Marie-FranceVign´eras PGL (2 ,F ) ACriterionforIntegralStructuresandCoefficientSystemsontheTreeof

Volume 4, Number 4 (Special Issue: In honor of Jean Pierre, Part 1 of 2) 1—29, 2008

A Criterion for Integral Structures

and Coefficient Systems on the Tree of P GL (2 , F )

Marie-France Vign´ eras

A Jean-Pierre Serre, avec admiration et amiti´ e

R´ esum´ e On donne un crit`ere n´ecessaire et suffisant pour que l’homologie de degr´e 0 d’un syst`eme de coefficients GL(2, F )-´equivariant, de R-modules li- bres de type fini, sur l’arbre de P GL(2, F ) soit un R-module libre, lorsque F est un corps p-adique et R un anneau complet de valuation discr`ete. Ceci per- met de caract´eriser les repr´esentations lisses p-adiques enti`eres de GL(2, F ), de construire des structures enti`eres remarquables, d´efinies par un syst`eme de coef- ficients, et par r´eduction de montrer que les repr´esentations de GL(2, F ) sur un corps fini de caract´eristique p qui se rel`event ` a la caract´eristique 0, sont isomor- phes ` a l’homologie de degr´e 0 d’un syst`eme de coefficients GL(2, F)-´equivariant, d’espaces vectoriels de dimension finie, sur l’arbre. Par exemple, prenons un car- act`ere mod´er´ement ramifi´e p-adique χ

⊗ χ

du tore diagonal T (F) de GL(2, F ), tel que χ

(p

)χ

(p

) soit une unit´e p-adique, qχ

(p

) et χ

(p

) soient des entiers p-adiques, p

´etant une uniformisante de F et q l’ordre du corps r´esiduel; alors la s´erie principale de GL(2, F ) induite lisse non normalis´ee de χ

⊗ χ

est enti`ere avec une structure enti`ere remarquable explicite; toute repr´esentation irr´eductible supersinguli`ere de GL(2, Q

) sur un corps fini s’obtient comme r´eduction d’une telle structure enti`ere avec χ

(p) non entier.

Received December 28, 2006.

Introduction

Let p be a prime number, q a power of p, let F be a local non archimedean field of characteristic 0 or p, with ring of integers O

and residual field k

= F

, and let G be the group of F -points of a reductive connected F -group. When E/F is a finite extension, an irreducible locally algebraic E-representation of G is the tensor product V

⊗

V

of a smooth one V

and of an algebraic one V

, uniquely determined ([Prasad] th. 1). The problem of existence of integral structures in V

⊗

V

(and their classification modulo commensurability) is crucial for the p-adic local Langlands correspondence expected to relate p-adic continuous finite dimensional E-representations of the absolute Galois group Gal

and Banach admissible E-representations of G. The classification of irreducible representations of G over a finite field k of characteristic p remains a mystery when G 6= GL(2, Q

), and the reduction of integral structures is a fundamental open problem. It is either “obvious” or “very hard” to see if V

⊗

V

is integral, and to determine the reduction of an integral structure. It is obvious that a non trivial algebraic representation V

is not integral, or that a smooth cuspidal irreducible representation V

with an integral central character is integral.

From now on, G = GL(2, F ).

We fix a local non archimedean field E of characteristic 0 and residual field k

of characteristic p and we suppose V

trivial when F is not contained in E, in particular when the characteristic of F is p. We will present the first local integrality criterion for V

⊗

V

, by a purely representation theoretic method, not relying on the theory of (φ, Γ)-modules as in [Co04], [Co05], [BeBr] or on rigid analytic geometry as in [Br]. The idea is to realise V

⊗

V

as the 0-homology of a G-equivariant coefficient system on the tree [Se77] (an easy generalization of a general result of Schneider and Stuhler [SS97] for complex finitely generated smooth representations).

Let X be the Bruhat-Tits tree of P GL(2, F ) with the natural action of G

[Se77]. A fundamental system of G in X consists of an edge σ

and of a vertex

σ

of σ

. For i = 0, 1, we denote by K

the stabilizer in G of σ

; the intersection

K

∩ K

has index 2 in K

, we choose t ∈ K

not in K

∩ K

, and we denote

by ε the non trivial character of K

/(K

∩ K

). Let R be a commutative ring.

A G-equivariant coefficient system L of R-modules on X is determined by its restriction to the vertex σ

and to the edge σ

, i.e. by a diagram

r : L

→ L

where r is a R(K

∩ K

)-morphism from a representation of K

on an R-module L

to a representation of K

on an R-module L

. The word “diagram” was intro- duced by Paskunas [Pas] in his beautiful construction of supersingular irreducible representations of GL(2, F ) on finite fields of characteristic p. The boundary map from the oriented 1-chains to the 0-chains gives an exact sequence of RG-modules

0 → H

(L) → ind

(L

⊗ ε) → ind

L

→ H

(L) → 0,

where the middle map is the composite of the natural maps ind

(L

⊗ ε) → ind

L

→ ind

L

= ind

(ind

o∩K1

L

) → ind

L

, and H

(L) is the i-th homology of L for i = 0, 1.

The natural RK

-equivariant map w

: L

→ H

(L) is injective, and the natural map w

◦ r : L

→ L

→ H

(L) is K

-equivariant (lemma 1.2). If H

(L) = 0, then clearly r is injective.

0.1 Basic proposition: local criterion of integrality. Let r : L

→ L

be a diagram defining a G-equivariant coefficient system L of R-modules on X .

1) H

(L) = 0 if and only if r is injective.

2) Suppose that

- R is a complete discrete valuationg ring of fractions field S, - L

is a free R-module of finite rank,

- r is injective,

and let L ⊗

S = V, r

= r ⊗

id

: V

→ V

. Then, the map H

(L) → H

(V) is injective and the R-module H

(L) is free, when the equivalent conditions are satisfied :

a) r

(V

) ∩ L

= r(L

),

b) the map V

/L

→ V

/L

is injective,

c) r(L

) is a direct factor in L

.

From now on, R is a complete discrete valuationg ring of fractions field S, of uniformizer x and residue field k = R/xR; for example, the ring O

of fractions field E, uniformizer p

and residue field k

. An S-representation V of a group H is called R-integral when V contains an S-basis which generates a R-module L which is H-stable; one calls L an R-integral structure of V ; the reduction L = L ⊗

k of L is a k-representation of H.

Remarks (i) If S

/S is a finite extension, the integral closure R

of R in S

is complete discrete valuationg ring of fractions field S

([Se] II §2 prop.3); the diagram L

→ L

satisfies the properties of 2) if and only if its scalar extension by ? ⊗

R

satisfies these properties.

(ii) When the properties of 2) are true, H

(L) identifies with an R-integral structure of H

(V) such that (lemma 1.4.bis)

H

(L) ∩ V

= L

.

(iii) When the properties of 2) are true, the reduction H

(L) = H

(L) ⊗

k is the 0-th homology H

(L) of the G-equivariant coefficient system L = L ⊗

k.

We have the exact sequence of SG-modules:

0 → ind

V

⊗ δ → ind

V

→ H

(V) → 0, of RG-modules:

0 → ind

L

⊗ δ → ind

L

→ H

(L) → 0, of kG-modules:

0 → ind

L

⊗ δ → ind

L

→ H

(L) → 0.

0.2 Corollary Let V be a G-equivariant coefficient system of S-vector spaces on X defined by a diagram r : V

→ V

with r injective. The S-representation H

(V) of G is R-integral if and only if there exists an R-integral structure L

of the representation V

of K

such that L

= L

∩ V

is stable by t (considering V

embedded in V

).

When this is true, r restricts to a diagram L

→ L

defining an G-equivariant

coefficient system L of R-modules on X , such H

(L) identifies with an R-integral

structure of the representation of G on H

(V).

From now on, r is injective and we forget r.

To determine if the S-representation V

of K

contains an R-integral structure L

such that L

∩ V

is invariant by t, one may proceed inductively as follows.

One reduces easily to V

= K

V

. Let i = 0 or i = 1 identified with an el- ement of Z/2Z. One starts with an arbitrary R-integral structure M

of the S-representation V

of K

. The RK

-module M

defined by

- if i = 1, then M

= K

M

,

- if i = 0, then M

= (M

∩ V

) + t(M

∩ V

),

is an R-integral structure of V

; we repeat this construction to get another R-integral structure z(M

) of the S-representation V

of K

:

- if i = 1, then z(M

) = (K

M

∩ V

) + t(K

M

∩ V

), - if i = 0, then z(M

) = K

((M

∩ V

) + t(M

∩ V

)).

We call z(M

) the first zigzag of M

; clearly M

⊂ z(M

).

0.3 Corollary Let V

→ V

be an injective diagram as in the corollary 0.2, V

= K

V

, let i = 0 or i = 1 and let M

be an arbitrary R-integral structure of the S-representation V

of K

. Then, the S-representation of G on H

(V) is R-integral if and only if the sequence of zigzags (z

(M

))

is finite.

From now on, we choose for σ

the vertex of the tree associated to the O

- module generated by the canonical basis of F

and for σ

the edge between σ

and tσ

where

t = 0 1 p

0

!

, p

uniformizer of O

.

Then K

= GL(2, O

)Z and K

=< IZ, t > is the group generated by IZ and t, where Z is the center of GL(2, F ), isomorphic to F

diagonally embedded, and I is the Iwahori group of matrices of GL(2, O

) congruent modulo p

to the upper triangular group B(F

) of GL(2, F

).

Set P

= O

p

. For an integer e ≥ 1, the e-congruence subgroup K(e) = 1 + P

P

P

1 + P

!

normalized by K

is contained in the group I (e) = 1 + P

P

P

1 + P

!

normalized by K

, generated by K(e) and tK(e)t

. The pro-p-Iwahori subgroup of I is I (1).

0.4 Proposition Let e be an integer ≥ 1, let V

be a smooth E-representation of G generated by its K(e)-invariants and let V

be an irreducible algebraic E- representation of G (hence F ⊂ E if V

is not trivial).

1) The locally algebraic E-representation V = V

⊗

V

of G is isomorphic to the 0-th homology H

(V) of the coefficient system V associated to the inclusion

V

⊗

V

→ V

⊗

V

.

2) Suppose that the E-vector space V

is finite dimensional. Then, the representation of G on V is O

-integral if and only if there exists an O

-integral structure L

of the representation of K

on V

⊗

V

such that L

= L

∩ (V

⊗

V

) is invariant by t. Then the 0-th homology L of the G-equivariant coefficient system on X defined by the injective diagram L

→ L

is an O

- structure of V .

Remarks a) By the corollary 0.3, when (V

⊗

V

) = K

(V

⊗

V

), one can suppose L

= K

L

in 2).

b) By the lemma 1.4bis, we have L

= L ∩ (V

⊗

V

) in 2).

We define the contragredient ˜ V = ˜ V

⊗

V

of V

⊗

V

as the tensor prod- uct of the smooth contragredient ˜ V

of V

and of the algebraic contragredient V

of V

.

0.5 Corollary A finite length locally algebraic E-representation of G is O

-integral if and only if its contragredient is O

-integral.

0.6 Remark A “moderately ramified” diagram:

- an R-representation L

of K

trivial on K(1) with Z acting by a character ω,

- an R-representation L

of K

which is a sum of characters as an I -representation,

- an RIZ-inclusion L

→ L

with image contained in L

,

is equivalent by “inflation” to a diagram:

- an R-representation Y

of GL(2, F

) with Z (F

) acting by a character, - a semi-simple R-representation Y

of T (F

),

- an RT (F

)-inclusion Y

→ Y

with image contained in Y

, - an operateur τ on Y

such that:

τ

is the multiplication by a scalar a ∈ E

,

τ permutes the χ-isotypic part and the χs isotypic part of Y

for any character χ of T(F

).

The action of GL(2, O

) on L

is the inflation of Y

, the action of I on L

is the inflation of Y

, the action of t on L

is given by τ , and a = ω(p

).

Let V be an R-integral finitely generated S-representation of a group H. There exists an R-integral structure L

of V which is a finitely generated RH-module;

two such R-integral structures L

, L

are commensurable: there exists a ∈ R non zero such that aL

⊂ L

, aL

⊂ L

. When the reduction L

is a finite length k-representation of H, the reduction L of an R-integral structure L of V commensurable to L

has finite length and same semi-simplification than L

. See [Vig96] I.9.5 Remarque, and [Vig96] I.9.6).

0.7 Lemma If the reduction L

is an irreducible k-representation of H, then the R-integral structures of V are the multiples of L

.

The integral structures constructed in the proposition 0.4 are finitely generated O

G-modules. We will explicit them when V

is a Steinberg representation, or when V = V

is a moderately ramified principal series.

The Steinberg representation Let B = N T be the upper triangular sub- group of G, with unipotent radical N and diagonal torus T . The Steinberg R-representation St

of G is the R-module of B-left invariant locally constant functions f : G → R modulo the constant functions, with G acting by right translations. By [BS] 2.6, we have

St

⊗

R = St

.

The Steinberg representation over any field of characteristic p is irreducible [BL], [Vig06]. The same definition and the same properties hold when G is replaced by GL(2, F

) [CE] 6.13, ex. 6. The Steinberg R-representation of GL(2, F

) will be denoted by st

. From the lemma 0.7 and [SS91] th. 8, one obtains:

0.8 Proposition The Steinberg R-representation St

of G is the 0-th ho- mology associated to the diagram which is the inflation of the Steinberg R- representation st

of GL(2, F

), the trivial character of T (F

) on st

≃ R, and the multiplication by −1 on st

(remark 0.6).

The S-Steinberg representation St

of G is integral, all R-integral structures are multiple of the Steinberg R-representation St

.

The irreducible algebraic representation Sym

of G of dimension k+1 is realized in the space F [X, Y ]

of homogeneous polynomials in X, Y of degree k ≥ 0 with coefficients in F; it has a central character z → z

. We denote by |?| the absolute value on an algebraic closure F

of F normalized by |p| = p

. Over any finite extension E of F [p

] contained in F

, the representation

Sym

⊗

| det(?)|

of G has an O

-integral central character. Let us consider the O

-module M

generated in O

[X, Y ]

by the monomials

X

Y

if i ≤ j and p

X

Y

if i > j,

and the image φ

in St

of the characteristic function of BI. One sees that the

O

-integral structure L

= O

φ

⊗

M

of the representation St

⊗

Sym

⊗

| det(?)|

of K

is equal to its zigzag z(L

) = K

L

∩ (St

⊗

Sym

⊗

| det(?)|

), using

tX = p

uY, tY = p

uX where u ∈ O

, and a small computation.

0.9 Proposition The locally algebraic representation St

⊗

Sym

⊗

| det(?)|

is O

-integral for any integer k ≥ 0; the 0-th homology of the G-equivariant co- efficient system on the tree defined by the diagram

L

= O

φ

⊗

M

→ L

= K

L

is an integral O

-structure.

When F = Q

, there are at least four different non trivial proofs of this propo- sition: three different local proofs have been given by Teitelbaum [T], Grosse- Kl¨ onne [GK1], Breuil [Br] (with some restrictions), Colmez [Co4], and one can use modular forms and the cohohomology of modular curves to get a global proof.

Principal series Let χ

⊗ χ

: T → E

be a character of T inflated to B.

The principal series ind

(χ

⊗ χ

) is the set of functions f : G → E satisfying f (hgk) = χ

(a)χ

(d)f (g) for all g ∈ G and h ∈ B with diagonal (a, d), and k in a small open subgroup of G, depending on f , with the group G acting by right translation. The central character χ

χ

is integral if and only if χ

(p

)χ

(p

) ∈ O

is a unit. When the E-character χ

⊗ χ

is moderately ramified, i.e. trivial on T (1 + P

), its restriction to T(O

) is the inflation of an E-character η

⊗ η

of T (F

), and the principal series is the 0-th homology of the G-equivariant coefficient system defined by the injective diagram

(ind

(χ

⊗ χ

))

→ (ind

(χ

⊗ χ

))

which is the inflation of the injective diagram

(ind

q)

(η

⊗ η

))

→ ind

q)

(η

⊗ η

) with the operator τ on (ind

q)

(η

⊗ η

))

= Ef

⊕ Ef

such that τ f

= χ

(p

)f

and τ

is the multiplication by χ

(p

)χ

(p

), where f

, f

have support B(F

), B(F

)sN (F

) and value 1 at id, s. Clearly,

Y

= O

f

⊕ O

χ

(p

)f

, is an O

-integral structure of (ind

q)

(η

⊗ η

))

stable by τ and Y

= GL(2, F

)Y

is an O

-integral structure of ind

q)

(η

⊗η

). When the central character χ

χ

is integral, (Y

, Y

, τ ) inflates to an injective diagram L

→ L

= K

L

defining a G-equivariant coefficient system L of free O

-modules of finite rank on X . An H

(G, I (1))-module is called O

-integral when it contains an E-basis which generates an O

-module stable by H

(G, I (1)).

0.10 Proposition When the E-character χ

⊗ χ

is moderately ramified

and χ

(p

)χ

(p

) ∈ O

is a unit, the following properties are equivalent:

a) the principal series ind

(χ

⊗ χ

) is O

-integral,

b) the H

(G, I(1))-module (ind

(χ

⊗ χ

))

is O

-integral, c) χ

(p

), χ

(p

)q are integral,

d) Y

= Y

,

e) the 0-th homology of L is an O

-integral structure L of ind

(χ

⊗ χ

).

When they are satisfied, we have L

= L

, L

= L

.

When χ

χ

is moderately unramified (trivial on 1 + P

), one reduces to χ

⊗ χ

moderately ramified by twist by a character. When F = Q

and χ

χ

is unramified (trivial on O

), the equivalence between c) and a) has been proved by Berger and Breuil [BeBr] using the theory of (φ, Γ)-modules.

0.11 Remarks and questions (i) When χ

(p

) is integral, the character χ

⊗χ

of T is O

-integral and conversely. In this banal case, L is the natural O

- integral structure of ind

(χ

⊗χ

) of functions with values in O

. Its reduction is the k

-principal series of G induced from the reduction χ

⊗ χ

of χ

⊗ χ

; when χ

6= χ

it is irreducible [BL], [Vig04], [Vig06] and the O

-integral structures of ind

(χ

⊗ χ

) are multiple of L, by the lemma 0.7.

When χ

= χ

, then ind

(χ

⊗ χ

) has length 2; one should be able to prove that the O

-integral structures of ind

(χ

⊗χ

) are O

G-finitely generated; when F = Q

, compare with [BeBr] 5.4.4 and [Co05] 8.5.

(ii) The module of B is |?|

⊗ |?|

where |p

|

= 1/q. The contragredient of ind

(χ

⊗ χ

) is ind

(χ

|?|

⊗ χ

|?|

) ([Vig96] I.5.11), hence the proposition 0.10 and the corollary 0.5 are compatible.

The representation

ind

(χ

⊗ χ

) ≃ ind

(χ

|?|

⊗ χ

|?|

)

is irreducible when χ

6= χ

, χ

|?|

; the condition is not symmetric because

the induction is not normalized. This isomorphism is also compatible with the

propositon 0.10.

(iii) Theoretically, there is no reason to restrict to the moderately ramified smooth case, but the computations become harder when the level increases or when one adds an algebraic part.

(iv) One should see c) as the limit at ∞ of the integrality local criterion. For ind

(χ

⊗ χ

) ⊗ Sym

⊗| det(?)|

, one should replace c) by:

χ

(p

)q

, χ

(p

)q

are integral.

This condition is automatic when the representation is integral; this can be seen by different, but equivalent, ways either via Hecke algebras or via exponents [E].

The representation of T on the 2-dimensional space of N -coinvariants (V

)

of the smooth part V

= | det(?)|

⊗ ind

(χ

⊗ χ

) is semi-simple equal to

| det(?)|

[(χ

⊗ χ

) ⊕ (χ

|?|

⊗ χ

|?|

)].

The character of T on the 1-dimensional space V

of N -invariants of the algebraic part V

= Sym

is ?

⊗ 1. The representation of T on (V

)

⊗

V

is

(χ

?

|?|

⊗ χ

|?|

) ⊕ (χ

?

|?|

⊗ χ

|?|

).

These characters are called the exponents of V . They are integral on the element 1 0

0 p

!

which dilates N if and only if χ

(p

)q

and χ

(p

)q

are integral.

Application to k-representations of G.

Let k be a finite field of characteristic p.

The reduction of the integral structure L of a principal series of G induced from an integral character, constructed in the proposition 0.10 is a principal series (remark 0.11 (i)), and one deduces:

0.12 Proposition Let µ

⊗ µ

be a k-character of T . The principal se- ries ind

(µ

⊗ µ

) is the 0-th homology of the G-equivariant coefficient system associated to the inclusion

(ind

(µ

⊗ µ

))

→ (ind

(µ

⊗ µ

))

,

equivalent by inflation to the data

- Y

= ind

q)

(η

⊗ η

), where η

⊗ η

inflates to the restriction of µ

⊗ µ

to T (O

),

- Y

and the operator τ of Y

such that

τ (f

) = µ

(p

)f

and τ

is the multiplication by µ

(p

)µ

(p

).

The reduction of the integral structure L of a principal series of G induced from a non integral character, constructed in the proposition 0.10 is never a principal series. The simple H

(G, I(1))-modules where the action of the center of H

(G, I(1)) is as null as possible are called supersingular; they are classified in [Vig04].

0.13 Proposition Let ind

(χ

⊗ χ

) be an O

-integral moderately ramified principal series of G and let L, L

, L

as in the proposition 0.10.

a) χ

(p

) is not integral if and only if L

is a simple supersingular H

(G, I(1))- module.

b) Any simple supersingular H

(G, I(1))-module arises in a) for some χ

⊗χ

. c) When L

= (L)

and L

is simple supersingular, the k

-representation L of G is irreducible supersingular.

d) L

= (L)

when F = Q

.

e) When L

6= (L

)

, then L is not admissible and has infinite length.

Remarks (i) When L

= (L)

, any integral O

-structure of ind

(χ

⊗ χ

) is a multiple of L by the lemma 0.7.

(ii) The k

-representation L of G is the 0-th homology of the G-equivariant coefficient system associated to τ , Y

→ Y

with τ, Y

, Y

described before the proposition 0.10. The k-representation Y

= GL(2, F

)Y

has the same semi- simplification than the reduction W of ind

q)

(η

⊗ η

). The inclusion

Y

⊂ (Y

)

is an equality when the length of W is 2; this is the case when q = p or when η

= η

.

From [Vig04], the map V → V

is a bijection between the irreducible k- representations of GL(2, Q

) and the simple H

(G, I(1))-modules. The proposi- tion 0.13 implies:

0.14 Corollary When F = Q

, any irreducible representation of G over a finite field of characteristic p, with a central character, is the reduction of an integral principal series.

The results on the integral structures were presented in 2005 in Tel-Aviv and in Montreal; the application to k-representations was presented in 2006 in Luminy.

At the Montreal conference, Elmar Grosse-Kl¨ onne suggested that the function i on the set of vertices of the Bruhat-Tits building of P GL(n, F ) that he invented [GK2] could be the key of a generalization of the local integrality criterion to n ≥ 2.

1 The tree [Se77] The vertices of the tree X of P GL(2, F ) are the similarity classes [L] of O

-lattices L in a 2-dimensional F -vector space. Two vertices z

, z

are related by an edge {z

, z

} when they admit representatives L

, L

such that p

L

⊂ L

⊂ L

. The vertex σ

:= [O

v

+O

v

] for the canonical basis v

, v

of F

-vector space, will be the origin of X . The group G = GL(2, F ) acts naturally on the tree. We have tσ

= [O

v

+ O

p

v

]. We denote σ

= {σ

, tσ

} the edge between from σ

and tσ

. The two orientations of the edge σ

are permuted by t.

The element t normalizes the Iwahori subgroup I and its congruence subgroups I(e) for e ≥ 1. We choose the combinatorial distance on X (the number of edges between two vertices). For any integer n ≥ 0, we denote by S

the sphere of vertices of distance n to σ

and by B(n) the ball of radius n. The action of G respects the distance.

Coefficient system

Let R be a commutative ring. A G-equivariant coefficient system of R-modules

V on the tree X consists of

- an R-module V

for each simplex σ of X ,

- a restriction linear map r

: V

→ V

for each edge τ containing the vertex σ,

- linear maps g

: V

→ V

for each simplex σ of X and each g ∈ G, compatible with the product of G and with the restriction: (gg

)

= g

g

, g

r

= r

g

.

In particular, the stabilizer of a simplex σ acts on V

and the restrictions r

, r

are equivariant by the intersection of the stabilizers of the vertices σ, σ

of τ . For any edge τ containing a vertex σ, there exists g ∈ G such that gτ = σ

, gσ = σ

. This is equivalent to the fact that a G-equivariant coefficient system is determined by its restriction to the vertex σ

and to the edge σ

, i.e. by a diagram

r : V

→ V

where V

= V

is an R-representation on K

, V

= V

is an R-representation of K

and r is an R-linear map which is K

∩ K

-equivariant [Pas].

We denote by X (o) the set of vertices, by X (1) the set of oriented edges (σ, σ

) (with origin σ) and by X

the set of non oriented edges {σ, σ

}.

The R-module C

(V ) of oriented 0-chains is the set of functions φ : X (o) → Q

σ∈X(o)

V

with finite support such that φ(σ) ∈ V

for any vertex σ.

The R-module C

(V) of oriented 1-chains is the set of functions ω : X (1) → Q

{σ,σ^′}∈X1

V

with finite support such that ω(σ, σ

) = −ω(σ

, σ) ∈ V

for any edge {σ, σ

}.

We denote by n(ω) = n the unique integer ≥ 1 such that the support of ω is contained in the ball B

and not in B

.

The boundary ∂ is the R-linear map sending an oriented 1-chain ω supported on one edge τ = {σ, σ

} to the 0-chain ∂ω supported on the vertices σ, σ

, with

∂ω(σ) = r

ω(σ, σ

), ∂ω(σ

) = r

ω(σ

, σ).

The key of the basic proposition is :

For any vertex σ ∈ S

with n ≥ 1, the neighbours of σ belongs to S

except

one which belongs to S

. In particular there is a unique edge τ = τ

starting

from σ and pointing toward σ

, and

(key) ∂ω(σ) = r

ω(τ

) for σ ∈ S

.

The group G acts on the R-module of oriented ∗-chains by (gω)(gσ) = g(ω(σ))

for any g ∈ G and any oriented ∗-chain ω. The boundary ∂ is G-equivariant.

The representation of G on C

(V) and on C

(V ) are isomorphic to the compactly induced representations ind

V

and ind

(V

⊗ ε) where ε : K

→ R

is the R-character of K

trivial on K

∩ K

such that ε(t) = −1. The 0-homology

H

(V ) = C

(V)

∂C

(V )

is an R-representation of G. The same is true with the 1-homology H

(V) = Ker ∂.

We identify naturally v

∈ V

with a 0-chain with support on the single vertex σ

. This induces a K

-equivariant linear map

w

: V

→ H

(X , V)

that we can “restrict” to V

to get a K

∩ K

-equivariant map w

r

: V

→ V

→ H

(X , V).

1.2 Lemma The map w

is injective and the map w

r

is K

-equivariant.

Proof. No 1-chain is supported on a single vertex and the key formula gives the injectivity. For v

∈ V

, if ω is the oriented 1-chain ω with support the edge σ

such that ω(σ

, tσ

) = tv

, we have

∂(ω) = ω

r

(tv

) − t(ω

r

(v

)).

This gives the K

-equivariance.

When the map

r

: V

→ V

is injective, then the map r

is injective for any vertex σ and any edge τ containing σ.

1.3 Lemma ∂ is injective if the map r

is injective.

Proof. Let ω 6= 0. Let σ ∈ S

such that the edge τ

(starting from σ and pointing towards the origin) belongs to the support of ω. As r

is injective,

∂(ω)(σ) = r

ω(τ

) does not vanish.

We suppose from now on that the map r

: V

→ V

is injective.

Let φ be a non zero 0-chain not supported on the origin. There exists a unique integer n(φ) ≥ 1 such that the support of φ is contained in the ball B

and not in B

. Let σ ∈ S

such that φ(σ) 6= 0; let τ be the edge pointing towards the origin, starting from σ. If φ(s) belongs to r

V

, then by the key formula, we can find an oriented 1-chain ω supported on the edge τ such that φ − ∂ω vanishes at σ. Obviously φ − ∂ω is equal to φ outside σ ∪ S

.

When the R-module image of V

in V

has a supplementary W

V

= W

⊕ r

(V

),

the image of V

in V

has a (non canonical) supplementary W

, V

= W

⊕ r

(V

).

We can find an oriented 1-chain ω supported on τ

such that (φ − ∂ω)(σ) ∈ W

. Hence any non zero element of H

(V) has a representative φ either supported on the origin, or such that n(φ) ≥ 1 and φ(σ) ∈ W

for any σ ∈ S

.

1.4 Proof of the basic proposition 0.1.

1) is a particular case of the lemma 1.2.

2) Equivalence of the properties a), b), c). It is obvious that V

∩ L

= L

is

equivalent to: the kernel of V

→ V

/L

is L

, is equivalent to: the quotient of

L

by L

is torsion free, and is equivalent to L

is a direct factor of L

because

L

is a free module of finite rank over the principal ring R.

The injectivity H

(L) → H

(V ) results from the condition b), the lemma 1.3 applied to the coefficient system V/L, the exactitude of the sequence H

(V/L) → H

(L) → H

(V).

One can also prove the injectivity as follows. If ω ∈ C

(V ) and ∂ω ∈ C

(L) then r

ω(τ

) ∈ L

for any vertex ? ∈ S

. By the condition a), and the injectivity of r

, we have ω(τ

) ∈ L

. One repeats the same argument to ω − ω

where ω

is the restriction of ω to the oriented edges between S

and S

.

The R-module H

(L) embedded in the countable S-vector space H

(V ) is R- free if and only if it contains no line because R is a local complete principal ring.

Let v be a non zero element of H

(L). We choose

- a representative φ of v such that φ is supported on σ

or such that n(φ) ≥ 1 and φ(σ) ∈ W

for any σ ∈ S

,

- a vertex σ ∈ S

such that φ(σ) 6= 0,

- an integer n ≥ 1 such that φ(σ) does not belong to x

L

for the chosen vertex σ ∈ S

.

Suppose that the line Sv is contained in H

(L). Then, there exists an oriented 1-cocyle ω ∈ C

(L) such that (φ + ∂(ω))(?) belongs to x

L

for any vertex

?. If n(ω) > n(φ), the key formula implies that ω(τ

) belongs to p

L(τ ) for any vertex ? ∈ S

, because the condition a) and the injectivity of r

imply r

L

∩ x

L

= r

(x

L

). The oriented 1-cocycle ω − ω

has the same property than ω. By decreasing induction on n(ω) we may suppose n(ω) ≤ n(φ). If φ is supported on σ

, then ω = 0 and φ(σ

) ∈ x

L

which is false. Otherwise, φ(?) + ω(τ

) ∈ x

L

for any ? ∈ S

. As φ(?) ∈ W

and ω(τ

) ∈ r

(V

), this is impossible.

1.4bis Lemma Let φ be a 0-chain with support on the single vertex σ

and let ω be an oriented 1-chain such that φ + ∂(ω) is integral. Then φ is integral.

Proof. As n(ω) ≥ 1, the restriction of ω on S

is integral by the key formula.

By a decreasing induction on n(ω), φ is integral.

1.5 Proof of the corollary 0.2