ε} is a disk bundle over X_{0,ε}. The intersection X_{ε} ∩ Y_{ε} is an oriented annulus bundle over X_{0,ε}.

Proof. The case of Y_{ε} is clear, because a point z = (z_{0} : z′) ∈ Y_{ε} if and only if z′ ∈ X_{0,ε} and z_{0} satisfies

(10.43) |z_{0}| < s max_{1≤k} |z_{k}| and |z_{0}| < min_{l_{1}≤j} r_{j,ε}^{−1}|π||z′(a_{j})|}.

For X_{ε} ∩ Y_{ε} note that since r_{0,ε} ≤ |π|, for every z′ ∈ X_{0,ε}

r_{0,ε} max_{1≤k} |z_{k}| < min_{l_{1}≤j} r_{j,ε}^{−1}|π||z′(a_{j})|. (10.44)

Thus the fiber of X_{ε} ∩ Y_{ε} above z′ is a non-empty annulus.

From (10.4), propositions 10.5 and 10.6 we get a long exact sequence

⋯ → H_{dR}^{k}(X′_{ε})
(10.45) → H_{dR}^{k}(X_{ε}) ⊕ H_{dR}^{k}(X_{0,ε}) → H_{dR}^{k−1}(X_{0,ε}) ⊕ H_{dR}^{k}(X_{0,ε}) → H_{dR}^{k+1}(X′_{ε}).

As before, cancelling the terms H_{dR}^{k}(X_{0,ε}) which map isomorphically onto each other, we get the Gysin long exact sequence

⋯ → H_{dR}^{k−2}(X_{0,ε}) → H_{dR}^{k}(X′_{ε}) → H_{dR}^{k}(X_{ε}) →^{ρ} H_{dR}^{k−1}(X_{0,ε}) → · · ·. (10.46)

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

⋯ → H_{dR}^{k−2}(X_{0}) → H_{dR}^{k}(X′) → H_{dR}^{k}(X) →^{ρ} H_{dR}^{k−1}(X_{0}) → ⋯ (10.47)

in which all the radii r_{i} ≤ |π|. From here we deduce that H_{dR}^{k}(X) is generated by logarithmic forms precisely as in the proof of proposition 10.8.

References

[Ber] Berthelot, P. : Finitude et pureté cohomologique en cohomologie rigide, Inv. Math. 128, 329-377 (1997)

[B-G-R] Bosch, S., Güntzer, U., Remmert, R. : Non-archimedean analysis, Springer-Verlag, Berlin (1984)

[B-T] Bruhat, F. , Tits, J. : Groupes réductifs sur un corps local I. Publ. Math. IHES 41, 5-252 (1972)

[Br] Brown, K. : Buildings, Springer-Verlag, Berlin (1989)

[Bu] Bump, D. : Automorphic forms and representations, Cambridge University Press, Cambridge (1997)

[Dr] Drinfel’d, V.G. : Elliptic modules, Math. USSR Sbornik 23, 561-592 (1974)

[G] Garland, H. : p-adic curvature and the cohomology of discrete subgroups of p-adic groups, Ann. of Math. 97, 375-423 (1973)

[Ga] Garrett, P. : Buildings and the classical groups, Chapman & Hall, London (1997)

[G-K] Große-Klönne, E. : de-Rham Kohomologie in der rigiden Analysis, Ph. D. thesis, Münster (1998)

[I-S] Iovita, A., Spiess, M.: Logarithmic differential forms on p-adic symmetric spaces, preprint, January 2000.

[Ki] Kiehl, R. : Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie, Inv. Math. 2, 256-273 (1967)

[M-W] Monsky, P., Washnitzer, G. : Formal cohomology I, Ann. of Math., 88, 181-217 (1968)

[Mo] Monsky, P. : Formal cohomology : II. The cohomology sequence of a pair, Ann. Math. 88, 218-238 (1968)

[Mus] Mustafin, G.A. : Nonarchimedean uniformization, Math USSR Sbornik 34, 187-214 (1978)

[O-S] Orlik, P., Solomon, L. : Combinatorics and topology of complements of hyperplanes, Inv. Math. 56, 167-189 (1980)

[O-T] Orlik, P., Terao. H. : Arrangements of hyperplanes, Springer-Verlag, Berlin (1992)

[S] Schneider, P. : The cohomology of local systems on p-adically uniformized varieties, Math. Ann. 293, 623-650 (1992)

[S-S] Schneider, P., Stuhler, U. : The cohomology of p-adic symmetric spaces, Inv. Math. 105, 47-122 (1991)

[S-S1] — : Resolutions for smooth representations of the general linear group over a local field, J. reine angew. Math. 436, 19-32 (1993)

54 EHUD DE SHALIT

[S-T] Schneider, P., Teitelbaum, J. : An integral transform for p-adic symmetric spaces, Duke Math. J. 86, 391-433 (1997)

[Se] Serre, J.-P. : Trees, Springer-Verlag, Berlin (1980)

Institute of Mathematics, Hebrew University, Giva’t-Ram, 91904, Jerusalem, ISRAEL

E-mail address: deshalit@math.huji.ac.il