Euler characteristics for p-adic Lie groups

Burt Totaro

Lazard [23] found definitive results about the cohomology of p-adic Lie groups such as GL_{n}Z_{p} with coefficients in vector spaces over Q_{p}. These results, applied to the image of a Galois representation, have been used many times in number theory. It remains a challenge to understand the cohomology of p-adic Lie groups with integral coefficients, and especially to relate the integral cohomology of these groups to the cohomology of suitable Lie algebras over the p-adic integers Z_{p}. In this paper, we do enough in this direction to compute a subtle version of the Euler characteristic, arising in the number-theoretic work of Coates and Howson ([14], [13]), for most of the interesting p-adic Lie groups.

The Euler characteristics considered in this paper have the following form. Let G be a compact p-adic Lie group with no p-torsion. Let M be a finitely generated Z_{p}-module on which G acts, and suppose that the homology groups H_{i}(G, M) are finite for all i. They are automatically 0 for i sufficiently large [27]. Then we want to compute the alternating sum of the p-adic orders of the groups H_{i}(G, M):

χ(G, M) := ∑_{i}(−1)^{i}ord_{p}|H_{i}(G, M)|,

where ord_{p}(p^{a}) := a. These Euler characteristics determine the analogous Euler characteristics for the cohomology of G with coefficients in a discrete “cofinitely generated” Z_{p}-module such as (Q_{p}/Z_{p})^{n}; see section 1 for details. If the module M is finite, Serre gave a complete calculation of these Euler characteristics in [29].

The first result on these Euler characteristics with M infinite is Serre’s theorem that χ(G, M) = 0 for any open subgroup G of GL_{2}Z_{p} with p ≥ 5, where M = (Z_{p})^{2} with the standard action of G [29]. This is the fact that Coates and Howson need for their formula on the Iwasawa theory of elliptic curves ([14], [13]). In fact, Serre’s paper [29] and the later paper by Coates and Sujatha [15] prove the vanishing of similar Euler characteristics for many p-adic Lie groups other than open subgroups of GL_{2}Z_{p}, but only for groups which are like GL_{2}Z_{p} in having an abelian quotient group of positive dimension. For example, it was not clear what to expect for open subgroups of SL_{2}Z_{p}.

We find that the above Euler characteristic, for sufficiently small open subgroups of SL_{2}Z_{p}, is equal to 0 for all p ≠ 3 and to −2 for p = 3. We also compute the Euler characteristic of these groups with coefficients in a representation of SL_{2} other than the standard one: it is again 0 except for finitely many primes p. The phenomenon simplifies curiously for larger groups (say, reductive groups of rank at least 2), as the following main theorem asserts: the Euler characteristic is 0 for all primes p and all representations of the group for which it makes sense. The proof of this theorem is completed at the end of section 9.

Theorem 0.1 Let p be any prime number. Let G be a compact p-adic Lie group of dimension at least 2, and let M be a finitely generated Z_{p}-module with G-action. Suppose that the homology of the Lie algebra 𝔤_{Q_{p}} of G acting on M ⊗ Q_{p} is 0; this is equivalent to assuming that the homology of any sufficiently small open subgroup G_{0} acting on M is finite, so that the Euler characteristic χ(G_{0}, M) is defined. (For 𝔤_{Q_{p}} reductive, this assumption is equivalent to the vanishing of the coinvariants of 𝔤_{Q_{p}} on M ⊗ Q_{p}.) Then the Euler characteristics χ(G_{0}, M) are the same for all sufficiently small open subgroups G_{0} of G (that is, all open subgroups contained in a certain neighborhood of 1).

The common value of these Euler characteristics is 0 if every element of the Lie algebra 𝔤_{Q_{p}} has centralizer of dimension at least 2 (example: 𝔤_{Q_{p}} reductive of rank at least 2). Otherwise, there is an element of 𝔤_{Q_{p}} whose centralizer has dimension 1 (example: 𝔤_{Q_{p}} = 𝔰𝔩_{2}Q_{p}), and then we give an explicit formula for the common value of the above Euler characteristics; in particular, this common value is not 0 for some choice of the module M.

Remarks. (1) The dimensions of centralizers play a similar role in the case of finite coefficient modules: if G is a compact p-adic Lie group with no p-torsion, then χ(G, M) = 0 for all finite p-torsion G-modules M if and only if every element of G has centralizer of dimension at least 1, by Serre [29], Corollary to Theorem C.

(2) There are simple sufficient conditions for G to be “sufficiently small” that χ(G, M) is equal to the value which we compute. For example, if M is a faithful representation of G, it suffices that G should act trivially on M/p if p is odd and on M/4 if p = 2. In fact, for the most natural p-adic Lie groups, we can avoid this assumption completely: Corollary 11.6 shows that χ(G, M) = 0 for all compact open subgroups G of a reductive algebraic group of rank at least 2 when p is big enough. In particular, such groups G (including SL_{n}Z_{p}, for example) need not be pro-p groups.

(3) It is somewhat surprising that Euler characteristics of this type are the same for all sufficiently small open subgroups, given that G has dimension at least 2. Other types of Euler characteristics tend instead to be multiplied by r when passing from G to a subgroup H of finite index r. Of course, these two properties are the same when the Euler characteristics of G and H are both 0.

(4) The theorem is false for G of dimension 1. In this case, for a given module M as above, G has an open subgroup isomorphic to Z_{p} such that

χ(p^{n}Z_{p}, M) = χ(Z_{p}, M) + n dim(M ⊗ Q_{p})

for all n ≥ 0. That is, the Euler characteristics for open subgroups need not attain a common value when G has dimension 1.

The key to the proof of Theorem 0.1 is to relate the homology of p-adic Lie groups to the homology of Lie algebras. Lazard did so for homology with Q_{p} coefficients. There is more to be discovered about the relation between group homology and Lie algebra homology without tensoring with Q_{p}, but at least we succeed in showing that the Euler characteristic of a p-adic Lie group (of dimension at least 2) is equal to the analogous Euler characteristic of some Lie algebra over Z_{p}. The proof in sections 8 and 9 sharpens Lazard’s proof that the group and the Lie algebra have the same rational cohomology, giving an explicit upper bound for the difference between the integral cohomology of the two objects. We compute these Euler characteristics for Lie algebras in sections 3 to 7, using in particular Kostant’s theorem on the homology of the “upper-triangular” Lie subalgebra of a semisimple Lie algebra over a field of characteristic zero [22]. Sections 1 and 2 give some preliminary definitions and results.

The rest of the paper goes beyond Theorem 0.1 in several directions. First, using the general results we have developed on the integral homology of p-adic Lie groups, Theorem 10.1 computes the whole homology with nontrivial coefficients of congruence subgroups, not just the Euler characteristic. Section 11 extends the earlier arguments to prove the vanishing of Euler characteristics for many p-adic Lie groups which are not pro-p groups, namely open subgroups of a reductive group of rank at least 2. The proof uses that for sufficiently large prime numbers p, all pro-p subgroups of a reductive algebraic group are valued in the sense defined by Lazard. A sharper estimate of the primes p with this property is given in section 12, using the Bruhat-Tits structure theory of p-adic groups. Finally, section 13 shows that the results of section 11 on vanishing of Euler characteristics do not extend to open subgroups of SL_{2}Z_{p}.

I am grateful for John Coates’s questions about these Euler characteristics. Jean-Pierre Serre, Ottmar Venjakob, and the referee made useful suggestions on the exposition.

Contents

1 Homology vs. cohomology 5

2 Euler characteristics for Lie algebras 7

3 Reductive Lie algebras in characteristic zero 12

4 The case of abelian Lie algebras 14

5 The case of reductive Lie algebras 15

6 Euler characteristics for Z_{p} and 𝔰𝔩_{2}Z_{p} 19

7 Euler characteristics for arbitrary Lie algebras 22

8 Filtered and graded algebras 33

9 Relating groups and Lie algebras 39

10 Cohomology of congruence subgroups 49

11 Euler characteristics for p-adic Lie groups which are not pro-p groups 52

12 Construction of valuations on pro-p subgroups of a semisimple group 61

13 Open subgroups of SL_{2}Z_{p} 69

1 Homology vs. cohomology

The main results of this paper are about the homology of groups or Lie algebras with coefficients in finitely generated Z_{p}-modules. We will explain here how to deduce analogous results for cohomology, or for coefficients in a module of the form (Q_{p}/Z_{p})^{n}.

Throughout the paper, a Lie algebra 𝔤 over a commutative ring R is always assumed to be a finitely generated free R-module. We use Cartan-Eilenberg [11], Chapter XIII, as a reference for the homology and cohomology of Lie algebras. The homology of a Lie algebra 𝔤 depends on the base ring R as well as on 𝔤, but (as is usual) we will not indicate that in the notation. One relation between homology and cohomology for Lie algebras is the naive duality:

Lemma 1.1 For any 𝔤-module M and any injective R-module I, there is a canonical isomorphism

Hom_{R}(H_{i}(𝔤, M), I) = H^{i}(𝔤, Hom_{R}(M, I)).

Typical cases are I = R, when R is a field, and I = Q_{p}/Z_{p}, when R = Z_{p}. Also, there is a canonical Poincaré duality isomorphism for any 𝔤-module M ([11], p. 288):

Lemma 1.2

H_{i}(𝔤, M) ≅ H^{n−i}(𝔤, ∧^{n}𝔤 ⊗_{R} M).

Either of these lemmas can be used to translate the results of this paper about Lie algebras from homology to cohomology.

A reference for the homology of profinite groups G is Brumer [9]. For a prime number p, let Z_{p}G denote the completed group ring of G over the p-adic integers,

Z_{p}G := \underleftarrow{lim} Z_{p}[G/U],

where U runs over the open normal subgroups of G. Define a pseudocompact Z_{p}G-module to be a topological G-module which is an inverse limit of discrete finite p-torsion G-modules. The category of pseudocompact Z_{p}G-modules is an abelian category with exact inverse limits and enough projectives. So we can define the homology groups H_{∗}(G, M) of a profinite group G with coefficients in a pseudocompact Z_{p}G-module M as the left derived functors of the functor H_{0}(G, M) = M_{G} := M/I(G)M, where I(G) = ker(Z_{p}G → Z_{p}). We have

H_{i}(G, M) = \underleftarrow{lim} H_{i}(G/U, M/I(U)M),

where U runs over the open normal subgroups of G ([9], Remark 1, p. 455). Furthermore, the category of pseudocompact Z_{p}G-modules is dual, via Pontrjagin duality

M^{∗} := Hom_{cont}(M, Q_{p}/Z_{p}),

to the category of discrete p-torsion G-modules ([9], Proposition 2.3, p. 448). The category of discrete p-torsion G-modules has enough injectives, and the cohomology of a profinite group G with coefficients in such a module can be defined as a right derived functor [28]. As a result, the homology theory of profinite groups G with coefficients in pseudocompact Z_{p}G-modules is equivalent to the better-known cohomology theory with coefficients in discrete p-torsion G-modules, via Pontrjagin duality:

Lemma 1.3

H_{i}(G, M)^{∗} = H^{i}(G, M^{∗}).

So the main results of this paper, about Euler characteristics associated to the homology of a p-adic Lie group with coefficients in a finitely generated Z_{p}-module, are equivalent to statements about the cohomology of such a group with coefficients in a discrete “cofinitely generated” Z_{p}-module such as (Q_{p}/Z_{p})^{n}.

2 Euler characteristics for Lie algebras

This section discusses some simpler situations where Euler characteristics can be shown to vanish. Most of the results and definitions here will be needed for the later results on Euler characteristics for p-adic Lie groups.

A simple fact in this direction is that a compact connected real Lie group G, viewed as a real manifold, has Euler characteristic 0 unless the group is trivial, in which case the Euler characteristic is 1. This fact can be reformulated as a statement about Euler characteristics in Lie algebra homology, by E. Cartan’s theorem that

H_{∗}(G, R) = H_{∗}(𝔤, R)

for a compact connected Lie group G with Lie algebra 𝔤 over the real numbers [10]. There is a much more general vanishing statement about Euler characteristics of Lie algebras, as follows.

Proposition 2.1 Let 𝔤 be a finite-dimensional Lie algebra over a field k, and let M be a finite-dimensional representation of 𝔤. Then the Euler characteristic

χ(𝔤, M) := ∑_{i} dim_{k}H_{i}(𝔤, M)

is equal to 0 if 𝔤 ≠ 0, and to the dimension of M if 𝔤 = 0.

Proof. For 𝔤 = 0, H_{0}(𝔤, M) = M and the higher homology is 0. For 𝔤 ≠ 0, we consider the standard complex which computes the Lie algebra homology H_{∗}(𝔤, M) ([11], p. 282):

→ ∧^{2}𝔤 ⊗_{k} M → 𝔤 ⊗_{k} M → M → 0,

where

d((x_{1} ∧ ⋯ ∧ x_{r}) ⊗ m) = ∑_{i}(−1)^{i}(x_{1} ∧ ⋯ ∧ x̂_{i} ∧ ⋯ ∧ x_{r}) ⊗ x_{i}m

+ ∑_{i<j}(−1)^{i+j}([x_{i}, x_{j}] ∧ x_{1} ∧ ⋯ ∧ x̂_{i} ∧ ⋯ ∧ x̂_{j} ∧ ⋯ ∧ x_{r}) ⊗ m

for x_{1}, …, x_{r} ∈ 𝔤 and m ∈ M. Since 𝔤 and M are finite-dimensional, this is a bounded complex of finite-dimensional vector spaces. The basic fact about Euler characteristics is that, in this situation, the Euler characteristic of the homology of this complex (that is, of H_{∗}(𝔤, M)) is equal to the alternating sum of the dimensions of the vector spaces in the complex. Thus, if we let n be the dimension of 𝔤, the Euler characteristic is

∑_{i=0}^{n}(−1)^{i}\binom{n}{i}dim M,

which is 0 for n > 0. QED

The same argument applies to the homology of Lie algebras over a discrete valuation ring Γ with coefficients in a Γ-module of finite length, as the following proposition says. We have in mind the case of a Lie algebra over the p-adic integers Z_{p} acting on a finite Z_{p}-module. A Z_{p}-module A is finite if and only if it has finite length, and in that case

ord_{p}|A| = length_{Z_{p}}A.

Proposition 2.2 Let 𝔤 be a Lie algebra over a discrete valuation ring Γ, and let M be a Γ-module of finite length on which 𝔤 acts. Then the Euler characteristic

χ(𝔤, M) := ∑_{i}(−1)^{i}length_{Γ}H_{i}(𝔤, M)

is equal to 0 if 𝔤 ≠ 0, and to length_{Γ}M if 𝔤 = 0.

Proof. For any 𝔤-module M, the homology groups H_{∗}(𝔤, M) are the homology of the standard complex

→ ∧^{2}𝔤 ⊗_{Γ} M → 𝔤 ⊗_{Γ} M → M → 0.

For M of finite length, as we assume, this is a bounded complex of Γ-modules of finite length. So the basic fact about Euler characteristics says that the Euler characteristic χ(𝔤, M) is equal to the alternating sum of the lengths of the Γ-modules in the complex, which is 0 for 𝔤 ≠ 0 by the same calculation as in the proof of Proposition 2.1. QED

We now consider a more subtle situation, which is essentially the main topic of this paper. Let 𝔤 be a Lie algebra over a discrete valuation ring Γ, the case of interest being Γ = Z_{p}. Let M be a finitely generated Γ-module on which 𝔤 acts, and suppose that H_{∗}(𝔤, M) ⊗ F = 0, where F is the quotient field of Γ. Then the homology groups H_{i}(𝔤, M) are Γ-modules of finite length, and we can try to compute the Euler characteristic

χ(𝔤, M) := ∑_{i}(−1)^{i}length_{Γ}H_{i}(𝔤, M).

In this situation, the standard complex which computes H_{∗}(𝔤, M),

→ ∧^{2}𝔤 ⊗_{Γ} M → 𝔤 ⊗_{Γ} M → M → 0,

does not consist of Γ-modules of finite length, and so the basic fact about Euler characteristics is not enough to determine χ(𝔤, M).

We do, however, have the following results on “independence of M” and “independence of 𝔤.” For part (2), we need to define relative Lie algebra homology for Lie algebras 𝔥 ⊂ 𝔤 and a 𝔤-module M. Namely, let H_{∗}(𝔤, 𝔥; M) be the homology of the mapping cone of the map of chain complexes C_{∗}(𝔥, M) → C_{∗}(𝔤, M) which compute the homology of 𝔥 and 𝔤. Then there is a long exact sequence

H_{j}(𝔥, M) → H_{j}(𝔤, M) → H_{j}(𝔤, 𝔥; M) → H_{j−1}(𝔥, M).

Proposition 2.3 Let 𝔤 be a Lie algebra over a discrete valuation ring Γ.

(1) Suppose 𝔤 ≠ 0. Let M_{1} and M_{2} be 𝔤-modules, finitely generated over Γ, which become isomorphic after tensoring with F. Then H_{∗}(𝔤, M_{1}) ⊗ F = 0 if and only if H_{∗}(𝔤, M_{2}) ⊗ F = 0, and if either condition holds then

χ(𝔤, M_{1}) = χ(𝔤, M_{2}).

(2) Suppose that 𝔤 has rank at least 2 as a free Γ-module. Let M be a 𝔤-module which is finitely generated over Γ. Let 𝔥 ⊂ 𝔤 be an open Lie subalgebra, meaning a Lie subalgebra such that the Γ-module 𝔤/𝔥 has finite length. Then the relative Lie algebra homology groups H_{∗}(𝔤, 𝔥; M) have finite length as Γ-modules, and the corresponding Euler characteristic χ(𝔤, 𝔥; M) is 0. It follows that H_{∗}(𝔤, M) ⊗ F = 0 if and only if H_{∗}(𝔥, M) ⊗ F = 0, and if either condition holds then

χ(𝔤, M) = χ(𝔥, M).

The assumption that 𝔤 has rank at least 2 as a free Γ-module is essential in statement (2). Indeed, for the Lie algebra 𝔤 = Z_{p} acting on a finitely generated Z_{p}-module M such that the space of coinvariants of 𝔤 on M ⊗ Q_{p} is 0 (so that these Euler characteristics are defined), χ(p^{n}𝔤, M) is equal to χ(𝔤, M) + n dim(M ⊗ Q_{p}), not to χ(𝔤, M). A general calculation of χ(𝔤, M) for 𝔤 isomorphic to Z_{p} can be found in Proposition 6.1.

Proof. Since H_{∗}(𝔤, M)⊗F = H_{∗}(𝔤, M ⊗F), we have the first part of statement (1). Furthermore, we can multiply a given 𝔤-module isomorphism M_{1}⊗F → M_{2}⊗F by a suitable power of a uniformizer π of Γ to get a 𝔤-module homomorphism M_{1} → M_{2} which becomes an isomorphism after tensoring with F. That is, the kernel and cokernel have finite length. Then (1) follows from Proposition 2.2.

Since H_{∗}(𝔤, M)⊗F = H_{∗}(𝔤⊗F, M ⊗F), the vanishing of H_{∗}(𝔤, 𝔥; M)⊗F follows from the isomorphism 𝔥 ⊗ F ≅ 𝔤 ⊗ F. So the Γ-modules H_{∗}(𝔤, 𝔥; M) have finite length, and the Euler characteristic χ(𝔤, 𝔥; M) is defined. Furthermore, Proposition 2.2 shows that χ(𝔤, M_{tors}) = χ(𝔥, M_{tors}) = 0, and so χ(𝔤, 𝔥; M_{tors}) = 0. Therefore, to show that χ(𝔤, 𝔥; M) = 0, it suffices to show that χ(𝔤, 𝔥; M/M_{tors}) = 0. That is, we can assume that the finitely generated Γ-module M is free.

The map of chain complexes C_{∗}(𝔥, M) → C_{∗}(𝔤, M) associated to the inclusion 𝔥 ⊂ 𝔤 has the form (∧^{∗}𝔥) ⊗_{Γ} M → (∧^{∗}𝔤) ⊗_{Γ} M. Since M is a finitely generated free Γ-module, this map is injective. So the relative Lie algebra homology H_{∗}(𝔤, 𝔥; M) is the homology of the cokernel complex C_{∗}(𝔤, 𝔥; M) of this map. Here C_{j}(𝔤, 𝔥; M) is a Γ-module of finite length with

length_{Γ}C_{j}(𝔤, 𝔥; M) = length_{Γ}(∧^{j}(𝔤)/ ∧^{j} (𝔥)) rank_{Γ}M
= \binom{n − 1}{j − 1}length_{Γ}(𝔤/𝔥) rank_{Γ}M,

where n denotes the rank of the Lie algebras 𝔤 and 𝔥 as free Γ-modules. The formula for length_{Γ}(∧^{j}(𝔤)/ ∧^{j} (𝔥)) which I have used here applies to any inclusion of a free Γ-module 𝔥 of rank n into another, 𝔤, of the same rank. It follows from the special case where 𝔤/𝔥 ≅ Γ/π. That special case can be proved by writing out a basis for ∧^{j}(𝔤)/ ∧^{j} (𝔥).

Since C_{∗}(𝔤, 𝔥; M) is a bounded complex of Γ-modules of finite length, we have

χ(𝔤, 𝔥; M) = χ(C_{∗}(𝔤, 𝔥; M))
= ∑_{j}(−1)^{j}\binom{n − 1}{j − 1}length_{Γ}(𝔤/𝔥) rank_{Γ}M
= 0,

using the assumption that the dimension n of 𝔤 is at least 2.

The statements in Proposition 2.3(2) about the homology of 𝔤 and 𝔥 follow from those about H_{∗}(𝔤, 𝔥; M) by the long exact sequence before the proposition. QED

The following definition makes sense thanks to Proposition 2.3.

Definition 2.4 Let 𝔤_{Q_{p}} be a Lie algebra of dimension at least 2 over Q_{p}. Let M_{Q_{p}} be a finite-dimensional 𝔤_{Q_{p}}-module such that H_{∗}(𝔤_{Q_{p}}, M_{Q_{p}}) = 0. Define χ_{fin}(𝔤_{Q_{p}}, M_{Q_{p}}) to be the Euler characteristic χ(𝔤, M) for any integral models 𝔤 and M of 𝔤_{Q_{p}} and M_{Q_{p}}. That is, 𝔤 is a Lie algebra over Z_{p} and M is a 𝔤-module, finitely generated as a Z_{p}-module, such that tensoring up to Q_{p} gives the Lie algebra 𝔤_{Q_{p}} and its module M_{Q_{p}}.

A slight extension of this definition is sometimes useful. Let K be a finite extension of the p-adic numbers, with ring of integers o_{K}. Let 𝔤_{o_{K}} be a Lie algebra over o_{K}, and let M_{o_{K}} be a finitely generated o_{K}-module on which 𝔤_{o_{K}} acts. Suppose that H_{∗}(𝔤_{o_{K}}, M_{o_{K}}) ⊗_{o_{K}} K = 0. Then we can define an Euler characteristic, extending our earlier definition for Lie algebras over the p-adic integers, by

χ(𝔤_{o_{K}}, M_{o_{K}}) = [K : Q_{p}]^{−1} ∑_{j}(−1)^{j}ord_{p}|H_{j}(𝔤_{o_{K}}, M_{o_{K}})|.

This rational number does not change if we tensor the Lie algebra 𝔤_{o_{K}} and the module M_{o_{K}} with o_{L} for some larger p-adic field L, as a result of the flatness of o_{L} over o_{K}. Also, for a Lie algebra of rank at least 2 as an o_{K}-module, Proposition 2.3 shows that this number only depends on the Lie algebra and its module after tensoring with K, so we have an invariant χ_{fin}(𝔤_{K}, M_{K}). Combining this with the previous observation shows that the following invariant is well defined.

Definition 2.5 Let 𝔤_{\overline{Q}_{p}} be a Lie algebra of dimension at least 2 over the algebraic closure of Q_{p}, and let M_{\overline{Q}_{p}} be a finite-dimensional 𝔤_{\overline{Q}_{p}}-module such that H_{∗}(𝔤_{\overline{Q}_{p}}, M_{\overline{Q}_{p}}) = 0. Define χ_{fin}(𝔤_{\overline{Q}_{p}}, M_{\overline{Q}_{p}}) to be the rational number χ(𝔤_{o_{K}}, M_{o_{K}}) for any models of 𝔤_{\overline{Q}_{p}} and M_{\overline{Q}_{p}} over the ring of integers o_{K} of some finite extension K of Q_{p}.

3 Reductive Lie algebras in characteristic zero

The following lemma is a reformulation of the basic results on the cohomology of reductive Lie algebras in characteristic zero, due to Chevalley-Eilenberg [12] and Hochschild-Serre [20]. By definition, a finite-dimensional Lie algebra 𝔤 over a field K of characteristic zero is called reductive if 𝔤, viewed as a module over itself, is a direct sum of simple modules. Equivalently, 𝔤 is the direct sum of a semisimple Lie algebra and an abelian Lie algebra. Beware that if 𝔤 is reductive but not semisimple, finite-dimensional 𝔤-modules are not all direct sums of simple modules, contrary to what the name “reductive” suggests.

Lemma 3.1 Let 𝔤 be a reductive Lie algebra over a field K of characteristic zero. Then any finite-dimensional 𝔤-module M splits canonically as a direct sum of modules all of whose simple subquotients are isomorphic. Also, if the space M_{𝔤} of coinvariants or the space M^{𝔤} of invariants is 0, then H^{∗}(𝔤, M) and H_{∗}(𝔤, M) are 0.

Proof. Let 𝔤 be a reductive Lie algebra over a field K of characteristic zero. Hochschild and Serre ([20], Theorem 10, p. 598), extending Chevalley and Eilenberg, showed that if M is a nontrivial simple 𝔤-module of finite dimension, then H^{∗}(𝔤, M) = 0. I will only describe the proof for 𝔤 semisimple. In that case, the Casimir operator in the center of the enveloping algebra U𝔤 acts by 0 on the trivial module K, and by a nonzero scalar on every nontrivial simple module M; so

H^{∗}(𝔤, M) = Ext_{U𝔤}^{∗}(K, M) = 0.

In particular, for 𝔤 reductive and a nontrivial simple 𝔤-module M, H^{1}(𝔤, M) = 0, which says that there are no nontrivial extensions between the trivial 𝔤-module K and a nontrivial simple 𝔤-module. So every finite-dimensional 𝔤-module M splits canonically as the direct sum of a module with all simple subquotients trivial and a module with all simple subquotients nontrivial.

Therefore, if M is a finite-dimensional 𝔤-module with M_{𝔤} = 0 or M^{𝔤} = 0, then all simple subquotients of M are nontrivial, and so H^{∗}(𝔤, M) = 0 by Hochschild and Serre’s theorem. The analogous statement for homology follows from naive duality, Lemma 1.1, which says that

H_{i}(𝔤, M)^{∗} = H^{i}(𝔤, M^{∗}),

where M^{∗} denotes the dual of the vector space M. Finally, the splitting of any finitedimensional 𝔤-module as a direct sum of modules all of which have the same simple subquotient follows from the vanishing of Ext_{𝔤}^{1}(S_{1}, S_{2}) = H^{1}(𝔤, Hom_{K}(S_{1}, S_{2})) for any two non-isomorphic simple modules S_{1} and S_{2}. That vanishing follows from what we have proved about the vanishing of cohomology, since

H^{0}(𝔤, Hom_{K}(S_{1}, S_{2})) = Hom_{𝔤}(S_{1}, S_{2})
= 0.

QED

4 The case of abelian Lie algebras

Let K be a finite extension of the field of p-adic numbers, with ring of integers o_{K}. We will show that the Euler characteristics we consider are 0 for any abelian Lie algebra over o_{K} of rank at least 2 as an o_{K}-module. (They are not 0 for a Lie algebra of rank 1 as an o_{K}-module, as I mentioned after the statement of Proposition 2.3.) This is the first step in a sequence of generalizations, the next step being Theorem 5.1 which proves the same vanishing for all reductive Lie algebras of rank at least 2.

Proposition 4.1 Let 𝔥 be an abelian Lie algebra of the form (o_{K})^{r} for some r ≥ 2. Let M be a finitely generated o_{K}-module with 𝔥-action such that M_{𝔥} ⊗ K = 0. Then the homology groups H_{∗}(𝔥, M) are finite and the resulting Euler characteristic χ(𝔥, M) (defined in section 2) is 0.

Proof. At first let 𝔥 = (o_{K})^{r} for any r. Let 𝔥_{K} = 𝔥 ⊗_{o_{K}} K. For any 𝔥_{o_{K}}- module M, finitely generated over o_{K}, such that the coinvariants of 𝔥_{K} on M_{K} are 0, Lemma 3.1 shows that H_{∗}(𝔥_{K}, M_{K}) = 0. It follows that H_{j}(𝔥, M) is a finite o_{K}-module for all j.

Now suppose that 𝔥 has rank r ≥ 2 as an o_{K}-module; we want to show that χ(𝔥, M) = 0. By Definition 2.5, which only makes sense for a Lie algebra of dimension at least 2, it is equivalent to show that χ_{fin}(𝔥_{\overline{Q}_{p}}, M_{\overline{Q}_{p}}) = 0, given that (M_{\overline{Q}_{p}})_{𝔥_{\overline{Q}_{p}}} = 0. By Lemma 3.1, the assumption (M_{\overline{Q}_{p}})_{𝔥_{\overline{Q}_{p}}} = 0 implies that M_{\overline{Q}_{p}} is an extension of nontrivial simple 𝔥_{\overline{Q}_{p}}-modules; so it suffices to show that χ_{fin}(𝔥_{\overline{Q}_{p}}, M_{\overline{Q}_{p}}) = 0 for a nontrivial simple 𝔥_{\overline{Q}_{p}}-module M_{\overline{Q}_{p}}. Since 𝔥_{\overline{Q}_{p}} is abelian, such a module has dimension 1 by Schur’s lemma. Changing the definition of the original Lie algebra 𝔥, we know that there is some p-adic field K and some models 𝔥 and M over o_{K} for 𝔥_{\overline{Q}_{p}} and the 1-dimensional module M_{\overline{Q}_{p}}, and we are done if we can show that χ(𝔥, M) = 0.

Since 𝔥 has rank r ≥ 2 as an o_{K}-module, there is a Lie subalgebra 𝔩 ⊂ 𝔥 of rank r − 1 which is a direct summand as an o_{K}-module and which acts nontrivially on M. Since M has rank 1 over o_{K}, it follows that the coinvariants of 𝔩 on M ⊗ K are 0. So the homology groups H_{∗}(𝔩, M) are finite as shown above. Consider the Hochschild-Serre spectral sequence

E_{ij}^{2} = H_{i}(𝔥/𝔩, H_{j}(𝔩, M)) ⇒ H_{i+j}(𝔥, M),

where 𝔥/𝔩 ≅ o_{K}. We have χ(𝔥/𝔩, N) = 0 for any finite 𝔥/𝔩-module N, by Proposition 2.2, and so this spectral sequence shows that χ(𝔥, M) = 0. (This concluding argument is a version for Lie algebras of the argument in Coates-Sujatha about the Euler characteristic of a p-adic Lie group which maps onto Z_{p} [15].) QED

5 The case of reductive Lie algebras

The rank of a reductive Lie algebra over a field K of characteristic zero is defined to be the dimension of the centralizer of a general element; the standard definition is equivalent ([6], Ch. VII, sections 2 and 4). The rank does not change under field extensions. For example, for the Lie algebra 𝔤_{K} of a reductive algebraic group G over K, the rank of 𝔤_{K} is the rank of G over the algebraic closure of K.

Theorem 5.1 Let 𝔤 be a Lie algebra over Z_{p} such that 𝔤_{Q_{p}} is reductive of rank at least 2. Let M be a finitely generated Z_{p}-module with 𝔤-action such that the coinvariants of 𝔤_{Q_{p}} on M_{Q_{p}} are 0. Then the homology groups H_{∗}(𝔤, M) are finite and the resulting Euler characteristic χ(𝔤, M) is equal to 0.

The optimal generalization of this statement is Theorem 7.1, which proves the same vanishing for all Lie algebras over Z_{p} in which every element has centralizer of dimension at least 2. See section 6 for the calculation of Euler characteristics, which are sometimes nonzero, for reductive Lie algebras of rank 1.

Proof. We will prove the analogous statement for Lie algebras 𝔤 over the ring of integers o_{K} of any finite extension K of Q_{p}, not just over Z_{p}. The homology groups H_{∗}(𝔤, M) are finitely generated o_{K}-modules. Since we assume that (M_{K})_{𝔤_{K}} = 0, Lemma 3.1 gives that H_{∗}(𝔤_{K}, M_{K}) = 0, and so the homology groups H_{∗}(𝔤, M) are in fact finite o_{K}-modules. Thus, since 𝔤 has rank at least 2 as an o_{K}-module, the Euler characteristic χ_{fin}(𝔤_{K}, M_{K}) is defined by Definition 2.4, and we want to show that it is 0. Since this Euler characteristic is unchanged under finite extensions of K, according to Definition 2.5, we can extend the field K so as to arrange that the reductive Lie algebra 𝔤_{K} has a Borel subalgebra 𝔟_{K} defined over K ([6], Ch. VIII, section 3). By Lemma 3.1 again, the assumption that (M_{K})_{𝔤_{K}} = 0 implies that M_{K} is an extension of nontrivial simple 𝔤_{K}-modules, so it suffices to show that χ_{fin}(𝔤_{K}, M_{K}) = 0 when M_{K} is a nontrivial simple 𝔤_{K}-module.

Let 𝔲_{K} be the commutator subalgebra of the Borel subalgebra 𝔟_{K}, so that 𝔟_{K}/𝔲_{K} ≅ K^{r} where r is the rank of 𝔤. We are assuming that r is at least 2. Let 𝔤 and M be models over o_{K}, which we take to be finitely generated free as o_{K}- modules, for 𝔤_{K} and M_{K}. Our goal is to show that χ(𝔤, M) = 0. Let 𝔟 = 𝔤 ∩ 𝔟_{K} and 𝔲 = 𝔤 ∩ 𝔲_{K}; these are Lie subalgebras of 𝔤 over o_{K}. The quotient Lie algebra 𝔟/𝔲 is isomorphic to (o_{K})^{r}, where r is at least 2.

To analyze H_{∗}(𝔤, M), we use two spectral sequences, both of homological type in the sense that the differential d_{r} has bidegree (−r, r − 1). First, there is the spectral sequence defined by Koszul and Hochschild-Serre for any subalgebra of a Lie algebra, which we apply to the integral Borel subalgebra 𝔟 ⊂ 𝔤 ([20], Corollary to Theorem 2, p. 594):

E_{kl}^{1} = H_{l}(𝔟, M ⊗_{o_{K}} ∧^{k}(𝔤/𝔟)) ⇒ H_{k+l}(𝔤, M).

In fact, Hochschild and Serre construct the analogous spectral sequence for cohomology rather than homology, and only over a field, but the same construction works for the homology of Lie algebras over a commutative ring R when (as here) the inclusion 𝔟 ⊂ 𝔤 is R-linearly split. Next we have the Hochschild-Serre spectral sequence associated to the ideal 𝔲 ⊂ 𝔟:

E_{ij}^{2} = H_{i}(𝔟/𝔲, H_{j}(𝔲, M ⊗_{o_{K}} ∧^{k}(𝔤/𝔟))) ⇒ H_{i+j}(𝔟, M ⊗_{o_{K}} ∧^{k}(𝔤/𝔟)).

Combining these two spectral sequences gives a formula for the Euler characteristic χ(𝔤, M) which is correct if the right-hand side is defined (that is, if the homology groups of 𝔟/𝔲 acting on the modules shown are finite):

χ(𝔤, M) = ∑_{j,k}(−1)^{j+k}χ(𝔟/𝔲, H_{j}(𝔲, M ⊗ ∧^{k}(𝔤/𝔟))).

Since 𝔟/𝔲 ≅ o_{K}^{r} with r ≥ 2, Proposition 4.1 says that the Euler characteristic χ(𝔟/𝔲, N) is defined and equal to 0 for any finitely generated o_{K}-module N with 𝔟/𝔲-action such that (N_{K})_{𝔟/𝔲} = 0. So to show that χ(𝔤, M) = 0, as we want, it suffices to prove the following statement, which fortunately follows from Kostant’s theorem [22].

Proposition 5.2 Let 𝔤 be a reductive Lie algebra over a field K of characteristic zero which has a Borel subalgebra 𝔟 defined over K, and let 𝔲 be the commutator subalgebra of 𝔟. Let M be a nontrivial simple 𝔤-module. Then

H_{∗}(𝔲, M ⊗_{K} ∧^{∗}(𝔤/𝔟))_{𝔟/𝔲} = 0.

Proof. When we extend scalars from K to its algebraic closure, the 𝔤-module M becomes a direct sum of nontrivial simple modules. So it suffices to prove the proposition for K algebraically closed. In this case, the center of 𝔤 acts on M by scalars, and acts trivially by conjugation on 𝔤/𝔟 and on 𝔲. If the center of 𝔤 acts nontrivially on M, then it acts nontrivially by scalars on H_{∗}(𝔲, M ⊗ ∧^{∗}(𝔤/𝔟)), and so the coinvariants of 𝔟/𝔲 (which includes the center of 𝔤) on these groups are 0. Thus we can assume that the center of 𝔤 acts trivially on M. Then, replacing 𝔤 by its quotient by the center, we can assume that the Lie algebra 𝔤 is semisimple. In this case, there is a canonical simply connected algebraic group G over K with Lie algebra 𝔤. Let B be the Borel subgroup of G with Lie algebra 𝔟, and let U ⊂ B be its unipotent radical. Choosing a maximal torus T ⊂ B, we define the negative roots to be the weights of T acting on 𝔲.

Kostant’s theorem, which can be viewed as a consequence of the Borel-Weil-Bott theorem, determines the weights of the torus B/U ≅ T acting on H_{∗}(𝔲, M) for any simple 𝔤-module M in characteristic zero [22]. The result is that the total dimension of H_{∗}(𝔲, M) is always equal to the order of the Weyl group W. More precisely, let λ ∈ X(T) = Hom(T, G_{m}) be the highest weight of M, in the sense that all other weights of M are obtained from λ by repeatedly adding negative roots. Then, for any weight μ, the μ-weight subspace of H_{j}(𝔲, M) has dimension 1 if there is an element w ∈ W such that j is the length of w and μ = w · λ; otherwise the μ-weight subspace of H_{j}(𝔲, M) is 0. Here the notation w · λ refers to the dot action of W on the weight lattice X(T):

w · λ := w(λ + ρ) − ρ,

where ρ denotes half the sum of the positive roots ([21], p. 179).

Since the weights of T occurring in 𝔲 are exactly the negative roots, the weights occurring in ∧^{∗}(𝔲) are exactly the sums of some set of negative roots. It follows easily that the set S of weights occurring in ∧^{∗}(𝔲) is invariant under the dot action of W on the weight lattice X(T).

Clearly the intersection of S with the cone X(T)^{+} of dominant weights is the single weight 0. So if λ is any nonzero dominant weight, then λ is not in S. Since S is invariant under the dot action of W on X(T), it follows that (W · λ) ∩ S is empty. By Kostant’s theorem, as stated above, it follows that for any nontrivial simple 𝔤-module M, the T -module H_{∗}(𝔲, M) has no weights in common with ∧^{∗}(𝔲).

The weights of T that occur in the B-module 𝔤/𝔟 are the negatives of those that occur in the B-module 𝔲. So the previous paragraph implies that

H_{∗}(𝔲, M ⊗ μ)_{B/U} = (H_{∗}(𝔲, M) ⊗ μ)_{B/U}

= 0

for any weight μ occurring in ∧^{∗}(𝔤/𝔟). Since the simple B-modules are the 1dimensional B/U-modules, the B-module ∧^{∗}(𝔤/𝔟) has a filtration with graded pieces the weights μ as above. It follows that

H_{∗}(𝔲, M ⊗ ∧^{∗}(𝔤/𝔟))_{B/U} = 0.

Since we are in characteristic zero, it is equivalent to say that the coinvariants of the Lie algebra 𝔟/𝔲 are 0. This proves Proposition 5.2 and hence Theorem 5.1. QED

6 Euler characteristics for Z_{p} and 𝔰𝔩_{2}Z_{p}

Since Theorem 5.1 proves the vanishing of the Euler characteristic we are considering for reductive Lie algebras of rank at least 2, it is natural to ask what happens in rank 1. If 𝔤 is a Lie algebra over Z_{p} such that 𝔤_{Q_{p}} is reductive of rank 1, then 𝔤_{\overline{Q}_{p}} is isomorphic to \overline{Q}_{p} or 𝔰𝔩_{2}\overline{Q}_{p}. In this section, we determine the Euler characteristic for all representations of such a Lie algebra 𝔤. In particular, for the irreducible representation of 𝔰𝔩_{2}Z_{p} of a given highest weight, the Euler characteristic is 0 for all but finitely many prime numbers p. For example, for the standard representation M = (Z_{p})^{2} of 𝔤 = 𝔰𝔩_{2}Z_{p}, the Euler characteristic χ(𝔤, M) is 0 for p ≠ 3 and −2 for p = 3.

Let us first compute Euler characteristics for a Lie algebra 𝔤 over Z_{p} such that 𝔤_{\overline{Q}_{p}} ≅ \overline{Q}_{p}. Clearly 𝔤 is isomorphic to Z_{p}.

Proposition 6.1 Let 𝔤 be the Lie algebra of rank 1 as a Z_{p}-module with generator x. Let M be a finitely generated Z_{p}-module with 𝔤-action. Then the homology groups H_{∗}(𝔤, M) are finite if and only if x ∈ End(M) is invertible on M ⊗ Q_{p}. If this is so, then

χ(𝔤, M) = ord_{p}(det x)

where we view x as an endomorphism of M ⊗ Q_{p}.

Proof. The only homology groups for 𝔤 ≅ Z_{p} acting on M are H_{0}(𝔤, M) = M_{𝔤} and H_{1}(𝔤, M), which is isomorphic to M^{𝔤} ⊗_{Z_{p}} 𝔤 by Poincaré duality (Lemma 1.2). For any 𝔤-module M, these two groups tensored with Q_{p} are 0 if and only if (M ⊗ Q_{p})_{𝔤} = 0 and (M ⊗ Q_{p})^{𝔤} = 0, which means precisely that x is invertible on M ⊗ Q_{p}.

Suppose that the 𝔤-module M is a finitely generated Z_{p}-module and that x is invertible on M ⊗ Q_{p}. To prove that χ(𝔤, M) = ord_{p}(det x), it suffices to prove it when M is a finitely generated free Z_{p}-module. Indeed, we have χ(𝔤, N) = 0 for every finite 𝔤-module N by Proposition 2.2, so that

χ(𝔤, M) = χ(𝔤, M_{tors}) + χ(𝔤, M/M_{tors})
= χ(𝔤, M/M_{tors}).

Given that M is a finitely generated free Z_{p}-module, with x invertible on M ⊗ Q_{p}, we have M^{𝔤} ⊂ (M ⊗ Q_{p})^{𝔤} = 0, and so H_{1}(𝔤, M) = 0. Thus

χ(𝔤, M) = ord_{p}|H_{0}(𝔤, M)|
= ord_{p}|M/xM|
= ord_{p} det x.

QED

Now let 𝔤 be a Lie algebra over Z_{p} such that 𝔤_{\overline{Q}_{p}} ≅ 𝔰𝔩_{2}\overline{Q}_{p}. For example, 𝔤 could be the Lie algebra 𝔰𝔩_{2}Z_{p} of 2 × 2 matrices over Z_{p} of trace 0, or an open Lie subalgebra of the Lie algebra 𝔰𝔩_{1}D = [D, D] ⊂ D over Q_{p} associated to the nontrivial quaternion algebra D over Q_{p}. In any case, Definition 2.5 shows that, since dim 𝔤 ≥ 2, the integer χ(𝔤, M) (assuming H_{∗}(𝔤, M)⊗Q_{p} = 0) only depends on the module M_{\overline{Q}_{p}} for 𝔰𝔩_{2}\overline{Q}_{p}. Since 𝔰𝔩_{2}\overline{Q}_{p} is semisimple over a field of characteristic zero, every finite-dimensional 𝔰𝔩_{2}\overline{Q}_{p}-module is a direct sum of simple modules. The simple modules are the symmetric powers S^{a}V_{\overline{Q}_{p}} of the standard module V_{\overline{Q}_{p}} = \overline{Q}_{p}^{2}, a ≥ 0, and Lemma 3.1 shows that H_{∗}(𝔰𝔩_{2}\overline{Q}_{p}, S^{a}V_{\overline{Q}_{p}}) = 0 if and only if a > 0. So we only need to compute the Euler characteristic χ_{fin}(𝔰𝔩_{2}\overline{Q}_{p}, S^{a}V_{\overline{Q}_{p}}) for the integers a > 0. The answer is:

Proposition 6.2 For any prime number p and any positive integer a,

χ_{fin}(𝔰𝔩_{2}\overline{Q}_{p}, S^{a}V_{\overline{Q}_{p}}) = 2(ord_{p}a − ord_{p}(a + 2)).

Thus, for a given a > 0, this Euler characteristic is 0 for almost all prime numbers p, in particular for all p > a + 2.

Proof. By Definition 2.5, it suffices to compute the Euler characteristic for a single model over Z_{p} of the Lie algebra and the module. We will compute χ(𝔰𝔩_{2}Z_{p}, S^{a}V) where V = (Z_{p})^{2} is the standard representation of 𝔰𝔩_{2}Z_{p}. It is possible to compute the homology groups of 𝔰𝔩_{2}Z_{p} acting on S^{a}V explicitly, but the actual homology groups are considerably more complicated than the Euler characteristic. We will therefore use another approach which gives the simple formula for the Euler characteristic more directly.

We will imitate, as far as possible, the proof of Theorem 5.1 that these Euler characteristics are 0 for reductive Lie algebras of rank at least 2. The difference is that for an abelian Lie algebra over Z_{p} of rank at least 2 as a Z_{p}-module, the Euler characteristic is 0 when it is defined (Proposition 4.1), whereas this is not true for the Lie algebra Z_{p}. We can instead use Proposition 6.1 to compute Euler characteristics for the Lie algebra Z_{p} explicitly.

Let 𝔤 = 𝔰𝔩_{2}Z_{p} and M = S^{a}V, where a > 0. Let 𝔟 be the subalgebra of uppertriangular matrices in 𝔤, and 𝔲 the subalgebra of strictly upper-triangular matrices in 𝔤. Using the spectral sequences

E_{kl}^{1} = H_{l}(𝔟, M ⊗_{Z_{p}} ∧^{k}(𝔤/𝔟)) ⇒ H_{k+l}(𝔤, M)

and

E_{ij}^{2} = H_{i}(𝔟/𝔲, H_{j}(𝔲, M ⊗_{Z_{p}} ∧^{k}(𝔤/𝔟))) ⇒ H_{i+j}(𝔟, M ⊗_{Z_{p}} ∧^{k}(𝔤/𝔟)),

we derive a formula for the Euler characteristic χ(𝔤, M) which is correct if the righthand side is defined (that is, if the homology groups of 𝔟/𝔲 acting on the modules shown are finite):

χ(𝔤, M) = ∑_{j,k}(−1)^{j+k}χ(𝔟/𝔲, H_{j}(𝔲, M ⊗_{Z_{p}} ∧^{k}(𝔤/𝔟))).

Here 𝔟/𝔲, 𝔲, and 𝔤/𝔟 all have rank 1 as Z_{p}-modules, so the sum runs over 0 ≤ j, k ≤ 1. A moment’s calculation shows that 𝔲 ⊂ 𝔰𝔩_{2}Z_{p} acts trivially on 𝔤/𝔟, so that the formula can be rewritten as:

χ(𝔤, M) = ∑_{j,k}(−1)^{j+k}χ(𝔟/𝔲, H_{j}(𝔲, M) ⊗_{Z_{p}} ∧^{k}(𝔤/𝔟)).

By Kostant’s theorem (as in the proof of Proposition 5.2), the homology groups H_{j}(𝔲, M) ⊗_{Z_{p}} Q_{p} are 1-dimensional, and the standard generator

H = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix})

of the Lie algebra 𝔟/𝔲 ≅ Z_{p} acts on H_{0}(𝔲, M) ⊗ Q_{p} by multiplication by −a and on H_{1}(𝔲, M) ⊗ Q_{p} by multiplication by a + 2, where M = S^{a}V. Also, the generator H of 𝔟/𝔲 acts on 𝔤/𝔟 ≅ Z_{p} by multiplication by −2. As a result, using that a > 0, Proposition 6.1 shows that the Euler characteristics in the previous paragraph’s formula are defined, and gives the following result:

χ(𝔤, M) = ord_{p}(−a) − ord_{p}(−a − 2) − ord_{p}(a + 2) + ord_{p}(a)
= 2(ord_{p}a − ord_{p}(a + 2)).

QED

7 Euler characteristics for arbitrary Lie algebras

In this section we show that for any Lie algebra 𝔤 over Z_{p} in which the centralizer of every element has dimension at least 2, the Euler characteristic χ(𝔤, M) is 0 whenever it is defined (Theorem 7.1). Conversely, for any Lie algebra 𝔤 over Z_{p} in which the centralizer of some element has dimension 1, we compute χ(𝔤, M) here whenever it is defined, in particular observing that this Euler characteristic is nonzero for some M (Theorem 7.4).

Theorem 7.1 Let 𝔤 be a Lie algebra over Z_{p} such that the centralizer of every element has dimension at least 2. Then the Euler characteristic χ(𝔤, M) is 0 whenever it is defined, that is, for all finitely generated Z_{p}-modules M with 𝔤-action such that H_{∗}(𝔤, M) ⊗ Q_{p} = 0.

Proof. The Lie algebra 𝔤_{Q_{p}} obtained by tensoring 𝔤 with Q_{p} clearly also has the property that the centralizer of every element has dimension at least 2. Its structure is described well enough for our purpose by the following lemma.

Lemma 7.2 Let 𝔤 be a Lie algebra over a field K of characteristic zero such that the centralizer of every element has dimension at least 2. Then 𝔤 satisfies at least one of the following three properties.

(1) 𝔤 maps onto a semisimple Lie algebra 𝔯 of rank at least 2.

(2) 𝔤 maps onto a semisimple Lie algebra 𝔯 of rank 1 with some kernel 𝔲, and there is an element x ∈ 𝔤 whose image spans a (1-dimensional) Cartan subalgebra 𝔥 in 𝔯 and whose centralizer in 𝔲 is not 0.

(3) 𝔤 maps onto a 1-dimensional Lie algebra 𝔯 = 𝔥 with some kernel 𝔲, and there is an element x ∈ 𝔤 whose image spans 𝔥 and whose centralizer in 𝔲 is not 0.

Proof. The quotient of 𝔤 by its maximal solvable ideal, called the radical rad(𝔤), is semisimple ([4], Ch. 5, section 2 and Ch. 6, section 1). If 𝔯 := 𝔤/rad(𝔤) has rank at least 2 then we have conclusion (1). Suppose it has rank 1. Then there is an element x in 𝔤 whose image spans a Cartan subalgebra 𝔥 in 𝔯, these being 1-dimensional. Since a Cartan subalgebra in a semisimple Lie algebra is its own centralizer, the centralizer of x in 𝔤 is contained in the inverse image of 𝔥 in 𝔤, an extension of 𝔥 by 𝔲 := rad(𝔤). Since the centralizer of x in 𝔤 has dimension at least 2, the centralizer of x in 𝔲 is not 0, thus proving (2).

Otherwise, 𝔤/rad(𝔤) is 0, which means that 𝔤 is solvable. Since the centralizer of 0 in 𝔤 has dimension at least 2, 𝔤 is not 0, and so it maps onto a 1-dimensional Lie algebra 𝔥 in some way. Let 𝔲 be the kernel. Let x be any element of 𝔤 whose image spans 𝔥. Since the centralizer of x in 𝔤 has dimension at least 2, the centralizer of x in 𝔲 is not 0, and we have conclusion (3). QED

To prove Theorem 7.1, it suffices to show that χ(𝔤, M) = 0 if 𝔤_{Q_{p}} satisfies any of the three conditions of Lemma 7.2. We first need the following lemma.

Lemma 7.3 Let 𝔤 be a Lie algebra over a field K of characteristic zero which maps onto a reductive Lie algebra 𝔯. Let 𝔲 be the kernel. If M is a finite-dimensional 𝔤-module such that H_{∗}(𝔤, M) = 0, then H_{∗}(𝔯, H_{∗}(𝔲, M)) = 0. In particular, the coinvariants of 𝔯 on H_{∗}(𝔲, M) are 0.

Proof. Consider the Hochschild-Serre spectral sequence

E_{ij}^{2} = H_{i}(𝔯, H_{j}(𝔲, M)) ⇒ H_{i+j}(𝔤, M).

We are assuming that the E_{∞} term of the spectral sequence is 0, and we want to show that the E_{2} term is also 0. Let l be the smallest integer, if any, such that H_{0}(𝔯, H_{l}(𝔲, M)) is not 0. The differential d_{r} in the spectral sequence has bidegree (−r, r −1), so all differentials are 0 on this group since they would map to homology in negative degrees. Moreover, Lemma 3.1 shows that all the homology groups of the characteristic-zero reductive Lie algebra 𝔯 acting on H_{j}(𝔲, M) are 0 for j < l. So no differentials in the spectral sequence can go into or out of H_{0}(𝔯, H_{l}(𝔲, M)), contradicting the assumption that the E_{∞} term of the spectral sequence is 0. So in fact H_{0}(𝔯, H_{j}(𝔲, M)) is 0 for all j, and by Lemma 3.1 again it follows that the whole E_{2} term of the spectral sequence is 0. QED

We now prove Theorem 7.1 for 𝔤_{Q_{p}} satisfying condition (1) in Lemma 7.2. Let 𝔲_{Q_{p}} be the kernel of 𝔤_{Q_{p}} → 𝔯_{Q_{p}}. Let 𝔯 be the image of the integral Lie algebra 𝔤 in 𝔯_{Q_{p}}, and let 𝔲 be the intersection of 𝔤 with 𝔲_{Q_{p}}. It follows from Lemma 7.3 that the integral Hochschild-Serre spectral sequence,

H_{∗}(𝔯, H_{∗}(𝔲, M)) ⇒ H_{∗}(𝔤, M),

has finite E_{2} term, and in particular that the coinvariants of 𝔯_{Q_{p}} on H_{j}(𝔲, M) ⊗ Q_{p} are 0 for all j. Since 𝔯_{Q_{p}} is reductive of rank at least 2, Theorem 5.1 says that the Euler characteristic χ(𝔯, H_{j}(𝔲, M)) is 0 for all j. Then the spectral sequence implies that χ(𝔤, M) = 0.

Cases (2) and (3) can be treated at the same time. In both cases, 𝔤_{Q_{p}} maps onto a reductive Lie algebra 𝔯_{Q_{p}} of rank 1 with some kernel 𝔲_{Q_{p}}, and there is an element x of 𝔤 whose image in 𝔯_{Q_{p}} spans a Cartan subalgebra 𝔥_{Q_{p}} and whose centralizer in 𝔲_{Q_{p}} is not 0. Let 𝔯 be the image of the integral Lie algebra 𝔤 in 𝔯_{Q_{p}} and let 𝔲 be the kernel of 𝔤 mapping to 𝔯_{Q_{p}}.

By the Hochschild-Serre spectral sequence for the extension of 𝔯 by 𝔲, we have

χ(𝔤, M) = ∑_{j}(−1)^{j}χ(𝔯, H_{j}(𝔲, M))

provided that the right-hand side makes sense. We are assuming that H_{∗}(𝔤_{Q_{p}}, M_{Q_{p}}) = 0. By Lemma 7.3, it follows that H_{∗}(𝔯_{Q_{p}}, H_{∗}(𝔲_{Q_{p}}, M_{Q_{p}})) = 0, which means that the right-hand side in the above formula does make sense. Moreover, by Proposition 2.3(1), for a finitely generated Z_{p}-module N with 𝔯-action such that H_{∗}(𝔯_{Q_{p}}, N_{Q_{p}}) = 0, the Euler characteristic χ(𝔯, N) only depends on N_{Q_{p}} as an 𝔯_{Q_{p}}-module (since 𝔯 has rank at least 1 as an o_{K}-module). In fact, it only depends on the class of N_{Q_{p}} in the Grothendieck group Rep_{≠1}(𝔯_{Q_{p}}) of 𝔯_{Q_{p}}-modules with all simple subquotients nontrivial. (This works even for 𝔯 of rank 1 as an o_{K}-module, so that Proposition 2.3(2) does not apply, because we are fixing 𝔯 ⊂ 𝔯_{Q_{p}} and only considering the dependence of these Euler characteristics on N.)

Thus we have a well-defined homomorphism

χ : Rep_{≠1}(𝔯_{Q_{p}}) → Z,

and the above formula for χ(𝔤, M) says that χ(𝔤, M) is the image of the alternating sum χ(𝔲_{Q_{p}}, M_{Q_{p}}), as an element of Rep_{≠1}(𝔯_{Q_{p}}), under this homomorphism. So Theorem 7.1 is proved if we can show that the element χ(𝔲_{Q_{p}}, M_{Q_{p}}) is 0 in the Grothendieck group Rep_{≠1}(𝔯_{Q_{p}}). Since this Grothendieck group injects into the usual Grothendieck group Rep(𝔯_{Q_{p}}), it suffices to show that χ(𝔲_{Q_{p}}, M_{Q_{p}}) is 0 in the latter group. The Grothendieck group of 𝔯_{Q_{p}} injects into that of the Cartan subalgebra 𝔥_{Q_{p}} ⊂ 𝔯_{Q_{p}} spanned by the given element x ∈ 𝔤_{Q_{p}}, so it suffices to show that χ(𝔲_{Q_{p}}, M_{Q_{p}}) is 0 in Rep(Q_{p}x).

But here we can use the standard complex that defines Lie algebra homology to see that

χ(𝔲_{Q_{p}}, M_{Q_{p}}) = (∑_{j}(−1)^{j} ∧^{j} 𝔲_{Q_{p}})M_{Q_{p}}

in the representation ring Rep(Q_{p}x). We are given that x has nonzero centralizer in 𝔲_{Q_{p}}, so 𝔲_{Q_{p}} is equal in the representation ring of Q_{p}x to 1+V for some representation V. The operation

∧_{−1}V := ∑_{j}(−1)^{j} ∧^{j} (V)

takes a representation V to an element of the corresponding Grothendieck group, transforming sums into products. Since ∧_{−1}1 = 0, it follows that ∧_{−1}𝔲_{Q_{p}} = 0 in Rep(Q_{p}x). Therefore χ(𝔲_{Q_{p}}, M_{Q_{p}}) = 0 in Rep(Q_{p}x), as we needed. QED Theorem 7.4 Let 𝔤 be a Lie algebra over Z_{p} which contains an element with centralizer of dimension 1. We will now compute the Euler characteristic χ(𝔤, M) whenever it is defined, that is, for all finitely generated Z_{p}-modules M with 𝔤-action such that H_{∗}(𝔤, M) ⊗ Q_{p} = 0. It is nonzero for some M.

Proof. The structure of 𝔤_{Q_{p}} is described by the following lemma.

Lemma 7.5 Let 𝔤 be a Lie algebra over a field K of characteristic zero which contains an element with centralizer of dimension 1. Then 𝔤 satisfies at least one of the following two properties.

(1) 𝔤 maps onto a 1-dimensional Lie algebra 𝔥 with some kernel 𝔲, and there is an element x ∈ 𝔤 whose image spans 𝔥 and whose centralizer in 𝔲 is 0.

(2) 𝔤 maps onto a semisimple Lie algebra 𝔯 of rank 1 with some kernel 𝔲, and there is an element x ∈ 𝔤 whose image spans a (1-dimensional) Cartan subalgebra 𝔥 in 𝔯 and whose centralizer in 𝔲 is 0.

Proof. Since the dimension of the centralizer is upper-semicontinuous in the Zariski topology on 𝔤, the general element of 𝔤 has centralizer of dimension 1.

If 𝔤 is solvable, then it maps onto a 1-dimensional Lie algebra 𝔥 with some kernel 𝔲. Let x be a general element of 𝔤 in the sense that the image of x spans 𝔥 and the centralizer of x in 𝔤 has dimension 1. Then the centralizer of x in 𝔲 is 0, proving statement (1).

Otherwise, 𝔤 maps onto some nonzero semisimple Lie algebra 𝔯. If 𝔯 has rank at least 2, then every element of 𝔯 has centralizer of dimension at least 2 in 𝔯. It follows that the linear endomorphism ad x of 𝔤 has rank at most dim(𝔤) − 2 for all x ∈ 𝔤. So every x has centralizer of dimension at least 2 in 𝔤, contrary to our assumption. So 𝔯 has rank 1. Let x be a general element of 𝔤 in the sense that the image of x spans a (1-dimensional) Cartan subalgebra 𝔥 of 𝔯 and the centralizer of x in 𝔤 has dimension 1. Then we have statement (2). QED

In fact, we need to strengthen Lemma 7.5 to say that, in both cases (1) and (2), any element of 𝔤 whose image spans 𝔥 has centralizer in 𝔲 equal to 0. This is a consequence of Lemma 7.6, as follows. (In case (2), we apply Lemma 7.6 with 𝔤 replaced by the inverse image of 𝔥 in 𝔤.)

Lemma 7.6 Let 𝔤 be a Lie algebra over a field K of characteristic zero which maps onto a 1-dimensional Lie algebra 𝔥 with some kernel 𝔲. If there is an element x of 𝔤 whose image spans 𝔥 and whose centralizer in 𝔲 is 0, then every element y of 𝔤 whose image spans 𝔥 has centralizer in 𝔲 equal to 0. Moreover, for any such element y, the element ∧_{−1}𝔲 = ∑_{i}(−1)^{i} ∧^{i} 𝔲 in the Grothendieck group Rep(Ky) is not 0.

Proof. We first need the following elementary lemma.

Lemma 7.7 Let a_{1}, …, a_{n} be elements of a field K of characteristic zero. Let S_{even} be the set of sums ∑_{i∈I} a_{i} ∈ K for subsets I ⊂ {1, …, n} of even order, and let S_{odd} be the analogous set of odd sums, both sets being considered with multiplicities. Then S_{even} = S_{odd} if and only if a_{i} = 0 for some i.

Proof. If a_{i} = 0 for some i, then the bijection from the set of even subsets of {1, …, n} to the set of odd subsets by adding or removing the element i does not change the corresponding sum of a_{j}’s. To prove the converse, we use the following identity of formal power series:

(e^{x_{1}} − 1) ⋯ (e^{x_{n}} − 1) = (x_{1} + ⋯) ⋯ (x_{n} + ⋯)
= x_{1} ⋯ x_{n} + terms of higher degree.

We can also write

(e^{x_{1}} − 1) ⋯ (e^{x_{n}} − 1) = ∑_{j=0}^{n}(−1)^{n−j} ∑_{1≤i_{1}<···<i_{j}≤n} e^{x_{i_{1}}+···+x_{i_{j}}} .

Equating terms in degree n, we find that

∑_{j=0}^{n}(−1)^{n−j} ∑_{1≤i_{1}<···<i_{j}≤n}(x_{i_{1}} + ⋯ + x_{i_{j}})^{n} = n!x_{1} ⋯ x_{n}.

This is now an identity of polynomials with integer coefficients. Plugging in the values a_{1}, …, a_{n} ∈ K, we find that the left-hand side is 0, since the set (with multiplicities) of sums of an even number of the a_{i}’s is equal to the corresponding set of sums for an odd number of the a_{i}’s. So the right-hand side is 0. Since K has characteristic zero, n! is not 0 in K, and so one of the a_{i}’s is 0. QED (Lemma 7.7)

We can now prove Lemma 7.6. Since we have an element x of 𝔤 whose image spans 𝔥 and whose centralizer in 𝔲 is 0, the eigenvalues of x on 𝔲 (in a suitable extension field of K) are all nonzero. By Lemma 7.7, the set with multiplicities of even sums of the eigenvalues of x on 𝔲 is not equal to the set of odd sums. Equivalently, ∧_{−1}𝔲 is not zero in the Grothendieck group Rep(Kx).

But the complex computing Lie algebra homology shows that the element χ(𝔲) := ∑_{i}(−1)^{i}H_{i}(𝔲, K) in Rep(𝔥) can be identified with ∧_{−1}𝔲 in Rep(Kx). The point is that the exact sequence of Lie algebras 0 → 𝔲 → 𝔤 → 𝔥 → 0 determines an action of 𝔥 on the homology of 𝔲, and hence an element χ(𝔲) of Rep(𝔥), whereas we need to choose an element x giving a splitting of the exact sequence in order to get an action of Kx ≅ 𝔥 on 𝔲 itself and hence to define ∧_{−1}𝔲 in Rep(Kx). Since ∧_{−1}𝔲 is nonzero in Rep(Kx), the element χ(𝔲) is nonzero in Rep(𝔥). So ∧_{−1}𝔲 is nonzero in Rep(Ky) for any element y of 𝔤 whose image spans 𝔥. By the easy direction of Lemma 7.7, it follows that the eigenvalues of y on 𝔲 are all nonzero. Equivalently, the centralizer of y in 𝔲 is 0. QED (Lemma 7.6)

We return to the proof of Theorem 7.4. If 𝔤 has rank 1 as a Z_{p}-module, then the theorem follows from Proposition 6.1, so we can assume that 𝔤 has rank at least 2 as a Z_{p}-module. Let 𝔯_{Q_{p}} = 𝔥_{Q_{p}} in case (1). Then, in both cases (1) and (2) of Lemma 7.5, let 𝔯 be the image of the integral Lie algebra 𝔤 in the reductive quotient 𝔯_{Q_{p}} and let 𝔲 be the kernel of 𝔤 mapping to 𝔯_{Q_{p}}.

Given a finitely generated Z_{p}-module M with 𝔤-action such that H_{∗}(𝔤, M) ⊗ Q_{p} = 0, Lemma 7.3 shows that H_{∗}(𝔯, H_{∗}(𝔲, M)) ⊗ Q_{p} = 0 and in particular that the coinvariants of 𝔯_{Q_{p}} on H_{∗}(𝔲, M) ⊗ Q_{p} are 0. Therefore the Euler characteristic χ(𝔤, M) is given by the formula

χ(𝔤, M) = ∑_{j}(−1)^{j}χ(𝔯, H_{j}(𝔲, M)).

As in the proof of Theorem 7.1, let Rep_{≠1}(𝔯_{Q_{p}}) denote the Grothendieck group of 𝔯_{Q_{p}}-modules with all simple subquotients nontrivial. Then the above formula says that χ(𝔤, M) is the image of the alternating sum χ(𝔲_{Q_{p}}, M_{Q_{p}}), as an element of Rep_{≠1}(𝔯_{Q_{p}}), under a homomorphism

χ : Rep_{≠1}(𝔯_{Q_{p}}) → Z.

Now the Lie algebra 𝔯_{Q_{p}} is either 1-dimensional or else semisimple of rank 1, so we have computed the homomorphism χ : Rep_{≠1}(𝔯_{Q_{p}}) → Z in Propositions 6.1 and 6.2.

Thus, to complete the calculation of χ(𝔤, M), it suffices to compute the element χ(𝔲_{Q_{p}}, M_{Q_{p}}) in Rep_{≠1}(𝔯_{Q_{p}}). As in the proof of Theorem 7.1, this Grothendieck group injects into the usual Grothendieck group Rep(𝔯_{Q_{p}}), so it suffices to compute χ(𝔲_{Q_{p}}, M_{Q_{p}}) as an element of the latter group. We can choose a splitting of the Lie algebra extension

0 → 𝔲_{Q_{p}} → 𝔤_{Q_{p}} → 𝔯_{Q_{p}} → 0,

since 𝔯_{Q_{p}} is either 1-dimensional or semisimple. Given such a splitting, 𝔯_{Q_{p}} acts on 𝔲_{Q_{p}}. We can then compute the element χ(𝔲_{Q_{p}}, M_{Q_{p}}) in the Grothendieck group Rep(𝔯_{Q_{p}}) using the definition of Lie algebra homology via the standard complex:

χ(𝔲_{Q_{p}}, M_{Q_{p}}) = (∑_{j}(−1)^{j} ∧^{j} 𝔲_{Q_{p}})M_{Q_{p}}
= (∧_{−1}u_{Q_{p}})M_{Q_{p}}

in the representation ring Rep(𝔯_{Q_{p}}). In particular, we see that χ(𝔲_{Q_{p}}, M_{Q_{p}}) and hence χ(𝔤, M) only depend on 𝔲_{Q_{p}} and M_{Q_{p}} as 𝔯_{Q_{p}}-modules.

We regard this as a calculation of χ(𝔤, M). To complete the proof of Theorem 7.4, we need to show that there is some 𝔤-module M, finitely generated over Z_{p}, such that χ(𝔤, M) is defined but not equal to 0. We know that these properties only depend on M_{Q_{p}} as a 𝔤_{Q_{p}}-module. We will take M to be a representation of the quotient Lie algebra 𝔯, in the notation we have been using, so that 𝔯_{Q_{p}} is either 1-dimensional or else semisimple of rank 1. It is enough to find a representation M_{o_{K}} of 𝔯_{o_{K}} for some finite extension K of Q_{p} such that χ(𝔤_{o_{K}}, M_{o_{K}}) is not zero. Indeed, we can then view M_{o_{K}} as a representation of 𝔤 over Z_{p}. We have

H_{∗}(𝔤, M_{o_{K}}) = H_{∗}(𝔤_{o_{K}}, M_{o_{K}})

by inspection of the standard complex defining Lie algebra homology, and so χ(𝔤, M_{o_{K}}) is not zero, giving a representation of 𝔤 over Z_{p} with nonzero Euler characteristic as we want.

We first consider case (1) of Lemma 7.5, where 𝔯_{Q_{p}} has dimension 1. In this case, I claim that there is an 𝔯_{o_{K}}-module M_{o_{K}} which is free of rank 1 over o_{K}, for some finite extension K of Q_{p}, such that χ(𝔤_{o_{K}}, M_{o_{K}}) is defined and not 0; that will prove the theorem in this case. Let x be an element of 𝔤 which maps to a generator of 𝔯 = im(𝔤 → 𝔯_{Q_{p}}), which is isomorphic to Z_{p}. For a finite extension K of Q_{p}, an 𝔯_{o_{K}}-module of rank 1 is defined by an element b ∈ o_{K}, which gives the action of the generator x.

By the general description of how to compute χ(𝔤_{o_{K}}, M_{o_{K}}) which we have given, we have

χ(𝔤_{o_{K}}, M_{o_{K}}) = ∑_{j}(−1)^{j}χ(𝔯_{o_{K}}, ∧^{j}𝔲_{o_{K}} ⊗ M_{o_{K}})

= ∑_{j}(−1)^{j}ord_{p} det(x| ∧^{j} 𝔲_{K} ⊗ M_{K}),

provided that x is invertible on ∧^{j}𝔲_{K} ⊗ M_{K} for all j. Here we are using Proposition 6.1. We choose the p-adic field K to be any one which contains the eigenvalues a_{1}, …, a_{n} of x on 𝔲; these are all in the ring of integers o_{K}, because 𝔲 is a finitely generated Z_{p}-module. Then the eigenvalues of x on ∧^{j}𝔲_{K} ⊗ M_{K} are the numbers

b + a_{i_{1}} + ⋯ + a_{i_{j}},

for 1 ≤ i_{1} < ⋯ < i_{j} ≤ n. In particular, these eigenvalues are all nonzero, for all 0 ≤ j ≤ n, if we choose b outside finitely many values, as we now decide to do. Then x acts invertibly on ∧^{j}𝔲_{K} ⊗ M_{K}, and so the above formula for χ(𝔤_{o_{K}}, M_{o_{K}}) is justified.

The above formula then gives, more explicitly:

χ(𝔤_{o_{K}}, M_{o_{K}}) = ord_{p} ∏_{j=0}^{n} ∏_{1≤i_{1}<···<i_{j}≤n}(b + a_{i_{1}} + ⋯ + a_{i_{j}})^{(−1)^{j}}.

Let f(b) be the rational function of b in this formula, whose p-adic order is χ(𝔤_{o_{K}}, M_{o_{K}}). By Lemmas 7.5 and 7.6, we know that ∧_{−1}𝔲 is not zero in the Grothendieck group Rep(Q_{p}x). Equivalently, the set with multiplicities of even sums of a_{1}, …, a_{n} is not equal to the set of odd sums, and so the rational function f(b) is not constant. The zeros and poles of this rational function are in the ring of integers o_{K}. Taking a b ∈ o_{K} which is close but not equal to one of these zeros or poles, we can arrange that f(b) is not a p-adic unit. That is, for the rank-1 𝔤_{o_{K}}-module M associated to b, χ(𝔤_{o_{K}}, M_{o_{K}}) is not 0. As mentioned earlier, it follows that χ(𝔤, M_{o_{K}}) is not 0, where M_{o_{K}} is viewed as a Z_{p}-module. Theorem 7.4 is proved in case (1) of Lemma 7.5.

We now prove Theorem 7.4 in case (2) of Lemma 7.5. Here 𝔤_{Q_{p}} is an extension of a semisimple Lie algebra 𝔯_{Q_{p}} of rank 1 by another Lie algebra 𝔲_{Q_{p}}. As mentioned earlier, we can fix a splitting of this extension, and then 𝔯_{Q_{p}} acts on 𝔲_{Q_{p}}. Let 𝔥_{Q_{p}} be the Cartan subalgebra given by case (2) of Lemma 7.5. No matter which splitting of the extension we have chosen, Lemma 7.6 shows that ∧_{−1}𝔲_{Q_{p}} is nonzero in the Grothendieck group of 𝔥_{Q_{p}}-modules. A fortiori, it is nonzero in the Grothendieck group of 𝔯_{Q_{p}}-modules.

We want to find a 𝔤-module M, finitely generated over Z_{p}, such that χ(𝔤, M) is defined and not 0. Let K be a finite extension of Q_{p} such that 𝔯_{K} is isomorphic to 𝔰𝔩_{2}K. It suffices to find a 𝔤_{o_{K}}-module M_{o_{K}}, finitely generated over o_{K}, such that χ(𝔤_{o_{K}}, M_{o_{K}}) is defined and not 0, in view of the isomorphism H_{∗}(𝔤, M_{o_{K}}) = H_{∗}(𝔤_{o_{K}}, M_{o_{K}}). We will take M_{o_{K}} to be a module over the quotient Lie algebra 𝔯_{o_{K}} (the image of 𝔤_{o_{K}} in 𝔯_{K}). It suffices to find an 𝔯_{o_{K}}-module M, finitely generated over o_{K}, such that (∧^{i}𝔲_{K})M_{K} has no trivial summands as an 𝔯_{K}-module for all 0 ≤ i ≤ n and (∧_{−1}𝔲_{K})M_{K} has nonzero image under the homomorphism

χ : Rep_{≠1}(𝔯_{K}) → Z.

It is somewhat difficult to construct 𝔯_{o_{K}}-modules, even though we know that 𝔯_{K} is isomorphic to 𝔰𝔩_{2}K. (For example, let H, X, Y be the usual basis vectors for 𝔰𝔩_{2}Q_{2}. Then the integral form of the Lie algebra 𝔰𝔩_{2}Q_{2} which is spanned by H/2, X, and Y has no action on (Z_{2})^{2} which gives the standard representation of 𝔰𝔩_{2}Q_{2} after tensoring with Q_{2}.) The obvious example of an 𝔯_{o_{K}}-module is the adjoint representation 𝔯_{o_{K}}; by taking symmetric powers of 𝔯_{o_{K}} and decomposing over K, we find that for every m ≥ 0, there is an 𝔯_{o_{K}}-module M, finitely generated over o_{K}, such that M_{K} is a simple module over 𝔯_{K} ≅ 𝔰𝔩_{2}K of highest weight 2m. We can get more if the 𝔯_{K}-module 𝔲_{K} has a summand with odd highest weight, since we know that 𝔲_{K} comes from an 𝔯_{o_{K}}-module 𝔲_{o_{K}} which is finitely generated over o_{K}. In that case, by tensoring 𝔲_{o_{K}} repeatedly with 𝔯_{o_{K}} and decomposing over K (using the Clebsch-Gordan formula, as stated below), we find that every simple 𝔯_{K}-module comes from an 𝔯_{o_{K}}-module which is finitely generated over o_{K}. To sum up, let c be 1 if the 𝔯_{K}-module 𝔲_{K} has a summand with odd highest weight, and 2 otherwise; then we have shown that for every m ≥ 0 there is an 𝔯_{o_{K}}-module M_{o_{K}}, finitely generated over o_{K}, such that M_{K} is a simple module over 𝔯_{K} ≅ 𝔰𝔩_{2}K of highest weight cm.

We have thereby reduced to the following question. Let V be the standard 2dimensional representation of 𝔰𝔩_{2}K. Given that ∧_{−1}𝔲_{K} = ∑_{i}(−1)^{i} ∧^{i} 𝔲_{K} is nonzero in the representation ring Rep(𝔰𝔩_{2}K), find an integer m ≥ 0 such that (∧^{i}𝔲_{K})S^{cm}V has no trivial summands for 0 ≤ i ≤ dim 𝔲_{K} and (∧_{−1}𝔲_{K})S^{cm}V has nonzero image under the homomorphism

χ : Rep_{≠1}(𝔰𝔩_{2}K) → Z.

The Clebsch-Gordan formula for representations of 𝔰𝔩_{2}K says that

S^{a}V · S^{b}V = S^{a−b}V + S^{a−b+2}V + ⋯ + S^{a+b}V

for 0 ≤ b ≤ a. This makes it clear that for 0 ≤ i ≤ dim 𝔲_{K}, (∧^{i}𝔲_{K})S^{cm}V has no trivial summands for m sufficiently large. Now let j be the largest natural number such that the multiplicity of S^{j}V in ∧_{−1}𝔲_{K} is not zero; there is such a j, since ∧_{−1}𝔲_{K} is not 0. If we can choose m such that j +cm+2 is divisible by a sufficiently large power of p, then (∧_{−1}𝔲_{K})S^{cm}V has nonzero image under the homomorphism

χ : Rep_{≠1}(𝔰𝔩_{2}K) → Z,

by the Clebsch-Gordan formula together with the formula for that homomorphism in Proposition 6.2:

χ(S^{a}V) = 2(ord_{p}a − ord_{p}(a + 2)).

There is no trouble choosing such an m if p is odd. If p = 2, we can do it unless j is odd and c = 2. But that cannot happen, since c = 2 means that the highest weights of 𝔯_{K} ≅ 𝔰𝔩_{2}K on 𝔲_{K} are all even, which would imply that the weights j occurring in ∧_{−1}𝔲_{K} were also even. So we can always find an m as needed, proving Theorem 7.4. QED

8 Filtered and graded algebras

In section 9, we will explain how to relate Euler characteristics for Lie algebras over the p-adic integers to Euler characteristics for p-adic Lie groups. In this section we develop the homological algebra needed for that proof. In particular, we need the spectral sequence defined under various hypotheses by Serre ([26], p. II-17) and May [24], relating Tor over a filtered ring to Tor over the associated graded ring. We set up the spectral sequence here under fairly weak hypotheses. We also need a relative version of that spectral sequence.

We begin with some general homological definitions. For any ring S, we have the groups Tor_{j}^{S}(A, B) for any right S-module A and left S-module B. Given a ring homomorphism R → S, we can view A and B as R-modules as well and consider the resulting Tor groups. Our first step is to define relative groups Tor_{j}^{S,R}(A, B) in this situation which fit into a long exact sequence

Tor_{j}^{R}(A, B) → Tor_{j}^{S}(A, B) → Tor_{j}^{S,R}(A, B) → Tor_{j−1}^{R}(A, B).

To do this, let R_{∗} be a free resolution of A as a right R-module, and let S_{∗} be a free resolution of A as a right S-module. Since R_{∗} is a complex of projective Rmodules, there is an R-linear homomorphism of chain complexes R_{∗} → S_{∗}, unique up to homotopy, which gives the identity map from H_{0}(R_{∗}) = A to H_{0}(S_{∗}) = A. This homomorphism determines a Z-linear homomorphism of chain complexes from R_{∗} ⊗_{R} B to S_{∗} ⊗_{S} B. We define Tor_{∗}^{S,R}(A, B) to be the homology of the mapping cone of the map of chain complexes R_{∗}⊗_{R}B → S_{∗}⊗_{S} B. These groups Tor_{∗}^{S,R}(A, B) fit into a long exact sequence as we wanted.

We now turn to the spectral sequence which relates Tor over filtered rings to Tor over the associated graded ring. We will need to modify this spectral sequence to apply to the above relative Tor groups. The proof below is essentially Serre’s argument in [26], p. II-17.

Proposition 8.1 Let Ω be a complete filtered commutative ring, Ω = Ω^{0} ⊃ Ω^{1} ⊃ ⋯, with gr Ω noetherian. Let R be a complete filtered Ω-algebra, R = R^{0} ⊃ R^{1} ⊃ ⋯, with gr R right noetherian. Let A be a complete filtered right R-module with gr A finitely generated over gr R, and let B be a complete filtered left R-module with gr B finitely generated as a gr Ω-module (not just as a gr R-module). Then there is a spectral sequence

E_{ij}^{1} = Tor_{i+j}^{gr R}(gr A, gr B)_{degree −i} ⇒ Tor_{i+j}^{R}(A, B).

This is a homological spectral sequence, meaning that the differential d_{r} has bidegree (−r, r − 1) for r ≥ 1. The groups Tor_{∗}^{R}(A, B) are finitely generated Ω-modules, complete with respect to a filtration whose associated graded groups are the E_{∞} term of the spectral sequence.

In our applications, Ω will be Z_{p}, and gr A and gr B will both be finitely generated gr Ω-modules. We state Proposition 8.1 under the above weaker (asymmetrical) assumptions only because the proof happens to work that way.

Proof. Since gr R is noetherian and gr A is finitely generated over gr R, there is a resolution G_{∗} of the graded module gr A over gr R by finitely generated free graded modules. We will use completeness of R and A to lift G_{∗} to a filtered free resolution R_{∗} of the filtered module A over R. Serre’s Lemma V.2.1.1, p. 545 in [23], is closely related, but we will prove what we need directly. For each i ≥ 0, let R_{i} be a filtered free R-module with generators in degrees so that gr R_{i} ≅ G_{i}. The surjection G_{0} → gr A lifts to a filtered R-linear map R_{0} → A by freeness of R_{0}. It is surjective, by the following lemma.

Lemma 8.2 Let f : A → B be a homomorphism of filtered abelian groups, A = A^{0} ⊃ A^{1} ⊃ ⋯ and B = B^{0} ⊃ B^{1} ⊃ · · ·. Suppose that A is complete (A → \underleftarrow{lim} A/A^{n} is an isomorphism), B is separated (B → \underleftarrow{lim} B/B^{n} is injective) and gr A → gr B is surjective. Then A → B is surjective and B is complete.

Proof (repeated from [23], Prop. I.2.3.13, p. 415). For any b ∈ \underleftarrow{lim} B/B^{n}, we can use surjectivity of gr A → gr B to define an element a ∈ \underleftarrow{lim} A/A^{n}, step by step, which maps to b in \underleftarrow{lim} B/B^{n}. That is, \underleftarrow{lim} A/A^{n} → \underleftarrow{lim} B/B^{n} is surjective. Since A → \underleftarrow{lim} A/A^{n} is an isomorphism, the map A → \underleftarrow{lim} B/B^{n} is surjective. Therefore B → \underleftarrow{lim} B/B^{n} is surjective as well as injective, so B is complete. It also follows that A → B is surjective. QED

We continue the proof of Proposition 8.1. By Lemma 8.2, the lift R_{0} → A is surjective. Suppose, inductively, that we have defined an exact sequence of filtered R-modules

R_{i} → ⋯ → R_{0} → A → 0

which lifts the exact sequence

G_{i} → ⋯ → G_{0} → gr A → 0.

Let K_{j} = ker(R_{j} → R_{j−1}) for 0 ≤ j ≤ i, with its filtration as a submodule of R_{j}, and let K_{−1} = A with its given filtration. Then the natural map gr R_{i} → gr K_{i−1} is surjective; this is clear for i = 0, and for i > 0 it follows from injectivity of the map gr K_{i−1} → gr R_{i−1} and surjectivity of the map

gr R_{i} → ker(gr R_{i−1} → gr R_{i−2}).

We have an exact sequence of filtered R-modules,

0 → K_{i} → R_{i} → K_{i−1} → 0.

Here K_{i} has the filtration induced from R_{i} by definition. Moreover, surjectivity of gr R_{i} → gr K_{i−1} implies that the filtration of K_{i−1} is also the one induced from R_{i}, that is, that (R_{i})^{j} → (K_{i−1})^{j} is surjective for all j; use Lemma 8.2 to prove this, noting that R_{i} is complete since it is a finitely generated free filtered R-module. It follows that the sequence

0 → gr K_{i} → gr R_{i} → gr K_{i−1} → 0

is exact. Since gr K_{i−1} ⊂ gr R_{i−1} by definition of the filtration on K_{i−1}, it follows that gr K_{i} = ker(gr R_{i} → gr R_{i−1}). So we have a surjection gr R_{i+1} → gr K_{i}, which we can lift to a filtered R-linear map R_{i+1} → K_{i}. This map is surjective by Lemma 8.2. So we have an exact sequence

R_{i+1} → R_{i} → ⋯ → R_{0} → A → 0

of filtered R-modules, lifting the exact sequence

G_{i+1} → G_{i} → ⋯ → G_{0} → gr A → 0.

This completes the induction. Thus, we have shown that G_{∗} lifts to a filtered free resolution R_{∗} of the complete filtered right R-module A.

In Proposition 8.1, we are also given a complete filtered left R-module B with gr B finitely generated as a gr Ω-module. Then R_{∗} ⊗_{R} B is a filtered complex of Ω-modules, with homology equal to Tor_{∗}^{R}(A, B). Its associated graded complex is G_{∗} ⊗_{gr R} gr B, which has homology equal to Tor_{∗}^{gr R}(gr A, gr B). We define the spectral sequence of Proposition 8.1 to be the spectral sequence associated to the filtered complex R_{∗}⊗_{R}B. The strong assumption on B is used to guarantee the convergence of the spectral sequence of this filtered complex, via the following lemma. QED (Proposition 8.1)

Lemma 8.3 Let Ω = Ω^{0} ⊃ Ω^{1} ⊃ ⋯ be a complete filtered ring with gr Ω noetherian, M_{∗} a homological complex (meaning that d has degree −1) of complete filtered Ω-modules with gr M_{j} finitely generated over gr Ω for each j ∈ Z. Then the spectral sequence of this filtered complex converges:

E_{ij}^{0} = gr^{−i}M_{i+j} ⇒ H_{i+j}M.

This is a homological spectral sequence, meaning that the differential d_{r} has bidegree (−r, r − 1) for r ≥ 0. The groups H_{k}M are finitely generated Ω-modules, complete with respect to a filtration whose associated graded groups are the E_{∞} term of the spectral sequence.

Proof. We refer to Cartan-Eilenberg [11], Chapter XV, as a reference for the spectral sequence of a filtered complex, although the gradings there (for a cohomological complex) are the negatives of ours. For each i, j ∈ Z, we have subgroups

0 ⊂ B_{ij}^{1} ⊂ B_{ij}^{2} ⊂ ⋯ ⊂ Z_{ij}^{2} ⊂ Z_{ij}^{1} ⊂ E_{ij}^{0} = gr^{−i}M_{i+j},

with E_{ij}^{r} = Z_{ij}^{r}/B_{ij}^{r}. Explicitly,

Z_{ij}^{r} = im({x ∈ M_{i+j}^{−i} : dx ∈ M_{i+j−1}^{−i+r}} → gr^{−i}M_{i+j})
B_{ij}^{r} = im({dx ∈ M_{i+j}^{−i} : x ∈ M_{i+j+1}^{−i−r+1}} → gr^{−i}M_{i+j}).

Moreover, for each k ∈ Z, E_{∗,k−∗}^{0} = gr M_{k} is a finitely generated module over gr Ω, and the B^{r}’s and Z^{r}’s are all submodules. Since gr Ω is noetherian, the increasing sequence of submodules

B_{∗,k−∗}^{1} ⊂ B_{∗,k−∗}^{2} ⊂ ⋯ ⊂ gr M_{k}

eventually terminates. That is, all differentials into total degree k are 0 after the rth term of the spectral sequence, for some r < ∞ depending on k. By the same statement for k − 1, it follows that all differentials out of total degree k are also 0 after some point. So there is an r = r(k) < ∞ such that E_{∗,k−∗}^{r} = E_{∗,k−∗}^{∞}.

Under the weaker assumption that for each i, j ∈ Z there is an r such that all differentials starting with d_{r} are zero on E_{ij}^{r}, together with completeness of the M_{k}’s, Boardman shows that the filtration induced by each group M_{k} on its subquotient H_{k}M is complete, with associated graded groups equal to the E_{∞} term of the spectral sequence, in [1], Theorem 7.1, the remark after it, and Theorem 9.2. Since gr H_{k}M is a subquotient of H_{k}M for each k and gr Ω is noetherian, gr H_{k}M is a finitely generated gr Ω-module. It follows that H_{k}M is a finitely generated Ω-module. QED

Now we set up the relative version of the above spectral sequence, the last general homological result we need here. Let Ω, R, A, B be as in Proposition 8.1. Suppose that we also have a homomorphism R → S of complete filtered Ω-algebras such that gr S is noetherian and A and B are S-modules.

Proposition 8.4 There is a spectral sequence

E_{ij}^{1} = Tor_{i+j}^{gr S,gr R}(gr A, gr B)_{degree −i} ⇒ Tor_{i+j}^{S,R}(A, B).

Here the groups Tor_{∗}^{S,R}(A, B) are finitely generated Ω-modules, complete with respect to a filtration whose associated graded groups are the E_{∞} term of the spectral sequence.

Proof. Start with a graded finitely generated free resolution G_{∗} of gr A as a gr R-module and a graded finitely generated free resolution H_{∗} of gr A as a gr Smodule. As in the definition of relative Tor groups, above, there is a graded gr Rlinear homomorphism G_{∗} → H_{∗}, unique up to homotopy, which gives the identity map from H_{0}(G_{∗}) = gr A to H_{0}(H_{∗}) = gr A.

As in the construction of this spectral sequence for a single ring (Proposition 8.1), we can lift G_{∗} to a filtered free resolution R_{∗} of A as an R-module and H_{∗} to a filtered free resolution S_{∗} of A as an S-module. The new point here, using completeness again, is that the homomorphism G_{∗} → H_{∗} of complexes of graded gr R-modules lifts to a homomorphism R_{∗} → S_{∗} of complexes of filtered R-modules. (We can argue as in the proof of Proposition 8.1, or we can just refer to [23], Lemma V.2.1.5, p. 548.) Then Tor_{∗}^{S,R}(A, B) is defined as the homology of the mapping cone of the map of chain complexes R_{∗} ⊗_{R} B → S_{∗} ⊗_{S} B. This mapping cone is now a filtered complex, with associated graded complex being the mapping cone of the map of chain complexes G_{∗} → H_{∗}. The homology of the latter mapping cone is therefore Tor_{∗}^{gr S,gr R}(gr A, gr B), and the spectral sequence we want is the usual spectral sequence of a filtered complex. It converges in the required sense by Lemma 8.3. QED

9 Relating groups and Lie algebras

We now explain how the results so far about Euler characteristics for Lie algebras over the p-adic integers imply analogous results for a large class of p-adic Lie groups, what Lazard called p-valued groups. For example, the group GL_{n}Z_{p} is not of this type, but any closed subgroup of the congruence subgroup ker(GL_{n}Z_{p} → GL_{n}(Z/p)) for p odd, or of ker(GL_{n}Z_{2} → GL_{n}(Z/4)) for p = 2, is p-valued. Groups of this type are in particular torsion-free pro-p groups.

For completeness, we recall Lazard’s definition of p-valued groups. First ([23], p. 428), define a filtration ω of a group G to be a function

ω : G → (0, ∞]

such that, for x, y ∈ G,

ω(xy^{−1}) ≥ min(ω(x), ω(y))
ω(x^{−1}y^{−1}xy) ≥ ω(x) + ω(y).

It follows in particular that G_{ν} := {x ∈ G : ω(x) ≥ ν} and G_{ν+} := {x ∈ G : ω(x) > ν} are normal subgroups of G. A filtered group G is said to be complete if G = \underleftarrow{lim} G/G_{ν}. For a fixed prime number p, a filtration ω of a group G is called a valuation (and G is called p-valued) if

ω(x) < ∞ for all x ≠ 1 in G
ω(x) > (p − 1)^{−1}
ω(x^{p}) = ω(x) + 1

for x ∈ G ([23], p. 465). Then gr G := ⊕G_{ν}/G_{ν+} is a Lie algebra over the graded ring Γ := gr Z_{p} = F_{p}[π] with π in degree 1 ([23], pp. 464–465). The action of π on gr G corresponds to taking the pth power of an element of G. The Lie algebra gr G is torsion-free, hence free, as a Γ-module. The dimension of a p-valued group G is defined to be the rank of the free Γ-module gr G. In this paper, p-valued groups will be assumed to be complete and of finite dimension. Such a group is automatically a p-adic Lie group ([23], Theorem III.3.1.7, p. 478).

Let v : Z_{p} → [0, ∞] be the standard valuation, which we sometimes call ord_{p}, so that v(p) = 1. By definition, a valuation on a Z_{p}-module M is a function w from M to [0, ∞] such that

w(x) < ∞ for all x ≠ 0 in M
w(x − y) ≥ min(w(x), w(y))
w(ax) = v(a) + w(x)

for a ∈ Z_{p} and x, y ∈ M ([23], Def. I.2.2.2, p. 409). We define a valuation on a Q_{p}-vector space V to be a function w from M to (−∞, ∞] which satisfies the same three properties; this definition generalizes to vector spaces over any p-adic field K using the standard valuation v = ord_{p} on K. A valuation on a Z_{p}-module M extends to a valuation on the vector space M ⊗_{Z_{p}} Q_{p} in a natural way, and we define

div M = {x ∈ M ⊗_{Z_{p}} Q_{p} : w(x) ≥ 0}.

Let Sat M be the completion of div M with respect to the filtration w. A valued Z_{p}-module M is called saturated if the natural homomorphism M → Sat M is an isomorphism. For a p-valued group G with valuation ω, we say that an action of G on a valued Z_{p}-module or Q_{p}-vector space M is compatible with the valuations if

w((g − 1)x) ≥ ω(g) + w(x)

for g ∈ G and x ∈ M.

Here is the basic theorem. Corollary 9.3 gives the main applications of this statement.

Theorem 9.1 Let G be a p-valued group. Suppose that the given valuation of G takes rational values, and that G has dimension at least 2. Let M be a finitely generated free Z_{p}-module with G-action. Suppose that M admits a valuation with rational values, compatible with the valuation of G, and that M is saturated for this valuation. The Lie algebra 𝔤_{Q_{p}} of G over Q_{p} acts on M ⊗ Q_{p}; let 𝔤 be any Lie algebra over Z_{p} such that 𝔤 ⊗ Q_{p} = 𝔤_{Q_{p}} and such that 𝔤 acts on M. Then the homology groups H_{∗}(G, M) are finite in all degrees if and only if the groups H_{∗}(𝔤, M) are finite in all degrees, and if this is so, then

χ(G, M) = χ(𝔤, M).

The proof relies on Proposition 2.3, which says that for a Lie algebra over a discrete valuation ring Γ whose rank as a Γ-module is at least 2, the Euler characteristics we are considering do not change upon passage from one Lie algebra to an open Lie subalgebra. The idea here is to think of both G and the Lie algebra 𝔤 as “subgroups of finite index” in the same thing, a ring which Lazard calls the saturation of the group ring of G. We relate Tor groups over this ring to Tor groups over its associated graded ring, which is essentially the universal enveloping algebra of a Lie algebra over the polynomial ring Γ = F_{p}[π]. Once we reduce to a question about such Lie algebras, we can apply Proposition 2.3, since Γ is a discrete valuation ring as a graded ring.

Proof. Let G be a p-valued group. Let Z_{p}G denote the completed group ring of G,

Z_{p}G := \underleftarrow{lim} Z_{p}[G/U]

where U runs over the open normal subgroups of G. (Lazard uses the name Al G for this ring.) Then the given valuation of G determines a complete filtration of the ring Z_{p}G which is also a valuation on Z_{p}G as a Z_{p}-module. The associated graded ring of Z_{p}G is the universal enveloping algebra of the Lie algebra gr G over the graded ring Γ := gr Z_{p} = F_{p}[π], by Theorem III.2.3.3, p. 471, in [23]. Explicitly, Lazard first defines a filtration on the naive group ring Z_{p}[G] as the infimum w of all filtrations of Z_{p}[G] as a Z_{p}-algebra which satisfy

w(g − 1) ≥ ω(g)

for all g ∈ G. He then identifies the completed group ring Z_{p}G in the sense defined above with the completion of Z_{p}[G] with respect to this filtration.

If M is a finitely generated free Z_{p}-module with G-action and with a valuation compatible with that on G, then M is a filtered Z_{p}G-module. To check this, observe that M induces a filtration w_{M} on the naive group ring Z_{p}[G] by

w_{M}(f) = inf_{x∈M−{0}} [w(f(x)) − w(x)] .

Then the above filtration w on Z_{p}[G] clearly satisfies w ≤ w_{M}, which says exactly that M is a filtered Z_{p}[G]-module. If M is a finitely generated free Z_{p}-module with G-action and a valuation compatible with that on G, then M is complete for its filtration and hence is a filtered module over the completed ring Z_{p}G.

Under the assumptions of Theorem 9.1, gr G and gr M are finitely generated free Γ-modules with all degrees rational. So they are concentrated in degrees (1/e)Z for some positive integer e.

The spectral sequence of Proposition 8.1, applied to the Z_{p}G-modules Z_{p} and M, has the form

Tor_{∗}^{Ugr G}(Γ, gr M) ⇒ Tor_{∗}^{Z_{p}G}(Z_{p}, M).

In that proposition, we assumed that the rings and modules were filtered by the integers, but we can apply the proposition to filtrations in (1/e)Z, as here, by rescaling the filtrations. By Brumer ([9], Lemma 4.2, p. 455, and Remark 1, p. 452), the homology of a compact p-adic Lie group G with coefficients in a pseudocompact Z_{p}G-module M is equal to Tor_{∗}^{Z_{p}G}(Z_{p}, M), since Z_{p}G is noetherian by [23], Prop. V.2.2.4, p. 550. So the spectral sequence can be rewritten as:

H_{∗}(gr G, gr M) ⇒ H_{∗}(G, M).

Here the initial term is the homology of gr G as a Lie algebra over Γ. Under our assumptions, gr G and gr M are finitely generated free Γ-modules. This spectral sequence also appears in the paper by Symonds and Weigel [31] in the case of F_{p}G-modules M.

We use the spectral sequence to compute the cohomology with nontrivial coefficients of congruence subgroups in Theorem 10.1. It is strange that no such direct relation is known between integral homology for p-adic Lie groups and for Lie algebras over Z_{p}; we have instead a relation between p-adic Lie groups and Lie algebras over Γ = F_{p}[π].

There are many cases in which the spectral sequence can be used to compute the Euler characteristic χ(G, M), assuming that the homology groups H_{∗}(G, M) are finite. It does not work in the generality of our assumptions here, however, because it is possible for H_{∗}(G, M) to be finite while H_{∗}(gr G, gr M) is not. For example, if G is a p-valued open subgroup of SL_{n}Z_{p} and M = (Z_{p})^{n} is the standard module, with the standard valuations on G and M as in the proof of Corollary 9.3, then H_{∗}(G, M) is always finite for n ≥ 2, but H_{∗}(gr G, gr M) is finite if and only if p does not divide n − 1 or n + 1.

So we consider instead the more general spectral sequence of Proposition 8.4, for the homomorphism Z_{p}G → Sat Z_{p}G of complete filtered rings, where the saturation of a valued Z_{p}-module such as Z_{p}G is defined before the statement of Theorem 9.1. Since M has a valuation compatible with the action of G, the ring div Z_{p}G acts on div M, compatibly with the filtrations, and so the completion Sat Z_{p}G acts on Sat M. Since we assume M is saturated, the action of Z_{p}G on M extends to Sat Z_{p}G.

For any valued Z_{p}-module N, it is easy from the definition of Sat N to check that

gr Sat N = (gr N ⊗_{F_{p}[π]} F_{p}[π, π^{−1}])_{degrees ≥0}.

If the valuation on G takes integer values, then one can show that gr Sat Z_{p}G is the universal enveloping algebra of a Lie algebra over Γ; in general, one can draw a similar conclusion after extending scalars as follows.

We know that gr G is concentrated in degrees (1/e)Z. Let K be a finite extension of Q_{p} with the same residue field F_{p} such that the maximal ideal of the ring of integers o_{K} is generated by an element π_{K} with valuation v(π_{K}) = 1/e. We have gr o_{K} = F_{p}[π_{K}] where π_{K} has degree 1/e, and there is a natural inclusion gr Z_{p} = F_{p}[π] ⊂ F_{p}[π_{K}]. The definitions of valued Z_{p}-modules and their saturations extend to o_{K}-modules in a natural way. We therefore have

gr Sat o_{K}G = (gr o_{K}G ⊗_{F_{p}[π_{K}]} F_{p}[π_{K}, π_{K}^{−1}])_{≥0}
= (gr Z_{p}G ⊗_{F_{p}[π]} F_{p}[π_{K}, π_{K}^{−1}])_{≥0}
= (U(gr G) ⊗_{F_{p}[π]} F_{p}[π_{K}, π_{K}^{−1}])_{≥0}.

Defining a graded Lie algebra 𝔰 over F_{p}[π_{K}] by tensoring gr G up from F_{p}[π] to F_{p}[π_{K}], we can say that

gr Sat o_{K}G = (U𝔰 ⊗_{F_{p}[π_{K}]} F_{p}[π_{K}, π_{K}^{−1}])_{≥0}.

Let 𝔱 be the saturation of 𝔰, defined by

𝔱 = (𝔰 ⊗_{F_{p}[π_{K}]} F_{p}[π_{K}, π_{K}^{−1}])_{≥0}.

Since 𝔰 is a graded free F_{p}[π_{K}]-module of rank n, so is 𝔱. Since 𝔰 is concentrated in degrees (1/e)Z and π_{K} has degree 1/e, the generators of 𝔱 are all in degree 0. Finally, 𝔱 is a Lie algebra over F_{p}[π_{K}] in an obvious way. It follows that gr Sat o_{K} is the universal enveloping algebra of 𝔱.

As a result, the spectral sequence of Proposition 8.4 has the form

Tor_{∗}^{U𝔱,U𝔰}(F_{p}[π_{K}], gr M_{o_{K}}) ⇒ Tor_{∗}^{Sat o_{K}G,o_{K}G}(o_{K}, M_{o_{K}}).

Again, in the proposition, we assumed that the filtrations were indexed by the integers, but we can apply the proposition when the filtrations are indexed by (1/e)Z, as here, by rescaling the filtrations. In a somewhat simpler notation, we can rename the groups in this spectral sequence as:

H_{∗}(𝔱, 𝔰; gr M_{o_{K}}) ⇒ H_{∗}(Sat o_{K}G, G; M_{o_{K}}).

Here, for the augmented algebra Sat o_{K}G over o_{K}, we write H_{∗}(Sat o_{K}G, M_{o_{K}}) to mean Tor_{∗}^{Sat o_{K}G}(o_{K}, M_{o_{K}}), by analogy with the definitions of group homology and Lie algebra homology. The homomorphism of Lie algebras 𝔰 → 𝔱 over F_{p}[π_{K}] is an injection from one free F_{p}[π_{K}]-module of finite rank to another, and the 𝔱module gr M_{o_{K}} is also a free F_{p}[π_{K}]-module of finite rank. Since G has dimension at least 2, the Lie algebras 𝔰 and 𝔱 have rank at least 2 as free F_{p}[π_{k}]-modules. By Proposition 2.3, the relative Lie algebra homology groups H_{∗}(𝔱, 𝔰; gr M_{o_{K}}) are finite, and the resulting Euler characteristic is 0. Then the above spectral sequence shows that the groups H_{∗}(Sat o_{K}G, G; M_{o_{K}}) are also finite and that the resulting Euler characteristic is 0. These groups are just the analogous groups over Z_{p} tensored up to o_{K}, so we deduce the same conclusions for the groups H_{∗}(Sat Z_{p}G, G; M). So H_{∗}(G, M) is finite if and only if H_{∗}(Sat Z_{p}G, M) is finite; and if either condition holds, then

χ(G, M) = χ(Sat Z_{p}G, M).

To analyze the homology of Lie algebras over Z_{p} by the above methods, which as written apply to complete rings, we first need the following lemma.

Lemma 9.2 Let 𝔤 be a valued Lie algebra over Z_{p}. That is, 𝔤 is a filtered Lie algebra over Z_{p} which is valued as a Z_{p}-module. As always, assume that 𝔤 is free of finite rank over Z_{p}. For any complete filtered 𝔤-module M, we can view M as a module over the completion U𝔤^{∧} of the universal enveloping algebra, and we have

H_{∗}(𝔤, M) = H_{∗}(U𝔤^{∧}, M).

Proof. Let

→ U𝔤 ⊗_{Z_{p}} ∧^{2}𝔤 → U𝔤 ⊗_{Z_{p}} 𝔤 → U𝔤 → Z_{p} → 0

be the standard resolution of Z_{p} as a U𝔤-module. Clearly these modules are filtered in a natural way. The point is that this is a resolution in the filtered sense, meaning that not only this complex but also the subcomplexes of elements of filtration ≥ ν, for all real numbers ν, are exact. Indeed, the Z_{p}-linear homotopies that prove exactness of the standard complex are compatible with the filtrations (V.1.3.7, p. 545, in [23]).

It follows that these Z_{p}-linear homotopies are defined on the completion of this complex, and so this completion is exact. It clearly has the form

→ U𝔤^{∧} ⊗_{Z_{p}} ∧^{2}𝔤 → U𝔤^{∧} ⊗_{Z_{p}} 𝔤 → U𝔤^{∧} → Z_{p} → 0.

So, for any U𝔤^{∧}-module M, H_{∗}(𝔤, M) and H_{∗}(U𝔤^{∧}, M) are computed by the same complex

→ ∧^{2}𝔤 ⊗_{Z_{p}} M → 𝔤 ⊗_{Z_{p}} M → M → 0.

QED

If 𝔤 is a sufficiently small open Lie subalgebra over Z_{p} of the Lie algebra of G over Q_{p}, then 𝔤 inherits a valuation from G, and we have Sat U𝔤 = Sat Z_{p}G by the proof of Theorem V.2.4.9, p. 562 in [23]. In particular, we have a homomorphism from U𝔤^{∧} to Sat Z_{p}G. As in the argument for groups, let o_{K} be an extension of Z_{p} such that a uniformizer π_{K} has valuation 1/e. After tensoring up to o_{K}, the graded homomorphism associated to U𝔤^{∧} → Sat Z_{p}G maps the universal enveloping algebra of one graded Lie algebra over F_{p}[π_{K}], 𝔯 := gr 𝔤 ⊗_{F_{p}[π]} F_{p}[π_{K}], to that of another, the saturation 𝔱 of 𝔯 as above. The Lie algebra homomorphism 𝔯 → 𝔱 is again an injection from one finitely generated free F_{p}[π_{K}]-module to another. So the argument for groups applies, again using that the dimension is at least 2, to show that H_{∗}(U𝔤^{∧}, M) is finite if and only if H_{∗}(Sat Z_{p}G, M) is finite, and if either condition holds then

χ(U𝔤^{∧}, M) = χ(Sat Z_{p}G, M).

By Lemma 9.2, we can replace H_{∗}(U𝔤^{∧}, M) in these statements by H_{∗}(𝔤, M).

Having related Euler characteristics for both the group G and the Lie algebra 𝔤 to those for Sat Z_{p}G, we have the relation between 𝔤 and G that we wanted. We had to assume above that the Lie algebra 𝔤 was sufficiently small, but that implies the same result for any open Lie subalgebra over Z_{p} of the Lie algebra of G over Q_{p} which acts on M, by Proposition 2.3. Theorem 9.1 is proved. QED

Corollary 9.3 Let p be any prime number. Let G be a compact p-adic Lie group of dimension at least 2, and let M be a finitely generated free Z_{p}-module with Gaction. Suppose that the image of G in Aut(M) is sufficiently small in the sense that either (1) this image is a pro-p group and p > rank(M) + 1, or else (2) G acts trivially on M/p if p is odd, or on M/4 if p = 2. Also assume that there is some faithful G-module (which could be M), finitely generated and free as a Z_{p}-module, which satisfies (1) or (2).

Let 𝔤 be any Lie algebra over Z_{p} such that 𝔤 ⊗ Q_{p} is the Lie algebra 𝔤_{Q_{p}} of G and such that 𝔤 acts on M. Then the homology groups H_{∗}(G, M) are finite if and only if the groups H_{∗}(𝔤, M) are finite, and if either condition holds then

χ(G, M) = χ(𝔤, M).

Proof. According to Theorem 9.1, it suffices to show that G has a valuation and that M has a compatible valuation which is saturated, both taking rational values. If M is faithful as well as satisfying (1) or (2), then Lazard constructed the required valuations of G and M; we will recall his definitions in the following paragraphs. In general, if there is some faithful module N which satisfies (1) or (2), then the minimum of the filtrations of G associated to M and N is a valuation of G which is compatible with the valuation of M, as we want. (To be precise, if M is not faithful, the filtration of G associated to M alone satisfies all the properties of a valuation, as defined before Theorem 9.1, except that it takes the value ∞ on the kernel of G → Aut(M). The minimum just mentioned is a genuine valuation of G.)

For (2), we use the obvious integral valuations. That is, after choosing a basis for M, G becomes a subgroup of GL_{n}Z_{p} whose image in GL_{n}Z/p is trivial, and we define

ω(a) = min_{1≤i,j≤n} v(a_{ij} − δ_{ij})

for a ∈ G and

w(x) = min_{1≤i≤n} v(x_{i})

for x ∈ M = (Z_{p})^{n}. The stronger assumption for p = 2 is needed to ensure that ω is a valuation of G (see the definition before Theorem 9.1).

In case (1), use the rational valuation of G defined in section III.3.2.7, pp. 484– 486, of [23]. We will generalize this construction to groups other than GL_{n} in Proposition 12.1. Namely, since G is a pro-p subgroup of GL_{n}Z_{p}, it is conjugate to a subgroup of the Sylow p-subgroup Iw_{u} ⊂ GL_{n}Z_{p} of matrices whose image in GL_{n}Z/p is strictly upper-triangular. (We call this subgroup Iw_{u} since it is the prop, or pro-unipotent, radical of an Iwahori subgroup of GL_{n}Q_{p}, the group of matrices in GL_{n}Z_{p} whose image in GL_{n}Z/p is non-strictly upper triangular.) So it is enough to define valuations on the group Iw_{u} and its standard module M = (Z_{p})^{n}. Since we assume n < p − 1, there is a rational number α such that α > (p − 1)^{−1} and (n − 1)α < 1 − (p − 1)^{−1}. Choose such an α and a finite extension K of Q_{p} with an element a ∈ K such that v(a) = α. Let D denote the diagonal matrix d_{ij} = a^{i−n}δ_{ij}; this differs from Lazard’s definition by a constant factor, which makes no difference in defining the valuation on Iw_{u}. Namely, let w be the standard valuation of the algebra M_{n}o_{K}, w(X) = min v(x_{ij}), and define a valuation of Iw_{u} by

ω(X) = w(D^{−1}XD − 1).

Lazard shows that this is a valuation of Iw_{u}. Similarly, let w be the standard valuation on M = (Z_{p})^{n}, w(m) = min v(m_{i}), and define a new valuation w′ of M by

w′(m) = w(D^{−1}m).

It is immediate that this valuation is compatible with that on G, in the sense defined before Theorem 9.1. Also, from our choice of D, M is saturated for this valuation. QED

Theorem 0.1 follows from Theorem 7.1, Theorem 7.4, and Corollary 9.3 when the G-module M is a free Z_{p}-module. To include arbitrary finitely generated Z_{p}- modules M in Theorem 0.1, we use that χ(G, A) = 0 for all p-adic Lie groups G of positive dimension which are pro-p groups and all finite Z_{p}G-modules A, by [28], I.4.1, exercise (e).

10 Cohomology of congruence subgroups

In this section we show how to use the spectral sequence arising in the proof of Theorem 9.1 to compute the whole homology with nontrivial coefficients of certain congruence subgroups, not just an Euler characteristic. See Corollary 10.2 for the special case of congruence subgroups of SL_{n}Z_{p}. In contrast to the results on Euler characteristics, we need to assume that p does not divide n − 1 or n + 1 in Corollary 10.2.

Theorem 10.1 Let 𝔤 be a Lie algebra over Z_{p}, M a finitely generated free Z_{p}- module on which 𝔤 acts. Suppose that the homology groups H_{∗}(𝔤_{F_{p}}, M_{F_{p}}) are 0. This holds for example if 𝔤_{Q_{p}} is semisimple, M_{Q_{p}} is a nontrivial simple 𝔤_{Q_{p}}-module, and p does not divide the eigenvalue of the Casimir operator, scaled to lie in the Z_{p}-algebra U𝔤, on M_{Q_{p}}.

Let G_{r} be the group defined by the Baker-Campbell-Hausdorff formula from the Lie algebra p^{r}𝔤, where r ≥ 1 if p is odd and r ≥ 2 if p = 2. Then the abelian group H_{i}(G_{r}, M) is isomorphic to the direct sum of \binom{n−1}{i}rank(M) copies of Z/p^{r}, where n is the rank of 𝔤 as a free Z_{p}-module. Also, H_{i}(p^{r}𝔤, M) is isomorphic to the same group.

Moreover, for any group H which acts compatibly on 𝔤 and M, we have

H_{i}(G_{r}, M) = r ∑_{j=0}^{i}(−1)^{i−j}(∧^{j}𝔤_{F_{p}} ⊗_{F_{p}} M_{F_{p}})

in the Grothendieck group of finite p-torsion H-modules.

Proof. Since r ≥ 1 if p is odd and r ≥ 2 if p = 2, G_{r} is a p-valued group by Lazard [23], IV.3.2.6, pp. 518–519. The valuation is defined by: ω(x) = a if x ∈ G_{r} corresponds to an element of p^{a}𝔤 − p^{a+1}𝔤. Since M is a 𝔤-module, the standard saturated valuation on M, where w(x) = a if x ∈ M lies in p^{a}M − p^{a+1}M, is compatible with the valuation of G_{r}.

In the proof of Theorem 9.1, we defined a spectral sequence

H_{∗}(gr G_{r}, gr M) ⇒ H_{∗}(G_{r}, M).

Here gr G_{r} is the Lie algebra π^{r}gr 𝔤 over Γ = gr Z_{p} = F_{p}[π]. The complex for computing the Lie algebra homology H_{∗}(π^{r}gr 𝔤, gr M) has the form

→ π^{r}gr 𝔤 ⊗_{Γ} gr M → gr M → 0.

It can be identified in an obvious way with the complex defining H_{∗}(gr 𝔤, gr M),

→ gr 𝔤 ⊗_{Γ} gr M → gr M → 0,

but with the differentials multiplied by π^{r}.

Clearly the Lie algebra gr 𝔤⊗_{Γ}F_{p} is equal to 𝔤_{F_{p}} and gr M ⊗_{Γ}F_{p} is equal to M_{F_{p}}. We assumed that H_{∗}(𝔤_{F_{p}}, M_{F_{p}}) = 0, and it follows that H_{∗}(gr 𝔤, gr M ⊗_{Γ} F_{p}) = 0 (since these homology groups are defined by the same complex). By the universal coefficient theorem, using that H_{∗}(gr 𝔤, gr M) is a finitely generated Γ-module, it follows that H_{∗}(gr 𝔤, gr M) = 0.

So the complex defining H_{∗}(gr G_{r}, gr M) is obtained from an exact complex by multiplying all the differentials by π^{r}. Since the Γ-modules in the complex are torsion-free, multiplying by π^{r} does not change the kernels, but the images are multiplied by π^{r}. Thus we have a canonical isomorphism

H_{i}(gr G_{r}, gr M) ≅ ker(d_{i}) ⊗_{Γ} Γ/π^{r},

where d_{i} : ∧^{i}gr 𝔤 ⊗_{Γ} gr M → ∧^{i−1}gr 𝔤 ⊗_{Γ} gr M is a differential in the complex defining H_{∗}(gr 𝔤, gr M). By exactness of the latter complex, we have

ker(d_{i}) = ∑_{j=0}^{i}(−1)^{i−j}(∧^{j}gr 𝔤 ⊗_{Γ} gr M)

in the Grothendieck group of finitely generated Γ-modules. So we know the rank of the free Γ-module ker(d_{i}), and it follows that H_{i}(gr G_{r}, gr M) is isomorphic to the direct sum of \binom{n−1}{i}rank(M) copies of Γ/π^{r}, where n is the rank of 𝔤 as a free Z_{p}- module. This is a graded Γ-module with generators in degree ir, as we see by going through the above identifications. We can also describe these homology groups less precisely but more canonically. Any group H which acts compatibly on 𝔤 and M automatically preserves the filtrations of 𝔤 and M, so it acts on the above spectral sequence, and we have

H_{i}(gr G_{r}, gr M) = r ∑_{j=0}^{i}(−1)^{i−j}(∧^{j}𝔤_{F_{p}} ⊗_{F_{p}} M_{F_{p}})

in the Grothendieck group of finite H-modules.

The differentials d_{k} in the spectral sequence

E_{ij}^{1} = H_{i+j}(gr G_{r}, gr M)_{degree −i} ⇒ H_{i+j}(G_{r}, M)

have bidegree (−k, k − 1). Since H_{i}(gr G_{r}, gr M) is concentrated in degrees from ir to ir + r − 1, the spectral sequence degenerates at E_{1}. It follows that the abelian group H_{i}(G_{r}, M) is isomorphic to the direct sum of \binom{n−1}{i}rank(M) copies of Z/p^{r}. Again, for any group H which acts compatibly on 𝔤 and M, it follows from the above results that

H_{i}(G_{r}, M) = r ∑_{j=0}^{i}(−1)^{i−j}(∧^{j}𝔤_{F_{p}} ⊗_{F_{p}} M_{F_{p}})

in the Grothendieck group of finite H-modules.

The proof of Theorem 9.1 gives a similar spectral sequence

H_{∗}(π^{r}gr 𝔤, gr M) ⇒ H_{∗}(p^{r}𝔤, M)

which degenerates by the same argument. So we get the same description of H_{∗}(p^{r}𝔤, M). QED

Corollary 10.2 Let G_{r} = ker(SL_{n}Z_{p} → SL_{n}Z/p^{r}), where r ≥ 1 if p is odd and r ≥ 2 if p = 2. Let M = (Z_{p})^{n} be the standard representation of G_{r}. Suppose that p ∤ (n − 1) and p ∤ (n + 1). Then the abelian group H_{i}(G_{r}, M) is isomorphic to the direct sum of \binom{n^{2}−2}{i}n copies of Z/p^{r}. Moreover, the group SL_{n}Z/p^{r} acts on H_{∗}(G_{r}, M) in a natural way, and we have

H_{i}(G_{r}, M) = r ∑_{j=0}^{i}(−1)^{i−j}(∧^{i}𝔰𝔩_{n}F_{p} ⊗_{F_{p}} M_{F_{p}})

in the Grothendieck group of finite p-torsion SL_{n}Z/p^{r}-modules.

Proof. To deduce the first statement from Theorem 10.1, we need to check that H_{∗}(𝔰𝔩_{n}F_{p}, M_{F_{p}}) = 0 if p ∤ n − 1 and p ∤ n + 1.

Let 𝔤 be the Lie algebra 𝔰𝔩_{n}Z, with its standard module M = Z^{n}. The Casimir operator c in the center of the enveloping algebra U𝔤_{Q} acts on M_{Q} by multiplication by (n^{2} − 1)/2n^{2}, say by formula (25.14) in Fulton and Harris [17], p. 418. Writing out c in terms of a basis for 𝔤 = 𝔰𝔩_{n}Z shows that c′ := 2n^{2}c lies in the integral enveloping algebra U𝔤. Clearly it acts by n^{2} − 1 on M. It is also clear that c′ maps to an element in the center of U𝔤_{F_{p}}, which acts by a nonzero scalar on M_{F_{p}} if n^{2} − 1 ≢ 0 (mod p), that is, if p ∤ (n − 1) and p ∤ (n + 1). It follows that H_{∗}(𝔤_{F_{p}}, M_{F_{p}}) = 0 if p ∤ (n − 1) and p ∤ (n + 1), as claimed.

So Theorem 10.1 applies, and we have the computation of H_{∗}(G_{r}, M) as an abelian group. The theorem also computes H_{i}(G_{r}, M) as an element in the Grothendieck group Rep(SL_{n}Z_{p}) of finite p-torsion SL_{n}Z_{p}-modules, since SL_{n}Z_{p} acts compatibly on 𝔰𝔩_{n}Z_{p} and on M_{Z_{p}}. Since H_{i}(G_{r}, M) and the expression on the right are in fact SL_{n}Z/p^{r}-modules, we deduce the same equality in the Grothendieck group Rep(SL_{n}Z/p^{r}) of finite p-torsion SL_{n}Z/p^{r}-modules, because the restriction map

Rep(SL_{n}Z/p^{r}) → Rep(SL_{n}Z_{p})

is injective. Indeed, Rep(SL_{n}Z/p^{r}) ≅ Rep(F_{p}[SL_{n}Z/p^{r}]) is detected by restriction to cyclic subgroups of order prime to p by Brauer [16], and these all lift to SL_{n}Z_{p} since the kernel of SL_{n}Z_{p} → SL_{n}Z/p^{r} is a pro-p group. QED

11 Euler characteristics for p-adic Lie groups which are not pro-p groups

Here at last we prove the vanishing of the Euler characteristics we have been considering for some p-adic Lie groups such as SL_{n}Z_{p} which are not pro-p groups. See Corollary 11.6 for some more explicit consequences of the following theorem.

Theorem 11.1 Let G_{Q_{p}} be a connected reductive algebraic group whose rank over \overline{Q_{p}} is at least 2, and let M_{Q_{p}} be a finite-dimensional G_{Q_{p}}-module with no trivial summands. Let G be a compact open subgroup of G(Q_{p}) and let M be a G-invariant lattice in M_{Q_{p}}. Suppose that there is a Sylow p-subgroup G_{p} ⊂ G with a valuation and that M has a compatible saturated valuation, both taking rational values. Then the homology groups H_{∗}(G, M) are finite and the resulting Euler characteristic χ(G, M) is 0.

Proof. Since G_{Q_{p}} is a reductive group in characteristic zero, representations of G_{Q_{p}} are completely reducible, and so the assumption on M_{Q_{p}} implies that the coinvariants of G_{Q_{p}} on M_{Q_{p}} are 0. Since G_{Q_{p}} is connected, it follows that the coinvariants of its Lie algebra 𝔤_{Q_{p}} on M_{Q_{p}} are 0. It follows that H_{∗}(K, M) is finite for all open subgroups K of G, by Lemma 3.1 and Lazard’s theorem that H_{∗}(K, M) ⊗ Q_{p} injects into H_{∗}(𝔤_{Q_{p}}, M_{Q_{p}}) ([23], V.2.4.10, pp. 562–563).

For any finite group F, let a(Z_{p}F) denote the Green ring, the free abelian group on the set of isomorphism classes of indecomposable Z_{p}F -modules that are finitely generated and free over Z_{p}. Conlon’s induction theorem says that for any finite group F, there are rational numbers a_{K} such that

Z_{p} = ∑_{K} a_{K}Z_{p}[F/K]

in a(Z_{p}G) ⊗ Q, where K runs over the set of p-hypoelementary subgroups of F, that is, extensions of a cyclic group of order prime to p by a p-group ([16], Theorem 80.51). (The name “hyperelementary” is also used for these subgroups.) In fact, although we do not need it here, there is an explicit formula for the rational numbers a_{K}, using Gluck’s formula for the idempotents in the Burnside ring tensored with the rationals [19]:

Z_{p} = ∑_{H} \frac{1}{|N_{F} (H)|} ∑_{K⊂H} |K|μ(K, H)Z_{p}[F/K].

Here the first sum runs over the conjugacy classes of p-hypoelementary subgroups H ⊂ F, and μ denotes the Möbius function on the partially ordered set of subgroups of F. Boltje’s paper [2] uses Gluck’s formula for similar purposes in Proposition VI.1.2 and the remarks afterward. For example, for the group F = S_{3} and p = 2, the above formula gives the identity

Z_{2} = −\frac{1}{2}Z_{2}S_{3}/1 + Z_{2}S_{3}/⟨(12)⟩ + \frac{1}{2}Z_{2}S_{3}/⟨(123)⟩.

Returning to the p-adic Lie group G, we know that there is an open normal subgroup H of G contained in the given Sylow p-subgroup G_{p}, for example the intersection of the conjugates of G_{p}. We apply Conlon’s induction theorem to the finite group G/H to get an equality

Z_{p} = ∑_{K} a_{K}Z_{p}[G/K]

in the Green ring of G/H-modules, where K runs over the p-hypoelementary subgroups of G containing H (that is, K is an extension of a cyclic group of order prime to p by a pro-p group). If we multiply this equation by a suitable positive integer and move terms with a_{K} negative to the other side of the equation, it states the existence of an isomorphism between two explicit G/H-modules, which we can view as an isomorphism between the same groups viewed as G-modules.

It follows that, for the given Z_{p}G-module M and all j ≥ 0, we have

H_{j}(G, M) = ∑_{K} a_{K}H_{j}(G, M ⊗_{Z_{p}} Z_{p}[G/K])

= ∑_{K} a_{K}H_{j}(K, M).

This is an equality in the Grothendieck group tensored with Q of finite abelian groups with respect to direct sums. It follows that

χ(G, M) = ∑_{K} a_{K}χ(K, M).

So, to show that the Euler characteristic χ(G, M) is 0, it suffices to show that χ(K, M) = 0 for all open p-hypoelementary subgroups K of G. Since such a subgroup satisfies all the properties we assumed of G, we can assume from now on that G is itself p-hypoelementary. That is, the Sylow p-subgroup G_{p} is normal in G and the quotient group Z := G/G_{p} is cyclic of order prime to p, and we want to show that χ(G, M) = 0.

The extension

1 → G_{p} → G → Z → 1

splits, and so G is a semidirect product Z ⋉ G_{p}. Also, H_{∗}(G, M) is equal to the coinvariants of Z acting on H_{∗}(G_{p}, M), and taking the coinvariants of Z is an exact functor on Z_{p}Z-modules. So it suffices to show that χ(G_{p}, M) = 0 in Rep(Z), the Grothendieck group of finite p-torsion Z-modules. We are given that M is a (Z ⋉ G_{p})-module and that G_{p} and M have compatible valuations. Replace the given valuation of G_{p} by the minimum of its conjugates under the action of Z on G_{p}. This is again a valuation of G_{p}, now Z-invariant, and still compatible with the given valuation of M since it is less than or equal to the original valuation of G_{p}.

Let 𝔤 be any Z-invariant Lie subalgebra over Z_{p} of the Lie algebra 𝔤_{Q_{p}} such that 𝔤 ⊗ Q_{p} = 𝔤_{Q_{p}}. To see that one exists, start with any Z-invariant Z_{p}-lattice in 𝔤_{Q_{p}}, and then multiply it by a big power of p. Propositions 11.2 and 11.4 will imply that χ(G_{p}, M) = 0 in Rep(Z), thus proving Theorem 11.1.

Proposition 11.2 In the above notation, we have χ(𝔤, M) = 0 in the Grothendieck group Rep(Z) of finite p-torsion Z-modules.

Proof. Let g be a generator of the finite cyclic group Z. Since G = Z ⋉G_{p} is an open subgroup of the connected reductive algebraic group G(Q_{p}), g is an element of finite order in G(Q_{p}), hence a semisimple element. So g is contained in some maximal torus T_{Q_{p}}, not necessarily split. Over some finite extension K of Q_{p}, T_{K} is contained in a Borel subgroup B_{K}. Therefore, when g acts on the Lie algebra 𝔤_{K}, it acts trivially on the Cartan subalgebra 𝔱_{K} and maps the Borel subalgebra 𝔟_{K} into itself. We will show that χ(𝔤_{o_{K}}, M_{o_{K}}) = 0 in the Grothendieck group of finite p-torsion Z-modules, which implies the statement of the proposition.

Briefly, the proofs of Proposition 4.1 and Theorem 5.1 work Z-equivariantly. The Z-equivariant analogue of Proposition 4.1 which we need is the following lemma.

Lemma 11.3 Let 𝔥 be an abelian Lie algebra of the form (o_{K})^{r} for some r ≥ 2. Let M be a finitely generated o_{K}-module with 𝔥-action such that M_{𝔥} ⊗ K = 0. Let Z be a group which acts trivially on 𝔥 and acts compatibly on M (in an obvious terminology, M is a (Z ×𝔥)-module). Then the homology groups H_{∗}(𝔥, M) are finite and the resulting Euler characteristic χ(𝔥, M) in the Grothendieck group Rep(Z) of finite Z-modules is 0.

Proof. First, we show that χ(𝔥, M) = 0 in Rep(Z) for any abelian Lie algebra 𝔥 of rank at least 1 as an o_{K}-module and any finite (Z × 𝔥)-module M. Indeed, in Rep(Z),

χ(𝔥, M) = ∑_{i}(−1)^{i} ∧^{i} 𝔥 ⊗_{o_{K}} M

= rank(∑_{i}(−1)^{i} ∧^{i} 𝔥)M

= 0 · M = 0,

where the first equality follows from the complex that computes Lie algebra homology, the second equality holds because Z acts trivially on 𝔥, and the third is because 𝔥 has rank at least 1 as an o_{K}-module.

Now suppose that 𝔥 has rank at least 2 as an o_{K}-module and that M is finitely generated over o_{K}, with M_{𝔥} ⊗ K = 0. We know that the Z-modules H_{∗}(𝔥, M) are finite by the corresponding nonequivariant statement, Proposition 4.1. The previous paragraph implies that the Euler characteristic χ(𝔥, M) in Rep(Z) only depends on the (Z × 𝔥_{K})-module M_{K} := M ⊗_{o_{K}} K, so we can use the notation χ_{fin}(𝔥_{K}, M_{K}) := χ(𝔥, M) in Rep(Z), generalizing Definition 2.4. Also, we can extend scalars as in Definition 2.5, so it suffices to show that χ_{fin}(𝔥_{\overline{Q}_{p}}, M_{\overline{Q}_{p}}) = 0 in Rep(Z) for all (Z × 𝔥_{\overline{Q}_{p}})-modules M_{\overline{Q}_{p}} such that the coinvariants of 𝔥_{\overline{Q}_{p}} on M_{\overline{Q}_{p}} are 0.

The simple (Z × 𝔥_{\overline{Q}_{p}})-modules are 1-dimensional by Schur’s lemma, and the assumption that the coinvariants of 𝔥_{\overline{Q}_{p}} on M_{\overline{Q}_{p}} are 0 means that all the simple subquotients of M_{\overline{Q}_{p}} as an 𝔥_{\overline{Q}_{p}}-module are nontrivial, by Lemma 3.1. So it suffices to show that χ_{fin}(𝔥_{\overline{Q}_{p}}, M_{\overline{Q}_{p}}) = 0 in Rep(Z) for a 1-dimensional (Z × 𝔥_{\overline{Q}_{p}})- module M_{\overline{Q}_{p}} which is nontrivial as an 𝔥_{\overline{Q}_{p}}-module. That is, it suffices to show that χ(𝔥, M) = 0 in Rep(Z) for any abelian Lie algebra 𝔥 of rank at least 2 over a p-adic ring of integers o_{K} and any (Z × 𝔥)-module M of rank 1 which is nontrivial as an 𝔥-module.

Since 𝔥 has rank at least 2 as an o_{K}-module, there is an o_{K}-submodule 𝔩 ⊂ 𝔥 such that 𝔥/𝔩 ≅ o_{K} and 𝔩 acts nontrivially on M. Then

χ(𝔥, M) = ∑_{i}(−1)^{i}χ(𝔥/𝔩, H_{i}(𝔩, N)).

Since Z acts trivially on 𝔥, it preserves 𝔩, and so this is an equality in Rep(Z). We have arranged that H_{i}(𝔩, N) is a finite (Z × 𝔥/𝔩)-module for all i, so the individual terms in this sum are 0 in Rep(Z) by the first paragraph of this proof. So χ(𝔥, M) = 0 in Rep(Z) as we want. QED

To complete the proof of Proposition 11.2, we need to show that χ(𝔤_{o_{K}}, M_{o_{K}}) = 0 in Rep(Z). We know that M is a (Z ⋉ 𝔤_{o_{K}})-module and that Z acts trivially on a Cartan subalgebra 𝔱_{K} and preserves a Borel subalgebra 𝔟_{K} containing 𝔱_{K}. Then it is clear that the following formula from the proof of Theorem 5.1 holds in Rep(Z):

χ(𝔤, M) = ∑_{j,k}(−1)^{j+k}χ(𝔟/𝔲, H_{j}(𝔲, M ⊗ ∧^{k}(𝔤/𝔟))).

We are assuming in Theorem 11.1 that the algebraic group G_{Q_{p}} has rank at least 2 over \overline{Q}_{p}, so the Lie algebra 𝔤_{K} has rank at least 2. That is, 𝔟_{K}/𝔲_{K} has dimension at least 2. Also, the group Z acts trivially on 𝔟_{K}/𝔲_{K} ≅ 𝔱_{K}, so we can apply Lemma 11.3 to show that all the terms in this sum are 0 in Rep(Z). So χ(𝔤, M) = 0 in Rep(Z). QED (Proposition 11.2)

Proposition 11.4 In the notation defined before Proposition 11.2, we have χ(𝔤, M) = χ(G_{p}, M) in the Grothendieck group of finite p-torsion Z-modules.

Proof. Since the valuation of G_{p} is Z-invariant, the proof of Theorem 9.1 works Z-equivariantly. The only point which is not obvious is that Proposition 2.3(2) works Z-equivariantly, given that the Lie algebra 𝔤 over a discrete valuation ring Γ in the proposition has (𝔤 ⊗_{Γ} F)^{Z} of dimension at least 2 over the field F = Γ[π^{−1}], as the following lemma asserts. That hypothesis will be valid for our Lie algebra gr G over the graded discrete valuation ring Γ = F_{p}[π] because 𝔤_{Q_{p}}^{Z} has dimension at least 2 over Q_{p}. Indeed, since the reductive algebraic group G_{Q_{p}} has rank at least 2 over \overline{Q}_{p}, every element of G(Q_{p}) (in particular, a generator of the cyclic group Z ⊂ G(Q_{p})) has centralizer of dimension at least 2.

Lemma 11.5 Let Γ be a discrete valuation ring with uniformizer π. Let 𝔤 be a Lie algebra over Γ which is a finitely generated free Γ-module, and 𝔥 ⊂ 𝔤 a Lie subalgebra of the same rank as a free Γ-module. Let M be a finitely generated free Γ-module with 𝔤-action. Finally, let Z be a group which acts compatibly on 𝔤, 𝔥, and M such that the trivial Z-module over the field F = Γ[π^{−1}] occurs with multiplicity at least 2 in 𝔤 ⊗ F. Then the relative Lie algebra homology groups H_{∗}(𝔤, 𝔥; M) have finite length as Γ-modules, and the corresponding Euler characteristic χ(𝔤, 𝔥; M) is 0 in the Grothendieck group of Γ[Z]-modules of finite length over Γ.

Proof. It suffices to consider the case where π𝔤 ⊂ 𝔥 ⊂ 𝔤. We apply that special case to the sequence of Z-invariant Lie subalgebras of 𝔤,

𝔤 ⊃ π𝔤 + 𝔥 ⊃ π^{2}𝔤 + 𝔥 ⊃ ⋯,

which eventually equals 𝔥.

For π𝔤 ⊂ 𝔥 ⊂ 𝔤, let B = 𝔤/𝔥 and A = ker(𝔤/π → 𝔤/𝔥). These are representations of the group Z over the field Γ/π which form an exact sequence

0 → A → 𝔤/π → B → 0.

We compute that, in the Grothendieck group Rep(Z) of Γ[Z]-modules of finite length over Γ,

∧^{i}𝔤/ ∧^{i} 𝔥 = ∑_{j=0}^{i} j(∧^{i−j}A ⊗_{Γ/π} ∧^{j}B).

This is proved using a canonical filtration of the finite-length Γ-module ∧^{i}𝔤/ ∧^{i} 𝔥 with quotients vector spaces over Γ/π. It follows that, in Rep(Z),

∑_{i}(−1)^{i} ∧^{i} 𝔤/ ∧^{i} 𝔥 = ∑_{j≤i}(−1)^{i}j(∧^{i−j}A)(∧^{j}B)

= ∑_{i,j}(−1)^{i} ∧^{i} (A) · j(−1)^{j} ∧^{j} B

= F_{1}(A)F_{2}(B), where we define

F_{1}(A) = ∑_{i}(−1)^{i} ∧^{i} A
F_{2}(A) = ∑_{i}(−1)^{i}i ∧^{i} A.

The operations F_{1} and F_{2} take a representation A of Z over Γ/π to an element of the corresponding Grothendieck group. For A of dimension n, the operation F_{1}(A) = ∧_{−1}(A) is related to the top gamma operation γ_{n} (the top Chern class with values in K-theory) and F_{2}(A) is related to the operation γ_{n−1}, in the terminology of λ-rings [18]. We do not need that terminology, but only the elementary properties that

F_{1}(A + B) = F_{1}(A)F_{1}(B)

and

F_{2}(A + B) = F_{1}(A)F_{2}(B) + F_{2}(A)F_{1}(B).

Also, F_{1}(1) = 0, so F_{1}(A + 1) = 0 for all representations A, and F_{2}(1) = −1 and F_{2}(2) = 0, so F_{2}(A + 2) = 0 for all representations A.

The relative Lie algebra homology H_{∗}(𝔤, 𝔥; M) is computed by a chain complex with Z-action, with ith group equal to (∧^{i}𝔤/ ∧^{i} 𝔥) ⊗_{Γ} M. From this it is immediate that the Γ-modules H_{∗}(𝔤, 𝔥; M) have finite length. Moreover, the previous paragraph shows that, in the Grothendieck group Rep(Z) of finite-length Γ[Z]-modules,

χ(𝔤, 𝔥; M) = ∑_{i}(−1)^{i}(∧^{i}𝔤/ ∧^{i} 𝔥) ⊗_{Γ} M
= F_{1}(A)F_{2}(B)M/π,

where B = 𝔤/𝔥 and A = ker(𝔤/π → 𝔤/𝔥).

We are assuming that the trivial Z-module over the field F occurs with multiplicity at least 2 in 𝔤 ⊗ F. It follows easily that the trivial Z-module over the field Γ/π occurs with multiplicity at least 2 in 𝔤/π. So it occurs either with multiplicity at least 1 in A or with multiplicity at least 2 in B. By the properties of the operations F_{1} and F_{2} listed above, either F_{1}(A) = 0 or F_{2}(B) = 0 in Rep(Z). So χ(𝔤, 𝔥; M) = 0 in Rep(Z), as we want. QED (Lemma 11.5 and hence Proposition 11.4).

Theorem 11.1 follows from Propositions 11.2 and 11.4, together with the analysis before Proposition 11.2. QED

Corollary 11.6 Let G_{Q_{p}} be a connected reductive algebraic group, of rank over \overline{Q_{p}} at least 2. Suppose that p is greater than the dimension of some faithful G_{Q_{p}}-module plus 1. (For G_{Q_{p}} semisimple, it suffices to assume instead the lower bound for p given in Proposition 12.1.) Let M_{Q_{p}} be a G_{Q_{p}}-module with no trivial summands. Let G be a compact open subgroup of G(Q_{p}), and let M be a G-invariant Z_{p}-lattice in M_{Q_{p}}. Then the homology groups H_{∗}(G, M) are finite and the resulting Euler characteristic χ(G, M) is 0.

Proof. The bound on p in Proposition 12.1 will imply that any Sylow psubgroup G_{p} of G admits a valuation, and that the vector space M_{Q_{p}} has a compatible valuation. Let us now prove the same statements when p is greater than the dimension of some faithful G_{Q_{p}}-module plus 1.

Let N_{Q_{p}} be a faithful G_{Q_{p}}-module with p > dim(N_{Q_{p}})+1. Since G is compact, it preserves some Z_{p}-lattice N in N_{Q_{p}}. Any Sylow p-subgroup G_{p} of G is a subgroup of some Sylow p-subgroup Iw_{u} ⊂ GL(N). Since p > rank(N) + 1, the proof of Corollary 9.3 shows that Iw_{u} has a valuation, which we can restrict to G_{p}, and that N has a compatible saturated valuation.

Since the group G_{Q_{p}} is reductive, any G_{Q_{p}}-module M_{Q_{p}} is a direct summand of some direct sum of tensor products N_{Q_{p}}^{⊗a} ⊗ (N_{Q_{p}}^{∗})^{⊗b} for a, b ≥ 0, by [25], II.4.3.2(a), p. 156. The valuation of N induces a vector space valuation

w : N_{Q_{p}}^{⊗a} ⊗ (N_{Q_{p}}^{∗})^{⊗b} → (∞, ∞],

which we can restrict to the subspace M_{Q_{p}}. Let

M_{0} = {x ∈ M_{Q_{p}} : w(x) ≥ 0}.

Then w is a saturated valuation on M_{0}. Since the valuation of G_{p} is compatible with that of N, it is compatible with that of M_{0}. By Theorem 11.1, given that M_{Q_{p}} has no trivial summand, we have χ(G, M_{0}) = 0. To deduce that χ(G, M) = 0 for all G-invariant lattices M in M_{Q_{p}}, we use Serre’s theorem that χ(G, A) = 0 for all finite Z_{p}G-modules A, since G is an open subgroup of a connected algebraic group over Q_{p} of dimension greater than zero ([29], Corollary to Theorem C). QED

For example, Corollary 11.6 implies that χ(G, M) = 0 for all open subgroups G of SL_{n}Z_{p} when M = (Z_{p})^{n} is the standard module, n ≥ 3, and p > n + 1.

12 Construction of valuations on pro-p subgroups of a semisimple group

In this section, we will improve the bound on p in Corollary 11.6, which says that for p sufficiently large, the Euler characteristics are zero for all compact open subgroups of a reductive group of rank at least 2. By the proof of Corollary 11.6, all we need is to give a weaker sufficient condition on a group G_{K} so that every closed pro-p subgroup of G(K) is p-valued. Proposition 12.1 will give such a weaker sufficient condition when the group G_{K} is semisimple. It may be interesting for other purposes to know that every closed pro-p subgroup of G(K) has a valuation; in particular, it follows that G(K) has no p-torsion. The proof combines the Bruhat-Tits structure theory of p-adic groups [32] with a generalization of Lazard’s construction of a valuation for pro-p subgroups of GL_{n}K (given in the proof of Corollary 9.3, above).

For the group SL_{n}Q_{p}, the bound in Corollary 11.6 is optimal: every closed pro-p subgroup of SL_{n}Q_{p} is p-valued if p > n+1. Indeed, if p = n+1 and n ≥ 2, then the cyclic group Z/p imbeds in SL_{n}Z and hence in SL_{n}Q_{p}. For other groups, however, we can do better. For example, let D be a division algebra of degree n (that is, of dimension n^{2}) over Q_{p}. Then the proof of Corollary 11.6 shows that every pro-p subgroup of SL_{1}D is p-valued if p > n^{2} + 1, but in fact p > n + 1 is enough, as the following proposition gives, using that SL_{1}D becomes isomorphic to SL_{n} over some unramified extension of Q_{p}. The need for an improvement in Corollary 11.6 is most apparent for the exceptional groups. For example, E_{8}(Q_{p}) has p-torsion if and only if p = 2, 3, 5, 7, 11, 13, 19, or 31 by [29], p. 492, but the proof of Corollary 11.6 shows only that every pro-p subgroup of E_{8}(Q_{p}) is p-valued (hence E_{8}(Q_{p}) has no p-torsion) if p > 248 + 1. The following proposition gives the optimal estimate that every pro-p subgroup of E_{8}(Q_{p}) is p-valued if p > 30 + 1, since 30 is the Coxeter number of E_{8}.

To state a sharp bound even for non-split groups, we need to define a generalization of the Coxeter number. Let G_{K} be an absolutely simple quasi-split group over a field K. (Quasi-split means that G_{K} has a Borel subgroup defined over K [3]). Such a group is described by its Dynkin diagram over the separable closure \overline{K} of K, of type A_{n}, B_{n}, C_{n}, D_{n}, E_{6}, E_{7}, E_{8}, F_{4}, or G_{2}, together with an action of the Galois group Gal(\overline{K}/K) on the Dynkin diagram. That is, the Galois group maps into the automorphism group of the Dynkin diagram, which has order 1 or 2, except that the automorphism group of the Dynkin diagram D_{4} is isomorphic to the symmetric group S_{3}. Equivalently, the Galois group acts on the root system, preserving the set of positive roots. Let α_{1}, …, α_{n} be the simple roots. We define the generalized Coxeter number h(G_{K}) to be the maximum over all positive roots α = ∑ r_{i}α_{i} of the numbers (1+∑ r_{i})|Gal(\overline{K}/K)α|, unless the Dynkin diagram is of type A_{n} with n even and the Galois action is nontrivial (so that the universal cover of G_{K} is a unitary group SU_{n+1}K with n + 1 odd); in that case, we define h(G_{K}) to be 2(n + 1). If G_{K} is split, meaning that the Galois action is trivial, then h(G_{K}) is the Coxeter number as defined in Bourbaki [5], VI.1, Proposition 31. Using the notation ^{i}X_{n} to denote a quasi-split group with Dynkin diagram of type X_{n} where the image of the Galois group has order i, we tabulate the numbers h(G_{K}) below.

G_{K}  h(G_{K})
1A_{n}  n + 1
2A_{n}  2n if n is odd
2A_{n} 2(n + 1) if n is even
B_{n}  2n
C_{n}  2n
1D_{n}  2n − 2
2D_{n}  2n
3D_{4}  12
6D_{4}  12
1E_{6}  12
2E_{6}  18
E_{7}  18
E_{8}  30
F_{4}  12
G_{2}  6

Proposition 12.1 Let G_{K} be a connected semisimple algebraic group over a p-adic field K. Let e_{K} be the absolute ramification degree of K. By Steinberg [30], G_{K} becomes quasi-split over some finite unramified extension E of K. Let E be any such extension. The universal covering of G_{E} is a product of restrictions of scalars of absolutely simple quasi-split groups H_{M} over finite extensions M of E. Suppose that

p > h(H_{M})e_{M} + 1

for all the simple factors H_{M}, where h(H_{M}) is the generalized Coxeter number as listed above. Then every pro-p subgroup G_{p} of the original group G(K) admits a valuation. Moreover, every representation of the algebraic group G_{K} admits a valuation compatible with that of G_{p}. Both valuations take rational values.

Proof. We first reduce to the case of a simply connected group. Let G̃_{K} → G_{K} be the universal covering of the semisimple group G_{K}, and let Z be its kernel, which is a finite subgroup of the center of G̃_{K}. From the above table, and the known centers of the simple algebraic groups, we see that any prime number p that divides the order of the center of a simply connected semisimple group is at most the maximum of the Coxeter numbers of its simple factors over the algebraic closure. So our assumption on p is more than enough to ensure that Z has order prime to p. By the exact sequence

Z(K) → G̃(K) → G(K) → H^{1}(K, Z(\overline{K})),

where the groups on the ends are abelian groups in which every element has order prime to p, we see that every pro-p subgroup of G(K) is the isomorphic image of some pro-p subgroup of G̃(K). Also, any representation of G_{K} can be viewed as a representation of G̃_{K}. So it suffices to prove the proposition with G_{K} replaced by G̃_{K}, that is, for G_{K} simply connected.

For a simply connected semisimple group G_{K} over a p-adic field K, Bruhat and Tits, generalizing earlier work by Iwahori-Matsumoto and Hijikata, defined a conjugacy class of compact open subgroups of G(K), the Iwahori subgroups Iw. A convenient reference is [32], 3.7. Write k for the finite residue field of K. For example, the group of matrices in SL_{n}o_{K} whose image in SL_{n}k is upper-triangular is an Iwahori subgroup of SL_{n}K. Bruhat and Tits showed that every compact subgroup of G(K) is contained in a maximal compact subgroup, and they classified the maximal compact subgroups C. In particular, every maximal compact subgroup contains an Iwahori subgroup. Moreover, using that G_{K} is simply connected, each maximal compact subgroup C is an extension of the k-points of some connected group G(k) over the finite residue field k by a pro-p group, by [32], 3.5.2. The inverse image of a Borel subgroup B(k) ⊂ G(k) in C is an Iwahori subgroup, by [32], 3.7. Let Iw_{u} denote the inverse image of a Sylow p-subgroup U(k) ⊂ B(k) ⊂ G(k) in C; then Iw_{u} is a pro-p group. We see that Iw_{u} is a Sylow p-subgroup of C. Also, all these subgroups Iw_{u} in different maximal compact subgroups are conjugate in G(K), since they are all Sylow p-subgroups of Iwahori subgroups. It is natural to call Iw_{u} the pro-p radical of an Iwahori subgroup, since it is normal in Iw and Iw/Iw_{u} is a finite group of order prime to p. It follows that any pro-p subgroup in the whole group G(K) is contained in some subgroup conjugate to Iw_{u}.

As a result, to prove the proposition, it suffices to define a valuation on one subgroup Iw_{u} ⊂ G(K), and to show that every representation of the algebraic group G_{K} admits a valuation compatible with that on Iw_{u}. Generalizing Lazard’s definition of a valuation on Iw_{u} ⊂ GL_{n}K (given in the proof of Corollary 9.3, above), the idea is to find an element a ∈ G(L) for some finite extension L of K and a Chevalley group G_{o_{L}} extending G_{L} such that a^{−1}(Iw_{u})a is contained in the subgroup of elements g ∈ G(o_{L}) with g ≡ 1 (mod m) for some m ∈ o_{L} with ord_{p}(m) > (p − 1)^{−1}. Then, taking a faithful representation V of G_{o_{L}}, a^{−1}(Iw_{u})a is contained in the subgroup of elements g ∈ GL(V) with g ≡ 1 (mod m) for some m ∈ o_{L} with ord_{p}(m) > (p − 1)^{−1}. So the group a^{−1}(Iw_{u})a has the valuation

ω(g) = ord_{p}(g − 1) ∈ (1/e_{L})Z.

Moreover, this valuation is compatible with the obvious valuation w on V. It follows that Iw_{u} ⊂ G(K) has a valuation defined by

ω′(g) = ω(a^{−1}ga),

and this is compatible with the valuation of V_{L} defined by

w′(x) = w(a^{−1}x).

Every representation of G_{L} (in particular, any representation of G_{K} tensored up to L) is a direct summand of a direct sum of tensor products of V_{L} and V_{L}^{∗}, by [25], II.4.3.2(a), p. 156. It follows that every representation of G_{K} has a valuation compatible with that of Iw_{u}, as we want. Thus, the proposition is proved if we can find an element a ∈ G(L) as above.

For this purpose, as mentioned in the proposition, we can choose a finite unramified extension E of K such that G_{E} is quasi-split, by Steinberg [30], Corollary 10.2(a), applied to the maximal unramified extension of K. The original pro-p subgroup Iw_{u} of G(K) is contained in the analogous subgroup of G(E), so it suffices to prove the same statement for G_{E} in place of G_{K}. Since G_{E} is simply connected and quasi-split, it is a product of restrictions of scalars of absolutely simple quasi-split groups H_{M} over finite extensions M of E, by [3], 6.21(ii). The pro-p subgroup Iw_{u} of G(E) is the product of the analogous subgroups for the simple factors, so it suffices to consider the simple factors.

That is, writing G_{K} in place of H_{M}, we are given an absolutely simple quasisplit group G_{K} over a p-adic field K such that p > h(G_{K})e_{K} + 1. Write Iw_{u} for the pro-p radical of an Iwahori subgroup of G(K), as above. To prove the proposition, it suffices to find an element a ∈ G(L) for some finite extension L of K and a Chevalley group G_{o_{L}} extending G_{L} such that a^{−1}(Iw_{u})a is contained in the subgroup of elements x ∈ G(o_{L}) with x ≡ 1 (mod m) for some m ∈ o_{L} with ord_{p}(m) > (p − 1)^{−1}.

For G_{K} of type ^{2}A_{n} with n even, so that G_{K} is the unitary group SU_{n+1}K associated to a quadratic extension L/K, we defined h(G_{K}) = 2(n + 1), while h(G_{L}) is the Coxeter number of SL_{n+1}L, that is, n + 1. The assumption that p > h(G_{K})e_{K} + 1 implies that p > h(G_{L})e_{L} + 1. Thus, if we can prove the above statement for G_{L} in place of G_{K}, then the statement for G_{K} follows. So we can assume from now on that G_{K} is not of type ^{2}A_{n} with n even. Equivalently, the relative root system Φ of G_{K} (defined below, or see Borel-Tits [3]) is reduced, that is, there is no root a such that 2a is also a root. (If G_{K} is split, its relative root system is reduced. Otherwise, G_{K} is of type ^{2}A_{2m},^{2} A_{2m−1},^{2} D_{n},^{2} E_{6},^{3} D_{4}, or ^{6}D_{4}, and then the relative root system is of type BC_{m}, C_{m}, B_{n−1}, F_{4}, G_{2}, G_{2}, respectively, of which only BC_{m} is non-reduced.)

To prove the above statement, we need a more explicit description of an Iwahori subgroup of G(K), following [8], section 4. Let S be a maximal split torus in G_{K}. Since G_{K} is quasi-split, the centralizer T of S is a maximal torus in G_{K}, and there is a Borel subgroup B defined over K that contains T. Let Φ ⊂ X^{∗}(S) be the set of roots of G_{K} relative to S, and let U_{a} ⊂ G(K) be the unipotent subgroup corresponding to a ∈ Φ. We know that the root system Φ is reduced because we have arranged that G_{K} is not of type ^{2}A_{n} with n even.

Also, let K̃ be the Galois extension of K which corresponds to the kernel of the action of the Galois group Gal(\overline{K}/K) on X^{∗}(T), let Φ̃ ⊂ X^{∗}(T) be the set of roots of GK̃ relative to T, and let Ũ_{α} ⊂ G(K) be the unipotent subgroup corresponding to α ∈ Φ̃. We can choose isomorphisms x_{α} : K → Ũ_{α} which satisfy the compatibility conditions with the action of the Galois group Gal_{K} on Φ̃ needed to form a “Chevalley-Steinberg system,” by [8], 4.1.3. These define a valuation of the root datum (T (K̃), Ũ_{α} : α ∈ Φ̃), meaning a set of functions φ̃_{α} : Ũ_{α} → (−∞, ∞] satisfying certain properties, by

φ̃_{α}(x_{α}(u)) = ord_{p}u.

In particular, the subsets Ũ_{α,c} = φ_{α}^{−1}([c, ∞]) and Ũ_{α,c+} = φ_{α}^{−1}((c, ∞]) are subgroups of Ũ_{α} for all real numbers c.

The Chevalley system of the split group GK̃ determines a model G_{o_{K}} of G_{K} over the ring of integers oK̃ which is a Chevalley group; see [8], proof of 4.6.15. By the construction, the obvious integral model Ũ_{α,0} ≅ oK̃ of Ũ_{α} ≅ K is a closed subgroup of G_{o_{K}}.

The Chevalley-Steinberg system also determines a valuation of the root datum (T, U_{a} : a ∈ Φ), by [8], 4.2. The definition is simplest in the case we need here, where Φ is reduced. Namely, any element u of U_{a} can be written uniquely as a product

u = ∏_{α∈A} ũ_{α}

where α runs over the set A of roots in Φ̃ that restrict to a, ordered in some fixed way. The functions φ_{a} : U_{a} → (−∞, ∞] are defined by

φ_{a}(u) = inf_{α∈A} φ̃_{α}(ũ_{α}).

Moreover, since Φ is reduced, the Chevalley-Steinberg system of G(K̃) induces an isomorphism

x_{a} : L_{a} → U_{a}

for every root a ∈ Φ, where L_{a} ⊂ K̃ is the extension field of K corresponding to the subgroup of Gal(K̃/K) which fixes some root α ∈ Φ̃ that restricts to a. By [8], 4.2.2, the valuation of U_{a} is given in terms of this isomorphism by

φ_{a}(x_{a}(u)) = ord_{p}u

for u ∈ L_{a}. Combining the two descriptions, we can say that the subgroup U_{a,0} ⊂ G(K) is contained in the subgroup of G(K̃) generated by U_{α,0} for roots α ∈ Φ̃ restricting to a. Likewise, U_{a,0+} ⊂ G(K) is contained in the subgroup of G(K̃) generated by U_{α,(c(α)e_{K})^{−1}} for roots α ∈ Φ̃ restricting to a, where c(α) := |Gal(K̃/K)α| = 1, 2, or 3. This uses that L_{a} is an extension of degree c(α) of K. We should add that the roots α of Φ̃ which restrict to a given element of Φ form a single orbit under the Galois group, by Borel and Tits [3], 6.4(2).

The reason for the above comments is that we can define an Iwahori subgroup Iw of G(K) as the subgroup generated by the subgroups U_{a,0} for all positive roots a ∈ Φ, U_{a,0+} for all negative roots a ∈ Φ, and the maximal compact subgroup H of T (K), by [7], 6.4.2 and 7.2.6 (where we take x to be the origin of the affine space A corresponding to the given valuation φ). Since U_{a,0} and U_{a,0+} are pro-p groups, they map trivially into the quotient group Iw/Iw_{u}, of order prime to p. So the pro-p radical Iw_{u} of Iw is the subgroup of G(K) generated by the subgroups U_{a,0} for positive roots a ∈ Φ, U_{a,0+} for negative roots a ∈ Φ, and the maximal pro-p subgroup H_{u} of T (K). By the previous paragraph, it follows that Iw_{u} is contained in the subgroup of G(K̃) generated by Ũ_{α,0} for positive roots α ∈ Φ̃, Ũ_{α,(c(α)e_{K})^{−1}} for negative roots α ∈ Φ̃, and the group H̃_{u} := {x ∈ T (K̃) : x ≡ 1 (mod πK̃)}.

Let α_{1}, …, α_{n} be the simple roots of Φ̃. We have assumed that p−1 > h(G_{K})e_{K}, which means (since G_{K} is not of type^{2}A_{n} with n even) that p−1 > (1+∑ r_{i})c(α) for all positive roots α = ∑ r_{i}α_{i} in Φ̃. Equivalently, for all positive roots α = ∑ r_{i}α_{i},

\frac{∑ r_{i}}{p − 1} < \frac{1}{c(α)e_{K}} − \frac{1}{p − 1}.

It follows that there is an element x ∈ X_{∗}(T) ⊗ Q such that ⟨x, α_{i}⟩ > 1/(p − 1) for all the simple roots α_{i} and ⟨x, α⟩ < (c(α)e_{K})^{−1} − (p − 1)^{−1} for all positive roots α.

We can find an element a ∈ T (L) for some finite extension L of K̃ whose absolute value is x ∈ X_{∗}(T) ⊗ Q. (We identify a cocharacter of T, f : G_{m} → T, with |f(p)|.) It follows that a^{−1}(Iw_{u})a is contained in the subgroup of G(L) generated by the subgroups U_{α,⟨x,α⟩} for positive roots α ∈ Φ̃, U_{−α,(c(α)e_{K})^{−1}−⟨x,α⟩} for negative roots −α, and H̃_{u} := {x ∈ T (K̃) : x ≡ 1 (mod πK̃)}. The inequalities on x imply that the first two subgroups are contained in the subgroups U_{α,1/(p−1)^{+}} for all roots α ∈ Φ̃. We check immediately from the table of values of h(G_{K}) that h(G_{K}) ≥ [K̃ : K], so that the assumption p−1 > h(G_{K})e_{K} implies that p−1 > eK̃, or in other words e−1 ; K̃ > (p − 1)^{−1}. It follows that the subgroup H̃_{u} is contained in {x ∈ T (K̃) : ord_{p}(x − 1) > (p − 1)^{−1}}.

In terms of the Chevalley group G_{o_{K}} extending G_{K} discussed earlier, these statements say that a^{−1}(Iw_{u})a is contained in the subgroup of elements x ∈ G(o_{L}) with x ≡ 1 (mod m) for some m ∈ o_{L} with ord_{p}(m) > (p − 1)^{−1}. This completes the proof, as explained earlier. QED

13 Open subgroups of SL_{2}Z_{p}

We will now show that the assumption that G_{Q_{p}} has rank at least 2 in Theorem 11.1 and Corollary 11.6 is essential. Some examples of nonzero Euler characteristics for open subgroups of SL_{2}Z_{p} follow already from Proposition 6.2, combined with Corollary 9.3. Those examples involve prime numbers p which are small compared to the representation considered. For example, if G is an open pro-p subgroup of SL_{2}Z_{p}, p ≥ 5, and M is the standard module M = (Z_{p})^{2}, then those results just say that χ(G, M) = 0. In this section, we will show that for any prime p ≥ 5, there is an open subgroup G in SL_{2}Z_{p}, necessarily not a pro-p group, such that χ(G, M) is not zero. We do this by computing all the homology groups H_{∗}(G, M) for a natural class of subgroups of SL_{2}Z_{p}.

Proposition 13.1 Let p ≥ 5 be a prime number, and G be the inverse image in SL_{2}Z_{p} of some subgroup Q of SL_{2}Z/p. Let M = (Z_{p})^{2} be the standard representation of G. Then the homology groups H_{∗}(G, M) are zero unless Q is either the trivial group, a cyclic group Z/3, a Sylow p-subgroup Z/p, or a semidirect product Z/3⋉Z/p. In those four cases, the homology groups H_{∗}(G, M) are F_{p}-vector spaces, zero except in degrees 0, 1, 2, of dimensions

2, 4, 2 if Q = 1
0, 2, 0 if Q ≅ Z/3
1, 2, 1 if Q ≅ Z/p
0, 2, 0 if Q ≅ Z/3 ⋉ Z/p.

So the Euler characteristic χ(G, M) is 0 unless Q is isomorphic to Z/3 or Z/3⋉Z/p, in which case it is −2.

In particular, for every p ≥ 5, the group SL_{2}Z/p contains a subgroup Q of order 3, and the Proposition implies that the inverse image G of Q in SL_{2}Z_{p} has χ(G, M) equal to −2, not zero. This is the counterexample described above. For p ≡ 1 (mod 3), SL_{2}Z/p also has a subgroup isomorphic to Z/3 ⋉ Z/p, giving another counterexample.

Proof. We first prepare to analyze subgroups Q of SL_{2}Z/p of order prime to p. Let G_{1} be the congruence subgroup

ker(SL_{2}Z_{p} → SL_{2}Z/p).

By Corollary 10.2, the homology groups H_{∗}(G_{1}, M) are F_{p}-vector spaces, zero except in degrees 0, 1, 2, of dimensions 2, 4, 2. Moreover, the action of SL_{2}Z/p on these groups is given by

H_{0}(G_{1}, M) = M_{F_{p}}
H_{1}(G_{1}, M) = 𝔰𝔩_{2}F_{p} − M_{F_{p}}
H_{2}(G_{1}, M) = ∧^{2}𝔰𝔩_{2}F_{p} − 𝔰𝔩_{2}F_{p} + M_{F_{p}}

in the Grothendieck group of finite p-torsion SL_{2}Z/p-modules. Using the representation theory of SL_{2}, we compute that these homology groups are M_{F_{p}}, S^{3}M_{F_{p}}, M_{F_{p}} in this Grothendieck group. In fact, these SL_{2}Z/p-modules are simple, since p ≥ 5, and so the homology groups H_{∗}(G_{1}, M) are actually isomorphic to these SL_{2}Z/p-modules. The restrictions of these modules to the diagonal torus (Z/p)^{∗} in SL_{2}Z/p have the form

H_{0}(G_{1}, M) = L^{−1} + L
H_{1}(G_{1}, M) = L^{−3} + L^{−1} + L + L^{3}
H_{2}(G_{1}, M) = L^{−1} + L,

where L is the standard 1-dimensional representation over F_{p} of the group (Z/p)^{∗}. The restrictions of these modules to a non-split torus ker(F_{p^{2}}^{∗} → F_{p}^{∗}) in SL_{2}Z/p have the same form, after extending scalars from F_{p} to F_{p^{2}}.

Let Q be a subgroup of SL_{2}Z/p of order prime to p, and let G be its inverse image in SL_{2}Z_{p}. Then the homology groups H_{∗}(G, M) are the coinvariants of Q acting on H_{∗}(G_{1}, M). Since every element of SL_{2}Z/p of order prime to p belongs to some torus, possibly non-split, the calculation of H_{∗}(G_{1}, M) shows that H_{∗}(G, M) = 0 if Q contains any elements of order not equal to 1 or 3. The Sylow 3-subgroup of SL_{2}Z/p is cyclic, so this leaves only the cases Q = 1 and Q ≅ Z/3. We read off from the calculation of H_{∗}(G_{1}, M) that the F_{p}-vector spaces H_{∗}(G, M) have dimension 2, 4, 2 for Q = 1 and 0, 2, 0 for Q ≅ Z/3, as we want.

Next, let Iw_{u} be the inverse image in SL_{2}Z_{p} of the strictly upper-triangular matrices in SL_{2}Z/p. Since p ≥ 5, the group Iw_{u} has a valuation as described in the proof of Corollary 9.3, and M has a compatible saturated valuation. So we have a spectral sequence

H_{∗}(gr Iw_{u}, gr M) ⇒ H_{∗}(Iw_{u}, M)

as in the proof of Theorem 9.1. Moreover, the diagonal subgroup Z_{p}^{∗} ⊂ SL_{2}Z_{p} normalizes Iw_{u}, preserving its valuation, and acts compatibly on M, so it acts on this spectral sequence. The Lie algebra homology is easy to compute, and the spectral sequence degenerates for degree reasons. The result is that the groups H_{∗}(Iw_{u}, M) are F_{p}-vector spaces of dimensions 1, 2, 1, on which Z_{p}^{∗} acts by

H_{0}(Iw_{u}, M) = L^{−1}
H_{1}(Iw_{u}, M) = L^{−3} + L^{3}
H_{2}(Iw_{u}, M) = L.

Here L is the standard 1-dimensional representation over F_{p} of the quotient group (Z/p)^{∗} of Z_{p}^{∗}.

Let Q be any subgroup of SL_{2}Z/p of order a multiple of p. By conjugating Q, we can assume that it contains the Sylow p-subgroup U ≅ Z/p of strictly uppertriangular matrices. The normalizer of U in SL_{2}Z/p is the Borel subgroup B = (Z/p)^{∗} ⋉ Z/p. Let G be the inverse image of Q in SL_{2}Z_{p}; then H_{∗}(G, M) is a quotient of the coinvariants of Q ∩ (Z/p)^{∗} on H_{∗}(Iw_{u}, M). Using the calculation of H_{∗}(Iw_{u}, M), it follows that H_{∗}(G, M) = 0 unless Q ∩ (Z/p)^{∗} has order 1 or 3. So suppose that Q ∩ (Z/p)^{∗} has order 1 or 3. Then Q is contained in the normalizer B of U in SL_{2}Z/p, since any subgroup of SL_{2}Z/p which contains U but is not contained in B must contain two distinct subgroups of order p, hence the subgroup they generate, which is the whole group SL_{2}Z/p. Thus, either Q = U or Q ≅ Z/3⋉U, and we can read off H_{∗}(G, M) in these two cases as the coinvariants of Q/U on H_{∗}(Iw_{u}, M). The dimensions of the F_{p}-vector spaces H_{∗}(G, M) are 1, 2, 1 for Q = U and 0, 2, 0 for Q ≅ Z/3 ⋉ U. QED

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