ETALE APPROXIMATIONS AND THE MOD l COHOMOLOGY OF GL_{n}

MARIAN FLORIN ANTON

1. Introduction

Let l be an odd prime number and O a ring of integers that contains the l^{th} roots of unity, has only one prime ideal above l and its Picard group has no l-torsion. Let A be the ring O[\frac{1}{l}] regarded as a fixed subring of the field ℂ of complex numbers and consider the etale approximation map BGL_{n}(A) ⟶^{χ_{n}} X_{n}(A) at the prime l (see 11). Here, BGL_{n}(A) means the classifying space of the group of n × n invertible matrices over A.

In this setting, it is conjectured (see [5]) that for n = ∞ the map χ_{n} induces an isomorphism on mod l cohomology. The goal of this article is to study the unstable version of this conjecture. Our main result is the following:

Theorem 1. The map χ_{n} does not induce an isomorphism on mod l cohomology for n sufficiently large.

Date: October 5, 1998.

As an example of a prime l and a ring A for which χ_{2} does induce an isomorphism we can cite our earlier result [1] 5.2:

Theorem 2. If A = ℤ[\frac{1}{3}, \sqrt[3]{1}] then χ_{2} induces an isomorphism on mod 3 cohomology.

The article is organized as follows. In §2, we prepare a number theoretical ingredient needed later. In §3 we study the etale approximation BGL_{n}(A) ⟶^{χ_{n}} X_{n}(A) at the prime l and give a homotopical description of X_{n}(A) (see 21). The proof of theorem 1 is given in §4 based on 28.

The author is very grateful to William Dwyer for many enlightening discussions about this subject.

Notation 3. Throughout the entire article we reserve the letter A for a ring that satisfies the prescribed hypotheses. Also, we denote H^{∗}(−, 𝔽_{l}) by H^{∗}(−) whenever it is not otherwise stated, and we keep the notation 2r_{2} for the number of complex embeddings of A.

2. A Useful Lemma

The goal of this section is to prove lemma 4 that will be needed later. Its proof is independent from the rest of the article.

Lemma 4. There is a finite set of prime ideals 𝔭_{1}, 𝔭_{2}, ..., 𝔭_{r} in the ring A such that the natural map

A^{×}/(A^{×})^{l} → ∏_{i=1}^{r} Â_{𝔭_{i}}^{×}/(Â_{𝔭_{i}}^{×})^{l}

is a group isomorphism, where Â_{𝔭} is the 𝔭-adic completion of A with respect to the prime ideal 𝔭 and r = r_{2} + 1.

Proof. Let s_{1}, s_{2}, ..., s_{n} be a set of representatives in A^{×} for all the nontrivial elements of A^{×}/(A^{×})^{l}. We can choose this set to be finite in agreement with the Dirichlet Unit Theorem. We claim that for each i between 1 and n, there exists at least one prime ideal 𝔮_{i} of A such that s_{i} is not an l-power in Â_{𝔮_{i}}.

Indeed, suppose by contradiction that for a given i between 1 and n, and every prime ideal 𝔮 of A, s_{i} is an l^{th} power in Â_{𝔮}. In other words, if K is the field of fractions of A, then for each prime ideal 𝔮 of A there exists at least one prime ideal 𝔔 of A[\sqrt[l]{s_{i}}] over 𝔮, such that

{ \atop K[\sqrt[l]{s_{i}}]}_{𝔔} = K̂_{𝔮}

where on the left hand side it is the completion with respect to 𝔔 and on the right hand side with respect to 𝔮. Observe that, because \sqrt[l]{1} is in K, the field extension K[\sqrt[l]{s_{i}}]/K is a Galois extension and the Galois group of this extension after completing with respect to 𝔔 is the

4 MARIAN FLORIN ANTON

decomposition group D_{𝔔/𝔮} of 𝔔 over 𝔮, [13] p.179. Therefore, we can reformulate our conclusion by saying that in particular, the following set of prime ideals

\{𝔮 : 𝔮 is unramified in A[\sqrt[l]{s_{i}}] with D_{𝔔/𝔮} = 1 for some 𝔔\}

has the density 1. But, according to the Tchebotarev Theorem, the density of this set of prime ideals is [K[\sqrt[l]{s_{i}}] : K]^{−1} and this number is < 1 because s_{i} is not an l^{th} power in K. This contradiction shows that the assumption made at the begining is false, proving our claim.

Also, we have that ([13], p.146)

Â_{𝔮_{i}}^{×} = 𝔽_{q_{i}}^{×} × pro 𝔮_{i}-group

where 𝔽_{q_{i}} is the residual field of A at 𝔮_{i}, and l ≢ 0 mod 𝔮_{i}, because l is a unit in A. Therefore, the image of s_{i} via the canonical map A → Â_{𝔮_{i}} generates Â_{𝔮_{i}}^{×}/(Â_{𝔮_{i}}^{×})^{l} ≈ ℤ/l. In this way, it follows not only that the natural map

A^{×}/(A^{×})^{l} → ∏_{i=1}^{n} Â_{𝔮_{i}}^{×}/(Â_{𝔮_{i}}^{×})^{l}

is injective, but also that it is a linear map between two vector spaces over 𝔽_{l}, given by an r × n matrix, where r = r_{2} + 1 is given by the Dirichlet Unit Theorem. By finding an r × r invertible minor of this matrix we can choose in fact a finite set of prime ideals 𝔭_{1},..., 𝔭_{r} among 𝔮_{1},..., 𝔮_{n}, such that the corestriction of the above map is an isomorphism as stated in our lemma.

3. The Etale Approximation to BGL_{n}(A) The goal of this section is to describe the etale approximation X_{n}(A)

to BGL_{n}(A) at the prime l which comes together with a natural map

χ_{n} : BGL_{n}(A) → X_{n}(A). (5)

After recalling the concepts of “etale topological type” and “etale approximation” we show that X_{n}(A) can be expressed up to homotopy in terms of X_{n}(𝔽_{p})’s for various residual fields 𝔽_{p}’s of the ring A.

A simplicial object in a category C is a contravariant functor from the category Δ of standard simplices \overline{n} = \{0, 1, 2, ..., n\}, n ≥ 0, to C. With the natural transformations of functors as morphisms, the simplicial objects of C form a new category simplicial-C. A pro-object in a category C is a functor from a small left filtering category to C ([3], III, 8.1). Similarly, the pro-objects of C form a new category pro-C.

The etale topological type as defined in Friedlander [7], p. 36, is a covariant functor ( )_{et} from the category of locally noetherian simplicial schemes to the category of pro-simplicial sets. Recall that the category of schemes (as defined in [8], p. 74) contains the opposite category of

6 MARIAN FLORIN ANTON

commutative rings with identity. In particular, if a noetherian ring R is regarded as an affine scheme X = Spec(R) and a scheme X as a simplicial scheme represented by the constant functor \overline{n} ↦ X, we can apply ( )_{et} to the ring R. Also, the category of pro-simplicial sets can be regarded as a generalization of the category of CW -complexes [2]. In this context, we have the following basic examples ([7], 4.5, [2], p. 115, 125, [6], 3.2).

6. pt → (ℂ)_{et} is a homotopy equivalence, where pt is a point.

7. (𝔽_{q})_{et} → (R)_{et} is a homotopy equivalence induced by the canonical map from a discrete valuation ring R to its residual finite field 𝔽_{q}.

8. S^{1} → (𝔽_{q})_{et} induces an isomorphism on mod l cohomology if the finite field 𝔽_{q} has the characteristic ≠ l and the given map sends the generator of π_{1}(S^{1}) to the Frobenius element of π_{1}((𝔽_{q})_{et}). Here, S^{1} is the one dimensional sphere.

Let L = ℤ[\frac{1}{l}] and GL_{n,L} the group scheme over Spec(L) defined by the ring L[x_{ij}; 1 ≤ i, j ≤ n][t]⧸(t · det(x_{ij}) − 1). Next, let BGL_{n,L} be the (classifying) simplicial scheme over Spec(L) defined by sending \overline{n} to the n-fold fibre product of GL_{n,L} with itself over Spec(L) [7], p. 8. As in [5], p. 250 we can naturally associate a pro-object of Kan fibrations {T_{β} → V_{β}} to the fibrewise l-completion of (BGL_{n,L})_{et}

ETALE APPROXIMATIONS 7

over (L)_{et}. Let U → (L)_{et} be a map of pro-simplicial sets, and think of U as a pro-simplicial set sending each object α of a left filtering category to the simplicial set U_{α}. For each induced diagram U_{α} → V_{β} ← T_{β} of simplicial sets we define the simplicial set Hom(U_{α}, T_{β})_{V_{β}} by sending \overline{n} to the set of natural transformations U_{α} ⊗Δ[n] → T_{β} over V_{β}. (U_{α} ⊗ Δ[n] is the simplicial set sending \overline{m} to the disjoint union of copies of U_{α,m}, the image of \overline{m} via U_{α}, indexed by maps \overline{m} → \overline{n} in the category Δ). Finally, we define as in [5], p. 250,

(9) Hom_{l}(U, (BGL_{n,L})_{et})_{(L)_{et}} ≡ ho\underleftarrow{lim}_{β} \underrightarrow{lim}_{α} Hom(U_{α}, T_{β})_{V_{β}}

This is a simplicial set with a distinguished point induced by the “identity” section Spec(L) → GL_{n,L} of the group scheme GL_{n,L} over Spec(L).

Notation 10. The connected component of the distinguished point in 9 will be denoted by X_{n}(U).

In particular, if R is a noetherian L-algebra, then we get a map (R)_{et} → (L)_{et} which allows us to define X_{n}((R)_{et}). This simplicial set will be simply denoted by X_{n}(R). With this notation, according to [5] 2.5 and p. 278, there exists a transformation

BGL_{n}(R) ⟶^{χ_{n}} X_{n}(R)

which is natural with respect to the functoriality of both sides in R.

Definition 11. The pair (χ_{n}, X_{n}(R)) is called the “etale approximation” of BGL_{n}(R) at the prime l.

Recall that the goal of this section is to describe X_{n}(A). In order to do this, the first step is to start with the fixed embedding A ⊂ ℂ and, by elementary field theory, to embed each Â_{𝔭_{i}} in ℂ, i = 1, ..., r, in such a way that the following diagrams are commutative:

\begin{matrix}A & ⟶ & ℂ \\ ↓ & & ∥ \\ Â_{𝔭_{i}} & ⟶ & ℂ \end{matrix} (12)

where the prime ideals 𝔭_{i}’s of A are those the existence of which is proved in 4 (in particular, r = r_{2} + 1).

By applying the etale topological type functor ( )_{et} to all of these commutative diagrams, we get commutative diagrams of pro-spaces:

\begin{aligned}(A)_{et} & ⟵ & (ℂ)_{et} \\ ↑ & & ∥ \\ (Â_{𝔭_{i}})_{et} & ⟵ & (ℂ)_{et} \end{aligned} (13)

where (ℂ)_{et} is contractible by 6. Hence, we get a map

(A)_{et} ← ⋁_{i=1}^{r}(Â_{𝔭_{i}})_{et} (14)

from the homotopy push out of all of the maps (ℂ)_{et} → (Â_{𝔭_{i}})_{et}, i = 1, ..., r, to (A)_{et}.

Proposition 15. The above map 14 induces an isomorphism on the mod l cohomology.

Proof. Both pro-spaces in 14 being connected, the isomorphism on the zero dimensional cohomology is trivial. Also, recall that for any noetherian ring R we have a natural isomorphism ([7], p. 49)

H^{∗}((R)_{et}) ≈ H_{et}^{∗}(Spec(R), 𝔽_{l})

and the following short exact sequence ([12] 4.11)

1 → R^{×}/(R^{×})^{l} → H_{ét}^{1}(Spec(R), 𝔽_{l}) → l-torsion of P ic(R) → 0

where 𝔽_{l} is the constant sheaf and H_{et}^{∗} is the etale cohomology (see [12], p. 84). Because in our case, P ic(R) does not have l-torsion for R = A or Â_{𝔭} with 𝔭 any prime ideal of A, by the assumptions made about A (see also [13] p. 74), it follows that

H^{1}((A)_{et}) ≈ A^{×}/(A^{×})^{l} ≈ 𝔽_{l}^{r_{2}+1} (16)

10 MARIAN FLORIN ANTON

and

H^{1}(⋁_{i=1}^{r}(Â_{𝔭_{i}})_{et}) ≈ ∏_{i=1}^{r} H^{1}((Â_{𝔭_{i}})_{et}) ≈ ∏_{i=1}^{r} Â_{𝔭_{i}}^{×}/(Â_{𝔭_{i}}^{×})^{l}

Hence, the map induced by 14 on the first mod l cohomology can be identified by naturality with the map of lemma 4 which is an isomorphism by construction.

In order to finish the proof, it is enough to show that both prospaces in 14 have trivial mod l cohomology in dimensions higher than 1. While this fact is obvious for ⋁_{i=1}^{r}(Â_{𝔭_{i}})_{et}, according to 7 and 8, for (A)_{et} it follows from Mazur [11]. Indeed, let β be the unique prime ideal of O over l (see the Introduction). Then we have the following exact sequence ([11], p. 540)

... → H_{et}^{i}(Spec(O), 𝔽_{l}) → H_{et}^{i}(Spec(A), 𝔽_{l}) → E_{β}^{3−i−1} → ... (17)

where ([11], p. 540)

E_{β}^{i} ≈ \{ \begin{matrix}𝔽_{l} & if i = 0 \\ Ô_{β}^{×}/(Ô_{β}^{×})^{l} & if i = 1 \\ 0 & otherwise \end{matrix}

and ([11], p. 539)

H_{et}^{i}(Spec(O), 𝔽_{l}) ≈ \{ \begin{matrix}𝔽_{l} & if i = 0, 3 \\ O^{×}/(O^{×})^{l} & if i = 2 \\ 0 & otherwise \end{matrix}

ETALE APPROXIMATIONS 11

in agreement with the fact that P ic(O) has no l-torsion and O contains \sqrt[l]{1}. Moreover, the map E_{β}^{0} → H_{et}^{3}(Spec(O), 𝔽_{l}) which apears in 17 is surjective (compare with [11], p. 241) and hence it immediately follows that H_{et}^{i}(Spec(A), 𝔽_{l}) = 0 for i > 2.

Finally, O^{×}/(O^{×})^{l} ≈ 𝔽_{l}^{r_{2}} and Ô_{β}^{×}/(Ô_{β}^{×})^{l} ≈ 𝔽_{l}^{2r_{2}+1} by the Dirichlet Unit Theorem and [13], p. 146. Hence, 17 and 16 give the following exact sequence

0 → 𝔽_{l}^{r_{2}+1} → 𝔽_{l}^{2r_{2}+1} → 𝔽_{l}^{r_{2}} → H_{et}^{2}(Spec(A), 𝔽_{l}) → 𝔽_{l} → 𝔽_{l} → 0

from which it follows that H_{et}^{2}(Spec(A), 𝔽_{l}) = 0 as it was stated.

The second step is to apply the etale approximation functor X_{n} to all of the commutative diagrams 13 and to study the induced map

X_{n}(A) → X_{n}(Â_{𝔭_{1}}, ..., Â_{𝔭_{r}}; ℂ) (18)

from X_{n}(A) to the homotopy pull back of all of the maps X_{n}(Â_{𝔭_{i}}) → X_{n}(ℂ), i = 1, ..., r, denoted by X_{n}(Â_{𝔭_{1}}, ..., Â_{𝔭_{r}}; ℂ). In order to do this, we can use the following known results:

Proposition 19. ([4] 2.7) Let R be a notherian ℤ[\frac{1}{l}]-algebra containing \sqrt[l]{1} and let U → (R)_{et} be a map of pro-simplicial sets which induces

12 MARIAN FLORIN ANTON

an isomorphism on mod l cohomology. Then the induced etale approximation map

X_{n}(R) → X_{n}(U)

is a homotopy equivalence.

Proposition 20. ([1] 2.13) There are natural homotopy equivalences of 𝔽_{l}- complete simplicial sets

(𝔽_{l})_{∞}BGL_{n}(𝔽_{p}) → X_{n}(Â_{𝔭})
(𝔽_{l})_{∞}BGL_{n}(ℂ_{top}) → X_{n}(ℂ)

where 𝔭 is any prime ideal of A and 𝔽_{p} is the residual field of A at 𝔭.

Here, (𝔽_{l})_{∞}X means the 𝔽_{l}- completion of the simplicial set X in the sense of Bousfield and Kan [3]. It follows that

Proposition 21. The above map 18 is a homotopy equivalence.

Proof. The idea of the proof is to factorize the map 18 as follows

X_{n}(A) → X_{n}(⋁_{i=1}^{r}(Â_{𝔭_{i}})_{et}) → X_{n}(Â_{𝔭_{1}}, ..., Â_{𝔭_{r}}; ℂ)

where the first map is the etale approximation map induced by 14, and according to 15 and 19 it is a homotopy equivalence, while the second map is given by the universal property of the homotopy pull back.

ETALE APPROXIMATIONS 13

All we have to prove is that the second map is also a homotopy equivalence. This fact can be immediately deduced from [1] 2.9 by induction on r.

Corollary 22. If χ_{n} induces an isomorphism on the mod l cohomology then it induces a homotopy equivalence

(𝔽_{l})_{∞}BGL_{n}(A) → X_{n}(A)

Proof. According to 21 and [3], I, §5, it is enough to show that

X_{n}(Â_{𝔭_{1}}, ..., Â_{𝔭_{r}}; ℂ)

is an 𝔽_{l}-complete simplicial set. This fact follows at once from [1] 2.13 by induction on r.

4. Proof of Theorem 1

Suppose that χ_{n}^{∗} is an isomorphism for n = n_{0}. According to 22, it follows that χ_{n_{0}} induces a homotopy equivalence

[Bμ_{m}, (𝔽_{l})_{∞}BGL_{n_{0}}(A)] ≈ [Bμ_{m}, X_{n_{0}}(A)] (23)

where [ , ] stands for the homotopy classes of unpointed maps, m is any natural number, and μ_{m} is the cyclic group of order l^{m}.

14 MARIAN FLORIN ANTON

Next, observe that by 21 we have [Bμ_{m}, X_{n_{0}}(A)] ≈ [Bμ_{m}, X_{n_{0}}(Â_{𝔭_{1}}, ..., Â_{𝔭_{r}}; ℂ)] (24)

Moreover, we can think of the space of unpointed maps

Map(Bμ_{m}, X_{n_{0}}(Â_{𝔭_{1}}, ..., Â_{𝔭_{r}}; ℂ))

as the homotopy pull back of all of the maps

Map(Bμ_{m}, X_{n_{0}}(Â_{𝔭_{i}})) → Map(Bμ_{m}, X_{n_{0}}(ℂ))

induced by 12. Lemma 25. The induced map

[Bμ_{m}, X_{n_{0}}(Â_{𝔭_{1}}, ..., Â_{𝔭_{r}}; ℂ)] → ∏_{i=1}^{r}[Bμ_{m}, X_{n_{0}}(Â_{𝔭_{i}})]

is injective. Proof. By using the homotopy exact sequence associated to a pull back, it is enough to check that Map(Bμ_{m}, X_{n_{0}}(ℂ)) is simply connected and this was shown in [4], 3.6.

Finally, by 20, we have [Bμ_{m}, X_{n_{0}}(Â_{𝔭_{i}})] ≈ [Bμ_{m}, (𝔽_{l})_{∞}BGL_{n_{0}}(𝔽_{p_{i}})] (26) and also, by [14], p. 124, all of the groups GL_{n_{0}}(A) and GL_{n_{0}}(𝔽_{p_{i}}), i = 1, 2, ..., r, satisfy the finiteness condition of the following

ETALE APPROXIMATIONS 15

Theorem 27. (Lee [10]) Let Φ be a group of virtually finite cohomological dimension, and let μ be a finite l-group. Then the natural map

Rep(μ, Φ) → [Bμ, (𝔽_{l})_{∞}BΦ]

is a bijection.

Here Rep(μ, Φ) is the set of conjugacy classes of group homomorphisms μ → Φ.

From 23, 24, 25, 26 and 27 it follows that the map

τ_{m,n_{0}} : Rep(μ_{m}, GL_{n_{0}}(A)) → ∏_{i=1}^{r} Rep(μ_{m}, GL_{n_{0}}(𝔽_{p_{i}}))

induced by the canonical maps A → A/𝔭_{i} ≈ 𝔽_{p_{i}} is injective for any m ≥ 0. At this point, we conclude the proof of theorem 1 by referring to the following

Lemma 28. There exists n_{1} such that for every n_{0} ≥ n_{1} we can find at least one m_{0} for which the above map τ_{m,n_{0}} is not injective for m = m_{0}.

Proof. We distinguish between two cases: Case 1. A is a principal ideal domain; and Case 2. A is not such a ring.

Case 1. Suppose by contradiction that for each n_{1} there exists n_{0} ≥ n_{1} such that τ_{m,n_{0}} is injective for all m ≥ 0. Let m ≥ a be fixed for a moment, where a ≥ 1 is the maximum exponent having the

16 MARIAN FLORIN ANTON

property that A contains the l^{a}-th root of unity. Denote A[ \sqrt[l^{m}]{1}] by A_{m} and let P be any prime ideal of A_{m} not dividing p_{1} · ... · p_{r}. Choose an embedding of μ_{m} in A_{m}^{×} and use it to define a module structure of P, A_{m} over the groupal ring A[μ_{m}] by letting μ_{m} act on P, A_{m} by multiplication. Let n_{1} be the degree of the minimal polynomial of \sqrt[l^{m}]{1} over A and n_{0} = n_{1} + n′, where n′ ≥ 0 is chosen such that τ_{m,n_{0}} is injective. Let A^{n′} be the trivial A[μ_{m}]-module of rank n′ as a free module over A. Because A is a principal ideal domain, both P and A_{m} are free modules over A of the same rank n_{1}. Therefore, the A[μ_{m}]- modules M = P ⊕ A^{n′} and N = A_{m} ⊕ A^{n′} can be regarded as elements of Rep(μ_{m}, GL_{n_{0}}(A)).

Also, for each i between 1 and r, P has an index in A_{m} relative prime to p_{i}, by the way P was chosen. Therefore, 𝔽_{p_{i}} ⊗_{A} P is isomorphic to 𝔽_{p_{i}} ⊗_{A} A_{m} as 𝔽_{p_{i}}[μ_{m}]-modules, and hence τ_{m,n_{0}}(M) = τ_{m,n_{0}}(N). Meanwhile, it is known that l is divisible by (1 − \sqrt[l^{m}]{1}) in A_{m} and l is a unit in A. Therefore, by choosing a generator σ of μ_{m} we have (1−σ)M = P and (1−σ)N = A_{m}. Because τ_{m,n_{0}} is injective, it follows that P is isomorphic to A_{m} as A[μ_{m}]-modules. In other words, P is a principal ideal in A_{m}. By the Tchebotarev Theorem, we conclude that A_{m} must be a principal ideal domain. But this is not true for m sufficiently large, according to [9], p. 298, and [15], p. 44.

ETALE APPROXIMATIONS 17

Case 2. Let I be a nonprincipal prime ideal of A not dividing p_{1}·...·p_{r}. The existence of such an ideal I follows from the Tchebotarev Theorem. Suppose that I^{h} is principal for some exponent h > 1. Consider M := I^{h−1}⊕I and N := A⊕A as A[μ_{1}]-modules by letting μ_{1} act by multiplication on the second and trivially on the first summands. Observe that M is a free module of rank 2 over A because A is a Dedekind ring, and in this way we can regard M and N as elements in Rep(μ_{1}, GL_{2}(A)). Because I has index prime to p_{i}, we conclude that if τ_{1,2} is injective, then I must be a principal ideal, contradiction.

References

[1] Anton, M.F.: On a conjecture of Quillen at the prime 3, to appear in J. Pure Appl. Algebra

[2] Artin, M.; Mazur, B.: Etale Homotopy, Lecture Notes in Math. 100, Springer, Berlin, 1969

[3] Bousfield, A.K.; Kan, D.M.: Homotopy Limits, Completions and Localizations, Lecture Notes in Math. 304, Springer, Berlin, 1972

[4] Dwyer, W.G.: Exotic cohomology for GL_{n}(ℤ[\frac{1}{2}]), Proc. Amer. Math. Soc. 126 (1998), 2159-2167

[5] Dwyer, W.G.; Friedlander, E.M.: Algebraic and etale K-theory, Trans. Amer. Math. Soc. 272 (1985), 247-280

[6] Dwyer, W.G.; Friedlander, E.M.: Topological models for arithmetic, Topology 33 (1994), 1-24

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[7] Friedlander, E.M.: Etale Homotopy of Simplicial Schemes, Princeton Univ. Press, Princeton, 1982

[8] Hartshorne, R.: Algebraic Geometry, Springer, New York, 1993

[9] Lang, S.: Cyclotomic Fields I and II, Springer, New York, 1990

[10] Lee, C-N.: On the unstable homotopy type of the classifying space of a virtually torsion-free group, Math. Z. 224 (1997), 253-253

[11] Mazur, B.: Notes on etale cohomology of number fields, Ann. Scient. Éc. Norm. Sup., 4^{e} série, 6 (1973), 521-552

[12] Milne, J.S.: Etale Cohomology, Princeton Univ. Press, Princeton, 1980

[13] Neukrich, J.: Algebraische Zahlentheorie, Springer, Berlin, 1992

[14] Serre, J-P.: Cohomologie des groups discrets, Princeton Univ. Press, Princeton, 1971, pp.77-169

[15] Washington, L.: Introduction to Cyclotomic Fields, Springer, Berlin, 1982

University of Notre Dame, Notre Dame, IN 46556, U.S.A., and Institute of Mathematics of the Romanian Academy, Bucharest, Romania

  E-mail address: manton1@nd.edu