PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 2159–2167 S 0002-9939(98)04279-8

EXOTIC COHOMOLOGY FOR GL_{n}(ℤ[1/2])

W. G. DWYER

(Communicated by Thomas Goodwillie)

Abstract. We show that for n = 32 the mod 2 group cohomology of GL_{n}(ℤ[1/2]) is not detected on diagonal matrices.

§1. Introduction

Let Λ denote the ring Z[1/2], G_{n} the group GL_{n}(Λ) of invertible n × n matrices over Λ, and D_{n} the group of diagonal matrices in G_{n}. The inclusion D_{n} → G_{n} induces a classifying space map ι_{n} : BD_{n} → BG_{n} and a cohomology homomorphism

ι_{n}^{∗} : H^{∗}(BG_{n}; F_{2}) → H^{∗}(BD_{n}; F_{2}) .

Say that the mod 2 cohomology of G_{n} is detected on diagonal matrices if ι_{n}^{∗} is injective. In [15, 14.7] Quillen made a conjecture which specializes in the case of the ring Λ to the following statement (see [9, p. 51]):

1.1 Conjecture. For any n ≥ 1 the mod 2 cohomology of G_{n} is detected on diagonal matrices.

There is some evidence for this conjecture. Mitchell [14] and Henn [10] have proved it for n ≤ 3. Voevodsky has announced a proof of the mod 2 Quillen-Lichtenbaum Conjecture for Z, and from [5] and [14] it follows that ι_{n}^{∗} is injective on the image of H^{∗}(BGL(Λ); F_{2}) → H^{∗}(BG_{n}; F_{2}). In particular, 1.1 is true in the stable range.

The aim of this paper, though, is to give a disproof of Conjecture 1.1.

1.2 Theorem. The mod 2 cohomology of G_{32} is not detected on diagonal matrices.

Given the remarks above, it is a consequence of 1.2 that there exists an element in the cohomology of G_{32} which is not in the image of H^{∗}(BGL(Λ); F_{2}). In fact, 1.2 is proved by a technique distantly related to the one which Quillen uses in [15, p. 592] to show that for various other number rings S and primes p the restriction map H^{∗}(BGL(S); F_{p}) → H^{∗}(BGL_{n}(S); F_{p}) is not surjective.

Further developments. Very recently, Henn and Lannes have improved upon 1.2 by showing that the mod 2 cohomology of G_{14} is not detected on diagonal matrices. Moreover, Henn proves in [11, 0.6] that the Poincaré series p_{n}(t) of the kernel of ι_{n}^{∗} has a pole at t = 1 of order n − n_{0} + 1, where n_{0} is the smallest natural number

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors November 14, 1996 and, in revised form, December 20, 1996.

1991 Mathematics Subject Classification. Primary 55N99, 20G16, 19D02, 57T99.

This research was partially supported by National Science Foundation Grant DMS95–05024.

such that ι_{n_{0}}^{∗} is not injective. Combining these results shows that for any n ≥ 14 the cohomology of G_{n} is far from being detected on diagonal matrices.

A generalization. Theorem 1.2 is a consequence of a result which is slightly more general. Recall that if P and G are groups and α, β : P → G are homomorphisms, then α is said to be conjugate to β if there is an element g ∈ G such that gαg^{−1} = β. Let ρ_{R} : G_{n} → GL_{n}(R) and ρ_{F_{3}} : G_{n} → GL_{n}(F_{3}) be the obvious homomorphisms. Two homomorphisms α, β : P → G_{n} are said to become conjugate over R (resp. become conjugate over F_{3}) if ρ_{R}α and ρ_{R}β (resp. ρ_{F_{3}}α and ρ_{F_{3}}β) are conjugate.

1.3 Theorem. Suppose that the mod 2 cohomology of G_{n} is detected on diagonal matrices. Let P be a finite 2-group with homomorphisms α, β : P → G_{n}. Then α is conjugate to β if and only if α becomes conjugate to β over R and over F_{3}.

To obtain 1.2 from 1.3 let μ_{n} denote the group of 2^{n}-th roots of unity. The smallest n with the property that the ideal class group of Λ(μ_{n}) is nontrivial is 6, and the degree of Λ(μ_{6}) over Λ (equivalently, the rank of Λ(μ_{6}) as a Λ-module) is ϕ(2^{6}) = 2^{5} = 32. Let P = μ_{6}, considered as a subgroup of Λ(μ_{6})^{×}, and let I be a nonprincipal ideal in Λ(μ_{6}). It is then not hard to use the multiplicative actions of P on I and on Λ(μ_{n}) itself to construct two nonconjugate homomorphisms P → G_{32} which become conjugate over R and over F_{3}.

The proof of 1.3 is homotopy theoretic. If X and Y are spaces, let [X, Y] denote the set of (unpointed) homotopy classes of maps X → Y; if P and G are groups, let {P, G} denote the set of conjugacy classes of homomorphisms P → G. The classifying space functor gives a bijection {P, G} ≅ [BP, BG] . For each n ≥ 1 we construct a space X_{n} together with a map

χ_{n} : BG_{n} → X_{n}

such that the following three statements hold.

1.4 Proposition. If the mod 2 cohomology of G_{n} is detected on diagonal matrices, then χ_{n}^{∗} : H^{∗}(X_{n}; F_{2}) → H^{∗}(BG_{n}; F_{2}) is an isomorphism.

1.5 Proposition. If χ_{n}^{∗} is an isomorphism, then for any finite 2-group P the map χ_{n} · (–) : [BP, BG_{n}] → [BP, X_{n}] is a bijection.

1.6 Proposition. Let P be a finite 2-group with homomorphisms α, β : P → G_{n}. Then χ_{n} · (Bα) is homotopic to χ_{n} · (Bβ) if and only if α becomes conjugate to β over R and over F_{3}.

Together, these imply 1.3.

Section 2 contains a description of the machinery which is used to construct the space X_{n}, §3 has proofs of the above three propositions, and §4 contains the derivation of 1.2 from 1.3. Throughout the paper, unspecified homology and cohomology is to be taken with mod 2 coefficients. The symbol B̂G denotes the 2-completion [1] of the classifying space of G. The usual topological groups GL_{n}(R) and GL_{n}(C) are denoted GL_{n}^{top}(R) and GL_{n}^{top}(C).

The idea for this paper originated in conversations with S. Mitchell, and the approach depends heavily on his calculations from [14]. The construction of X_{n} goes back to work with E. Friedlander [4]. Some of the arguments below can be generalized, but we have decided to concentrate on proving 1.2.

EXOTIC COHOMOLOGY FOR GL_{n}(ℤ[1/2]) 2161

§2. Étale homotopy theory

The space X_{n} promised in §1 is constructed using étale homotopy theory [8], which gives a covariant mechanism for assigning a space (more accurately a prospace) A_{ét} to any reasonable scheme or simplicial scheme A. We build X_{n} as a certain space of maps between two étale homotopy types (2.6), in imitation of the way in which GL_{n}(Λ) can be described as a certain set of maps between schemes.

Étale homotopy types. Here are some examples of étale homotopy types. In the examples, the symbol ℓ stands for a prime number.

2.1 Fields and complete local rings. If k is a field, then Spec(k)_{ét} is a prospace of type K(π, 1), where π is the Galois group over k of the separable algebraic closure of k. If S is a complete local ring with residue class field k, then the map Spec(k)_{ét} → Spec(S)_{ét} induced by S → k is an equivalence. For instance, Spec(C)_{ét} is contractible, Spec(R)_{ét} is equivalent to BZ/2, and both Spec(F_{3})_{ét} and Spec(Z_{3})_{ét} are equivalent to the profinite completion of a circle.

If S is a commutative ring, let GL_{n,S} denote the rank n general linear group scheme over S. Applying the usual bar construction to GL_{n,S} gives a classifying object BGL_{n,S} [8, 1.2], which is a simplicial scheme. See [8, §8] for the following three examples.

2.2 General linear groups over algebraically closed fields. If k is an algebraically closed field of characteristic zero, the pro-space (BGL_{n,k})_{ét} is equivalent to the profinite completion of the space BGL_{n}^{top}(C). If k is the algebraic closure of a finite field, then the ℓ-completion of (BGL_{n,k})_{ét} at any prime ℓ not equal to char(k) is equivalent to the Bousfield-Kan ℓ-completion tower {(Z/ℓ)_{s}BGL_{n}^{top}(C)}_{s}.

2.3 General linear groups over other fields. If k is a field of characteristic zero with algebraic closure k̄, then the natural sequence

(BGL_{n,k̄})_{ét} → (BGL_{n,k})_{ét} → Spec(k)_{ét}

is a fibration sequence of pro-spaces. In effect, if π is the Galois group of k̄ over k, (BGL_{n,k})_{ét} is the Borel construction of the natural action of π on (BGL_{n,k̄})_{ét}. If k is a finite field with algebraic closure k̄ and ℓ ≠ char(k), there is a similar fibration sequence

{(Z/ℓ)_{s}(BGL_{n,k̄})_{ét}}_{s} → {(Z/ℓ)_{s}^{•}(BGL_{n,k})_{ét}}_{s} → Spec(k)_{ét},

where {(Z/ℓ)_{s}^{•}(–)}_{s} denotes fibrewise ℓ-completion over Spec(k)_{ét}.

2.4 General linear groups over number rings. Suppose that R is a number ring, and that S is the ring R[1/ℓ]. Let k̄ denote either the algebraic closure of the quotient field of S, or the algebraic closure of one of the residue class fields of S (note that none of these residue fields have characteristic ℓ). Then the natural sequence

{(Z/ℓ)_{s}(BGL_{n,k̄})_{ét}}_{s} → {(Z/ℓ)_{s}^{•}(BGL_{n,S})_{ét}}_{s} → Spec(S)_{ét}.

is a fibration sequence. There are identical fibration sequences if S is replaced by a complete local ring of residue characteristic different from ℓ.

2.5 Number rings. The étale homotopy type of a number ring is not easy to pin down; see [5, 2.1] for a description of its untwisted “integral” homology. We will settle for a partial description of Spec(Λ)_{ét}, where, as usual, Λ = Z[1/2]. Choose an embedding Z_{3} → C. There are induced commutative diagrams of rings and of pro-spaces

\begin{aligned}C & ⟵ & R \\ ↑ & & ↑ \\ Z_{3} & ⟵ & Λ \end{aligned}  Spec(C)_{ét} ⟶ Spec(R)_{ét} ↓  ↓ Spec(Z_{3})_{ét} ⟶ Spec(Λ)_{ét} .

Since Spec(C)_{ét} is contractible, the diagram induces a map

Spec(Z_{3})_{ét} ∨ Spec(R)_{ét} → Spec(Λ)_{ét} .

Pick an equivalence BZ/2 → Spec(R)_{ét}, and a map S^{1} → Spec(Z_{3}) which sends the generator of π_{1}S^{1} to the Frobenius automorphism of F̄_{3} over F_{3}. What results is a map

S^{1} ∨ BZ/2 → Spec(Λ)_{ét} .

By [5], this map induces an isomorphism on mod 2 cohomology (cf. [6, §2]).

2.6 Étale approximations to BGL_{n}(S). Let S be an algebra over Λ. The homomorphism Λ → S induces a map Spec(S)_{ét} → Spec(Λ)_{ét}. We set ℓ = 2 and let BGL_{n}^{ét}(S) denote the basepoint component of the space which in [4, 2.3] is called Hom_{ℓ}(Spec(S)_{ét}, (BGL_{n,Λ})_{ét})_{Λ}; in other words, BGL_{n}^{ét}(S) is the basepoint component of the function space of maps over Spec(Λ)_{ét} from Spec(S)_{ét} to the fibrewise 2-completion of (BGL_{n,Λ})_{ét}. (The function space is pointed because the map BGL_{n,Λ} → Spec(Λ) has a natural section provided by the basepoint of the fibrewise bar construction.) Since BGL_{n}(S) can be identified as the basepoint component of the space of maps Spec(S) → BGL_{n,Λ} over Spec(Λ) [4, 4.2], functoriality gives a map

χ_{n,S} : BGL_{n}(S) → BGL_{n}^{ét}(S)

(see [4, 2.5, pf. of 4.2]). We will describe this map below in some particular cases.

2.7 Remark. Suppose that B is a space (i.e. a trivial pro-space) together with a map B → Spec(Λ)_{ét}. Let “holim” denote the homotopy inverse limit functor from pro-spaces to spaces [8, §6], and E be the fibration over B obtained by the following homotopy pullback diagram:

E ⟶ holim{(Z/2)_{s}^{•}(BGL_{n,Λ})_{ét}}_{s} ↓  ↓ B ⟶  holim Spec(Λ)_{ét} .

The homotopy fibre of E → B is B̂GL_{n}^{top}(C). Essentially by definition, the space Hom_{ℓ}(B, (BGL_{n,Λ})_{ét})_{Λ} is equivalent to the space of sections of the map E → B.

It follows from a spectral sequence argument [4, 2.11] that if S is a Λ-algebra and f : B → Spec(S)_{ét} is a map of pro-spaces which induces an isomorphism on mod 2 cohomology, then f induces an equivalence

BGL_{n}^{ét}(S) →^{≃} Hom_{ℓ}(B, (BGL_{n,Λ})_{ét})_{Λ} .

2.8 The complex numbers. Since Spec(C)_{ét} is contractible, it follows from 2.7 that BGL_{n}^{ét}(C) is equivalent to B̂GL_{n}^{top}(C). The same calculation works for any separably closed field of characteristic not 2. The map χ_{n,C} is essentially the composite of the usual map BGL_{n}(C) → BGL_{n}^{top}(C) with the 2-completion map.

2.9 The real numbers. Since Spec(R)_{ét} is equivalent to BZ/2, it follows from 2.7 that BGL_{n}^{ét}(R) is equivalent to the basepoint component of the space of sections of a fibration over BZ/2 with B̂GL_{n}^{top}(C) as the fibre. By naturality, this fibration is the one associated to the action of Z/2 on B̂GL_{n}^{top}(C) by complex conjugation. It follows from [7] that the natural map from B̂GL_{n}^{top}(R) to this space of sections is an equivalence. The map χ_{n,R} is the essentially the composite of the usual map BGL_{n}(R) → BGL_{n}^{top}(R) with the 2-completion map.

2.10 The field F̄_{3}. Consider the commutative diagram

BGL_{n}(F̄_{3}) ⟶ colim_{n} BGL_{n}(F̄_{3}) = BGL(F̄_{3})
χ_{n,F̄_{3}} ↓  ↓^{colim_{n}(χ_{n,F̄_{3}} )}
BGL_{n}^{ét}(F̄_{3}) ⟶  colim_{n} BGL_{n}^{ét}(F̄_{3})

in which the colimit maps are induced by the usual matrix block inclusions. The right hand vertical arrow induces an isomorphism on mod 2 cohomology (see [4, 4.5, 8.6]). The horizontal maps induce surjections on mod 2 cohomology; for the upper one see [16] and for the lower one 2.7. By diagram chasing, χ_{n,F̄_{3}} induces a surjection on mod 2 cohomology. Since BGL_{n}(F̄_{3}) and BGL_{n}^{ét}(F̄_{3}) have mod 2 cohomology rings which are abstractly isomorphic ([16], 2.7) and finite in each dimension, χ_{n,F̄_{3}} must be an isomorphism on mod 2 cohomology. It follows that BGL_{n}^{ét}(F̄_{3}) is equivalent to B̂GL_{n}(F_{3}).

2.11 The field F_{3}. There is a map S^{1} → Spec(F_{3})_{ét} which sends a generator of π_{1}(S^{1}) to the Frobenius automorphism ψ of F̄_{3} over F_{3}. This map is an isomorphism on cohomology with finite coefficients, in particular, on mod 2 cohomology. It follows from 2.7 that BGL_{n}^{ét}(F_{3}) is equivalent to the space of sections of a fibration over S^{1} with fibre B̂GL_{n}^{top}(C) ≃ BGL_{n}^{ét}(F̄_{3}) (2.8). By naturality, this is the fibration associated to the action of ψ on BGL_{n}^{ét}(F̄_{3}), and so its space of sections is (by definition) the homotopy fixed point set (BGL_{n}^{ét}(F̄_{3}))^{hψ}. Consider the commutative diagram of spaces with an action of ψ (the action in the left hand column is trivial):

BGL_{n}(F_{3}) ⟶^{u} BGL_{n}(F̄_{3}) ^{χ_{n,F_{3}}} ↓ ^{χ_{n,F̄_{3}}} ↓ BGL_{n}^{ét}(F_{3}) ⟶^{v} BGL_{n}^{ét}(F̄_{3}) .

The space BGL_{n}^{ét}(F̄_{3}) is 2-complete (2.7). The map χ_{n,F̄_{3}} induces an equivalence B̂GL_{n}(F̄_{3}) → BGL_{n}^{ét}(F̄_{3}) (2.10). Quillen [16] shows that the map BGL_{n}(F_{3}) → (B̂GL_{n}(F̄_{3}))^{hψ} induced by the map u gives an isomorphism on mod 2 cohomology. It follows that the map

BGL_{n}(F_{3}) →^{χ_{n,F_{3}}} BGL_{n}^{ét}(F_{3}) →^{≃} (BGL_{n}^{ét}(F̄_{3}))^{hψ}

also induces an isomorphism on mod 2 cohomology, and that BGL_{n}^{ét}(F_{3}) is equivalent to B̂GL_{n}(F_{3}).

Remark. The conjecture that χ_{n,S} induces an isomorphism on mod 2 cohomology is a very strong unstable analogue of the mod 2 Quillen-Lichtenbaum Conjecture for the ring S. This conjecture is true for a finite field (cf. 2.11) or the algebraic closure of a finite field (cf. 2.10). It is unknown whether or not it is true for the fields R and C. The results in this paper show that it is false for the ring Λ.

§3. The space X_{n} and its properties

We define X_{n} to be the space BGL_{n}^{ét}(Λ) from 2.6, and χ_{n} : BG_{n} → X_{n} the map χ_{n,Λ} : BGL_{n}(Λ) → BGL_{n}^{ét}(Λ). In this section we prove 1.4, 1.5 and 1.6.

(Co)homology of X_{n}. Let BD_{•}, BG_{•} and X_{•} denote respectively the spaces ∐ BD_{n}, ∐ BG_{n} and ∐ X_{n}. The index n in these coproducts runs over all nonnegative integers, where for n = 0 the spaces involved are contractible. There are maps

BD_{•} →^{ι} BG_{•} →^{χ} X_{•}.

Under matrix block sum all three of these spaces are homotopy associative Hspaces; BG_{•} and X_{•} are homotopy commutative. The maps ι and χ respect the multiplications.

There is a natural identification

BD_{1} ≃ B(Z × Z/2) ≃ S^{1} × BZ/2.

Let e ∈ H_{1} S^{1} and β_{k} ∈ H_{k} BZ/2 be generators. We denote the classes e ⊗ β_{k−1} and 1 ⊗ β_{k} in H_{k} BD_{1} by a_{k} and b_{k} respectively. Let a_{k}^{G} = ι_{∗}(a_{k}), b_{k}^{G} = ι_{∗}(b_{k}), a_{k}^{X} = (χι)_{∗}(a_{k}), b_{k}^{X} = (χι)_{∗}(b_{k}). Since H_{∗} BD_{•} is the free F_{2}-algebra on the elements a_{k} (k ≥ 1) and b_{k} (k ≥ 0), it follows that the image of ι_{∗} is the subalgebra of H_{∗} BG_{•} generated by the classes a_{k}^{G} and b_{k}^{G}, while the image of (χι)_{∗} is generated by the classes a_{k}^{X} and b_{k}^{X}.

3.1 Proposition. The algebra H_{∗} X_{•} is the free commutative F_{2}-algebra on the classes a_{k}^{X} (k ≥ 1) and b_{k}^{X} (k ≥ 0) subject to the following relations:

1. (a_{k}^{X})^{2} = 0 for k odd, and

2. a_{k}^{X}b_{0}^{X} + a_{k−1}^{X}b_{1}^{X} + ⋯ + a_{1}^{X}b_{k−1}^{X} = 0 for k even.

3.2 Lemma. There is a homotopy fibre square

X_{n}  ⟶ BGL_{n}^{ét}(R) ↓  ↓ BGL_{n}^{ét}(Z_{3}) ⟶ BGL_{n}^{ét}(C) .

Proof. This is a consequence of 2.5 and 2.7.

3.3 Remark. The square from 3.2 can up to homotopy be rewritten in the following way:

X_{n}  ⟶ B̂GL_{n}^{top}(R) ↓  ^{u}↓ B̂GL_{n}(F_{3}) ⟶^{v} B̂GL_{n}^{top}(C) .

The rewriting is justified by 2.9, 2.8, 2.1 and 2.7. By naturality the map u is the usual one induced by the map R → C. The map v is a little more problematical, but it is clear from 2.5 that the restriction of v to the diagonal matrices in GL_{n}(F_{3}) is induced by the ordinary diagonal inclusion {±1}^{n} → BGL_{n}(C). The fact that the mod 2 cohomology of BGL_{n}(F_{3}) is detected on diagonal matrices [14, §3] [16] means that it is easy to compute the map induced by v on mod 2 homology or cohomology.

Proof of 3.1. This is essentially given by Mitchell in [14, 4.6]; his space JK(Z) can be identified with colim_{n} X_{n}, where the maps in the colimit are the block inclusions given by product with a point in X_{1}. The role of diagram [14, 4.1] is played by the homotopy fibre square from 3.3 above.

3.4 Proposition. The classes a_{k}^{G}, k ≥ 1, and b_{k}^{G}, k ≥ 0, in H_{∗} BG_{•} satisfy the analogs of relations 1 and 2 from 3.1.

Proof. See [14, pf. of 7.1]. The place to look for these relations is in H_{∗} BGL_{2}(Λ), and Mitchell computes this homology explicitly [14, §6].

Proof of 1.4. By 3.1 and 3.4, the homology map (χ_{n}ι_{n})_{∗} is surjective and its kernel is equal to the kernel of (ι_{n})_{∗}. By duality, the cohomology map (χ_{n}ι_{n})^{∗} is injective and its image is equal to the image of ι_{n}^{∗}. This implies that ι_{n}^{∗} is injective if and only if χ_{n}^{∗} is an isomorphism.

Maps into X_{n}. Let P be a finite 2-group. We are interested in studying the set [BP, X_{n}] of unbased homotopy classes of maps from BP to X_{n}. The homotopy fibre square from 3.3 gives a map

(3.5) [BP, X_{n}] → [BP, B̂GL_{n}(F_{3})] × [BP, B̂GL_{n}^{top}(R)] .

3.6 Lemma. The map 3.5 is injective.

Proof. It follows from 3.3 that there is a homotopy fibre square of mapping spaces

Map(BP, X_{n})  ⟶ Map(BP, B̂GL_{n}^{top}(R)) ↓  ↓ Map(BP, B̂GL_{n}(F_{3})) ⟶ Map(BP, B̂GL_{n}^{top}(C)) .

By an elementary argument, it is enough to show that each component of the space Map(BP, B̂GL_{n}^{top}(C)) is 1-connected. By [7] the space Map(BP, B̂GL_{n}^{top}(C)) is equivalent to the 2-completion of the disjoint union ∐_{ρ} BC(ρ(P)), where ρ runs through a set of representatives for the conjugacy classes of homomorphisms P → GL_{n}(C), and C(ρ(P)) is the centralizer of ρ(P). By elementary representation theory each one of these centralizers is isomorphic to a product of complex general linear groups, and so is connected.

The following theorem is derived in [13, §1] from Carlsson’s work in [2] and [3]. We state it only for the prime 2, although it holds for any prime.

3.7 Theorem. Let Γ be a group of virtually finite cohomological dimension, and let P be a finite 2-group. Then the natural map

{P, Γ} ≅ [BP, BΓ] → [BP, B̂Γ]

is a bijection.

Proof of 1.5. The space X_{n} is 2-complete (3.3), so the fact that χ_{n}^{∗} is an isomorphism implies that X_{n} is equivalent to B̂G_{n}. Since G_{n} is a group of virtually finite cohomological dimension [17, p. 124], the result follows from 3.7.

Proof of 1.6. Consider the commutative diagram

[BP, BGL_{n}(R)] ⟵^{Bρ_{R}} [BP, BG_{n}] ⟶^{Bρ_{F_{3}}} [BP, BGL_{n}(F_{3})] ↓  ↓  ↓ [BP, BGL_{n}^{ét}(R)] ⟵ [BP, X_{n}] ⟶ [BP, BGL_{n}^{ét}(F_{3})] .

By 2.9, 2.11, 3.7 and [7], the right and left vertical arrows are bijections. The result follows from 3.6.

§4. Exploiting the class group

In this section we derive 1.2 from 1.3 by showing how class groups of cyclotomic extensions of Λ can be used to construct the necessary homomorphisms from finite 2-groups into GL_{32}(Λ).

Recall that μ_{n} denotes the multiplicative group of 2^{n}-th roots of unity. We make a distinction between Λ[μ_{n}], which is the group ring of μ_{n} over Λ, and Λ(μ_{n}), which is the integral closure of Λ in the field obtained from Q by adjoining μ_{n}. There is a surjection Λ[μ_{n}] → Λ(μ_{n}) (cf. [18, 2.6]) which is not an isomorphism unless n = 0. Because of this surjection, though, two modules over Λ(μ_{n}) are isomorphic as Λ(μ_{n})-modules if and only if they are isomorphic as Λ[μ_{n}]-modules.

4.1 Remark. If M is a Λ-module and α : μ_{n} → Aut(M) is a homomorphism, let M_{α} denote the Λ[μ_{n}]-module obtained by letting μ_{n} act on M via α. Given two homomorphisms α, β : μ_{n} → Aut(M), it is clear that α is conjugate to β if and only if the Λ[μ_{n}]-modules M_{α} and M_{β} are isomorphic.

4.2 Lemma. Suppose that I is a nonzero ideal in Λ(μ_{n}). Then there are isomorphisms of Λ(μ_{n})-modules

R ⊗_{Λ} I ≅ R ⊗_{Λ} Λ(μ_{n}),
F_{3} ⊗_{Λ} I ≅ F_{3} ⊗_{Λ} Λ(μ_{n}).

Proof. Since the quotient Λ(μ_{n})/I is a torsion group, the first isomorphism results from taking the inclusion I → Λ(μ_{n}) and tensoring with R. For the second, note that by the Chebotarev density theorem there are an infinite number of prime ideals in Λ(μ_{n}) isomorphic (as modules) to I. Up to isomorphism, then, I can be taken to be a prime ideal of residue characteristic different from 3. Tensoring the inclusion I → Λ(μ_{n}) with F_{3} again gives the required isomorphism.

Recall that Λ(μ_{n}) is free as a module over Λ of rank ϕ(2^{n}) = 2^{n−1} [18, 2.5]. The same is true of any nonzero ideal in Λ(μ_{n}).

4.3 Lemma. If Λ(μ_{n}) has a nonzero ideal which is not principal, then there exist two nonconjugate homomorphisms α, β : μ_{n} → G_{ϕ(2^{n})} which become conjugate both over R and over F_{3}.

Proof. Let I be such a nonzero ideal. Choosing bases for I and for Λ(μ_{n}) allows both to be identified (as Λ-modules) with Λ^{ϕ(2^{n})}. The multiplicative actions of μ_{n} on I and on Λ(μ_{n}) thus give two homomorphisms α, β : μ_{n} → GL_{ϕ(2^{n})}(Λ). These homomorphisms are not conjugate, because I and Λ(μ_{n}) are not isomorphic as Λ[μ_{n}]-modules (equivalently, as Λ(μ_{n})-modules). The homomorphisms are conjugate over R or over F_{3}, because (4.2) the two modules become isomorphic when tensored with R or with F_{3}.

Proof of 1.2. According to 4.3, it is enough to show that Λ(μ_{6}) is not a principal ideal domain. By [18, p. 353] the ideal class group of Z(μ_{6}) is cyclic of order 17. Since the ideal class group of Λ(μ_{6}) is the quotient of the ideal class group of Z(μ_{6}) by the subgroup generated by the prime ideals of Z(μ_{6}) which lie above 2, it is enough to show that there is only one prime P in Z(μ_{6}) above 2, and that P is principal. This is a standard calculation; see [18, 1.4] or [12, p. 73].

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Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

E-mail address: dwyer.1@nd.edu